Tamta Mamistvalovi radical polymerization of styrene Kinetic ...

Kinetic modeling of the in situ nitroxide mediated radical polymerization of styrene Tamta Mamistvalovi

Promotoren: prof. dr. Marie-Françoise Reyniers, prof. dr. ir. Guy Marin Begeleider: Lien Bentein Scriptie ingediend tot het behalen van de academische graad van Burgerlijk scheikundig ingenieur

Vakgroep Chemische proceskunde en technische chemie Voorzitter: prof. dr. ir. Guy Marin Faculteit Ingenieurswetenschappen Academiejaar 2007-2008

Kinetic modeling of the in situ nitroxide mediated radical polymerization of styrene door Tamta MAMISTVALOVI Scriptie ingediend tot het behalen van de academische graad van burgerlijk scheikundig ingenieur Academiejaar 2007–2008 Promotoren Prof. dr. M.-F. REYNIERS Prof. dr. ir. G.B. MARIN Scriptiebegeleider Ir. L. BENTEIN Faculteit Ingenieurswetenschappen Universiteit Gent Vakgroep Chemische Proceskunde en Technische Chemie Directeur: Prof. dr. ir. G.B. Marin

Abstract In Chapter 1, a short introduction is given about the in situ nitroxide mediated polymerization (NMP) and the objective of this Master thesis is proposed. The major factors of influence on the in situ NMP process are revealed. In Chapter 2, various types of computational methods are described and special attention is paid to Density Functional Theory (DFT) methods. Also, the methods for thermodynamic and kinetic calculations are enlightened. In Chapter 3, a level of theory study is performed on a test set of molecules and reactions in order to find the best calculation method for the following calculations. In Chapter 4, the selected calculation method is applied to various nitrone/AIBN/styrene systems. Based on the results, experimentally observed behavior of the nitrones is explained. In Chapter 5, formal kinetic model simulations for the regular NMP of styrene are performed and the influence of the different kinetic and thermodynamic parameters on the controllability of the process and the polymerization rate are discussed. Finally, in Chapter 6, the conclusions are summarized.

Keywords NMP, nitroxide, ab initio, polymerization, in situ, styrene, modeling

Voorwoord In eerste instantie wil ik alle mensen bedanken die op een of andere manier bijgedragen hebben bij het tot stand komen van dit werk. In het bijzonder wil ik mijn promotoren prof. dr. Marie-Fran¸coise Reyniers en prof. dr. ir. Guy B. Marin bedankt voor de geboden mogelijkheden en de nodige bijstand. Mijn grootste en oprechtste dank gaat naar mijn begeleider, ir. Lien Bentein, waar ik elke minuut op kon rekenen voor hulp, advies, alsook voor het nodige duwtje in de rug. Ik bewonder het ijzeren geduld en doorgedreven motivatie waarmee zij mij bijgestaan heeft. Enorm bedankt Lien, voor je tijd en energy die je in dit werk gestoken hebt! Ik wens je succes in elk aspect van je leven. Ik wil eveneens mijn medestudenten Annelies, Stefanie, Raf en Tom bedanken, die ik hoogstpersoonlijk verantwoordelijk houd voor de toffe sfeer in ons labo. Dankzij jullie teamspirit en eeuwige bereidheid om te helpen hebben ik alle Latex perikkelen overleefd. Verder wil ik alle personeel in de S5 bedanken voor een bemoedigend woord of een glimlach. Tenslotte wil ik mijn ouders en Rebecca bedanken voor al de steun en onvoorwaardelijk geloof in mij die ze tijdens mijn studies hebben geuit. Ook al mijn vrienden die mij hebben gesteund verdienen een grote merci, voor hun luisterend oor, en een schouderklopje, ook al verstonden ze niet waar ik het over had. Hartelijk dank aan iedereen die aan dit mooi resultaat een steentje heeft bijgedrage! Veel leesplezier!

Tamta Mamistvalovi, juni 2008

Kinetic modeling of the in situ nitroxide mediated polymerization of styrene Tamta Mamistvalovi Coach: Ir. Lien Bentein Promoters: Prof. dr. Marie-Françoise Reyniers, Prof. dr. ir. Guy B. Marin Abstract— In this work, the in situ nitroxide mediated polymerization (NMP) of styrene is studied. First, a level of theory study is performed by applying various (DFT) calculation methods to a test set of structures and reactions, which are similar to those involved in the in situ NMP process of styrene. Next, the optimal calculation method is applied to some elementary reactions of the in situ NMP process and thermodynamic and kinetic data is derived. This data is used in order to describe the influence of the nitrone structure on the polymerization rate and polymerization control. In addition, simulations based on a formal kinetic model of the NMP of styrene are performed to gain more insight in the influence of thermodynamic and kinetic parameters on the NMP process. Finally, a first step is made toward the development of a more fundamental model for the in situ NMP of styrene. Keywords—ab initio, NMP, in situ, styrene, polymerization, modeling

combination/dissociation reactions and equilibrium coefficients. Kinetic calculations involving the transition state theory were also performed and reaction rate coefficients were obtained. Prereaction (1)

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I. I NTRODUCTION

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ONTROLLED radical polymerization (CRP) is a promising polymerization technique, because it produces well– defined, end–functionalized polymers. An interesting and recently developed type of CRP is the in situ nitroxide mediated radical polymerization (in situ NMP). It follows the same initial concept as regular NMP, yet here the nitroxides are formed in the polymerization medium itself, based on relatively cheap precursors, such as nitrones (see Fig. 1). So, this process requires no purification steps and is therefore cheaper and environmentally friendlier than the NMP process. Unfortunately, the factors of influence on the in situ NMP process are not yet fundamentally understood. Therefore, ab initio and kinetic modeling must be applied, in order to provide deeper insight in this polymerization technique. II. C OMPUTATIONAL METHOD In order to find a cost-efficient computational method to model the in situ NMP process, a level of theory study is performed, where various density functional theory (DFT) methods, such as B3LYP, B3P86, MPW1PW91, BB1K, BMK and BHandHLYP, were applied to a test set of molecules and reactions similar to those involved in an actual in situ NMP process of styrene. Composite methods, such as G3B3 and CBS–QB3 were also investigated, but merely as benchmarks. In addition, various basis sets were examined, namely, the 6–311g tripleξ basis set, the 6–311g** basis set which additionally includes polarization functions on all atoms and, finally, the 6–311++g** basis set that includes diffuse functions as well. These methods were applied for the study of geometrical structures of the molecules in the test set, as well as for thermodynamic calculations, such as the calculation of the enthalpies of reaction for reT. Mamistvalovi is with the Chemical Engineering Department, Ghent University (UGent), Gent, Belgium. E-mail: [email protected] .

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Fig. 1. In situ NMP mechanism: I–I: initiator; I•: initiator radical; R, R’: alkyl groups; P•: propagating radical; St: styrene; n: number of monomers; kp , kc , kd : rate coefficients of resp. propagation, coupling and dissociation

III. R ESULTS A. Level of theory study Based on the study of the optimized geometries of the structures in the test set, it was found that the 6–311g basis set overestimates the bond lengths. The computationally more expensive 6–311++g** basis set yields values which are close to those obtained with the 6–311g** basis set. B3LYP/6–311g** is found to yield the best results for geometry calculations. The standard heats of formation are best predicted by the composite methods, however, these methods are only applicable for the smallest molecules, hence, they will be only used as benchmarks for the (small) molecules when no experimental data is available. The best DFT alternatives are provided by the MPW1PW91/6–311g** and the BMK/6-311g** methods,

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C. Modeling of the NMP of styrene

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Fig. 2. Nitrones used for in situ NMP: (1) N-tert-butyl-α-isopropylnitrone (nitrone 1), (2) N-tert-butyl-α-tert-butylnitrone (nitrone 2), (3) C-phenyl-Ntert-butylnitrone (PBN), (4) N-α-diphenylnitrone (DPN), (5) 5,5-dimethyl1-pyrroline-N-oxide (DMPO)

yet the standard molecular entropy is best described by the B3LYP/6–311g** method. The results obtained using the 6– 311g basis set are significantly worse and the 6–311++g** basis set does not perform significantly better to justify its higher computational cost. Moreover, the BMK/6-311g** method performed best for the reaction enthalpy calculations and also well for the prediction of the rate coefficients for the addition/β–scission reactions. In this Master thesis, obtaining accurate energy and kinetic calculations are most important. Hence, the BMK/6–311g** is chosen to be the most cost–efficient method and will be applied in the following calculations. B. Applying the selected method to various nitrone / AIBN / styrene systems The chosen method BMK/6–311g** was applied to the nitrone/AIBN/styrene systems, for the nitrones depicted in Fig. 2. For each of the elementary reactions of these systems the reaction enthalpy and the reaction entropy was calculated, yielding the equilibrium coefficient Kc . The aim was to explain the experimental observations reported for these nitrones in literature, such as the polymerization rate, the evolution of the number average molecular mass with conversion and the polydispersity index. This was done by evaluating the relative amounts of nitroxide and alkoxyamine formed in the prereaction step by the various nitrones. These relative amounts can be estimated by considering different effects, which influence the ease of the nitroxide and alkoxyamine formation. First, the structural effects of the nitrones were examined, namely the steric hindrance, as increasing steric hindrance will decrease the amount of nitroxide and, hence, alkoxyamine formed. Second, the thermodynamic effects, namely the exothermicity of the calculated reaction enthalpies were investigated. Finally, polarization effects were taken into account by analyzing the calculated Mulliken charges in the nitrones. The most important thermodynamic value, is the equilibrium coefficient Kc of reaction (6) of Fig. 1. In the theoretical calculations, P• is modeled by a styryl radical. The calculated value for Kc equals 5.77 10−8 kmol m−3 , as the experimentally reported value is 8.70 10−10 kmol m−3 . These values differ with a factor 102 , so the calculated data should be looked at as a rather indicative measure. Taking the influence of the chain length (P•) into account in the calculation may also have an influence on the obtained values. Besides the thermodynamic data, kinetic data was obtained where possible, yet this is not easy, due to the size of the systems under study. Based on our analysis, C–phenyl–N–tert– butylnitrone (PBN) (see Fig. 2 (3)) was found to yield the best results in terms of control and polymerization rate. This is in agreement with the experimental findings.

The ranges for the thermodynamic and the kinetic values obtained with the ab initio calculations were plugged in a formal kinetic model for the regular NMP of styrene in order to gain deeper insight in the influence of each kinetic parameter and of the equilibrium coefficient Kc on the polymerization controllability and the rate of the polymerization process. From simulations, it was found that the equilibrium coefficient has an important influence on both control and the rate of the process, and the optimal trade-off between control on the one hand, and polymerization rate on the other hand is found in the range of 10−11 10−10 kmol m−3 . For a Kc of 10−11 kmol m−3 , the rate coefficient of termination by recombination should be in the range of 107 -108 m3 kmol−1 s−1 in order to have a high conversion and a low polydispersity index. The effect of the self–initiation of styrene was also examined and it was found that this may worsen the control of the NMP process, hence, this effect should be inhibited as much as possible. Finally, it was found that the rate coefficient of transfer to monomer determines whether a polymerization, or an oligomerization process occurs. In order to maintain a polymerization process, the rate coefficient of transfer to monomer must be lower than 10−1 m3 kmol−1 s−1 . A first step was made toward a fundamental kinetic model of the prereaction of the in situ NMP of styrene. IV. C ONCLUSION Based on a very small number of experimental findings, there appears to be a deviation between experimentally obtained and calculated values for reactions involved in the in situ NMP process of the nitrone/AIBN/styrene system. Qualitatively – however – the results give a good relative measure, when different structures are compared, that makes it possible to explain the experimentally observed polymerization behavior. The influence of the nitrone structure was investigated theoretically and compared to experimental findings. PBN was found to be the best choice for a good controllability and polymerization rate for the in situ NMP of styrene. From formal kinetic model simulations on regular NMP, a qualitative insight in the influence of different thermodynamic and kinetic parameters was obtained. However, a model for in situ NMP is yet to be developed. V. F UTURE WORK In the first place, an extension of the formal kinetic model of NMP will have to be done in order to be able to describe the in situ process. Also, diffusional effects on the polymerization and other reactions, such as transfer to dimer and the formation of hydroxylamines, should be accounted for. Next, a more fundamental model, describing (in situ) NMP should be developed. To obtain sufficient kinetic and thermodynamic data to perform simulations, more ab initio calculations on elementary reactions will have to be performed. In a next step, the kinetic model of the in situ NMP polymerization of styrene could be expanded toward other monomers.

List of symbols

Symbol

Description

Units

a A E Exc g G h H ˆ H IA , IB , IC k K ka kB kc Kc kd kda kp ktc ktd kth ktrm M, Mm M Mn , Mw , Mz n N p P P• P −X q Q r R R R–X s S Sm T Tni U0 V Vee Vme Vxc x X X•

bond angle pre-exponential factor in the Arrhenius’ law energy exchange–correlation energy degeneracy Gibbs free energy Planck’s constant enthalpy hamiltonian operator principal moments of inertia rate coefficient equilibrium coefficient rate coefficient of activation Boltzmann’s constant rate coefficient of recombination equilibrium coefficient based on concentrations rate coefficient of dissociation rate coefficient of deactivation rate coefficient of propagation rate coefficient of termination by recombination rate coefficient of termination by disproportionation rate coefficient of thermal initiation rate coefficient of transfer to monomer molecular mass monomer moments of molecular mass distribution number total number of particles pressure terminated propagating radical chain alkoxyamine molecular partition function canonical partition function bond length universal gas constant active dormant spin multiplicity entropy molecular entropy temperature kinetic energy of non–interacting electrons internal energy volume electron–electron repulsion operator nuclear–electron interaction operator one–electron operator conversion nitrone nitroxide

◦

s−1 J mol−1 J mol−1 J mol−1 J s J mol−1 kg m2 s−1 s−1 J K −1 m3 mol−1 s−1 mol m−3 s−1 m3 mol−1 s−1 m3 mol−1 s−1 m3 mol−1 s−1 m3 mol−1 s−1 m6 mol−2 s−1 m3 mol−1 s−1 kg mol−1 kg mol−1 Pa ˚ A J mol−1 K −1 J mol−1 K −1 J mol−1 K −1 K J mol−1 J mol−1 m3 -

Symbol

Description

Units

* + χ ξ σ Φ ∇ δ ψ τ µ λ ω ν

polarization function diffuse function percentage of HF exchange in the functional basis function biscribing a type of orbital number of pure rotations orbital nabla operator difference wave function moment of the distribution of the dormant species moment of the distribution of the terminated species moment of the distribution of the active species vibrational frequency number

mol m3 mol m3 mol m3 s−1 -

Superscript

Description

0 ‡ + −

standard state movement along the reaction path or activated complex forward reaction backward reaction

Subscript

Description

react, r ‡ 0 a A, B calc elec exp f m n prod, p r rot trans vib

reactants activated complex, without the movement along the reaction path initial activation molecules calculated electronic experimental formation molar chain length products reaction rotational translational vibrational

Acronym

Description

BDE CRP FRP LRP MAD NMP PDI QMHO TS ZPE

bond dissociation energy controlled radical polymerization free radical polymerization living radical polymerization mean average deviation nitroxide mediated polymerization polydispersity index quantum mechanical harmonic oscillator transition state zero point energy

CONTENTS

i

Contents Nederlandse samenvatting 1 Introduction

1 18

1.1

Polystyrene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2

Living/Controlled Radical Polymerization . . . . . . . . . . . . . . . . . . 19

1.3

Nitroxide mediated radical polymerization . . . . . . . . . . . . . . . . . . 22

1.4

In situ nitroxide mediated radical polymerization . . . . . . . . . . . . . . 25

1.5

Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5.1

Kinetic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.5.2

Thermodynamic parameters . . . . . . . . . . . . . . . . . . . . . . 34

1.6

Theoretical calculations

1.7

Objective of this Master thesis . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Computational methods

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

40

2.1

Hartree–Fock calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2

Post-Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.1

Configuration interaction method (CI) . . . . . . . . . . . . . . . . 42

2.2.2

Møller-Plesset perturbation method (MPn) . . . . . . . . . . . . . . 43

2.2.3

Coupled cluster method (CC) . . . . . . . . . . . . . . . . . . . . . 44

2.2.4

Composite methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3

Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4

Basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5

Statistical thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.6

Thermodynamic calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 59

CONTENTS

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2.6.1

For molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.6.2

For reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.7

Enthalpies and Free energies of reaction . . . . . . . . . . . . . . . . . . . . 60

2.8

Kinetic calculations: Conventional transition state theory . . . . . . . . . . 61

3 Level of Theory Study

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3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2

Definition of test set, methods and basis sets . . . . . . . . . . . . . . . . . 65

3.3

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4

Heats of formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5

Entropies of formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6

Dissociation/recombination reactions . . . . . . . . . . . . . . . . . . . . . 96

3.7

Addition/β–scission reactions . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4 Applying the selected DFT method to reactions involved in the in situ NMP process

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4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.2

Influence of the nitrone structure . . . . . . . . . . . . . . . . . . . . . . . 108

4.3

4.2.1

Substituents in α-position . . . . . . . . . . . . . . . . . . . . . . . 111

4.2.2

Substituents in β-position . . . . . . . . . . . . . . . . . . . . . . . 116

4.2.3

Structural effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5 Modeling of (in situ) NMP

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5.1

Formal kinetic model for NMP . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2

Simulation results for NMP of styrene . . . . . . . . . . . . . . . . . . . . . 127

5.3

First step toward a fundamental model for in situ NMP . . . . . . . . . . . 144

5.4

5.3.1

Prereaction with AIBN and PBN . . . . . . . . . . . . . . . . . . . 144

5.3.2

Addition of styrene . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6 Conclusion

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CONTENTS

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Appendix A

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Appendix B

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NEDERLANDSE SAMENVATTING

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Nederlandse samenvatting 1

Introductie

Polystyreen is een hard, amorf polymeer, dat grootschalig gebruikt wordt in de verpakkingsindustrie. Industrieel wordt het vooral gesynthetiseerd op basis van vrije radicalaire polymerisatie (FRP). Echter, door de stijgende vraag naar goed gedefineerde materialen, werd de techniek van levende/gecontroleerde radicalaire polymerisatie ontwikkeld (LRP/CRP). Daar waar bij levende polymerisatietechnieken (LP), zoals anionische polymerisatie, terminatie- en ketenoverdrachtreacties uitgesloten worden, komen deze bij gecontroleerde radicalaire polymerisatie (CRP) wel nog voor, zij het in beperkte mate door de lage radicaalconcentratie. Een veelbelovende CRP techniek werd ontwikkeld, namelijk de nitroxide gemedieerde polymerisatie (NMP). Figuur 1 licht het basisprincipe van deze techniek toe. Bij de deactiveringsreactie wordt er een labiele C–ON binding gevormd tussen het propagerend radicaal (actieve species) en het nitroxide (mediërend radicaal), wat het alkoxyamine oplevert (slapende species). Dissociatie van deze binding activeert het propagerend radicaal en een klein aantal monomeereenheden kunnen aan de propagerende polymeerketen geaddeerd worden vooraleer deze weer gedeactiveerd wordt door de recombinatie met een nitroxide. Wanneer het dynamisch evenwicht tussen de actieve en de slapende ketens verschoven wordt in de richting van de slapende ketens (d.i. een zeer lage evenwichtscoëfficiënt Kc = kkdc ≈ 10−11 kmol m−3 ) is de ogenblikkelijke concentratie aan radicalen in het systeem zeer laag, wat de kans op ketenoverdrachtreacties, alsook de kans op terminatiereacties door recombinatie en disproportionering, beperkt. Bovendien groeien de polymeerketens aan nagenoeg dezelfde globale snelheid, wat voor de lage polydispersiteitsindex (PDI) zorgt. Een nieuwe ontwikkeling in het kader van NMP is het zogenaamde ’in situ’ NMP proces.


2

K = kd/k c

Alkoxyamine (dormant species)

dead polymer

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Propagating radical (active species)

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Figuur 1: Basismechanisme van de nitroxide gemedieerde polymerisatie (NMP); P • = propagerend radicaal, X • = nitroxide (actieve species), P − X = alkoxyamine (slapende species), M = monomeer; kd , kc , kp , kt = snelheidscoëfficiënten voor dissociatie- (activering), recombinatie(deactivering), propagatie- en terminatiereacties respectievelijk

Daar waar bij gewone NMP het nitroxide apart bereid moet worden, wordt bij in situ NMP het nitroxide in het reactiemedium zelf, dus ’in situ’ gegenereerd op basis van precursoren, doorgaans nitronen. Aangezien de precursoren voor dit proces goedkoop zijn, en er geen zuiveringsstappen nodig zijn om enkel het nitroxide te verkrijgen, is het ’in situ’ NMP proces goedkoper en milieuvriendelijker dan het gewone NMP proces. Het verschil tussen beide mechanismen wordt toegelicht aan de hand van Figuur 2. Bij NMP dissocieert de initiator en het hierbij gevormde radicaal initieert op zijn beurt een monomeereenheid, waarna de polymerisatie verloopt volgens het basisprincipe van NMP, dat hierboven werd toegelicht. Bij ’in situ’ NMP daarentegen, wordt de polymerisatiestap voorafgegaan door een zogenaamde prereactie. Daarbij wordt een initiator en een nitrone samengebracht in een reactiemengsel bij een temperatuur lager dan de polymerisatietemperatuur. De bedoeling is om het nitrone door een additie van de gedissocieerde initiator eerst om te zetten tot een nitroxide en vervolgens door een recombinatie van de gevormde nitroxide met een ander initiator radicaal tot een alkoxyamine. Na deze prereactie wordt er geen zuiveringsstap uitgevoerd. Het monomeer wordt aan de reactiemedium toegevoegd. Door temperatuurverhoging dissocieert het alkoxyamine met de vorming van een initiatorradicaal en een nitroxide. Het initiatorradicaal activeert het monomeer, waarna het proces verloopt volgens het standaard NMP principe. Hoewel het principe van ’in situ’ NMP niet moeilijk is, worden initiator/nitrone paren die in staat zijn het polymerisatieproces te controleren toch erg moeizaam gevonden, omdat er veel factoren zijn die een invloed hebben op het al dan niet succesvol verlopen van een ’in situ’ NMP proces met een bepaald initiator/nitrone paar. Uit de literatuurstudie blijkt dat de structuur van zowel de initiator als het nitrone de waarde van de evenwichtscoëfficiënt Kc bepalen (zie Figuur 1) (Sciannamea et al., 2005),


3

NMP Mechanism

In situ NMP Mechanism

Initiator decomposition

Prereaction

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Figuur 2: Het NMP mechanisme (links) en het ’in situ’ NMP mechanisme (rechts): I − I = initiator; I • = initiatorradicaal; R, R0 , R00 = alkylgroepen; P • = propagerend radicaal; St = styreen; n = aantal monomeren; kp = snelheidscoëfficiënt van de propagatiestap


4

en bijgevolg het hele polymerisatieproces sturen. Bovendien speelt de reactietijd van de prereactiestap ook een belangrijke rol. Een duur van 4 uur werd optimaal gevonden (Sciannamea et al., 2007). Daarnaast heeft de molaire verhouding tussen het nitrone en de initiator in het systeem een invloed op de controle van het proces, evenals de temperaturen waarbij de prereactie en de polymerisatie uitgevoerd worden. Een temperatuur van 85◦ C voor de prereactie en 110◦ C voor de polymerisatie werden voorgesteld (Sciannamea et al., 2007). Om fundamenteel inzicht te verkrijgen in hoeveel de bovengenoemde factoren het ’in situ’ NMP proces be¨ınvloeden, dient het proces kinetisch gemodelleerd te worden. Een onderdeel van deze modellering is het ab initio modelleren van elementaire reacties die voorkomen in het ’in situ’ NMP proces. Hiertoe moet eerst een goede berekeningsmethode gekozen worden, wat het onderwerp is van de volgende sectie.

2

Computationele methode

Computationele methodes leveren thermodynamische en kinetische data op die experimenteel vaak niet beschikbaar zijn. Deze methodes steunen op verschillende theorieën. De eerste theorie die ontwikkeld werd, is de Hartree–Fock (HF) theorie en heeft als grootste nadeel dat ze elektronencorrelatie niet in rekening brengt. Bijgevolg faalt deze methode voor energieberekeningen en zal het niet aangewend worden in dit werk. Verscheidene post–Hartree–Fock (post–HF) methoden trachten aan de tekortkomingen van de HF methode tegemoet te komen door een correlatieterm te introduceren. Tot deze groep behoren ook de composietmethoden, zoals G3B3 en CBS–QB3, die in dit werk toegepast worden. Aangezien composietmethoden erg rekenintensief zijn, zijn ze slechts toepasbaar op de kleinste moleculen. Wegens de grote accuraatheid van deze berekeningen, werden deze methoden aangewend als referentie voor de kleine moleculen waar geen experimentele gegevens voor beschikbaar waren. Ten slotte werden verschillende densiteitsfunctionaaltheorie (DFT) methoden onderzocht. Wegens de computationele mogelijkheid van deze methoden om grotere moleculen te berekenen (minder rekenintensief dan post–HF methoden), stond het a priori vast dat de optimale methode om de ’in situ’ NMP te modelleren moest gezocht worden in de categorie van de DFT methoden. Naast verscheidene functionalen is er ook gekeken


5

Tabel 1: De combinaties van functionalen (eerste kolom) en de basissets (eerste rij) die bestudeerd worden in dit werk

B3LYP B3P86 MPW1PW91 BB1K BMK BHandHLYP G3B3b CBS-QB3b

6-311g

6-311g**

6-311++g**

+ + + + + +

+ + + + + +

+ + + + + +

cbsb7a 6-31g(d)

+ +

a

Deze basisset is van de vorm 6-311g(2d,d,p): het verbetert de flexibiliteit van de polarisatiefuncties vergeleken met de 6-311g** basis set. b Basisset gebruikt voor de optimalisatiestap, in combinatie met de B3LYP functionaal.

naar verschillende basissets. Als kleinste basisset werd de 6–311g basisset onder de loep genomen. Daarnaast werd ook de 6–311g** basis set onderzocht, die additioneel polarisatiefuncties bevat op alle atomen. Tenslotte werd de invloed van het toevoegen van diffuse functies beschouwd, met de 6–311++g** basis set. Deze functionalen en basis sets werden toegepast op een testset van moleculen en reacties, die typerend zijn voor de ’in situ’ NMP van styreen. De verkregen resultaten worden besproken in de volgende sectie. Een overzicht van alle combinaties van functionalen en methoden die onderzocht zijn in deze Master thesis wordt gegeven in Tabel 1.

3

’Level of theory’ studie: keuze van de computationele me-

thode In deze sectie wordt een geschikte methode gezocht om de in situ NMP te beschrijven. Hiertoe wordt een testset van moleculen en reacties opgesteld, die gelijkaardig zijn aan deze die voorkomen in het eigenlijke ’in situ’ NMP proces. De testset van moleculen is te vinden in Figuur 3. De reacties die beschouwd worden, zijn enerzijds dissociatie-/recombinatiereacties en anderzijds additie-/β-scissie reacties. De reacties die deel uitmaken van de testset worden getoond in Figuur 4. De moleculen worden getest


6

op geometrische karakteristieken en op thermodynamische gegevens zoals de standaard vormingsenthalpie. Van de reacties worden, in de mate van het mogelijke, thermodynamische en kinetische data afgeleid, zoals respectievelijk reactie-enthalpie en snelheidscoëfficiënten. Indien er experimentele data beschikbaar zijn, worden de verkregen resultaten eraan getoetst. 3.1

Geometrie

Er zijn weinig experimentele gegevens beschikbaar voor de geometrieën uit de testset. Bijgevolg worden globale tendensen van de methoden met betrekking tot de geoptimaliseerde geometrieën geanalyseerd. Daaruit blijkt dat de 6–311g basisset de bindingslengte overschat. De computationeel veeleisende 6–311++g** basisset geeft waarden die dicht liggen bij de waarden die verkregen worden op basis van de 6–311g** basisset. Doorgaans wordt de bindingslengte onderschat door BB1K en de BHandHLYP functionalen. Van alle onderzochte methoden lijkt de B3LYP/6–311g** methode het best in staat om geometrieën accuraat te beschrijven. 3.2

Thermodynamica

Voor de structuren uit de test set werden standaard vormingsenthalpieën en standaard moleculaire entropieën bepaald. De berekende waarden worden vergeleken met de experimentele waarden, en waar geen experimentele waarden voorhanden zijn, wordt er vergeleken met de waarden verkregen met de G3B3 composiet methode. De 6–311g basisset geeft de grootste afwijkingen van de experimentele waarden. De resultaten verkregen op basis van de 6–311++g** basisset zijn iets beter dan die verkregen met de 6–311g** basisset, maar het verschil is niet significant genoeg om de aanzienlijk hogere computationele kost te rechtvaardigen. Voor de standaard vormingsenthalpieën verschillen de resultaten die verkregen worden door verschillende methoden aanzienlijk. Dit wordt ge¨ıllustreerd door Figuur 5, waarop de afwijkingen tussen de berekende en de experimentele standaard vormingsenthalpieën staan weergegeven. De beste resultaten worden gevonden met de MPW1PW91/6–311g** en de BMK/6–311g** methoden. Voor de standaard moleculaire entropieën is er minder verschil tussen de waarden


7

Initiator related structures O

Hydrocarbons

O

O

O O

O

benzoylperoxide (BPO)

(a)

(b) benzoyloxy radical O

O

O

O

O O (c )

(d) methylcarboxyl radical

diacetylperoxide

N

N

(m) 1-methylethylbenzene (n) 1-methylethylbenzene radical

(o) benzene

(p) phenyl radical

(q) ethylbenzene

(r) ethylbenzene radical

CN

CN CN (e) azobisisobutyronitrile (AIBN)

(f) cyanoisopropyl radical

(s) styrene

Alkoxyamine

Nitrone/nitroxide related structures H

H O

N

H

H (g)

(i)

N

O

H

hydroxylamine

N

(h) oxylamine radical

N

N

OH

O

TEMPO-H

(t)

(k) dimethylnitrone

TEMPO-styryl

(j) 2,2,6,6-tetramethylpiperidinyl-N-oxyl (TEMPO) O

O N

O

N

(l) C-phenyl-N-tert-butylnitrone (PBN)

Figuur 3: Structures studied in the level of theory study


O

8

O

O 2

O O

O

+ H

O

O

O O O

N

N

O

O

H

O + H

N H

H

+ H

+ N2

2 CN

N CN

CN

H

H

2

N

N

OH

O

+ H

N

+ H

+

O

N O

+

CH3

CN

O O

+

CN

N

N

+

CN CN O O

O N

O

+

O

+

O

N

O

O

O O

O O

+

N

N

+

Figuur 4: Bestudeerde reacties in de level of theory studie: dissociatie/recombinatie reacties (boven), additie/β-scissie reaction (bottom)

die berekend werden met de verschillende methoden. De beste resultaten voor de entropieberekeningen worden verkregen met de B3LYP/6–311g** methode. In de testset zijn ook dissociatie-/recombinatiereacties opgenomen en voor deze reacties werden de reactie-enthalpieën bepaald. De beste resultaten werden wederom verkregen met de BMK/6–311g** methode, zoals af te leiden is uit Tabel 2 met gemiddelde afwijkingen. Ook de evenwichtscoëfficiënten zijn berekend voor deze reacties, aan de hand van


9

(a) ethylbenzene

(b) styrene

600

400

6-311g 6-311g** 6-311++g**

500 Df H0exp-Df H0calc [kJ mol-1]

Df H0exp-Df H0calc [kJ mol-1]

500

600

300 200 100 0 -100

400

6-311g 6-311g** 6-311++g**

300 200 100 0 -100

-200

-200

-300

-300 an dH P LY

P LY

(d) hydroxylamine 300

6-311g 6-311g** 6-311++g**



BH

K BM

dH

400

1K 91 BB W 1P PW M

an

6 P8 B3 P LY B3

BH

K BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

(c) benzene 500

300 200 100 0

6-311g 6-311g** 6-311++g**

200 150 100 50

-100

0

-200

-50 BH

K

an

BM

dH

dH

P LY

P LY

1K 91 BB W 1P PW M 86

P LY

an

P B3

B3

BH

K BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

(e) phenyl radical

(f) diacetylperoxide

400 6-311g 6-311g** 6-311++g**



300 200 100 0

6-311g 6-311g** 6-311++g**

600 400 200

-100

0

-200

-200 BH an

K BM

dH

dH

P LY

P LY

1K 91 BB W 1P PW M

an

6 P8 B3 P LY B3

BH

K BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

Figuur 5: Afwijkingen van de berekende standaard vormingsenthalpieën van de experimentele waarden voor een aantal moleculen uit de testset


10

Tabel 2: Gemiddelde afwijkingen voor reactie-enthalpieën t.o.v. experimentele data in kJ mol−1

6-311g 6-311g**

B3LYP 31.410 25.917

B3P86 19.147 14.018

MPW1PW91 29.726 23.125

BB1K 53.179 28.218

BMK 32.310 8.531

BHandHLYP 26.606 39.720

de Gibbs vrije energieverandering tijdens de reactie (R is de universele gasconstante, T de temperatuur en p de druk; G is de Gibbs vrije energie, H de enthalpie en S de entropie; vr is het aantal reactanten, vp het aantal producten): Kc ∆r G0 Vm

3.2

−∆r G0 exp = RT 0 0 = ∆r H − T ∆r S RT = p Vmvr −vp

(1) (2) (3)

Kinetiek

Voor de additie/β–scissie reacties uit de testset werden snelheidscoëfficienten bepaald. Helaas is er voor slechts 1 reactie uit deze testset eveneens experimentele data beschikbaar. Op basis van die data werden de BB1K/6–311g** en de BMK/6–311g** methoden aanvaardbaar bevonden. Om de prestatie van de verschillende methoden voor de kinetische berekeningen te valideren is er bijkomende experimentele data nodig. Tenslotte dient er opgemerkt te worden dat de resultaten verkregen op basis van composietmethoden aanzienlijk beter zijn dan de resultaten op basis van DFT methoden. Wegens de beperking van de composietmethoden die reeds hierboven werd aangehaald, dient er een DFT alternatief gekozen te worden. Op basis van het voorgaande kan er besloten worden dat de BMK/6–311g** methode het beste DFT alternatief is. Met deze methode zullen dan ook de ’in situ’ NMP reacties bestudeerd worden.

4

Toepassing van de gekozen methode op de reacties in de ’in

situ’ NMP van styreen Voor de vijf types nitronen die weergegeven zijn in Figuur 6 werden een aantal elementaire reacties van de ’in situ’ NMP in het systeem nitrone/AIBN/styreen ab initio gemodelleerd.

nitrone 1 nitrone2 PBN DPN DMPO

reaction c

-1 +1 +1 +1 +1 +1

reaction d

reaction e


-1 -1 -1 -1 -1


reaction b

-1 -1 -1 -1 -1

2

reaction a

np − nr

-95.862 -52.419 -67.433 -121.017 -119.658

-88.042

-68.673 -61.681 -84.893 -128.734 -231.839

-96.767 -90.190 -111.947 -132.236 -140.977

∆r H kJmol−1 33.013

∆r H kJmol−1 32.879 -96.673 -90.103 -111.799 -132.086 -140.853 -68.604 -61.563 -84.773 -128.633 -231.442 -87.869 -95.980 -52.364 -67.445 -120.993 -119.661

Kc (kmol m−3 )np −nr 5.460 1015 1.210 104 3.465 102 1.020 107 7.354 109 1.234 1011 1.806 10−2 1.772 10−2 5.515 1.247 108 2.232 106 3.563 1041 1.618 10−12 4.587 10−6 5.343 10−8 4.335 10−17 7.060 10−17

85◦ C

103 101 105 108 109

3.263 10−12 3.561 10−6 5.771 10−8 1.515 10−16 2.394 10−16

4.364 1041

4.295 10−3 4.914 10−3 9.193 10−1 7.951 106 1.863 105

1.553 5.134 9.404 4.345 6.018

Kc (kmol m−3 )np −nr 1.202 1016

110◦ C

Tabel 3: De berekende thermodynamische gegevens voor de nitronen bij de verschillende reactiestappen

NEDERLANDSE SAMENVATTING 11


12

-

-

O

-

O

+

O

+

N

+

N

N

(c ) PBN

(b) nitrone 2

(a) nitrone1 -

O +

N

N+ -

O

(d) DPN

(e) DMPO

Figuur 6: Nitronen voor ’in situ’ NMP: (a) N-tert-butyl-α-isopropylnitrone (nitrone 1), (b) N-tertbutyl-α-tert-butylnitrone (nitrone 2), (c) C-phenyl-N-tert-butylnitrone (PBN), (d) N-αdiphenylnitrone (DPN), (e) 5,5-dimethyl-1-pyrroline-N-oxide (DMPO)

Daarbij zijn de reacties onderzocht die weergegeven zijn in Figuur 7. Er werd gepoogd de experimenteel waargenomen initiatie-efficiëntie (gedefinieerd als Mn,theoretisch /Mn,experiment ) te verklaren op basis van ab initio berekeningen. De initiatie–efficiëntie is enerzijds afhankelijk van de hoeveelheid alkoxyamine gevormd in de prereactiestap (reactie c in Figuur 7), en anderzijds van het gemak waarmee het gevormde alkoxyamine cyanoisopropylradicalen kan vrijstellen bij verhoogde temperaturen, deze cyanoisopropylradicalen kunnen dan de polymerisatie initiëren (terugwaartse reactie c en reactie d). Eerst wordt er gekeken naar het gemak waarmee de additie aan een nitrone doorgaat met de vorming van een nitroxide (reactie b in Figuur 7. Ab initio berekeningen geven thermodynamische en – waar mogelijk – ook kinetische gegevens. Om deze te verklaren, zijn er drie criteria onderzocht, die een indicatie geven voor het gemak waarmee een additiereactie met de vorming van een radicaal (nitroxide in dit geval) doorgaat. Ten eerste zal er gekeken worden naar de sterische hindering rond de dubbele bindingen van de nitronen. De additie aan de dubbele binding zal gemakkelijker zijn naarmate de binding sterisch minder afgeschermd is. Vervolgens wordt de exotermiciteit van de additiereactie onderzocht. De additie zal gemakkelijker doorgaan naarmate de reactie meer exotherm is. Ten slotte worden de polarisatie-effecten in kaart gebracht door middel van berekende Mulliken ladingen. Een partieel negatief geladen radicaal centrum reageert immers gemakkelijker met een partieel positief geladen dubbele binding en omgekeerd. Daarna zal de recombinatiereactie tussen het gevormde nitroxide en het cyanoisopropyl


13

N

reaction a

2

N

CN

+ N

CN

CN cyanoisopropyl

AIBN

-

O

O reaction b

2

+

N

N +

CN CN

PBN

CN

nitroxide

CN O

O reaction c

N

N +

CN

CN

CN alkoxyamine

CN

reaction d

+

CN

styrene

styryl derivative

CN

O reaction e

O N

CN

N

+

CN

CN

PNA

Figuur 7: Reacties betrokken in het ’in situ’ NMP proces waarvoor thermodynamische data verkregen is met de BMK/6-311g** methode


14

radicaal onderzocht worden en tenslotte wordt dezelfde reactie in de omgekeerde richting beschouwd. Het is deze dissociatiereactie die van groot belang is bij het verklaren van de initiatie–efficiëntie. Substituent in α-positie Eerst worden nitrone 1, nitrone 2 en PBN (zie Figuur 6) vergeleken. Op basis van de berekende Kc waarden kan verwacht worden dat in de reactie met PBN meer nitroxide gevormd zal worden dan met nitrone 1 en tenslotte nitrone 2, aangezien de evenwichtsligging voor PBN met meest naar de kant van het nitroxide verschoven is (zie Tabel 3). Dit zal verklaard worden aan de hand van de bovenvermelde criteria. De beschouwde structuren hebben dezelfde substituent in β–positie van het C–atoom van de dubbele binding, maar verschillende substituenten in de α–positie. PBN is het minst sterisch gehinderd wegens de planaire structuur van de fenylring, gevolgd door nitrone 1 en tenslotte nitrone 2. De prereactie gebeurt bij een temperatuur van 85◦ C. Bij deze temperatuur neemt de exothermiciteit van de additiereactie af in de volgorde PBN > nitrone 1 > nitrone 2 (zie Tabel 3, reactie b). Wanneer de polarisatie-effecten worden onderzocht, blijkt er dat het cyanoisopropylradicaal centrum een licht negatieve lading draagt. Zowel op PBN als op nitrone 2 is het C–atoom van de dubbele binding positief geladen. Bijgevolg zou vanuit dit standpunt de additie aan nitrone 2 en PBN gemakkelijker zijn dan aan nitrone 1. Het aandeel van elk van de beschouwde effecten in het geheel is onbekend. Men kan echter verwachten dat door de sterke sterische hindering in nitrone 2, de additiereactie met dit nitrone sterk bemoeilijkt zal worden, in verhouding tot het geval met nitrone 1 en PBN. Wanneer dezelfde criteria worden onderzocht voor de recombinatiereactie (reactie c in 7), zijn de polarisatie-effecten van minder belang, aangezien alle recombinatiecentra gelijkaardig gepolariseerd zijn. De sterische effecten zijn hier van groot belang, deze zijn analoog aan het geval voor de nitroxidevorming. Op basis van de thermodynamica kan besloten worden dat PBN de grootste hoeveelheid alkoxyamine zal vormen, gevolgd door nitrone 1 en tenslotte nitrone 2 (zie Tabel 3, reactie c). De terugwaardse reactie c, die de cyanoisopropylradicalen moet vrijstellen die de polymerisatiereactie moeten initiëren, gebeurt bij 110◦ C. Ook bij deze temperatuur zijn de sterische en de thermodynamische effecten dezelfde als in het geval van recombinatie (zie


15

Tabel 3, reactie c bij 110◦ C). Bijgevolg zal deze reactie gemakkelijker doorgaan voor nitrone 2 en nitrone 1 dan voor PBN. Toch wordt er op basis van de bovenstaande uiteenzetting verwacht dat de initiatie–efficiëntie hoger zal zijn dan voor de andere twee nitronen. Dit kan als volgt worden ingezien: initieel is er minder nitroxide gevormd door nitrone 1 en nitrone 2 dan met PBN. Er is bijgevolg ook minder alkoxyamine gevormd met nitronen 1 en 2 dan in het geval van PBN. Daarom zal PBN, ondanks de minder gunstige ligging van de evenwichtscoëfficiënt, toch kwantitatief meer monomeereenheden initiëren.

Substituent in β-positie De initiatie-efficiënties van PBN en DPN zullen met elkaar worden vergeleken. Deze structuren verschillen in hun substituent in β–positie. De sterische effecten zullen in dit geval minder een rol spelen bij de vorming van het nitroxide, aangezien de substituenten in α– positie gelijk zijn (Fischer, 2001). De polarisaties zijn eveneens gelijk voor deze structuren en worden bijgevolg niet beschouwd. Bijgevolg blijkt de reactie–enthalpie de bepalende factor voor reactiviteit te zijn. Uit de berekende reactie–enthalpie waarden blijkt dat de reactie met DPN exothermer is dan met PBN, wat ook te zien is aan de evenwichtscoëfficiënt, die hoger is voor DPN. Ook de vorming van het alkoxyamine gaat gemakkelijker voor DPN dan voor PBN. Dit komt ondermeer doordat DPN met de planaire structuur van de fenyl groep de radicaaldrager (zuurstofatoom) minder afschermt dan PBN, met een tert–butyl groep. Bovendien reageert DPN ook hier aanzienlijk exothermer dan DPN (6 grootteordes verschil). Hieruit kan besloten worden dat DPN veel meer alkoxyamines zal vormen dan PBN. Toch kan er verwacht worden dat PBN een hogere initiatie–efficiëntie zal hebben dan DPN. Immers, de gevormde alkoxyamines op basis van DPN zijn zodanig stabiel (Kc hoger dan 108 kmol m−3 ) t.o.v. het geval van PBN (Kc ≈ 10−1 kmol m−3 ), dat DPN minder geneigd zal zijn om cyanoisopropylradicalen vrij te stellen om het monomeer te initiëren. Bovendien is het styryl adduct dat gevormd wordt in reactie e veel stabieler voor DPN dan voor PBN, wat een lagere conversie in functie van de tijd zou moeten opleveren voor DPN dan voor PBN. Dit wordt eveneens experimenteel waargenomen.


16

Structurele effecten Tenslotte wordt de cyclische nitrone DMPO beschouwd. Voor deze nitrone is de sterische hindering erg laag, en de exothermiciteit van de nitroxidevorming is gelijkaardig aan die van DPN. De evenwichtscoëfficiënt voor reactie c (alkoxyaminevorming) is echter twee grootteordes lager, wat de iets hogere initiatie-efficiëntie van dit nitrone kan verklaren die experimenteel wordt waargenomen. Bij DMPO wordt er immers ook een grote hoeveelheid alkoxyamine gevormd, maar in tegenstelling tot het geval van DPN worden er hier ook meer cyanoisopropylradicalen vrijgesteld om de polymerisatie te initiëren. Doordat reactie e sterk naar links verschoven is voor DMPO, wordt ook hier een lagere conversie in functie van de tijd verwacht, vergeleken met PBN. Dit wordt eveneens experimenteel waargenomen.

5

Modellering van de gewone NMP van styreen

Om meer inzicht te krijgen in de invloed van de afzonderlijke kinetische parameters en van de evenwichtscoëfficiënt Kc op de controle en de snelheid van het NMP proces van styreen, werden er simulaties uitgevoerd van een formeel kinetisch model voor de gewone NMP van styreen met de ab initio berekende kinetische en thermodynamische waarden. Op basis van de uitgevoerde simulaties werd vastgesteld dat de grootte van de evenwichtscoëfficiënt Kc van groot belang is voor het verloop van het polymerisatieproces. Voor een optimaal evenwicht tussen polymerisatiesnelheid en controle moet Kc gelegen zijn tussen 10−11 - 10−10 kmol m−3 . Deze Kc is immers afhankelijk van de structuur van het nitroxide, waardoor een goede keuze van nitroxide van primordiaal belang is. Voor de snelheidscoëfficiënten voor terminatie door recombinatie en disproportionering wordt een aanvaardbare range van 107 - 108 m3 kmol−1 s−1 gevonden. De thermische initiatie van styreen werd eveneens onderzocht. Deze reactie moet zoveel mogelijk worden beperkt, omdat het de controle ondermijnt. Dit kan door in te spelen op de polymerisatietemperatuur. Tenslotte werd de invloed van de ketenoverdracht naar monomeer onderzocht. De snelheidscoëfficiënt voor ketenoverdracht naar monomeer blijkt te bepalen of er een polymerisatieproces plaats zal vinden of een oligomerisatieproces. Om van polymerisatie te kunnen spreken, moet deze snelheidscoëfficiënt lager zijn dan 10−1 m3 kmol−1 s−1 . De verkregen simulaties laten toe een kwalitatief beeld te schetsen van de invloeden van


17

verscheidene kinetische en thermodynamische parameters. Dit is een aanwijzing dat een gelijkaardig model voor het ’in situ’ NMP proces ook diepgaande inzichten zou verschaffen omtrent de belangrijkste invloedsfactoren. Een eerste stap in de richting van een fundamenteel kinetisch model is gezet in deze Master thesis, door alle belangrijke reacties die kunnen optreden tijdens de prereactie van het PBN/AIBN systeem in kaart te brengen. Ook de reacties met het styreen monomeer werden beschouwd.

6

Conclusie

In deze Master thesis werd een berekeningsmethode geselecteerd op basis van een uitgebreide level of theory studie, die toelaat om op een kwalitatieve wijze de experimentele observaties met betrekking tot de ’in situ’ NMP van styreen te verklaren. De invloed van de structuur van het nitrone op de controle en de snelheid van de polymerisatieproces werd onderzocht. Zoals ook bleek uit gerapporteerde experimentele resultaten, heeft PBN (nitrone) de beste kenmerken om in het ’in situ’ NMP proces aangewend te worden. Met behulp van simulaties van het formeel kinetisch model van de gewone NMP van styreen werden er kwalitatieve inzichten verkregen in de invloed van verschillende kinetische en thermodynamische parameters op het NMP proces. De evenwichtscoëfficiënt Kc werd als een van de belangrijkste parameters van het proces ge¨ıdentificeerd. Een gelijkaardig model voor ’in situ’ NMP zou eveneens verregaande inzichten kunnen verschaffen in dit polymerisatieproces. Idealiter kan ook een fundamenteel model ontwikkeld worden voor de ’in situ’ NMP van styreen. Hiertoe werd reeds de eerste stap gezet in deze Master thesis.

INTRODUCTION

18

Chapter 1

Introduction 1.1

Polystyrene

A vinyl compound, styrene (see Figure 1.1), can be converted into polystyrene (PS), that is an amorphous, brittle and hard polymer, used for packaging. In order to compensate for the brittleness of the homopolymer, co– and terpolymers were synthesized via radical polymerization. The most interesting terpolymer consists of acrylonitrile, butadiene and styrene (ABS) and can be applied to fabricate molded products for assembly in the automotive industry. The most widely used copolymer is styrene-butadiene rubber (SBR) e.g. for the production of tires. The monomer is susceptible to various polymerization techniques. On the one hand, styrene can undergo ionic polymerization methods such as cationic polymerization induced by AlCl3 at 80◦ C and anionic polymerization using alkali metals and their hydrides (Flory, 1953). On the other hand, polymerization can occur via a radical mechanism. Industrially most of the polystyrene (and derivatives) are produced by the free radical polymerization (FRP) technique. The drawback of the latter polymerization mechanism is the lack of control over the chain length and architecture and the difficulty encountered to obtain end functionalization (Matyjaszewski, 2002). Because of the growing demand for functionalized, well-defined materials, a new concept overcoming the former problems has been developed, namely the Living/Controlled Radical Polymerization (LRP/CRP). This promising technique is industrially attractive and will be further explained in the next section. The difference between living and Living/Controlled Radical Polymerization (LRP/CRP) will also be enlightened there.

1.2 Living/Controlled Radical Polymerization

19

Figure 1.1: Styrene molecule

1.2

Living/Controlled Radical Polymerization

Living/Controlled polymerization is a synthetic method to prepare polymers that are well-defined with respect to (Bisht and Chatterjee, 2001): 1. Topology 2. Terminal functionalities 3. Composition and arrangement of comonomers 4. Molecular mass predetermined by the ratio of concentration of reacted monomer to introduced initiator (DPn = ∆[M ]/[I]0 , with DPn = degree of polymerization, [M ] = monomer concentration, [I]0 = initial initiator concentration) 5. Narrow polydispersity Living polymerization is characterized by the absence of termination and chain transfer reactions. Ionic polymerizations, i.e. anionic and cationic polymerizations, are examples of living radical polymerizations, in the strict sense. The major disadvantage of these techniques is the sensitivity with respect to traces of oxygen, water or CO2 , that leads to the necessity of purification and drying of the polymerization system. Controlled radical polymerization is based on a radical mechanism. The termination and chain transfer reactions are curbed by a low concentration of radicals. This is achieved by a reversible deactivation of growing radicals/activation of these deactivated structures.


20

K = kd/k c kd P-X Alkoxyamine (dormant species)

kt

+M

dead polymer

kp

+

P

kc

Propagating radical (active species)

X Nitroxide

Figure 1.2: Basic mechanism of the nitroxide mediated polymerization (NMP); P • = propagating radical, X • = nitroxide, P − X = alkoxyamine, M = monomer; kd , kc , kp , kt = rate coefficients for dissociation (activation), coupling (deactivation), propagation and termination reactions respectively

The three major classes of controlled radical polymerization are the stable nitroxidemediated living free radical polymerization (NMP), the atom transfer radical polymerization (ATRP) and the reversible addition-fragmentation chain transfer (RAFT). In this Master thesis, the focus lies on the NMP mechanism. As shown in Figure 1.2, the basic mechanism is an activation/deactivation process. The deactivation reaction consists of the formation of a labile C–ON bond between the propagating radical (active species) and the nitroxide (radical scavenger), forming an alkoxyamine (dormant species). Dissociation of this bond activates the propagating radical and a limited amount of monomers can be added to the propagating chain before it is deactivated by recombination with the nitroxide. Whenever this equilibrium between active (propagating radical) and dormant species (alkoxyamine) is shifted toward the dormant species (i.e. when the equilibrium coefficient K =

kd kc

≈ 10−11 , see Figure 1.2), the in-

stantaneous concentration of the radicals is decreased and the extent of the irreversible termination reactions (such as combination and disproportionation) and transfer reactions is limited. Moreover, the polymers grow by the same overall rate, yielding a narrow polydispersity. A sufficiently fast initiation is necessary to initiate all chains quasi simultaneously. Though irreversible termination of radicals is inhibited, it still occurs, leading to an excess of nitroxide (persistent) radicals in the reaction medium. As time develops, the nitroxide accumulation becomes significant, thus the nitroxide/propagating radical coupling becomes more favored compared to the irreversible termination reaction. This is referred to as the persistent radical effect (PRE), first introduced by Fischer (1997). A cumyl–TEMPO system was studied experimentally in order to gain deeper insight in the PRE by Kothe et al. (1998). The rate coefficients and the mechanistic details of the


21

ln([M] o / [M])

living steady state termination

slow initiation

time

Figure 1.3: Schematic effect of slow initiation and unimolecular termination on the polymerization kinetics (Matyjaszewski, 2002)

reactions were determined. Recently Tang et al. (2006) reevaluated the original Fischer equations, making them valid for higher persistent radical concentrations than originally done by Fischer (1997). There is no strict criterion to identify a controlled process, yet in practice it is known that indications for well controlled systems may be (Matyjaszewski, 2002): A linear kinetic plot in semilogarithmic coordinates (ln[M ]0 /[M ] vs time), if the reac-

tion is first-order with respect to the monomer concentration. Acceleration on such a plot may indicate slow initiation, whereas, deceleration may indicate termination (see Figure 1.3). Straight lines indicate only a constant number of active sites and will also be present under steady state conditions, typical for any radical polymerization (RP). A linear evolution of the number average molecular mass (Mn ) with conversion. A

Mn lower than predicted by the [M ]/[I]0 ratio indicates either inefficient initiation or chain coupling (see Figure 1.4). Straight lines indicate only a constant number of all chains (dead and growing) and cannot detect unimolecular termination (or termination by disproportionation in RP). The polydispersity, defined as Mw /Mn , where Mw stands for the mass average molec-

ular mass, should decrease with conversion for systems with slow initiation and slow exchange between active and dormant species. Polydispersities increase with conver-

1.3 Nitroxide mediated radical polymerization time

22

slow initiation

Mn transfer

Monomer Conversion

Figure 1.4: Schematic effect of slow initiation and transfer to monomer on the polymerization kinetics (Matyjaszewski, 2002)

sion when the contribution of chain breaking reactions becomes significant. The end-functionality is not affected by slow initiation and exchange but is reduced

when chain breaking reactions (transfer, termination, etc.) become important. Over the past three decades a lot of work has been done by various groups with respect to the NMP process. The controllability of the NMP process has been improved systematically. The major developments will be discussed in the next section.

1.3

Nitroxide mediated radical polymerization

The earliest attempts to use nitroxides as polymerization mediators are situated in the early 1980s. Initially, bimolecular initiators were used, i.e. a radical initiator and a nitroxide. These two different agents were simultaneously introduced in the polymerization medium containing the monomer. Rizzardo and Solomon (1979) have investigated the ditert-butyl peroxalate/2,2,6, 6-tetramethylpiperidinyl-N-oxyl (TEMPO) bimolecular initiation of methyl acrylate at low temperatures, with no observation of polymerization. At increased temperatures, the polymerization was found to be out of control. Georges et al. (1993) have reported a styrene polymerization with a bimolecular benzoylperoxide (BPO)/TEMPO initiating system. This method was also studied in detail by Veregin et al. (1995) and found to be successful, yielding narrow polydispersities. Though the bimolecular initiation of the polymerization yielded promising results, there

1.3 Nitroxide mediated radical polymerization

23

Figure 1.5: Nitroxides for NMP: 2,2,6,6-tetramethylpiperidinyl-N-oxyl (TEMPO) (1), 2,2,5-trimethyl4-phenyl-3-azahexane-3-nitroxide (TIPNO) (2), N-tert-butyl-N-[1-diethylphosphono-(2,2dimethylpropyl)] nitroxide (DEPN) (3)

is little known about the concentration and the structure of the initiating species. This shortcoming has prompted the development of a single molecular initiating system. A single molecular initiator is an alkoxyamine derivative with a thermally unstable C–ON bond. When heated, an initiating radical and a mediating nitroxide radical are being released in equal amounts. Using a single molecular initiator allows to have full control of the structure and of the amount of the formed initiating species. There is no significant difference found in the polymerization rate between the unimolecular and the corresponding bimolecular systems, yet the unimolecular system offers better control over molecular mass and polydispersity (Hawker et al., 1996). As aforementioned, the first type of nitroxide being used in the NMP process was TEMPO. Though it allows for the polymerization of styrene to be controlled, there are still some major disadvantages when using this nitroxide as a mediator. At slightly increased temperatures (up to 60◦ C) TEMPO forms a stable C–ON bond with the propagating alkyl radical, thus high polymerization temperatures (120-140◦ C) and long reaction times (up to 72 hours) are needed for the polymerization process. Moreover, TEMPO can only be used for the polymerization of styrene, dienes and copolymerization of styrene with butyl acrylate or methyl methacrylate. The polymerization of acrylates is found to be out of control with TEMPO as a mediator. A great amount of nitroxides has been tested in the past decade (Hawker et al., 2001) and the great importance of the structure of the mediating nitroxide radical for the course of the polymerization has been revealed. This implies that a modification of the structure can make the NMP process compatible with more monomer families and imply more favor-

1.3 Nitroxide mediated radical polymerization

24

able reaction conditions. The best known second generation nitroxides are the phosphonated acyclic nitroxide N-tert-butyl-N-[1-diethylphosphono-(2,2-dimethylpropyl)] nitroxide (DEPN) (Benoit et al., 2000b) and 2,2,5-trimethyl-4-phenyl-3-azahexane-3-nitroxide (TIPNO) (see Figure 1.5). The use of these structures as mediators allows the polymerization of a great variety of monomers, such as acrylates, acrylamides, 1,3-dienes and acrylonitrile based monomers (Sciannamea et al., 2006). Besides the compatibility with the major groups of monomers, NMP offers some important advantages. Compared to other polymerization techniques, NMP has a high tolerance toward impurities (traces of oxygen, water or CO2 ). This bypasses the necessity of purification and drying of the polymerization medium. Moreover, the temperature control is relatively easy and it is found that, for bulk polymerization of styrene applying the NMP technique, no Trommsdorf effect is encountered (Pavlovskaya et al., 2002). Therefore no runaway reactions occur, which significantly reduces the reactor and operational costs. Finally, well-defined polystyrene random and block copolymers can be prepared in a more convenient way compared to the previously used anionic polymerization. This offers great opportunities toward the generation of new materials which cannot be prepared by other LRP techniques. However, the success of the NMP process is highly dependent on the structure and the availability of the used nitroxide. The preparation of these nitroxides is an expensive and time consuming task, making the NMP process unsuitable for industrial applications, especially with respect to the preparation of commodity polymers, such as polystyrene and acrylates. Therefore, a faster and cheaper method of preparation of nitroxides has emerged, namely the in situ NMP.

1.4 In situ nitroxide mediated radical polymerization

Initiator

25

Nitroxide

Monomer

System 1

O

O

O N O O

BPO

PBN

Styrene (St)

System 2 O N

N

N CN

CN AIBN

PBN

Styrene (St)

Figure 1.6: Systems studied by Pavlovskaya et al. (2002) and Sciannamea et al. (2007): System 1 = BPO/PBN/St and System 2 = AIBN/PBN/St

1.4

In situ nitroxide mediated radical polymerization

In situ NMP follows the same initial concept as regular NMP, yet in the former technique the nitroxides are formed in the polymerization medium itself, based on precursors, so no separation processes are required. Such precursors may be nitrones, nitroso compounds, sodium nitrite, nitric oxide, hydroxylamines and secondary amines. Special attention has been paid to nitrones for the CRP of various monomers. Nitrones form indeed stable nitroxides by reaction with radicals. No other precursors for in situ NMP have yet been reported in literature. In literature two main approaches were reported for the in situ NMP. Pavlovskaya et al. (2002) have reported a CRP in a system comprising BPO, C-phenyl-N-tert-butylnitrone (PBN) and styrene (St) and in another system comprising azobisisobutyronitrile (AIBN), PBN and styrene (St). The radical initiator, the nitrone and the monomer are brought in the polymerization medium simultaneously, yielding a polymerization in a controlled way, for both systems under consideration (see Figure 1.6). These two systems were also studied by Sciannamea et al. (2007). For the BPO/PBN/St system the polymerization is reported to be under control, but they did not observe control for the AIBN/PBN/St system. This is in contradiction with the findings of Pavlovskaya


26

NMP Mechanism

In situ NMP Mechanism

Initiator decomposition

Prereaction

I

I

2I

I

2I

I

O R’

O

I R’

N

N

R’

R’’

R’’

R’’

I

O

I

N

I

I

I

O

N R’’

Initiation

I

Initiation (T )

oxyamine

O R’

I

O

N

R’

N

+

R’’ I

I

+

I

I

R’’ I

+

I

I

Propagation

Propagation

P

P

P

I

+

I

n

I

n

+

I

n

n

n

O R’

P

N

O R’

R’’ R

+St

R’’ R

O

P

+

N

kp

R’

P

N

O R’

R’’ I

P

+

N

+St

R’’ I

kp

Figure 1.7: NMP mechanism (left) and in situ NMP mechanism (right): (I −I = initiator; I • = initiator radical; R, R0 , R00 = alkyl groups; P • = propagating radical; St = styrene; n = number of monomers)


27

et al. (2002). In order to obtain control for the AIBN/PBN/St system, a prereaction step has been introduced by Sciannamea et al. (2007). The principle of this latter approach is shown in Figure 1.7 on the right. Preceding the polymerization process, there is a prereaction step. During the prereaction, only the radical initiator and the nitrone are present in the reaction medium. The addition of the initiator radical to the nitrone yields a nitroxide, which can recombine with another initiating radical with the formation of an alkoxyamine. The aim is to convert all of the nitrone and initiator into alkoxyamines, so the equilibrium coefficient should be quite large. After the prereaction procedure, no purification step is required. The monomer is directly added in the prereaction medium and the temperature is risen. At higher temperatures the labile C–ON bond of the alkoxyamine breaks, yielding an initiating radical and a nitroxide radical. The initiator radical then initiates the monomer, subsequently the polymerization process proceeds according to the general NMP mechanism described previously in Section 1.3. Comparing the NMP process with the in situ NMP process, some differences should be noticed. While the preparation of the nitroxide used in the NMP process is an expensive, time consuming job, the nitroxide in the in situ NMP process is synthesized in a straightforward way from relatively cheap precursors and the preparation time is significantly reduced. Moreover, bypassing the purification steps for the nitroxide makes the in situ NMP process more environmentally friendly. These advantages make the in situ process interesting for industrial applications. Unfortunately, the in situ NMP process is more sensitive toward reaction conditions than regular NMP, e.g. duration of the prereaction, reaction temperature etc. Therefore it is important to have an insight in those conditions that might have an influence on the in situ NMP process. These conditions are discussed in the following paragraphs. Influence of the nitrone structure on the polymerization process

The structures of both the nitrone and the radicals formed in the polymerization medium dictate the structure of the in situ formed nitroxide and have a decisive effect on the result of the in situ NMP process. Several studies clearly indicated that the ability of the nitroxides to control the styrene polymerization process is parallel to the steric crowding


28

Figure 1.8: Nitrones for the in situ NMP: N-tert-butyl-α-isopropylnitrone (1), N-tert-butyl-α-tert-butylnitrone (2), N-α-diphenylnitrone (DPN)(3), 5,5-dimethyl-1-pyrroline-N-oxide (DMPO)(4), C-phenyl-N-tert-butylnitrone (PBN)(5)

around the N-O moiety of the nitroxides (Mannan et al., 2007). For example, the rate coefficient for the reversible deactivation (kc ) of polystyryl radicals by DEPN (see Figure 1.5 (3)) appears to be much lower than this rate coefficient of reversible deactivation by TEMPO radicals; this reflects the difficulty encountered by DEPN in trapping moderately reactive polystyryl radicals due to its hindering phosphonate substituent. Recently Sciannamea et al. (2005) have investigated 5 types of nitrones (see Figure 1.8) in an AIBN/nitrone/St system. For all the nitrones under consideration, a linear dependence of the number average molecular mass (Mn ) on conversion was found. Also the time dependence of ln([M ]0 /[M ]) was found to be linear. The initiator efficiency (f=Mn,theor /Mn,exp ) for systems with different nitrones was investigated. Comparing the structures of nitrone 1 and nitrone 2 shows that they differ in a substitution of an isopropyl group of the nitrone 1 by a tert-butyl group (nitrone 2). It was found that nitrone 2 yields higher molecular mass polymers than nitrone 1, thus the initiator efficiency of nitrone 2 is lower. This observation clearly indicates that the addition of the initiating radicals to nitrone 2 with formation of nitroxides and eventually alkoxyamines by recombination, is more sterically hindered than in the case of nitrone 1. Therefore less alkoxyamines are formed during the prereaction with nitrone 2, yielding higher molecular mass polymers. Substitution of both alkyl groups by phenyl groups (nitrone 3) improves the initiation


29

efficiency significantly. The initiator efficiency is comparable with that of nitrone 5. With nitrone 4 twice as high initiator efficiencies can be obtained as for nitrone 5. In line with the conclusions drawn from comparison of nitrone 1 and nitrone 2, it could be anticipated that using the less sterically hindered nitrone 3 and nitrone 5, the initiator efficiency would be higher. This is confirmed experimentally. Note that although a phenyl group contains more heavy atoms than a t-butyl group, the former causes less steric hindrance, due to the planarity of the structure. A kinetic model will be able to give more insight in the observations and the differences between the nitrones.The polydispersities were observed for the different nitrones. For nitrone 1, nitrone 2 and nitrone 4 a polydispersity index (PDI) of about 1.7 was found. For nitrone 3 the polydispersity increased strongly with conversion; at 30% conversion PDI was found to be about 5. With nitrone 5 a PDI of about 1.3 was found, and the polydispersity did not vary with conversion. In conclusion, it is found that the use of nitrone 5 yields relatively low polydispersity and relatively high initiator efficiency. This nitrone provided the best characteristics for the in situ polymerization process. When substituting the phenyl group of nitrone 5 with electron donor and electron acceptor groups, only minor changes in the polymerization behavior were observed (Sciannamea et al., 2005). Influence of the prereaction time on the alkoxyamine formation

During the prereaction step of the in situ NMP process, the nitrone has to react with the initiator radical with the formation of nitroxides and alkoxyamines. Hence, it can be anticipated that the duration of the prereaction step will have an influence on the amount of alkoxyamines formed in situ. Sciannamea et al. (2006) have investigated the influence of the PBN/AIBN prereaction time on the styrene polymerization process. PBN and AIBN have been prereacted for respectively 1, 2 and 4 hours, followed by addition of the styrene monomer and styrene polymerization at elevated temperatures. It was found that the chain length is under control regardless of the prereaction time. Furthermore, the initiator efficiency is higher when the prereaction time is shorter, thus lower molecular mass polymers are produced at shorter prereaction times. This observation is surprising, because lower molecular mass should indicate that more alkoxyamine (ini-


30

tiator) is available, which should be the case at longer prereaction times. This can be explained as follows: for the short prereaction times (1 and 2 hours), a burst effect is observed. This is due to the unreacted AIBN, still present in the polymerization medium. At elevated temperatures AIBN rapidly releases cyanoisopropyl radicals, which contribute to the monomer initiation. On the other hand, at 4 hours of prereaction an induction period occurs. This indicates that excess nitroxide has been formed during the prereaction step. This nitroxide has to react with self-initiated styryl radicals first, before the NMP polymerization process can start. The maximum amount of alkoxyamine is found to be formed after 4 hours of prereaction. The polydispersity decreases when the styrene conversion increases. This is commonly observed for polymerization processes under control (Matyjaszewski, 2002). The polydispersity is found to be higher when the prereaction time is shorter. This could be expected because at short prereaction times unreacted AIBN dissociates into cyanoisopropyl radicals (initiator radicals), yet no nitroxide is available for mediation. This leads to the formation of short dead polymers, increasing the polydispersity. Finally, based on the analysis of the electron spin resonance (ESR) spectra, it was found that the polystyryl radicals produced by autopolymerization of styrene do not contribute significantly to the whole process (Sciannamea et al., 2007). Hence, the nitroxides formed during the polymerization process predominantly result from reaction of cyanoisopropyl radicals with PBN. Based on the observations described above, it can be concluded that a 4–hour prereaction is optimal for the further polymerization process, as a maximum amount of alkoxyamine is formed. This leads to the lowest polydispersities and to an optimal control of the overall process. Influence of the nitrone/initiator molar ratio on the polymerization process

In the prereaction medium, the equilibrium between the formed nitroxide and alkoxyamine is aimed to be shifted toward the alkoxyamine, as the latter will act as the initiating species in the further polymerization process. The influence of the PBN/AIBN molar ratio on the yield of the in situ formed nitroxide and of the alkoxyamine has been studied at constant St/PBN molar ratio. PBN/AIBN molar ratios of 1/1, 2/1 and 4/1 have

1.5 Experimental data

31

been investigated (Sciannamea et al., 2006). For PBN/AIBN ration of 1/1 and 2/1 a comparable polydispersity index (PDI) of about 1.35 was found. Unlike the PBN/AIBN ratio of 4/1, that lead to a higher PDI of about 1.50. Remark here that for the PBN/AIBN ratio of 1/1 and 2/1 a linear dependence of molecular mass on monomer conversion and ln ([M]0 /[M]) on time was observed. However, when the PBN/AIBN ratio equals 4/1, ln ([M]0 /[M]) versus time is only linear for the first 3–4 hours and the molecular mass is higher than that observed in the other cases. This may indicate a lower radical concentration in case of the PBN/AIBN ratio of 4/1. Because the polymerization rate and polydispersity is quite similar for the PBN/AIBN molar ratios of 1/1 and 2/1, it is preferable to conduct the prereaction with the latter, because this way less initiator is required. Influence of the temperature, prereaction time and PBN/AIBN molar ratios on the equilibrium coefficient K

Experiments have been conducted for the PBN/AIBN molar ratio of 2/1 after a prereaction of 4 hours at four temperatures (90, 100, 110 and 120◦ C). The evolution of molecular mass with monomer conversion is found to be linear and independent of temperature. This observation is consistent with the formation of the maximum amount of alkoxyamine after a 4–hour prereaction and with a similar efficiency of the alkoxyamine toward initiation of styrene irrespective the temperature. The equilibrium coefficient K (about 9 · 10−10 kmol m−3 ) is similar to K values obtained for classical NMP based on preformed alkoxyamines (about 6 · 10−9 kmol m−3 ). As can be expected, the polymerization is faster as the temperature rises in line with a higher propagation rate coefficient (kp ) and a more important cleavage of the alkoxyamines. Only slight differences are observed at 110◦ C for the polymerization of styrene performed under the different prereaction conditions, such as different PBN/AIBN molar ratios and prereaction times. This observation suggests that the nitroxides and alkoxyamines formed under these conditions must be identical, and that the in situ NMP process is governed by the same active/dormant species equilibrium (Sciannamea et al., 2006).


32

R

R

O

N

N

R 1

N

O

O

R 5

2 : R = CH3 3 : R = C2H5 4 : R = n-C3H7

O

O

CN

7

6

8

O

O

N 9

O

O 10

Figure 1.9: Nitroxides and radicals studied by Moad and Rizzardo (1995)

1.5

N Experimental data

In this Master thesis a kinetic model for the in situ NMP process will be derived. In order to evaluate the model, experimental data is required. Unfortunately, the available data is scarce. 1.5.1

Kinetic parameters

Rate coefficients for the cross-coupling reaction of several carbon-centered radicals with various nitroxides (e.g. TEMPO, DEPN, TIPNO) and their temperature dependence have been reported. The temperature dependence of the rate coefficient of reversible deactivation (kc ) shows a non-Arrhenius behavior: it increases with increasing temperature at low temperatures and then approaches a limiting value. This behavior is most likely due to the large negative activation entropy contribution to the free energy term. For the rate coefficient of dissociation (kd ) the normal Arrhenius behavior is observed (Sobek et al., 2001; Marque et al., 2000). When considering the reversible trapping of polystyryl radicals by DEPN, the kc appears to be relatively low for a rate coefficient of recombination between two radicals; this re-


33

Table 1.1: Half-Lives (s) of Alkoxyamines in Ethyl Acetate Solution a

a

Nitroxide

Half-lives (s)

1 2 3 4 5

16800 46800 1980 1860 3900

Alkoxyamines based on nitroxides 1–5 and cyanoisopropyl radical (6) (see Figure 1.9), half-lives at 60◦ C

Table 1.2: Half-Lives (s) of Alkoxyamines in Ethyl Acetate Solutiona

Radicals

Half-lives (min)

6 7 8 9 10

1980 >60000b 4500 3300 7380

a

Alkoxyamines based on nitroxide 3 and radicals 6–10 (see Figure 1.9), half–lives at 60◦ C b At 80◦ C

flects the difficulty encountered by DEPN in trapping reactive polystyryl radicals owing to its hindering phosphonate substituent (Benoit et al., 2000a). Half-lives for a range of alkoxyamines based on initiator-derived radicals or low molecular mass propagating species have been measured experimentally by Moad and Rizzardo (1995). The values are dependent on the structures of both the nitroxide and the radical components: as shown in Table 1.1, homolysis rates increase with the ring size of the nitroxide fragment. An increase in size of the substituents in α-position to the nitroxide nitrogen also results in enhanced homolysis rates. Table 1.2 shows that the homolysis rate increases with increasing steric hindrance of the radicals, as could be expected. These measurements give a qualitative indication about the strength of the NO–C bond of the alkoxyamine. Several research groups have reported experimental values for kd of systems comprising styrene and various nitroxides. Some of those values are listed in Table 1.3. For the DEPN/St system, for example, the kd is found to be two to four times larger than for the


34

Table 1.3: kc and kd values for several NMP systems

system

kc (l mol−1 s−1 )

kd (s−1 )

T (K)

source

TEMPO/St TEMPO/PS DEPN/PS

7.6 · 107 8.0 · 107 5.7 · 105

1.6 · 10−3 8.0 · 10−4 3.4 · 10−3

393 393 393

Goto et al. (1997) Greszta and Matyjaszewski (1996) Benoit et al. (2000a)

TEMPO-based alkoxyamine at 120◦ C. For a more profound understanding of the rate coefficients of the NO-C bond cleavage (kd ) in TEMPO- and DEPN-based alkoxyamines, a multi-parameter analysis was performed by Bertin et al. (2005). The alkoxyamines were analyzed in terms of polarity, steric effects and radical stabilization contributions of the leaving alkyl radicals. It was found that the polar effect depends strongly on the structure of the nitroxyl fragment; that is, for the weakly polar nitroxyl fragment, the influence of the polarity of the leaving alkyl group on the homolysis is weak, and vice versa . It may be concluded that the rate coefficients increase with increasing electron withdrawal, steric hindrance and stability of the leaving alkyl radicals. In general, a large equilibrium coefficient for the reversible cleavage kd /kc leads to short polymerization times and a large product kd kc provides small polydispersities. Moreover, a large kd is required to obtain a linear increase of the degree of polymerization with increasing conversion already at low conversions (Sobek et al., 2001). 1.5.2

Thermodynamic parameters

Knowledge of the bond dissociation energies (BDEs) for the NO–C bond in alkoxyamines plays a pivotal role in understanding the polymerization dynamics. An insight into structural effects on the energetics is necessary in order to design new nitroxide derivatives to provide bond cleavage within a specific temperature range. The same value for BDE of TEMPO–H was reported by Mahoney et al. (1973) and Marsal et al. (1999), namely 291 kJ mol−1 . For the BDE of TEMPO–styryl several values were reported, which vary within a range of 15 kJ mol−1 around 127 kJ mol−1 . These variations are most likely due to the difference in experimental conditions. (Marsal et al., 1999; Skene et al., 1998; Ciriano et al., 1999). For instance, for values reported by Ciriano et al. (1999) different solvents have been used

1.6 Theoretical calculations

35

to conduct the experiments, namely trichlorobenzene (TCB), dimethyl sulfoxide (DMSO) and toluene (Li et al., 1995). Moreover, in some experiments an average value for BDE is reported for a temperature range of 50◦ C (Skene et al., 1998). Also Mannan et al. (2007) have investigated the stability of the NO–C bond of a nitroxide– styryl compound. The nitroxide was a TEMPO derivative, bearing bulky substituents. An activation energy (Eact ) of 124.5 kJ mol−1 and a pre–exponential factor (Aact ) of 1.4 · 1015 s−1 were reported. These values were in good agreement with the values found for other TEMPO–styryl systems (Skene et al., 1998; Ciriano et al., 1999; Goto et al., 1997). An equilibrium coefficient K of 2 · 10−11 kmol m−3 is reported for TEMPO–polystyrene (Goto et al., 1997; Greszta and Matyjaszewski, 1996). The K for TEMPO–styryl was also measured and this value was found to equal that for the TEMPO–polystyrene. This indicates that the length of the polymer chain does not have a significant effect on the reaction equilibrium. In addition, K values were reported for the DEPN/St system (Benoit et al., 2000a). It is found that the mean value of K is at least 2 orders of magnitude higher than the one determined for the TEMPO–mediated polymerization of styrene, indicating that DEPN–controlled systems permit a higher concentration of active radicals and yet exhibit a controlled character.

1.6

Theoretical calculations

The nitroxides TEMPO, DEPN and the nitrone PBN, have extensively been studied for the NMP of styrene. Moreover, DEPN was shown to be efficient also during the NMP of various acrylic monomers (Benoit et al., 2000a). Nevertheless, the development of appropriate nitroxides and nitrones is still a challenge. That is why molecular modeling could be of great help to estimate, before their experimental synthesis, the efficiency of new nitroxides and nitrones for NMP. Moreover, a rapid procedure to estimate a priori the BDE (NO–C) or the dissociation rate coefficient kd of new alkoxyamines may help organic and polymer chemists in the design of new suitable mediators. Finding these data through ab initio calculations can also be of great importance in the simulation/modeling of the polymerization process where experimental data is lacking. Recently, some groups have started to apply ab initio calculations to study the mechanism of bond cleavage in

O

I

O

N

R’

N

R’

+

R’’ I

I

1.6 Theoretical calculations O

O

P

N R’’

P

N

R’

+St

R’’

R’’

N O + R +St

I

Alkoxiamine N

Nitroxide

P

N

R’ N O R

36

St

O

I

R

I

R’’

O R

Nitroxide

N

+

RO

Figure 1.10: Competition between homolytic N–OC and NO–C bond cleavage in alkoxyamines

1

2

O

3

N

O

+ R

N

O

+ R

N

O

+ R

N

OR

O

N

OR

N

OR

Figure 1.11: Reactions studied by Marsal et al. (1999): 1. recombination/dissociation reaction of TEMPO–H system, 2. recombination/dissociation reaction of 4-oxo-TEMPO–R system, 3. recombination/dissociation reaction of di-tert-butyl-nitrone–R system with R = alkyl substituent.

alkoxyamines. The NO–C bond in alkoxyamine is usually considered as the most labile but, at high temperatures, the competitive N–OC homolytic bond cleavage must also be considered (see Figure 1.10). This pathway is not reversible and leads to two reactive radicals: an aminyl radical and an alkoxyl radical, which are both able to initiate radical O

O et O al. Gaudel-Siri

O

O

O

reactions.

(2006) have studied

O the

BDE of

O O O some

common

O N O nitroxidesCN and

N

CN

CN radicals,

using the semi-empirical PM3 and DFT methods. It was found that PM3 calculations O O H BDE, whereas the H overestimate the slightly BDE is underestimated by DFT calculations. N

O

H

N

O

N

N

No reliable results were H found, and it was emphasized that ion-pair-ion-pair Nrepulsions H

N

are not well represented by semi empirical methods.

OH

O

N O

O

+ O

N

1.6 Theoretical calculations

37

Table 1.4: Calculated activation energies for the AIBN two-bond dissociation (Sun et al., 2004)

Calculated Ea (kJ/mole)

Experimental Ea (kJ/mole)

Solvent

96 100

87–125 87–125

gas phase benzene

The NO–C vs. N–OC bond homolysis was also studied by Gigmes et al. (2006) by thermal degradation, electron spin resonance (ESR) spin trapping, mass spectrometry (MS) experiments and DFT calculations. Alkoxyamines with a primary or secondary alkyl group bound to the O–atom of the nitroxide function mainly underwent N–OC bond homolysis, because the cleavage temperature of alkoxyamines releasing a primary alkyl radical is very high. When the O–alkyl radical is a tertiary alkyl or benzyl group (cfr. styryl), NO–C bond cleavage occurred as the main process. For the reactions shown in Figure 1.11, ∆r H 0 (T ) has been computed by DFT methods by Marsal et al. (1999) (R represents alkyl substituents). The same reactions, yet using DEPN instead of TEMPO were also addressed. The B3LYP, B3P86 and B3PW91 hybrid functionals were used. The effects of polarization and diffuse valence atomic orbitals in the basis set were also examined. Good agreement with experiment was reached for the smallest hydroxylamines and alkoxyamines at respectively the B3P86/6-31G** (structures optimized with HF/6-31G**) and the B3P86/6-31G* (structures optimized with HF/6-31G*) level of theory. Experimental trends were very well reproduced by the theoretical calculations for TEMPO as well as DEPN. Besides the nitroxide and the alkoxyamine behavior, the understanding of the mechanism of the initiator decomposition is of great importance for the overall in situ NMP process. BPO and AIBN are by far the most commonly used initiators for the in situ NMP of styrene. Sun et al. (2004) and Lin et al. (2007) have reported a theoretical study on the thermal decomposition of AIBN. The ab initio study of the two research groups has lead to same conclusions concerning the mechanism: for the three most probable cleavage paths shown in Figure 1.12, the simultaneous cleavage of the N-C bond (Path 2) is found to be most favored based on quantum mechanical calculations. In addition, the calculated reaction enthalpies are about -33 kJ mol−1 , what shows that the two-bond dissociation process is thermally favorable. The calculated activation energies (UB3LYP/6-311g* method is used) are in good agreement with experimental values (see Table 1.4).

1.7 Objective of this Master thesis

38

Path 1a N

N

CN

CN

Path 2

Path 1b

+

N

+

CN

Path 3

N

N

N CN

+ N2 + CN

CN

CN

CN

Figure 1.12: Possible paths for AIBN decomposition: Path 1a and Path 1b: a two step process of each time one N-C bond homolytic cleavage (−N2 ...C−), Path 2: two-bond homolytic cleavage (−C...N2 ...C−), Path 3: one-bond ionic heterolytic cleavage.

1.7

Objective of this Master thesis

In this Master thesis a kinetic modeling of the in situ NMP of styrene will be performed. The ultimate goal is to obtain kinetic and thermodynamic data of elementary reactions involved in this polymerization process by means of ab initio calculations. In particular, the influence of the structure of the nitrone/initiator pair and the reaction conditions on the kinetics and the control of the polymerization process will be investigated. In order to meet the objective, four topics will be examined. First, a literature study concerning the andOsimilar systems will be perO O O O NMP of styrene O NMP and the in situ N the N formed. The main focus lies on gaining insight in the parameters of influence for O O

O

O O

O

CN

in situ NMP process of styrene, i.e. the structure of the nitrone and the initiator, the

C CN

reaction conditions, the presence of a prereaction step etc. Herefore particular attention O

O

H H be paid to the experimental will findings reported by the CERM of the ULg. A literature N

O

H

O

N

N

N

study concerning computational methods in chemistry will be performed as well (reported H H N

in Chapter 2), in order to gain insight in the theories and methods interesting to Oapply in this Master thesis. Second, a level of theory study will be performed as described in Chapter 3, in order N O

to find the best ab initio calculation method for the system under study. Various DFT functionals and basis sets will be examined on a test set of structures and elementary reactions. Thermodynamic quantities, such as enthalpies of formation, enthalpies of reaction, O

O

O 2

O O

O

O

O

O

O

+ O

1.7 Objective of this Master thesis

39

bond dissociation energies (BDE) and kinetic quantities, such as rates of addition and β–scission, will be calculated for the test set and compared with the experimental data and theoretical calculations gathered from literature. The most cost-efficient calculation method will be chosen for further calculations. Thirdly, in Chapter 4 the selected ab initio calculation method will be applied to some elementary reactions involved in the in situ NMP of styrene. Ab initio calculations will be used to model the influence of the nitrone structures on the thermodynamics and the kinetics of this polymerization process. Finally, a formal kinetic model for the NMP process will be developed, in order to investigate the influence of the equilibrium coefficient and various kinetic parameters, such as rate of termination and rate of thermal initiation, on the NMP of styrene. An initial step toward the development of a fundamental model of the in situ NMP of styrene with the PBN/AIBN pair will be derived. Chapter 5 concerns all model developments and performed simulations. The sixth and last chapter summarizes general conclusions and reveals ideas for future work.

COMPUTATIONAL METHODS

40

Chapter 2

Computational methods In situ NMP is a process with great potential toward industrial applications. Before the step toward the industry can be made, a deeper insight in the mechanism of this process is needed. Theoretical methods can provide thermodynamic and kinetic data, which is not always available experimentally. Hence, a deeper insight in the key features controlling the in situ NMP mechanism can be gained. Mainly Density Functional Theory (DFT) methods will be applied in this Master thesis, but in view of completeness, different computational methods will be enlightened, namely: Hartree–Fock calculations Post–Hartree–Fock calculations Density Functional Theory (DFT) calculations

For each computational method the underlying concept will be discussed shortly and the major advantages and drawbacks will be depicted. All methods and calculations explained afterwards are performed with the Gaussian03 program package (Frisch et al., 2004).

2.1

Hartree–Fock calculations

The first methodology developed is the Hartree–Fock Molecular Orbital Theory. The aim was to solve the electronic Schrödinger equation, given by: ˆ = EΨ HΨ

2.1 Hartree–Fock calculations

41

ˆ the Hamiltonian operator that returns the system energy E, as an eigenvalue. with H, Hence, Ψ is an eigenfunction, the many–electron determinental wave function. Namely, the fundamental postulate of quantum mechanics is that this so–called wave function ˆ Ψ, exists for any (chemical) system, and that appropriate operators (in this case: H), which act upon Ψ returns the observable properties (in this case: E) of the system. The electronic wave function Ψ also contains a lot of information, such as dipole moments, polarizability etc. The electronic Schrödinger equation is derived from the time–independent Schrödinger equation, after applying the Born–Opperheimer approximation. This approximation states – without going into mathematical detail – that the motion of the nuclei can be decoupled from the motion of the electrons, because of the significant difference in mss and thus in kinetic energy. Hence, electronic energies will be computed for fixed nuclear positions: the nuclear kinetic energy is taken to be independent of the electrons, correlation in the attractive electron–nuclear potential energy term is eliminated and the repulsive nuclear–nuclear potential energy term becomes a simply evaluated constant for a given geometry. This approximation lead to the existance of a Potential Energy Surface (PES), i.e. surface defined by Eel over all possible nuclear coordinates. Also the concept of equilibrium and transition state geometries were deduced from this approximation, as these are defined as critical points on the PES. Another fundamental assumption of the HF pheory is the so calles self–consistent field approximation, implying that all separate interactions between electrons are replaces by a static field of all of these (other) electrons. Subsequently, all electron correlation is ignored. Remark, however, that electron exchange is accounted for. The HF theory is the most basic of molecular orbital theories, so it has some significant drawbacks. Because the HF theory ignores electron correlation, it cannot realistically be used to compute heats of formation (Feller and Peterson, 1998). In general, the energy associated with any process involving a change in the total number of paired electrons is very poorly predicted at the HF level. Even if the number of paired electrons remains the same but the nature of the bonds is changed, the HF level can show a rather large error. One can imagine that accounting for electron correlation when calculating transition state (TS) energies is important. Yet Wiest et al. (1997) have analyzed TS structures for many different organic reactions using the HF level and found that, when a large basis set is used, the TS structures are of good quality. When studying structures with heteroatoms

2.2 Post-Hartree-Fock

42

bearing single ion pairs, basis sets including polarization functions must be used. Basis sets will be discussed in more detail in Section 2.4. Luckily, the HF level performance improves significantly, when conformational changes, such as rotation about a single bond, are studied. Also protonation/deprotonation energies are predicted well, when appropriate basis sets, accounting for polarization, are used. For minimum-energy structures, the HF geometries are usually very good when using basis sets of relatively modest size. Bond angles and dihedral angles are well predicted, with an average accuracy of 1.5◦ . The HF level is known to overestimate the occupation of bonding orbitals. This leads to predicted bond lengths which are too short. Finally, it must be noted that the HF level scales to O(N 4 ), with N being the amount of basis functions in the used basis set. However, in practice, linear scaling HF implementations are available. Though a lower scaling does not necessarily mean higher speed, it may give a good indication. To reduce the calculation time for geometric optimization, a good initial guess of the structure is of great importance. Summarizing the findings mentioned above, it can be concluded that HF calculations yield good results for geometry studies, yet are not designated for energy calculations.

2.2

Post-Hartree-Fock

As described in the previous section, the HF method yields good results for structural optimizations, yet neglects electron correlation. This method is therefore not suitable for energy calculations. To overcome this shortcoming, other methods have been developed, which take electron correlations into account. These methods are based on the HF method and referred to as post–Hartree–Fock methods (post–HF). A short overview of some post– HF methods is given: 2.2.1

Configuration interaction method (CI)

CI uses a wave function which is a linear combination of the HF determinant and determinants from excitations of electronsand thus accounts for electron correlation. If the expansion includes all possible configuration state functions, then this is a complete CI procedure, which exactly solves the electronic Schrödinger equation within basis set limit. As the complete CI expansion grows exponentially with the system size, the expansion


43

is usually truncated at some order, e.g. CISD, where singly and doubly excited determinants are considered. One of the most appealing features of CISD is that it is variational. According to the variational theorem this means that the energy derived by this method yields an upper bound on the exact energy. Unfortunately, CISD is found to be not ’size– consistent’. The term size consistency is defined loosely in the literature, but it can be intuitively understood as follows: a method is called siz–consistent if it gives an energy E = EA + EB for two non–interacting subsystems A and B (Cramer, 2005). A recent variation of CISD, spin–flip (SF) CISD was proposed by Krylov (2001) and proven to be variational as well as size–consistent. The scaling for CISD is O(N 6 ), which is considerably worse than HF, making this method impractical for all but the smallest systems. A more acceptable way to make truncated CI size–consistent was introduced by Pople et al. (1987). This approach also consists of the addition of higher excitation terms as CISD does, yet now these terms are quadratic in the expansion coefficients, which force size-consistency. In addition, a perturbative treatment of the triple excitations was proposed, giving rise to QCISD(T) theory. This addition has proven to be worthwhile and QCISD(TQ) has even been proposed to include quadruple excitations. QCISD scales as O(N 6 ) while QCISD(T) requires one iteration of O(N 7 ). The results obtained with QCISD and QCISD(T) are comparable to those obtained with coupled cluster methods, respectively CCSD and CCSD(T) (discussed in Section 2.2.3). 2.2.2

Møller-Plesset perturbation method (MPn)

ˆ as a small perThe Møller-Plesset Perturbation theory treats the exact Hamiltonian, H, ˆ 0 – the sum of the one-electron Fock operators. turbation of the HF Hamiltonian, H The electron correlation effects are added to the HF method by means of the Rayleigh– Schrödinger perturbation theory. The order at which the perturbation theory is truncated is defined by the n in the acronym MPn (with n = 2, 3, ...) For example, the MP2 method will contain doubly excited determinants. The main disadvantage of the MPn lies in the following: perturbation theory works best when the perturbation is small. But, in the case of MPn, the perturbation is the full electron-electron repulsion energy, which is a significant contribution to the total energy. Therefore, it cannot be assumed that MPn will give good results, describing the correlation energy. Moreover, the MPn method is


44

not variational. So, it is possible that for instance, the MP2 estimation for the correlation energy will be too large instead of too small. Important to remark is that MPn theory is size–consistent. Perturbation theory relies on the starting wave function being close to the exact wave function. When this is the case, convergence of the MP series is rapid. However, when bonds are stretched the MP series becomes oscillatory. Also, if an UHF (unrestricted HF) wave function with high spin is used, convergence can be extremely slow. Recent results have suggested that with large basis sets, divergence can occur even for systems where HF is a good starting point. For these reasons it can be expected that stand alone MPn calculations will become less popular in the future. Though MP2 is a relatively cheap way to account for correlation (scaling of O(N 5 )), the higher orders (e.g. O(N 7 ) for MP4) become comparatively very expensive, especially considering that a CC (discussed in Section 2.2.3) or QCI (discussed in section 2.2.1) calculation may be more accurate. 2.2.3

Coupled cluster method (CC)

In the CC theory approach (Cizek, 1966), the higher excitations are partially included, but their coefficients are determined by the lower order excitations. This way a non-linear simultaneous system is derived, requiring iterative solution. In practice, the increase in accuracy justifies the inclusion of double excitations next to the single excitations, leading to a so–called CCSD model, which scales to O(N6 ). Inclusion of triple excitations defines the CCSDT system, which scales to O(N8 ), making it extremely computationally expensive, and unattractive for all but the smallest molecules. An approach to estimate the effects of the connected triples using perturbation theory, called CCSD(T), has been proposed by Raghavachari and Head-Gordon (1989). This is the most effective CC method. CC methods are size–consistent, but – unfortunately – not variational. 2.2.4

Composite methods

With composite methods an energy is found, which is a combination of energies calculated with different levels of theory and basis sets. Several types of composite methods are used in practice, such as G2, G3 and CBS. The G3 and the CBS method will be considered in


45

more detail, as similar methods will be applied in this Master thesis. The G3 method comprises of eight steps, which are listed in Table 2.1. The last step (8) is an empirical correction procedure to account for core-valence correlations and to improve the performance of the model over systems having different numbers of unpaired spins. Over time several variations of G3 have been proposed to reduce the computational cost or to increase its accuracy for a particular set of chemical compounds. The G3B3 (Baboul et al., 1999b) model of Baboul et al. (1999a) is one of such variations. For the G3B3 model the geometries and zero-point energies are obtained with the B3LYP/6-31G(d) level (discussed in Section 2.3) instead of geometries from second-order perturbation theory (MP2(full)/6-31G(d)) and zero-point energies from Hartree–Fock theory (HF/6-31G(d)) (see Table 2.1). Generally, the results of G3 and G3B3 are similar, yet G3B3 method yields improved results for some cases where MP2 theory is deficient for geometries, such as CN, O+ 2 , and CCH. Besides the G3 model and its variants, the complete basis set (CBS) models have been developed by Petersson et al. (1998). The G3 and CBS models have certain similarities. In each, geometries and vibrational zero-point energies obtained at relatively low levels of theory are combined with higher level calculations of the total electronic energy to obtain a composite total molecular energy. The main difference between the G3 models and the CBS models is that, rather than assuming basis set incompleteness effects to be completely accounted for by additive corrections, results for different levels of theory are extrapolated to the complete-basis-set limit in defining a composite energy (Cramer, 2005). Four models of CBS exist, which are CBS-4, CBS-q, CBS-Q and CBS-APNO, in order of increasing accuracy, and hence, computational cost. A particular feature of CBS methods is that they include a correction for spin contamination in open shell species. However, studies of transition states for chemical reactions present special problems for methods such as CBS-Q as well as G3, which employ different methods for geometry and zero point energy (ZPE). The unrestricted Hartree-Fock (UHF) ZPE may refer to a different position along the reaction path from that of the single point higher level electronic energy calculation. As in case of the G3 method, some modifications have been made to the CBS-Q method, one of those yielding the CBS-QB3 method (Montgomery et al., 1999a). It combines the general design of the CBS-Q energy calculation with B3LYP density

2.3 Density Functional Theory

46

Table 2.1: Steps in G3 theory for molecules

Step

Calculation

1

HF/6-31G(d) geometry optimization

2

ZPE from HF/6-31G(d) frequencies

3

MP2(full)a/6-31G(d) geometry optimization (all subsequent calculations use this geometry)

4

E[MP4/6-31+G(d)]-E[MP4/6-31G(d)]

5

E[MP4/6-31G(2df, p)]-E[MP4/6-31G(d)]

6

E[QCISD(T)/6-31G(d)]-E[MP4/6-31G(d)]

7

E[MP2(full)/G3largeb]-E[MP2/6-31G(2df, p)] -E[MP2/6-31+G(d)]+E[MP2/6-31G(d)]

8

-0.006386 · (number of valence electron pairs) -0.002977 · (number of unpaired valence electrons)

E0 a b

0.8929 · (2)+E[MP4/6-31G(d)]+(4)+(5)+(6)+(7)+(8)

Core electrons were included in the correlation treatment. http://chemistry.anl.gov/compmat/g3theory.htm

functional theory optimized geometries and frequencies (Montgomery et al., 1999b). The accuracy and the speed of CBS-Q and CBS-QB3 methods are comparable to the G3 method. Both G3 and CBS methods yield outstanding results for the calculation of minimum energy geometries as well as other chemical properties. Unfortunately, the computational cost of these methods is so high, that the applicability of these methods is restricted to the smallest of molecules, generally compounds with a maximum of 6 heavy atoms can be computed.

2.3

Density Functional Theory

In recent years Density Functional Theory (DFT) has become the most popular method in quantum chemistry, accounting for the great majority of all calculations performed today. The reason for this preference is the extreme computational cost required to obtain chemical accuracy with the methods described in the previous sections. The fundamentals of the DFT methods are based on two theorems, which were formulated by Hohenberg–Kohn: the first of these demonstrates the existence of a one-to-one mapping


47

between the electron density and the wave function of a many-particle system. The second theorem proves that the ground state density minimizes the total electronic energy of the system. The main idea of the DFT approach is to replace the many-particle wave function (as used in methods based on the molecular orbital theory, such as the HF method) by the electron density as an observable. Whereas a wave function is dependent on 4n variables (three spatial variables and one spin variable, with n the number of electrons in the system under consideration), the density function depends only on three variables. This simplifies the calculation of geometries and other chemical properties to a great extent. The practical implementation of DFT is based on the equations of Kohn and Sham (1965). They took as a starting point, a fictitious system of non–interacting electrons that have for their overall ground–state density the same density as some real system of interest where the electrons do interact. The Kohn–Sham equations are given by: 1 (− ∇2 + Vme + Vee + Vxc ) Φi = i Φi 2 with Φi the orbitals (independent particle wave function). The first term corresponds with the kinetic energy of non–interacting electrons (Tni ). The second and the third term account for nuclear–electron interaction and electron–electron repulsion respectively. The term Vxc represents the one–electron operator for which the expectation value of the Kohn–Sham Slater determinant is Exc . Exc is called the ’exchange–correlation energy’ but contains the effects of quantum mechanical exchange and correlation, as well as the correlation for the classical self–interaction energy and also accounts for the difference in kinetic energy between the fictitious non–interacting system and the real one. Because there is no predefined form for the exchange-correlation functional, great amount of research has been done to find a function of density to adequately approximate this term. The particular theories used for this approximation have induced a subdivision within the field of DFT functionals and will be discussed in what follows (Sousa et al., 2007). Local density approximation (LDA)

LDA implicitly assumes that the exchange-correlation energy at any point in space is a function of the electron density at that point in space only and can be given by the


48

electron density of a homogeneous electron gas of the same density. The performance of LDA improves with the size of the system and it is especially suitable for systems having slowly varying densities. Generalized gradient approximation (GGA)

Any real system has a spatially varying electronic density. The GGA method takes this into account: the exchange-correlation energies depend on the density as well as on the density gradient. Two types of functionals have been developed based on the GGA method. The first one has an empirical nature and is based on numerical fitting procedures involving large sets of molecules. Exchange functionals that follow this philosophy include Becke88 (B), Perdew–Wang (PW) and modified–Perdew–Wang (mPW). These functionals yield good results for atomization energies and reaction barriers for molecules, yet fail for solid–state physics. The second type of exchange functionals is more fundamentally based on the principles derived from quantum mechanics. This type of functionals include Becke86 (B86) and Perdew. These functionals perform worse for the description of atomization energies and reaction barriers for molecules, yet yield better results for solid–state physics than the fitted functionals do. GGA correlation functionals include Perdew-Wang 91 (PW91) and Lee–Yang–Parr (LYP). Generally, the GGA methods perform much better than the LDA methods. Meta generalized gradient approximation (meta–GGA)

This method is based on the GGA method, yet additionally includes semi-local information beyond the first-order density gradient, which is present in the GGA methods. These methods represent a great improvement in the determination of atomization energies, but these methods are numerically less stable than the GGA methods. Some examples of the meta-GGA methods are B95, TPSS and VSXC. Hybrid generalized gradient approximation (H–GGA)

This method combines the exchange correlation of a GGA method with a percentage of Hartree-Fock exchange. The weight factors for each component are determined empirically. Hybrid functionals render a significant improvement for the most molecular


49

Table 2.2: Overview of thr DFT functionals used in this Master thesis

functional

type

χi

exchange functional

correlation functional

B3LYPa

H–GGAg

20

Becke88

Lee–Yang–Parr

B3P86b

H–GGAg

20

Becke88

Perdew86

BHandHLYPc

H–GGAg

50

Becke88

Lee–Yang–Parr

MPW1PW91d H–GGAg

25

modified–Perdew–Wang91

Perdew–Wang91

BB1Ke

HM–GGAh

42

Becke88

Becke95

BMKf

HM–GGAh 42

BMK

BMK

a

Becke (1993a), Becke (1988), Lee et al. (1988) Becke (1993a), Perdew (1986), c Becke (1993b), Becke (1988), Lee et al. (1988) d Adamo and Barone (1998) e Zhao et al. (2004) f Boese and Martin (2004) g hybrid generalized gradient approximation h hybrid meta generalized gradient approximation i percentage of HF exchange in the functional b

properties and have therefore become very popular in computational chemistry. Some widely used hybrid functionals are B3LYP, B3P86, B3PW91 and BHandHLYP. Hybrid meta generalized gradient approximation (HM–GGA)

This class of functionals is quite similar to the meta-GGA: it combines the exchange– correlation of a meta–GGA method with a percentage of Hartree–Fock exchange. Some examples of this class of functionals are BB1K, MPW1B95 and TPSSh. These functionals improve the results for the determination of atomization energies, as well as for the reaction barriers. The DFT functionals used in this Master thesis are summarized in Table 2.2. Also the type of exchange functional and correlation functional used in each method is indicated. The overall performance of DFT methods improves in the same order as itemized above, but results vary from functional to functional. Moreover the performance of a functional depends strongly on the property under evaluation. For instance, B3LYP functional is known to yield good results for minimum energy structural optimizations, whereas BB1K functional came out to be the best for H-transfer reaction kinetics (Sousa et al., 2007). Finally, it is important to highlight the key difference between the HF theory and DFT.

2.4 Basis set

50

Namely, HF is a deliberately approximate theory, whose development was in part motivated by an ability to solve the relevant equations exactly, while DFT is an exact theory, but the relevant equations must be solved approximately because a key operator (Vxc ,Exc ) has an unknown form (Cramer, 2005).

2.4

Basis set

Molecular orbitals in computational chemistry are expressed in terms of a set of atomic orbitals (AO) (basis set). Therefore, the choice of a basis set is of great importance, regardless of the type of functional. In theory, an infinite basis set has to be used to correctly calculate the electron energy. In practice, however, only a finite basis set can be implemented. Initially Slater type orbitals (STOs) have been used. However, it was found that a linear combination of Gauss type orbitals (GTOs) could approximate the STOs and required a lower computational cost. The linear combination of three GTOs was found to yield the best results, and was denoted STO-3G. This basis set is called a single–ξ or minimal basis set, because there is only one basis function available to describe each type of orbital. Basis sets with two or three functions to describe each AO are also being used, denoted respectively double–ξ and triple–ξ basis sets. This way the flexibility of the basis functions is being increased. It must be noted that, from a chemical point of view, it is more interesting to have a greater flexibility in the valence basis functions than in the core basis functions. Therefore split–valence basis sets have been developed (Pople et al., 1969), such as 6–31g and 6–311g. The nomenclature indicates the contraction scheme. To illustrate this, the 6– 311g basis set will be taken as an example. The first number indicates the number of primitives used in the contracted core functions (6). The numbers after the hyphen indicate the numbers of primitives used in the valence functions (three such numbers, in this case 311 means that this is a triple–ξ basis set). To describe atoms adequately, it is sufficient to have basis functions describing the necessary orbitals for all electrons of the atom. But when molecular systems are considered, far more mathematical flexibility is needed to predict the right geometry. This flexibility is usually added in the form of basis functions corresponding to one quantum number

2.5 Statistical thermodynamics

51

Table 2.3: Formal scaling behavior of various methods in function of the number of basis functions N in basis set

scaling behavior

methods

N3

DFT

N4

HF

N5

MP2

N6

MP3, CISD, CCSD, QCISD

N7

MP4, CQSD(T), QCISD(T)

N8

MP5, CISDT, CCSDT

of higher angular momentum than the valence orbitals. This functions are called the polarization functions, and are denoted with ’*’. A second star, ’**’ indicated that (p) polarization functions have also been added to H and He. Some systems, such as anions or systems with highly excited electronic states tend to be strongly spatially diffuse. If the basis set does not offer the flexibility to account for a weakly bound electron far from the core density, significant errors in calculating the molecular properties may occur. Therefore, the basis set has to include the so–called diffuse functions, ’+’, to account for this limitation. Only one plus, ’+’, indicates that diffuse functions have been added for all the heavy atoms, two plusses, ’++’, show that additionally diffusion functions were added to H and He atoms. In this Master thesis the following basis functions have been used: 6-311g, 6-311g** and 6-311++g**. Table 2.3 provides an overview of the scaling of different computational methods in function of the basis set size. This clearly illustrates that for big molecular systems, the choice of the basis set will be one of the crucial steps toward obtaining computational feasibility.

2.5

Statistical thermodynamics

In computational chemistry typically a single molecule is considered. In order to make the step toward realistic systems where an uncountable number of molecules is present (i.e. moving from a microscopic toward a macroscopic system), methods from statistic thermodynamics (ST) are used. The theory of ST will be discussed briefly. For a more detailed review, the reader is directed to Laidler (1987) and Cramer (2005).


52

Zero-point vibrational energy

In order to determine the lowest vibrational level (not equal to zero) a harmonic oscillator approximation is used. The zero-point vibrational energy (ZPE) will be given by the sum of all energies over all molecular vibrations. Further, the internal energy of a molecule at 0 K is defined as the sum of the electronic energy and the ZPE:

U0 = Eelec +

modes X i

1 hωi 2

(2.1)

where ω i is the computed vibrational frequency and h is Planck’s constant (6.6261 · 10−34 J s). It must be noted that the electronic energy is evaluated in a stationary point on the Born-Oppenheimer potential energy surface (see Section 2.1). Fundamental equations

The macroscopic molecular system under consideration is modeled as a canonical ensemble, where the total number of particles N , the volume V and the temperature T are constants. All equations derived in this section are based on this assumption. The central function of a canonical ensamble in statistical mechanics is the canonical partition function: X

Q(N, V, T ) =

e−Ei (N,V )/kB T

(2.2)

i

where i stands for all possible energy states of the system having energy Ei and kB is Boltzmann’s constant (1.3806 · 10−23 J K −1 ). Within the canonic ensemble, the following thermodynamic relations apply: U = kB T

2

∂lnQ ∂T

(2.3) N,V

H = U + PV

(2.4)

S = kB lnQ + kB T G = H − TS

∂lnQ ∂T

(2.5) N,V

(2.6)

where U is internal energy, H is enthalpy, P is pressure, S is entropy and G is Gibbs free energy.


53

The partition function Q has an extremely complex form when applied to realistic ensembles. Hence, some simplifying assumptions must be proposed. First, it is assumed that the canonical ensemble is an ideal gas, so that the particles are indistinguishable. Thus, the formulation of the partition function becomes [q(V, T )]N Q(N, V, T ) = N!

(2.7)

where q(V,T) is the molecular partition function. Furthermore, the ideal gas law states that P V = RT , where one mole of gas is considered and where R is the universal gas constant (8.3145J mol−1 K −1 ). Hence the P V -term in Equation 2.4 may be replaced by RT . Secondly, an additional assumption is made for the molecular partition function q, namely that the molecular energy can be expressed as a sum of electronic, translational, vibrational and rotational motions: q(V, T ) = qelec (T )qtrans (V, T )qrot (T )qvib (T )

(2.8)

This expression is substituted in Equation 2.7 and the natural logarithm is taken, because of its appearance in Equations 2.3 and 2.5. [qelec (T )qtrans (V, T )qrot (T )qvib (T )]N ln [Q(N, V, T )] = ln N! = N (ln[qelec (T )] + ln[qtrans (V, T )] + ln[qrot (T )] + ln[qvib (T )])

−ln (N !) ≈ N (ln [qelec (T )] + ln [qtrans (V, T )] + ln [qrot (T )] + ln [qvib (T )]) −N lnN + N

(2.9)

where the last step makes use of Stirling’s approximation for ln(N !) when N is large. Next the various components of the molecular partition function have to be expressed in an analytic form.


54

Molecular electronic partition function

The general expression for the electronic partition function is

qelec (T ) =

elec X

gi e−Ei /kB T

(2.10)

i

where gi is the degeneracy of the level i. For typical closed-shell singlet molecules the degeneracy of the ground state equals unity and the various excited states are so high in energy that at temperatures below thousands of degrees they make no significant contribution to the partition function, so this equation can be written as: qelec (T ) = e−Eelec /kB T

(2.11)

In practice, the ground state energy equals zero by definition. As a result the electronic partition function equals unity. On the other hand, when the ground state has a higher spin multiplicity than singlet, the exponential part of Eq. 2.10 still yields unity, yet the degeneracy equals 2s + 1, where s is the spin multiplicity. So in this case the partition function becomes: qelec (T ) = 2s + 1

(2.12)

The above derived approximations for the electronic partition function are sufficient for all calculations involved in this Master thesis. The expressions for internal energy and entropy (Equations 2.3 and 2.5) become: Uelec = 0

(2.13)

0 Selec = Rln (2S + 1)

(2.14)

Molecular translational partition function

The molecular translational partition function has the following form:

qtrans (V, T ) =

trans X i

gi e−Ei (V )/kB T

(2.15)


55

Assuming that the molecule acts as a particle in a cube with a side of length a, the energy level for this system is given by Ei (nx , ny , nz ) =

h2 (n2 + n2y + n2z ) 8M a2 x

(2.16)

where M is the molecular mass and each level is associated with nx , ny , nz ,the three unique quantum numbers. Substitution of Equation 2.16 in Equation 2.15 and replacing the summation by an infinite integral yields (this is legitimate because the energy levels for a particle in a cube lie extremely close to each other): qtrans (V, T ) =

2πM kB T h2

3/2 V

(2.17)

where V is the volume of the box under consideration. As the ideal gas was chosen as a model for the substance one is dealing with, the V in Eq. 2.17 can be replaced by RT /p. In order to compare thermodynamic values, a standard state for the volume or the pressure has to be chosen. Typically a pressure of 1 atm or a volume of 24.5 l at 298 K is chosen as a standard. The internal energy and entropy can be evaluated as: 3 RT 2( " ) 3/2 0 # 5 2πM kB T V + = R ln h2 NA 2

Utrans =

(2.18)

0 Strans

(2.19)

Equation 2.19 is derived from Equation 2.9, where the last two terms (−N lnN +N ) are included in the translational partition function. These terms are temperature independent, so they do not have an influence on Utrans . Molecular rotational partition function

The simplest approach to model molecular rotation is the so-called rigid-rotor approximation. Unfortunately, the general rigid-rotor Schrödinger equation for a molecule does not have an easy solution. However, by generalization of the classical mechanical rigid-rotor problem, a quantum mechanical approximation can be derived, which is generally quite


56

accurate. Within that approximation, the rotational partition function becomes √ qrot (T ) =

πIA IB IC σ

8π 2 kB T h2

3/2 (2.20)

where IA , IB and IC are the principal moments of inertia and σ is the number of pure rotations that carry the molecule into itself. With this form for the partition function, the expressions for internal energy and entropy become: 3 RT 2( " ) √ 3/2 # πIA IB IC 8π 2 kB T 3 = R ln + σid h2 2

Urot =

(2.21)

Srot

(2.22)

It must be noted that in order to calculate Urot and Srot very little molecular information is needed. Only the principal moments of inertia are needed, which is derived from the molecular structure. So any methodology able to predict accurate geometries can be used to construct the rotational partition functions and all other associated thermodynamic data. Molecular vibrational partition function

In order to derive an expression for the vibrational partition function, a sum of individual energies associated with each mode has to be taken (there are 3N − 6 modes for a nonlinear molecule and 3N − 5 modes for a linear molecule, with N the number of atoms).      X X X qvib (T ) =  e−Ej (1)/kB T   e−Ej (3)/kB T  ...  e−Ej (3N −6)/kB T  j(1)

j(2)

(2.23)

j(3N −6)

Each mode will be approximated as a quantum mechanical harmonic oscillator (QMHO), and the zero of energy is taken as the energy of the equilibrium structure plus the ZPE, so the energy of the zeroth vibrational energy level is zero for every mode. With this

2.6 Thermodynamic calculations

57

convention partition function for each mode can be written as QM HO qvib (T ) =

∞ X

e−khω/kB T

k=0

=

1 1−

(2.24)

e−hω/kB T

Substitution of Equation 2.24 in Equation 2.23 yields

qvib (T ) =

3N −6 Y i=1

1

(2.25)

1 − e−hωi /kB T

Applying this partition function provides the following equations for the vibrational components of the internal energy and entropy:

Uvib = R Svib = R

3N −6 X

k

B i=1 3N −6 X i=1

hωi hω (e i /kB T −1 ) hωi

kB T (ehωi /kB T − 1)

(2.26) hωi /kB T

− ln 1 − e

(2.27)

Some important remarks for the calculations of partition functions: External and internal symmetry numbers are implicitly taken into account within the

Gaussian partition functions. The scaling factors applied in these calculations are listed in Table 2.4. Only the Harmonic Oscillator approach is considered in this Master thesis.


58

Table 2.4: Scaling factors applied in calculations of partition functions by the methods considered in this Master thesis

method

scaling factor

B3LYPa

0.99

B3P86a

0.99

MPW1PW91a

0.99

BB1Ka

0.99

BMKa

0.99

BHandHLYPa 0.99 G3B3b

0.96 (default)

CBS–QB3c

0.99 (default)

a

close to the value of 0.9877 advised by Andersson and Uvdal (2005) for scaling of DFT/triple–ξ ZPEs b B3LYP/6–311g(d) c B3LYP/6–311g(d,d,p)

Enthalpy

wC

xH

atom H exp

w Cgraphite

x 2

(298.15 K)

H2

0

f

H (C w Hx Ny Oz

zO

yN

y 2

N2

z 2

O2

atom H cal (298.15

K)

; 298.15 K ) molecule

C w Hx Ny Oz Figure 2.1: Illustration of the calculation of standard enthalpies of formation from ab initio standard enthalpies of atomization (also presented by Saeys et al. (2003)) and experimental atomic enthalpies of formation


59

Table 2.5: Experimental atomization energies used for the calculation of enthalpies of formation (Luo, 2003)

2.6 2.6.1

element

atomization energy [kJmol−1 ]

H

217.998

C

716.680

N

472.680

O

249.180

Thermodynamic calculations For molecules

The standard enthalpy of formation at 298.15 K is calculated according to the following equation: 0 0 ∆f H 0 (Cw Hx Ny Oz ; 298.15K) = w∆atom Hexp (C; 298.15K) + x∆atom Hexp (H; 298.15K) 0 0 + y∆atom Hexp (N ; 298.15K) + z∆atom Hexp (O; 298.15K) 0 0 − (wHcal (C; 298.15K) + xHcal (H; 298.15K) 0 0 + yHcal (N ; 298.15K) + zHcal (O; 298.15K) 0 − Hcal (Cw Hx Ny Oz ; 298.15K))

(2.28)

This equation is schematically depicted in Figure 2.1. The experimental atomization enthalpies for the considered elements are taken from literature (Luo, 2003) and listed in 0 Table 2.5. The Hcal (298.15 K) are the standard enthalpies calculated for the atoms or 0 the compounds, ∆atom Hcal is thus the calculated atomization enthalpy of the compound

under study. The standard enthalpy of formation of a species is calculated for a species in the ideal gas state at standard pressure (1 bar) and a reference temperature of 298.15 K.

2.7 Enthalpies and Free energies of reaction

2.6.2

60

For reactions

Bond dissociation energies (BDE) are derived from these enthalpies of formation for components involved in the homolytic bond scission: BDE(R − X; 298.15 K) = ∆f H 0 (R; 298.15 K) + ∆f H 0 (X; 298.15 K) −∆f H 0 (R − X; 298.15 K)

(2.29)

Analogously, enthalpies of reactions in general can easily be deduced from the enthalpies of formation of the involved compounds.

2.7

Enthalpies and Free energies of reaction

Gibbs free energies for reactions can immediately be calculated from the previous results and derivations. The Gibbs free energy of reaction is obtained as follows: ∆r G0 =

X

=

X

∆f G0prod (T ) −

X

∆f G0react (T )

react

prod 0 ∆f Hprod (T ) −

0 ∆f Hreact (T )

react

prod

−T (

X

X prod

0 Sm prod (T )

−

X

0 Sm

react (T ))

(2.30)

react

0 with Sm the standard molar entropy for a molecule (calculated from Equation 2.5) and

∆f H 0 the enthalpy of formation, as defined in Equation 2.28. From this Gibbs free energy for reaction, the equilibrium coefficient Kc for reaction can be obtained by applying: −∆r G(T ) Kc (T ) = Vmnr −np (T ) exp( ) RT

"

m3 kmol

nr −np # (2.31)

with Vm the molar volume (=

RT p

for the ideal gas assumed), and nr , np the numbers of

reactants and products respectively.

2.8 Kinetic calculations: Conventional transition state theory

2.8

61

Kinetic calculations: Conventional transition state theory

Conventional transition state theory (CTST) (Eyring, 1935; Evans and Polanyi, 1935) uses the potential energy surface (PES) and the so-called activated complex in order to model chemical reactivity. The main idea is that in a chemical reaction, reactants and products can be situated in two valleys which are separated by a mountain pass. The highest point on this mountain pass is called an activated complex or a transition state. This point corresponds to a saddle point on the PES. In the CTST some assumptions are made (Laidler, 1987): Molecular systems that have surmounted the mountain pass in the direction of products

cannot turn back and form reactant molecules again. The reactants are in thermodynamic equilibrium with the activated complex, even when

the whole system is not at equilibrium. The energy distribution among the reactant molecules is in accordance with the Maxwell–Bontzmann distribution. The movement of the reactants in the direction of the reaction coordinate over the

mountain pass can be separated from all other movements associated with the transition state. The movement of the reactants over the mountain pass can be described as a classical

motion over the barrier, quantum effects are being ignored. In order to illustrate this theory, the expression for the rate coefficient will be derived for a typical bimolecular reaction (Wynne et al., 1935): K‡

k‡

A + B (AB)‡ → products

(2.32)

Considering the equilibrium between initial and activated states gives an equilibrium coefficient as: Kc‡ =

[AB ‡ ] [A][B]

(2.33)


62

where Kc‡ is the concentration equilibrium coefficient. According to statistical mechanics, the molecular equilibrium coefficient for a reaction is also given by −∆E0‡ RT

q‡ exp Kc‡ = qA qB

! (2.34)

where the q’s are the partition functions per unit volume, evaluated with respect to the zero–point level of the respective molecule. The energy −∆E0‡ is the molar energy difference between the activated complex in its ground state and the reactants, at the absolute zero with all substances in their ground states. If molecule A contains NA atoms and B contains NB atoms, the activated complex AB ‡ contains NA + NB atoms. Hence, the complex has 3 degrees of rotational freedom and 3(NA + NB ) − 6 degrees of vibrational freedom (assuming AB ‡ a non–linear structure), describing the partition function q ‡ . The motion surmounting the pass corresponds to a loose vibration without restoring force (the complex can form products without any restraints). This motion is described by: lim

ω→0

1 1 − exp

−hω kB T

=

1

1− 1−

−hω kB T

=

kB T hω

(2.35)

Next, the partition function corresponding with the movement along the reaction path is separated from the partition function of the other movements q‡ (referring to only 3(NA + NB ) − 7 degrees of vibrational freedom): q ‡ = q‡

kB T hω

(2.36)

Substitution of Equation 2.36 in 2.34 yields: BT q‡ khω [AB ‡ ] ‡ Kc = = exp [A][B] q A qB And thus: kB T q ‡ hω ω[AB ‡ ] = [A][B] = exp qA qB

−∆E0‡ RT

!

−∆E0‡ RT

(2.37)

! (2.38)


63

The left hand side of Equation 2.38 is the product of the concentration of complexes and frequency of their conversion into products, therefore this equals the rate of reaction, v. kB T q‡ exp v = [A][B] = hω qA qB

−∆E0‡ RT

! (2.39)

Comparing this with the rate coefficient for the overall reaction rate expression v = k[A][B] yields: k = k(T ) =

kB T q‡ exp hω qA qB

−∆E0‡

! (2.40)

RT

In this equation kB is Boltzmann’s constant, T is the temperature (Kelvin) and h is Planck’s constant. qA and qB are the molecular partition functions of the reactants and q‡ is the molecular partition function of the activated complex without the movement along the reaction path. ∆E0‡ is the zero point corrected electronic activation barrier. Other derivations are possible, the interested reader is referred to literature (Laidler, 1987). The dependence of the rate coefficient on the temperature is given by Arrhenius’ law: k(T ) = A.e

−EA RT

(2.41)

This can be also written as ln(k(T )) = ln(A) −

EA 1 . R T

(2.42)

In order to evaluate the pre–exponential factor (A) and activation energy (EA ), the rate coefficient k is calculated at different temperatures using Equation 2.40. The ln(k(T ))– values are plotted as a function of the reciprocal temperature. According to Eq 2.42 the slope of the best linear fit of these plotted values, gives an estimate for

EA . R

A value for

ln A is obtained from the y–intercept. This method is illustrated in Figure 2.2.


ln A

ln k

Ea slope = R

1/T Figure 2.2: Arrhenius plot

64

LEVEL OF THEORY STUDY

65

Chapter 3

Level of Theory Study 3.1

Introduction

In the past decades great interest for living/controlled polymerization techniques has emerged. NMP is one of those promising techniques. As already mentioned in the previous chapters, a new method of NMP, namely in situ NMP is a way of providing high level control over the polymerization process, yet being cost–efficient and environmentally friendly. Though there is some experimental data available concerning in situ NMP, there is no fundamental insight in the parameters which determine the success or failure of this polymerization process. In order to provide deeper insight in the in situ NMP process, the process is modeled by ab initio calculations. But first, a level of theory has to be identified, which describes the studied structures and reactions most adequately. Therefore, a level of theory study (LOTS) is performed in this Master thesis, and the results are presented next.

3.2

Definition of test set, methods and basis sets

In order to evaluate the performance of different levels of theory with respect to the in situ NMP process, a test set of molecules was chosen, based on the availability of reliable experimental data. Moreover, the chosen structures are representative of molecules that are involved in the actual in situ NMP process (see Figure 3.1).

3.2 Definition of test set, methods and basis sets

66

Initiator related structures O

Hydrocarbons

O

O

O O

O

benzoylperoxide (BPO)

(a)

(b) benzoyloxy radical O

O

O

O

O O (c )

(d) methylcarboxyl radical

diacetylperoxide

N

N

(m) 1-methylethylbenzene (n) 1-methylethylbenzene radical

(o) benzene

(p) phenyl radical

(q) ethylbenzene

(r) ethylbenzene radical

CN

CN CN (e) azobisisobutyronitrile (AIBN)

(f) cyanoisopropyl radical

(s) styrene

Alkoxyamine

Nitrone/nitroxide related structures H

H O

N

H

H (g)

(i)

N

O

H

hydroxylamine

N

(h) oxylamine radical

N

N

OH

O

TEMPO-H

(t)

(k) dimethylnitrone

TEMPO-styryl

(j) 2,2,6,6-tetramethylpiperidinyl-N-oxyl (TEMPO) O

O N

O

N

(l) C-phenyl-N-tert-butylnitrone (PBN)

Figure 3.1: Structures studied in the level of theory study

3.2 Definition of test set, methods and basis sets

O

O

67

O 2

O O

O

+ H

O

O

O O O

N

N

O

O

H

O + H

N H

H

+ H

+ N2

2 CN

N CN

CN

H

H

2

N

N

OH

O

+ H

N

+ H

+

O

N O

+

CH3

CN

O O

+

CN

N

N

+

CN CN O O

O N

O

+

O

+

O

N

O

O

O O

O O

+

N

N

+

Figure 3.2: Reactions studied in the level of theory study: dissociation/recombination reactions (top), addition/β-scission reactions (bottom)

In addition, a test set of reactions was defined, including the types of reactions commonly occurring in the in situ NMP process, such as addition/β -scission and dissociation/recombination reactions (see Figure 3.2). As it can be seen in Figures 3.1 and 3.2, the molecules in the test set are rather voluminous, and contain multiple heavy atoms (C, O, N). The aim is to gain thermodynamic information, as well as kinetic data. Keeping this in mind, considering various DFT meth-

3.3 Geometry

68

Table 3.1: The combinations of methods (first column) and basis sets (first row) studied in the level of theory study

6-311g

6-311g**

6-311++g**

B3LYP

+

+

+

B3P86

+

+

+

MPW1PW91

+

+

+

BB1K

+

+

+

BMK

+

+

+

BHandHLYP

+

+

+

cbsb7a 6-31g(d)

G3B3b CBS-QB3b

+ +

a

This basis set has the form of 6-311g(2d,d,p): it improves the flexibility of the polarization functions compared to 6-311g** b The basis set used for the optimization, in combination with the B3LYP functional.

ods (B3LYP, B3P86, MPW1PW91, BB1K, BMK, BHandHLYP) and composite methods (G3B3 and CBS-QB3) seem appropriate, knowing that composite methods are rather included as a benchmark for the smaller molecules, than as a possible applicable method, because of the size of the systems under study, as was already stated in Chapter 2. To obtain accurate results, a sufficiently large basis set must be used. In this LOTS a 6-311g basis set was used (see Section 2.4). Moreover, the influence of the polarization functions (6-311g**) and diffuse functions (6-311++g**) on the heavy atoms and the hydrogen atoms was assessed. A clear overview of the methods and basis set combinations studied in this work is given in Table 3.1.

3.3

Geometry

Geometries were computed for the molecules depicted in Figure 3.1 using the various methods and basis sets (see Table 3.1). In order to evaluate the results, experimental or computational data is required. However, for the structures under consideration, there is nearly no experimental and computational geometrical data available. Only for hydroxylamine (see Figure 3.3 ) some experimental data was found on the NIST (2007) website, and calculated data was reported by Marsal et al. (1999). Our results for hydroxylamine are shown in Table 3.2.

O

2

3.3 Geometry

69

C2

C1 N

First of all, it must be noticed that on average, all methods yield acceptable results for O

the N–H1 and O–H3 bond lengths. C3 C5

C4 (b)

H1 N

O

H3

H2

C2

C1 N O

Figure 3.3: Structure of hydroxylamine

(d)

The largest deviations from experiment are seen for the BB1K and BHandHLYP methods, with underestimations up to 0.011 and 0.013 ˚ A respectively. These methods yield underestimated bond lengths. The MPW1PW91 and the BMK method slightly underestimate bond lengths, whereas the B3LYP, B3P86, G3B3 and CBS–QB3 methods yield slightly overestimated values. The largest variation in bond length between the methods is observed for the NO–bond. Logically, the focus of the following discussion will lie on the NO–bond length. The largest deviations are seen for BB1K/6–311++g** (about 0.045 ˚ A) and BHandHLYP/6– 311++g** (about 0.04 ˚ A), again underestimating the experimental value. In general, the calculations using the 6–311++g** basis set yield poorer results than the other calculations (except for B3LYP). Only 4 calculations gave an overestimation for the NO–bond length, namely B3LYP/6–311g, B3P86/6–311g, MPW1PW91/6–311g and BMK/6–311g. This leads to the conclusion that applying a 6–311g basis set implies an overestimation of the actual bond length. In the case of BB1K and BHandHLYP however, this is compensated by the tendency of the functional to significantly underestimate the bond length. and with the 6–311g, hence, resulting in the best values observed with the 6–311g basis set for these functionals. The same effect is observed for the MPW1PW91, B3P86 and BMK methods, yet less pronounced. However, calculations with the 6-311g basis set using the B3LYP method lead to large deviations from the experimental value (≈ 0.032 ˚ A). For this method the 6-311g** basis set gives more satisfactory results. The results obtained with CBS–QB3 are comparable to the B3LYP/6–311g** as expected (see Table 3.1). The G3B3 values are not so good, resulting in large overestimations of the bond lengths. This was expected as a 6–31g(d) basis set is used for optimization (see Table 3.1).

3.3 Geometry

70

Table 3.2: Geometric data for hydroxylamine calculated with various levels of theory

r(NO)a r(NH1)a r(OH3)a a(ONH1)b B3LYP 6-311g

1.485

1.018

0.973

104.294

6-311g**

1.446

1.018

0.962

103.000

6-311++g**

1.445

1.019

0.963

104.168

6-311g

1.469

1.017

0.971

104.849

6-311g**

1.431

1.018

0.960

104.200

6-311++g**

1.429

1.018

0.961

104.469

6-311g

1.463

1.015

0.969

105.027

6-311g**

1.427

1.016

0.958

104.293

6-311++g**

1.424

1.016

0.959

104.640

6-311g

1.444

1.007

0.961

105.747

6-311g**

1.411

1.010

0.951

104.719

6-311++g**

1.408

1.009

0.952

105.070

6-311g

1.460

1.016

0.968

104.121

6-311g**

1.433

1.018

0.958

104.121

6-311++g**

1.431

1.016

0.959

104.377

6-311g

1.447

1.005

0.958

105.684

6-311g**

1.415

1.008

0.949

104.790

6-311++g**

1.413

1.008

0.951

105.101

G3B3

1.448

1.023

0.970

103.209

CBS-QB3

1.446

1.019

0.962

103.821

expc

1.453

1.016

0.962

107.010

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

a b c

bond length in ˚ A, bond angle in degrees experimental data provided by the NIST (2007) website

It is important to note that bond lengths calculated with the very computationally expensive 6-311++g** basis set are very close to those calculated with the more computationally

3.3 Geometry

71

C2

C1 N O H1 O

N

C

H2

C2

C1 N

OH (a)

(b)

(c )

C2

C1 N O (d)

Figure 3.4: Structure of aminoxyl radical (a), TEMPO–styryl (b), TEMPO-H (c) and TEMPO (d)

friendly 6-311g** basis set. Therefore, geometrically, there are no advantages when using the basis set with diffuse functions. The bond angle a(ONH1) is badly predicted in all cases, the underestimation being up to 4 degrees. On average, the BB1K and the BHandHLYP yield bond angles that are closest H1

to the experimental values, O H3whereas the B3LYP method gives the worst description of the N bond angle.H2 Based on the results listed in Table 3.2, it may be concluded that the values calculated for the bond lengths by B3LYP/6-311g** offer the best agreement with experimental results, yet the worst agreement for the bond angle. However, it must be kept in mind that experimental data lack for other structures. Hence the conclusions drawn above are merely qualitative indications for the performance of the various methods. In order to identify certain tendencies inherent to the methods under consideration, some typical bonds and bond angles have been identified in different molecules. The way these geometrical characteristics (bond lengths and angles) vary with the different methods will be discussed. From this point on, for each method, the results obtained with the 6-311g** basis set are taken as a reference. Nitroxide related structures

First, the bond lengths and bond angles in the nitroxide related structures shown in Figure 3.4 were evaluated. The deviations from the values obtained wilth the 6–311g** basis set are listed in Table 3.3 (a minus sign represents a value lower than the one obtained with the 6-311g** basis set and vice versa). For instance, the NC bond length (both NC1

3.3 Geometry

72

Table 3.3: Deviation of bond lengths and bond angles calculated with 6-311g and 6-311++g** basis sets, 6-311g** basis set as reference

basis set

TEMPO

aminoxyl radical

TEMPO-H

∆r(NC)a

6-311g

0.006

-

0.013

∆r(NO)a

6-311g

0.044

0.043

0.043

∆r(NH)a

6-311g

-

-0.006

-

∆r(OH)a

6-311g

-

-

0.014

-0.15

-

-0.115

0.15

-

0.056

6-311g

-

-

1.432

6-311++g**

-

-

-0.522

6-311g

-

-0.55

-

6-311++g**

-

0.1

-

∆a(ONC)b 6-311g 6-311++g** ∆a(NOH)b ∆a(ONH)b a b

bond lengths in ˚ A bond angles in degrees

and NC2) in TEMPO–H (see Figure 3.4 (d)), calculated using the 6–311g basis set is larger than that calculated by using the 6–311g** basis set by 0.013 ˚ A, for all methods. For all the calculated bond lengths the values predicted using the 6–311++g** basis set differ less than 0.001 ˚ A from those obtained using the 6–311g** basis set and are therefore not explicitly listed. For the bond angles the trend of the deviation is reported and the maximum value of the deviations for different methods is given. As for the different DFT methods, the same trend is observed as mentioned in the case of hydroxylamine: the values for the bond length obtained with the BB1K and the BHandHLYP functionals are the lowest, the BMK, MPW1PW91 and B3P86 functionals yield higher values, whereas the B3LYP method gives the highest values for the bond length. The calculated bond lengths and bond angles for the structures presented in Figure 3.4 are given in Table 3.4. Only the values obtained with the 6-311g** basis set are reported. The structural effects of protonation of the oxylamine radical and TEMPO are shown in Table 3.4. Upon formation of the O–H bond, the N–O bond is elongated. Meanwhile, the N–H1(H2) bond length in the oxylamine radical increases slightly, unlike the N–C1(C2)

3.3 Geometry

73

Table 3.4: Bond lengths and bond angles of the nitroxide related structures (see Figure 3.4)

r(NO)a r(NX)a,b r(OH)a,d a(ONX)b,c a(NOH)c TEMPO B3LYP

1.280

1.505

-

115.631

-

B3P86

1.272

1.493

-

115.704

-

MPW1PW91

1.269

1.492

-

115.714

-

BB1K

1.261

1.478

-

115.527

-

BMK

1.282

1.491

-

115.421

-

BHandHLYP

1.263

1.487

-

115.574

-

B3LYP

1.426

1.491

0.971

110.329

107.048

B3P86

1.411

1.480

0.970

110.153

106.799

MPW1PW91

1.407

1.478

0.967

110.330

107.108

BB1K

1.394

1.466

0.960

110.159

107.071

BMK

1.417

1.481

0.966

110.073

107.103

BHandHLYP

1.400

1.474

0.956

110.509

107.769

B3LYP

1.275

1.016

-

120.221

-

B3P86

1.267

1.015

-

120.276

-

MPW1PW91

1.265

1.013

-

120.342

-

BB1K

1.257

1.007

-

120.088

-

BMK

1.278

1.014

-

119.853

-

BHandHLYP

1.261

1.004

-

119.884

-

B3LYP

1.446

1.019

0.962

103.781

102.032

B3P86

1.431

1.018

0.960

104.175

102.192

MPW1PW91

1.427

1.016

0.958

104.293

102.408

BB1K

1.411

1.010

0.951

104.719

103.003

BMK

1.433

1.018

0.958

104.121

102.668

BHandHLYP

1.415

1.008

0.949

104.790

103.299

B3LYP

1.450

1.503

1.449

opzoeken

113.757

B3P86

1.434

1.493

1.439

opzoeken

113.298

MPW1PW91

1.429

1.492

1.436

opzoeken

113.481

BMK

1.436

1.495

1.428

opzoeken

113.666

BHandHLYP

1.418

1.488

1.428

opzoeken

114.424

TEMPO-H

aminoxyl radical

hydroxylamine

TEMPO–Styryl

˚ Bond length in A X = C1 , C2 in the case of TEMPO(–H) or TEMPO–Styryl X = H1 , H2 in the case of hydroxylamine c a(ONX) and a(NOH) = bond angle in degrees d in the case of TEMPO–styryl, this is r(OC)

a

b

3.3 Geometry

74

Table 3.5: Deviation of bond lengths and bond angles, obtained with the 6–311g basis set compared to the results obtained with the 6-311g** basis set

C-phenyl-N-tert-butylnitrone (PBN)

dimethylnitrone (DMN)

r(NC)a

0.004

0.004

r(NO)a

0.053

0.053

a(CNC)b

1.150

1.433

a(ONC)b

-0.655

-1.063

a b

bond lengths in ˚ A bond angles in degrees

bond length in TEMPO. The angles O–N–X (X = H, C) are smaller in the protonated structure. The geometric data reported in Table 3.4 for TEMPO–H and TEMPO–Styryl (see Figure 3.4 (c) and (d)) are compared, in order to evaluate the influence of a bulky substituent on the O–atom. It can be seen that the N–C1(C2) and the N–O bonds are elongated significantly and the a(NOC) angle in TEMPO–Styryl (about 113◦ ) is larger than the a(NOH) in TEMPO-H (about 107◦ ), which could be expected with regard to the steric hindrance of the styryl substituent. Nitrones

The geometries of the nitrones C–phenyl–N–tert–butylnitrone (PBN) and dimethylnitrone (DMN) will be discussed (see Figure 3.5). As for the nitroxide related structures, the performance of the 6-311g and the 6-311g** basis sets is compared. The systematic deviation of the 6-311g values from the 6-311g** values for the bond lengths and the bond angles is given in Table 3.5. The r(NC) and r(NO) values obtained with the 6-311g basis set are larger than those found with the 6-311g** basis set, which is in agreement with the earlier observations of related structures. As mentioned previously, results obtained with 6–311++g** again yield values close to the results with 6-311g**. The calculated values (obtained with the 6-311g** basis set) for the nitrones shown in Figure 3.5 are given in Table 3.6. The substitution of the two methyl groups of dimethylnitrone (DMN) (see Figure 3.5 (a)) for a tert-butyl and a phenyl group (PBN) does not have a large influence on the N–O and N–C2 bond lengths, yet the N–C1 bond is being stretched in PBN compared to DMN. This is due to the increased steric hindrance induced by the tert–butyl substituent. The r(C2C3) bond length in PBN is shorter than in DMN.

3.3 Geometry

75

O

O C1

C1

N C2

N C2

C3

C3

(b)

(a)

Figure 3.5: Structure of dimethylnitrone (DMN) (a) and PBN (b) O4

O1 C1

O1

C3

C2 O2

(a)

C1

O1

C3

C2 O2

O2 (b)

O4

O1

C3

C1

O3

C1

O3

C2 O2 (d)

( c)

C3 N1 C1 C2 N2

N3 C3

CN

N1 C1 C2 C4

(e)

(f)

Figure 3.6: Structure of BPO (a), benzoyloxy radical (b), diacetylperoxide (c), methylcarboxyl radical (d), AIBN (e) and cyanoisopropyl radical (f)

This is in agreement with the experimental observation that the sp3 -sp2 hybridization of C3–C2 in DMN yields a longer bond than the sp2 -sp2 hybridization of C3–C2 in PBN. The a(C1NC2) and a(ONC2) are slightly larger in PBN compared to DMN, due to the steric hindrance caused by the tert-butyl and the phenyl group in PBN. For DMN G3B3 and CBS-QB3 calculations are available. The values obtained with the B3LYP/6–311g** method are closest to those obtained with the composite methods, as expected from Table 3.1. Initiator related structures

The geometric data of the initiator related structures presented in Figure 3.6 will be discussed. From this point on only values obtained with the 6–311g** basis set are reported.

3.3 Geometry

76

Table 3.6: Calculated geometries for dimethylnitrone and PBN with various levels of theory (6– 311g** basis set)

r(NO)a r(NC1)a r(NC2)a r(C2C3)a a(C1NC2)b a(ONC2)b a(ONC1)b dimethylnitrone B3LYP

1.271

1.480

1.308

1.486

120.753

124.113

115.134

B3P86

1.263

1.470

1.305

1.479

120.606

124.085

115.309

MPW1PW91

1.261

1.468

1.303

1.479

120.616

124.108

115.276

BB1K

1.254

1.457

1.292

1.473

120.874

123.874

115.252

BMK

1.278

1.469

1.297

1.494

121.392

123.862

114.746

BHandHLYP

1.261

1.462

1.288

1.480

121.165

123.933

114.902

G3B3

1.276

1.478

1.310

1.488

121.002

124.047

114.951

CBS-QB3

1.272

1.479

1.307

1.486

120.843

124.079

115.078

PBN B3LYP

1.274

1.537

1.315

1.451

121.827

124.462

113.711

B3P86

1.265

1.524

1.311

1.445

121.514

124.640

113.846

MPW1PW91

1.262

1.521

1.309

1.446

121.528

124.649

113.823

BB1K

1.255

1.505

1.297

1.443

121.578

124.728

113.694

BMK

1.279

1.517

1.303

1.461

122.010

124.396

113.594

BHandHLYP

1.264

1.510

1.294

1.450

122.181

124.244

113.575

a b

bond length in ˚ A bond angle in degrees

3.3 Geometry

77

In Table 3.7 geometric data for benzoylperoxide (BPO), benzoyloxy radical, diacetylperoxide (DAP) and methylcarboxyl radical is reported. DAP resembles the structure of BPO, except for the two methyl groups instead of two phenyl groups (see Figure 3.6 (a) and (c)). The bond length of the single C–O bond r(C2O2) and the double C=O bond r(C2O1) are not strongly affected by the substitution of the methyl group in DAP by a phenyl group in BPO (stretching of about 0.002 ˚ A and 0.004 ˚ A respectively). However, there is a significant difference in the O2–O3 bond length, which is 0.010 ˚ A longer in DAP than in BPO. The bond angle a(C1C2O1) is larger in BPO than in DPN, which is not surprising, regarding the steric effects caused by the voluminous phenyl substituent. Steric hindrance is most likely also responsible for the increase (of about 5◦ ) in the dihedral angle of BPO relative to DAP. The methyl group in methylcarboxyl radical is replaced by a phenyl group in benzoyloxy radical. An important difference between the two structures is that the r(C2O1) and r(C2O2) bond lengths in the benzoyloxy radical are equal, whereas these bond lengths in methylcarboxyl radical differ. The difference is small when calculated with B3LYP, B3P86 and MPW1PW91 methods (about 0.001 ˚ A), but when BB1K, BMK or BHandHLYP methods are used, the difference in bond length becomes up to 0.150 ˚ A. The reason for the pronounced difference between the two C–O bond lengths is due to the Cs symmetry of methylcarboxyl radical, contrary to the C2v symmetry of the benzoyloxy structure. The a(C1C2O2) bond angle is more obtuse and a(O1C2O2) is more acute in the benzoyloxy radical, compared to these bond angles in the methylcarboxyl radical. This is logical, with regard to the sterically large phenyl group present in benzoyloxy structure. Upon dissociation (for both structures) the C2O1 bond length increases while the C2O2 bond length decreases. The C1C2O1 angle becomes more obtuse, in contrast to the C1C2O2 and O1C2O2 bond angles. In Table 3.9 structural data is provided for AIBN and the cyanoisopropyl radical (see Figure 3.6 (e) and (f)). The calculated bond lengths are in good agreement with bond lengths typically found experimentally.

1.481

1.474

1.495

1.478

1.473

1.467

1.468

1.461

1.480

1.461

1.472

1.473

MPW1PW91

BB1K

BMK

BHandHLYP

B3LYP

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

G3B3

CBS-QB3

c

b

1.261

1.266

1.246

1.250

1.245

1.254

1.257

1.261

1.182

1.188

1.184

1.192

1.194

1.195

1.261

1.266

1.246

1.250

1.245

1.254

1.257

1.261

1.368

1.372

1.366

1.378

1.380

1.389

110.224

109.924

110.010

109.753

109.663

109.656

-

-

-

-

-

-

-

-

124.430

124.539

124.878

124.593

124.723

124.563

124.506

124.430

benzoyloxy radical

1.387

1.402

1.386

1.405

1.412

1.428

bond length in ˚ A, structures shown in Figure 3.6 (a) and (b). bond angle in degrees dihedral angle d in degrees

1.481

B3P86

a

1.488

B3LYP

BPO

124.430

124.539

124.878

124.593

124.723

124.563

124.506

124.430

126.590

126.563

126.691

126.711

126.765

126.783

111.141

110.922

110.245

110.814

110.553

110.874

110.987

111.141

123.185

123.511

123.297

123.533

123.570

123.559

-

-

-

-

-

-

-

-

111.718

111.658

111.432

111.343

111.219

111.313

-

-

-

-

-

-

-

-

86.691

83.591

82.365

86.009

85.679

87.690

r(C1C2)a r(C2O1)a r(C2O2)a r(O2O3)a a(C1C2O1)b a(C1C2O2)b a(O1C2O2)b a(C2O2O3)b d(C2O2O3C3)c

Table 3.7: Geometries calculated for BPO and the benzoyloxy radical with various levels of theory (6–311g** basis set)

3.3 Geometry 78

1.497

1.488

1.511

1.494

1.091

1.091

1.092

1.086

1.094

1.085

1.093

1.093

MPW1PW91

BB1K

BMK

BHandHLYP

B3LYP

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

G3B3

CBS-QB3

c

b

1.256

1.262

1.190

1.243

1.228

1.251

1.253

1.256

1.179

1.185

1.181

1.189

1.190

1.191

1.257

1.262

1.320

1.250

1.258

1.251

1.253

1.257

1.367

1.371

1.365

1.376

1.378

1.386

108.337

108.102

108.253

107.878

107.764

107.656

-

-

-

-

-

-

-

-

124.057

124.120

127.105

124.891

126.025

124.298

124.204

124.052

127.98

127.86

128.11

128.01

128.04

127.99

124.030

124.118

112.036

123.857

122.723

124.242

124.184

124.034

methylcarboxyl radical

1.393

1.409

1.392

1.413

1.421

1.437

bond length in ˚ A, structures shown in Figure 3.6 (c) and (d). bond angle in degrees dihedral angle d in degrees

1.498

B3P86

a

1.506

B3LYP

diacetyl peroxide

111.902

111.746

120.849

111.252

111.252

111.447

111.599

111.902

123.676

124.025

123.625

124.102

124.182

124.346

-

-

-

-

-

-

-

-

111.831

111.757

111.436

111.46

111.366

111.515

-

-

-

-

-

-

-

-

82.724

79.889

78.456

81.437

81.055

82.77

r(C1C2)a r(C2O1)a r(C2O2)a r(O2O3)a a(C1C2O1)b a(C1C2O2)b a(O1C2O2)b a(C2O2O3)b d(C2O2O3C3)c

Table 3.8: Geometries calculated for diacetylperoxide and methylcarboxyl radical with various levels of theory (6–311g** basis set)

3.3 Geometry 79

c

b

a

1.152 1.151 1.143 1.148 1.139 1.168 1.168 1.166 1.158 1.162 1.155

B3P86

MPW1PW91

BB1K

BMKc

BHandHLYP

B3LYP

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

1.384

1.397

1.382

1.386

1.385

1.388

1.472

1.482

1.466

1.470

1.469

1.475

1.213

1.235

1.215

1.223

1.226

1.228

-

-

-

-

-

-

-

-

-

-

-

-

cyanoisopropyl radical

1.472

1.480

1.467

1.480

1.482

1.493

bond length in ˚ A, structures shown in Figure 3.6 (e) and (f). bond angle in degrees symmetry not accounted for in geometry optimization

1.152

B3LYP

AIBN

180.000

180.000

180.000

180.000

180.000

180.000

176.827

176.895

177.043

177.399

177.423

177.449

-

-

-

-

-

-

112.963

113.412

112.894

113.031

113.030

113.167

-

-

-

-

-

-

115.630

115.164

115.025

114.963

114.900

115.169

r(N1C1)a r(C1C2)a r(C2N2)a r(N2N3)a a(N1C1C2)b a(C1C2N2)b a(C2N2N3)b

Table 3.9: Geometries calculated for AIBN and the cyanoisopropyl radical with various levels of theory (6–311g** basis set)

3.3 Geometry 80

3.3 Geometry

81

Concluding remarks

In this section some detailed geometric data was presented for the structures studied in this LOTS. When the calculated data is compared to experimental findings for hydroxylamine, the B3LYP/6–311g** method is found to give the best results. Values obtained using the 6–311g basis set tend to overestimate the bond lengths. The expensive 6–311++g** basis set yields values remarkably close to the values obtained using the 6–311g** basis set. In general, BB1K and BHandHLYP methods are found to underestimate bond lengths, and are therefore not suitable for accurate geometry calculations. Based on these results, B3LYP/6–311g** seems to be the best DFT method from this test set to describe molecular geometries.

3.4 Heats of formation

82

O

O O O (1) ethylbenzene

(3) benzene

(5) diacetylperoxide

H N

O

H

H (2) styrene

(4) phenyl radical

(6) hydroxylamine

Figure 3.7: Structures for which the experimental and the composite method calculations of the heats of formation were available (subset 1)

3.4

Heats of formation

Geometry optimizations yield the minimal energy configuration for a molecule, after which frequency calculations can be performed. Based on the obtained data the heats of formation (∆f H 0 ) and entropies of formation (S0 ) can be calculated using the method described in Section 2.6. The same test set for molecules is still used (see Figure 3.1). In order to evaluate the calculated heats of formation, experimental data is needed. Here again, experimental data is scarce for the structures studied in this Master thesis. However, for a small subset of six molecules (shown in Figure 3.7) it was possible to obtain experimental data as well as results from composite method calculations. For another 4 molecules, only experimental data is available (subset 2, shown in Figure 3.9). Next, for 6 molecules (5 radicals and a nitrone, see Figure 3.11) no experimental data is available, but the results of the composite thethods are (subset 3). Finally, for 4 (larger) structures of the test set (see Figure 3.13), only DFT results are available (subset 4). Subset 1

The calculated values for this subset are presented in Table 3.11. In order to qualitatively evaluate the performance of the various DFT methods for the molecules shown in Figure 0 3.7, the difference between the experimental (∆f Hexp ) and the calculated standard heats 0 of formation (∆f Hcalc ) have been calculated, namely the mean average deviation (MAD)


83

Table 3.10: Mean average deviations (MAD) of standard heats of formation between calculated and experimental values for subset 1 (in kJmol−1 )

6-311g

6-311g**

6-311++g**

B3LYP

259.138

61.271

69.953

B3P86

37.925

-158.199

-152.610

MPW1PW91

225.407

27.390

23.254

BB1K

274.857

57.713

53.537

BMK

229.951

43.060

33.752

BHandHLYP

492.613

262.909

222.168

G3B3

4.853

CBS–QB3

8.898

is calculated for the different calculation methods and compared (see Table 3.10). It is observed that the best results are obtained using the composite G3B3 method, which has a MAD of less than 5 kJmol−1 . Unfortunately the G3B3 method can not be used for the calculations of most of the structures involved in the test set, because it is too computationally expensive. Therefore an ’optimal’ DFT alternative has to be identified. The 0 0 differences between ∆f Hexp and ∆f Hcalc are also plotted in Figure 3.8. It can be seen that

the most cost–efficient DFT alternatives are MPW1PW91/6–311g** and BMK/6–311g**. Generally, for DFT methods the 6–311g basis set performs worst. The 6–311++g** basis set offers slightly better results than the 6–311g** basis set, but this does not compensate the larger computational cost.

282.142

526.631

95.771

-73.837

115.188

229.602

138.078

382.842

-26.391

-456.558

136.945

248.969

152.431

396.308

-34.756

3. benzene

4. phenyl

5. hydroxylamine

6. diacetylperoxide

1. ethylbenzene

2. styrene

3. benzene

4. phenyl

5. hydroxylamine

6. diacetylperoxide

1. ethylbenzene

2. styrene

3. benzene

4. phenyl

5. hydroxylamine

c

b

a

Domalski and Hearing (1993) Lide (1997) NIST (2007)

-

421.870

2. styrene

6. diacetylperoxide

317.383

1. ethylbenzene

B3LYP

-

-91.606

183.422

-70.581

-48.477

-185.678

-685.491

-83.441

174.927

-79.813

-60.695

-199.550

-297.397

42.600

315.301

59.954

125.811

-3.587

B3P86

85.958

164.863

503.034

252.697

385.933

271.786

BB1K

-

-0.546

335.082

99.501

179.253

-

-

31.330

370.414

121.859

210.816

-

-351.407

29.496

348.261

98.571

180.078

56.408

6-311++g**

-455.207

6.964

324.482

88.019

164.083

51.129

6-311g**

-59.314

135.469

464.916

227.869

350.907

247.725

6-311g

MPW1PW91

-

-29.509

129.878

-

-

-497.940

-20.269

372.788

123.418

209.007

86.485

-90.049

107.436

496.835

246.553

373.862

260.200

BMK

-

63.851

559.425

316.665

469.001

-

-135.149

72.550

552.167

308.973

458.556

335.489

315.397

208.158

717.703

476.291

682.399

570.860

BHandHLYP

-533.308

-41.887

347.477

87.188

151.901

32.879

G3B3

-540.211

-46.190

355.571

90.753

159.208

38.965

CBS-QB3

Table 3.11: Calculated ∆f H 0 ’s with DFT and composite methods for subset 1 (in kJmol−1 )

-535.000

-50.000

339.900

82.865

147.507

29.860

exp

a

b

c

b

b

a

3.4 Heats of formation 84


85

(a) ethylbenzene

(b) styrene

600

400

6-311g 6-311g** 6-311++g**



500

600

300 200 100 0 -100

400

6-311g 6-311g** 6-311++g**

300 200 100 0 -100

-200

-200

-300

-300 an dH P LY

P LY

(d) hydroxylamine 300

6-311g 6-311g** 6-311++g**



BH

K BM

dH

400

1K 91 BB W 1P PW M

an

6 P8 B3 P LY B3

BH

K BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

(c) benzene 500

300 200 100 0

6-311g 6-311g** 6-311++g**

200 150 100 50

-100

0

-200

-50 BH

K

an

BM

dH

dH

P LY

P LY

1K 91 BB W 1P PW M 86

P LY

an

P B3

B3

BH

K BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

(e) phenyl radical

(f) diacetylperoxide

400 6-311g 6-311g** 6-311++g**



300 200 100 0

6-311g 6-311g** 6-311++g**

600 400 200

-100

0

-200

-200 BH an

K BM

dH

dH

P LY

P LY

1K 91 BB W 1P PW M

an

6 P8 B3 P LY B3

BH

K BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

Figure 3.8: Deviations of calculated heats of formation from experimental values for subset 1


86

O

O O O

(7) 1-methylethylbenzene

(9) BPO

O N N

N CN

CN

(10) AIBN

(8) PBN

Figure 3.9: Structures for which the experimental values for the heats of formation are available (subset 2) Table 3.12: Mean average deviations (MAD) of standard heats of formation between calculated and experimental values for subset 2 (in kJmol−1 )

B3LYP

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

6-311g

567.231

129.744

496.092

594.023

473.837

1056.441

6-311g**

147.636

303.800

75.029

133.661

93.352

567.022

6-311++g**

225.308

294.262

95.476

-

-

845.501

Subset 2

For another four molecules presented in Figure 3.9 DFT values for ∆f H 0 were calculated (see Table 3.13) and compared with experimental values. No composite methods were used for these structures due to the size of these molecules. Here again the deviations 0 0 of the ∆f Hcalc from the experimental ∆f Hexp have been plotted, see Figure 3.10. The

MADs of the several DFT methods from experimental values are listed in Table 3.12. The best performing methods are again MPW1PW91/6–311g** and BMK/6–311g**.

c

b

a

-

-156.387

-

-

-

-

-

-

76.226

-

-

-

-

-158.594

475.655b

128.094

-50.620

145.147

38.147

70.342

-

981.210b

6-311++g**

441.552

-202.675

97.355

37.571

430.686

481.510

269.101

BMK

619.491

565.143

283.934

6-311g**

908.083

403.421

484.132

262.417

BB1K

6-311g

MPW1PW91

Domalski and Hearing (1993) symmetry not accounted for in geometry optimization Lide (1997)

-

-590.066

-

61.419

10. AIBN

447.506

10. AIBN

-629.801

-7.922

-68.647

9. BPO

-318.354

9. BPO

175.825

8. PBN

-254.778

-294.271

109.547

7. 1-methylethylbenzene

520.094

214.305

907.309

10. AIBN

-30.956

8. PBN

530.894

9. BPO

64.723

-238.663

563.695

8. PBN

-30.761

B3P86


340.711


B3LYP

-

570.368

-

-

860.345

533.919

594.151

353.360

1398.200

1234.390

1044.160

622.701

BHandHLYP

Table 3.13: Calculated ∆f H 0 ’s with DFT methods for subset 2 (inkJmol−1 )

313.900c

-275.133a

30.900

4.020a

exp



88

(a) 1-methylethylbenzene 1000

6-311g 6-311g** 6-311++g**



600

(b) PBN

400

200

0

6-311g 6-311g** 6-311++g**

600 400 200 0 -200

-200

-400 an

K

dH P LY

(d) AIBN

6-311g 6-311g** 6-311++g**



BH

BM

P LY

1K 91 BB W 1P PW

dH

1500

M

an

K

6 P8 B3 P LY B3

BH

BM

1K 91 BB W 1P PW

M

6 P8 B3 P LY B3

(c) BPO

1000

500

0

800

6-311g 6-311g** 6-311++g**

600 400 200 0 -200 BH

K

an

BM

dH

dH

P LY

P LY

1K 91 BB W 1P PW

M

an

K

6 P8 B3 P LY B3

BH

BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

Figure 3.10: Deviations of calculated heats of formation from experimental values for subset 2


89

O H

O

(11) benzoyloxy radical

O

N CN

H

(13) cyanoisopropyl radical

(15) oxylamine radical

O N

O O (12) methylcarboxyl radical

(16) dimethylnitrone

(14) ethylbenzene radical

Figure 3.11: Structures for which the DFT and composite method values for the heats of formation are calculated (subset 3) Table 3.14: Mean average deviations (MAD) of standard heats of formation between DFT and G3B3 calculated values for subset 3 (in kJmol−1 )

B3LYP

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

6-311g

220.831

58.530

208.017

235.111

185.016

457.235

6-311g**

34.837

148.550

18.324

56.368

24.936

191.512

6-311++g**

48.960

152.866

16.803

100.759

-

-

Subset 3

For the molecules presented in Figure 3.11, most of which are radicals, no experimental values for heats of formation were available in literature. However, these structures are small enough to be calculated using composite methods. In Table 3.10 it was shown that the G3B3 composite method gives very good estimates of standard heats of formation. To evaluate the performance of the DFT methods in describing the heats of formation for these structures, the DFT results will be compared to the values obtained using the G3B3 method. The calculated values for ∆f H 0 are given in Table 3.15. The MAD values are provided in Table 3.14, and the graphs representing the deviation between the G3B3 and DFT values are given in Figure 3.12. The conclusions here are consistent with those made for former subsets of molecules. The 6–311g** basis set gives the best results, and MPW1PW91 and BMK are the best methods for ∆f H 0 calculations.

371.501 447.105 149.770 229.140 10.035 -178.593 207.490 249.839 53.534 33.667

13. cyanoisopropyl radical

14. ethylbenzene radical

15. H2NO

16. dimethylnitrone

11. benzoyloxy radical

12. methylcarboxyl radical



15. H2NO

16. dimethylnitrone

-168.099 270.115 47.829 41.839




15. H2NO

16. dimethylnitrone

-

12.948



308.003


B3LYP

-141.184

1.054

-41.692

-

-278.748

-

-144.446

6.764

-54.344

42.091

-285.406

-262.477

52.067

106.540

137.346

203.879

-92.040

34.256

B3P86

-

74.120

196.057

-

-164.448

-

297.847

218.796

408.032

395.985

82.242

110.663

197.632

217.518

-110.936

34.570

-

114.242

-

-

-89.891

-

6-311++g**

46.855

79.349

180.291

203.631

-172.381

-51.045

103.878

6-311g**

246.878

181.454

372.405

367.510

24.372

248.966

BB1K

6-311g

MPW1PW91

-

60.366

-

-

-

-

36.051

67.496

231.293

204.001

-179.947

-15.795

225.909

164.429

401.060

358.676

23.991

-

BMK

-

144.716

-

-

-

-

237.915

151.247

466.615

366.112

-15.345

-

464.701

260.566

694.717

557.590

260.576

698.740

BHandHLYP

-533.308

66.801

188.653

185.209

-190.650

-55.503

G3B3

Table 3.15: Calculated ∆f H 0 ’s with DFT and composite methods for subset 3 (in kJmol−1 )

-540.211

60.337

191.646

190.644

-193.974

-54.760

CBS–QB3



91

(a) benzoyloxy radical

Df H0G3B3-Df H0calc [kJ mol-1]

600

6-311g 6-311g** 6-311++g**

400 Df H0G3B3-Df H0calc [kJ mol-1]

700

(b) methylcarboxyl radical

500 400 300 200 100 0

6-311g 6-311g** 6-311++g**

300 200 100 0

-100 -200

-100 an

K

dH P LY

(d) ethylbenzene radical 500

6-311g 6-311g** 6-311++g**



BH

BM

P LY

1K 91 BB W 1P PW

dH

300

M

an

K

6 P8 B3 P LY B3

BH

BM

1K 91 BB W 1P PW

M

6 P8 B3 P LY B3

(c) cyanoisopropyl radical

200

100

0

-100

6-311g 6-311g** 6-311++g**

300 200 100 0 -100 -200 BH

K

an

BM

dH

dH

P LY

P LY

1K 91 BB W 1P PW

M

P LY

an

K

6 P8 B3

B3

BH

BM

1K 91 BB W 1P PW M

6 P8 B3 P LY B3

(e) H2NO radical

(f) dimethylnitrone

150

6-311g 6-311g** 6-311++g**



200

100 50

0

6-311g 6-311g** 6-311++g**

300 200 100 0 -100

-50

BH

K

an

BM

dH

dH

P LY

P LY

1K 91 BB W 1P PW

M

an

K

6 P8 B3 P LY B3

BH

BM

1K 91 BB W 1P PW

M

6 P8 B3 P LY B3

Figure 3.12: Deviations of calculated heats of formation using DFT methods from the values obtained using the G3B3 composite method for benzoyloxy radical (top), methylcarboxyl radical (middle), cyanoisopropyl radical (bottom), in kJ.mol−1


(17) 1-methylethylbenzene radical

92

N

N

N

OH

O

O

(18) TEMPO-H

(19) TEMPO

(20) TEMPO-Styryl

Figure 3.13: Structures for which only DFT method calculations of the heats of formation were performed

Subset 4

Finally, ∆f H 0 calculations were performed for the four structures shown in Figure 3.13. For these molecules no experimental values were available, and no composite method calculations could be conducted, because of their size. The calculated DFT values are given in Table 3.16. The expectation is that the values obtained with MPW1PW91/6– 311g** and BMK/6–311g** are the closest to the actual values of the heats of formation, namely in the range of 37–70 kJmol−1 for the 1–methylbenzene radical, -46 – -63 kJmol−1 for TEMPO, -77 – -105 kJmol−1 for TEMPO–H and 31 – 36 kJmol−1 for TEMPO–Styryl. In this section the calculated heats of formation using DFT methods were compared to experimental and G3B3 values, when available. It was found that the MPW1PW91/6– 311g** and BMK/6–311g** methods are most suitable for this purpose.

353.999 730.393 109.547 13.190 -21.998 172.290

19. TEM°PO-H

20. TEMPO-Styryl

17. 1-methylethylbenzene radical

18. TEMPO

19. TEM°PO-H

20. TEMPO-Styryl

51.205 16.153 236.451

18. TEMPO

19. TEM°PO-H

20. TEMPO-Styryl

-

369.584

18. TEMPO


340.711


B3LYP

-

-490.866

-442.397

-238.663

-630.777

-514.672

-465.802

-254.778

-81.869

-142.722

-114.834

-30.761

B3P86

-

-

-17.003

-

-

-84.920

-41.693

38.147

-

-

-

76.226

6-311++g**

31.562

-77.731

-46.468

37.571

-

317.893

339.834

283.934

6-311g**

584.792

298.650

308.889

262.417

BB1K

6-311g

MPW1PW91

-

-

-

-

35.512

-105.067

-63.647

70.342

538.733

241.924

261.355

269.101

BMK

Table 3.16: Calculated ∆f H 0 ’s with DFT methods (in kJmol−1 )

-

-

-

-

698.880

290.140

327.116

353.360

1350.110

725.815

742.983

622.701

BHandHLYP


3.5 Entropies of formation

94

0 Table 3.17: MAD for the standard molecular entropy Sm between experimental and DFT calculated −1 −1 values (in Jmol K )

3.5

B3LYP

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

MAD exp

2.402

2.443

3.555

4.190

4.009

4.439

MAD G3B3

1.351

2.162

1.988

5.513

6.679

3.837

MAD CBS–QB3

0.714

2.139

1.497

5.288

6.125

3.900

Entropies of formation

0 The standard molecular entropies (Sm ) were calculated for the structures given in Figure

3.1. The values obtained for one method using various basis sets (6–311g, 6–311g** and 6–311++g**) differ less than 1 Jmol−1 K −1 . Therefore only the values calculated with the 6–311g** basis set are reported in Table 3.18. The results for the 6–311g basis set is reported in Appendix A, whereas the results obtained with the 6–311++g** basis set are reported in the lab journal. For some structures experimental results were found in literature. In order to evaluate the performance of the composite and DFT methods for these calculations, MADs were calculated with respect to the experimental values. For G3B3 and CBS–QB3 the MAD was found to be 1.701 Jmol−1 K −1 and 1.087 Jmol−1 K −1 respectively. The MAD values of the DFT methods with respect to experimental values are given in Table 3.17. Because the composite methods yield lower MAD values than DFT methods (about 1.000–1.700 Jmol−1 K −1 ), composite methods were taken as a reference for the structures for which no experimental data was available. The MAD values of DFT methods with respect to G3B3 and CBS-QB3 methods are also given in Table 3.17. It can be concluded that all DFT methods give satisfactory results for the calculation of 0 standard molecular entropy Sm with maximal MAD values of less than 7 Jmol−1 K −1 .

The B3LYP/6-311g** method yields the best results.

486.654 355.598 297.056 330.243 234.538 432.295 319.507 461.876 231.014 440.086 347.766 268.575 356.774 382.844 360.782 392.028 288.504 590.906

AIBN

benzoyloxy radical



hydroxylamine

TEMPO-H

dimethylnitrone

PBN

aminoxyl radical

TEMPO

styrene

benzene

ethylbenzene

1-methylethylbenzene

ethylbenzene radical

1-methylethylbenzene radical

phenyl radical

TEMPO-Styryl

Domalski and Hearing (1993)

393.060

diacetylperoxide

a

535.534

BPO

B3LYP

588.655

288.491

391.802

360.414

382.798

357.473

268.563

348.251

438.040

232.407

460.763

319.377

431.634

234.137

331.318

303.549

355.406

486.331

391.085

535.678

B3P86

587.484

288.215

391.246

360.368

382.158

356.883

268.370

353.343

438.307

231.717

460.135

318.792

431.554

233.978

330.988

296.579

354.632

484.490

390.717

534.272

MPW1PW91

287.487

390.827

353.966

376.656

361.627

267.600

352.849

430.032

231.767

457.156

311.654

425.810

233.505

330.628

304.549

355.054

482.808

384.671

513.488

BB1K

576.258

288.173

397.074

352.414

378.472

354.632

268.023

340.331

439.446

230.897

455.458

309.854

420.500

234.066

330.770

292.001

355.456

378.789

527.044

BMK

577.300

286.424

387.568

357.050

378.041

353.862

266.596

349.293

432.086

230.085

454.261

315.758

425.312

233.534

328.653

298.574

477.394

386.847

522.409

BHandHLYP

288.504

360.782

356.786

268.575

348.050

228.219

320.256

234.990

331.231

298.353

358.318

396.763

G3B3

289.897

364.146

360.021

270.052

347.636

230.980

319.273

234.538

330.243

297.060

355.598

393.060

CBS-QB3

0 for the entire test set (in Jmol−1 K −1 ) Table 3.18: Calculated values for the standard molecular entropy Sm

388.570a

360.475

269.250

345.100a

236.180

exp

3.5 Entropies of formation 95

3.6 Dissociation/recombination reactions

O (a)

(b)

O

96

O 2

O O

O

O

O

N

(c ) CN

2 CN

N

(g)

O

N

H

H

2 O O

+ H

(f)

O

O

O + H

N H

H

+ N2

H

+ H

(h)

CN

N

N

OH

O

+ H (d) N

(i)

+ H

(e)

+

O

N O

Figure 3.14: The studied dissociation/recombination reactions

3.6

Dissociation/recombination reactions

The different levels of theory are used to calculate standard reaction enthalpies ∆r H 0 , standard reaction entropies ∆r S 0 and, hence, Gibbs free energies of reaction ∆r G0 , for dissociation and recombination reactions (shown in Figure 3.14), applying the equations mentioned in Section 2.6. The obtained Gibbs free energy of reaction was used to determine the equilibrium constant of recombination Krecomb , using the following equation: Krecomb = exp

−∆r G0 RT

(3.1)

with ∆r G0 the Gibbs free energy of reaction, R the gas constant and T the temperature. The former equation yields a dimensionless equilibrium coefficient. To evaluate the concentration–based equilibrium coefficient, there should be corrected for the stoichiometry of the reaction by correcting with the molar volume of an ideal gas (with Vm the molar volume, p the pressure, vr the number of reactants, vp the number of products): Kc =

Vmvr −vp

Vm =

RT p

exp

−∆r G0 RT

(3.2) (3.3)


97

Table 3.19: MAD values for reaction enthalpies of reactions (a), (b), (d), (e), (h), (i) of Figure 3.14 compared to experimental data

B3P86

MPW1PW91

BB1K

BMK

BHandHLYP

6-311g

31.410

19.147

29.726

53.179

32.310

26.606

6-311g**

25.917

14.018

23.125

28.218

8.531

39.720

energy

B3LYP

products A° + B°

reactants A-B reaction coordinate

Figure 3.15: A schematic reperesentation of the low activation energy for recombination reactions

The equilibrium coefficient calculated this way equals the one based on the following equation: Kc =

k+ k−

(3.4)

with k + the rate coefficient of the forward reaction and k − the rate coefficient of the backward reaction, calculated with equation 2.40. For recombination/dissociation reactions – however – no transition states are found, due to the very small activation energy for recombination, as shown in Figure 3.15. So, only Equation 3.1 will be applied. The method represented by Equation 3.4 will be used to study addition/dissociation reactions in the next section. Only for 6 reactions experimental data of ∆r H was reported in literature. No experimental data was found for the equilibrium coefficient. The MAD of the calculated ∆r H values from experimental data is reported in Table 3.19 for the DFT methods, based on 6 reactions (namely reactions (a), (b), (d), (e), (h), (i) of Figure 3.14). It is seen that – generally – the 6–311g** basis set yields better results than the 6–311g


98

basis set. The calculated values with the former basis set are listed in Tables 3.20–3.22. The results obtained with the 6–311g basis set are tabulated in Appendix A. The MADs of the results with the composite methods from the experimental values are 14.821 kJmol−1 and 15.384 kJmol−1 for respectively G3B3 and CBS–QB3. In Table 3.20 thermodynamic data is reported on the initiator related reactions (reactions (a), (b) and (c) in Figure 3.14). Here, the BMK method gives good results. Also the homolytic scission of a hydrogen–carbon bond is investigated (i.e. reactions (e)–(f) of Figure 3.14). Again, BMK offers good results, like the BB1K method, in this case. At last, the dissociation of a hydroxyl- and alkoxyamines is investigated (see Table 3.22). Based on the reported data, again, it is BMK that seems to perform best. Concluding, the best results for thermodynamic calculations on reactions are found with the BMK/6–311g** method.

4.111 1010

-50.275

1.59 1010

89.419

5.182 10−15

5.739 107

-99.372

-201.052

-39.428

1.99 108

18.533

-365.250

127.432

1.1324 10−21

Krecomb

∆r H

∆r S

∆r G

Krecomb

∆r H

∆r S

∆r G

c

b

a

-367.706

-20.212

-216.013

-114.679

9.356 10−17

99.367

-368.862

-10.609

1.47 1010

-50.087

-202.441

-110.445

7.486 109

-48.411

-174.992

-100.585

MPW1PW91

BMK

1.100 1014

-72.184

-183.868

-127.004

7.27 1014

-76.862

-205.213

-138.046

6.307 10−16

94.640

-369.707

-15.588

129.492 4.904 10−22

-371.121 45.610 c 2.482 10−7c

18.842 -178.732 c

5.11 108

-41.759

-210.301

-104.460

-

-

-

-

BHandHLYP

-7.679 c

reaction (c)

2.32 1012

-62.622

-224.427

-129.535

reaction (b)

1.274 1012

-61.138

-196.620

-119.760

reaction (a)

BB1K

Patai (1983) Luo (2003) symmetry was not accounted for for AIBN geometry optimization

Krecomb

-52.631

-36.342

∆r G

-175.134

-175.662

∆r S

-104.847

-88.716

B3P86

∆r H

B3LYP

-

-

-

-

3.84 1017

-92.395

-199.943

-152.008

-

-

-

-

G3B3

-

-

-

-

3.72 1017

-92.317

-201.060

-152.263

-

-

-

-

CBS–QB3

-

-

-

-

-

-

-

-130.733b

-

-

-124.700a

exp

Table 3.20: Calculated ∆r H 0 (in kJmol−1 ), ∆r S 0 (in Jmol−1 K −1 ), ∆r G0 (in kJmol−1 ) and Kc (in m3 kmol−1 ) for reactions (a), (b) and (c) of Figure 3.14

3.6 Dissociation/recombination reactions 99

-328.157

1058

-317.284

1057

∆r G

-432.626

1077

-462.762

-134.537

-422.650

1075

∆r H

∆r S

∆r G

-378.427

1067

∆r G

a

Yao et al. (2003)

5.273

1069

-123.792

∆r S

Krecomb

-388.662

-415.336

∆r H

3.285

-123.612

-425.517

2.977

Krecomb

1.673

-134.536

-472.738

1.012

Krecomb

8.137

-117.549

-118.616

∆r S

-363.204

-352.649

B3P86

∆r H

B3LYP

2.104

1066

-370.448

-123.696

-407.328

1.058

1074

-414.374

-134.453

-454.461

1.176

1056

-311.951

-118.093

-347.160

MPW1PW91

BMK

1.298

1.863

3.944

1068

-383.414

-128.779

-421.809

3.118

1068

-382.825

-133.210

-422.541

1076

-427.190

-134.758

-467.368

reaction (f )

2.183

1076

-427.588

-134.495

-467.688

1059

-329.297

-112.390

-362.806

reaction (e)

5.845

1058

-327.336

-106.947

-359.222

reaction (d)

BB1K

4.927

1066

-372.551

-124.135

-409.562

1.602

1075

-421.110

-134.436

-461.192

2.693

1056

-314.003

-117.796

-349.124

BHandHLYP

-

-

-

-

1.584

1078

-438.197

-134.453

-478.284

5.014

1060

-338.368

-118.733

-373.769

G3B3

-

-

-

-

9.733

1078

-442.701

-134.537

-482.813

1.463

1060

-335.314

-118.604

-370.676

CBS–QB3

-

-

-

-

-

-

-

-471.551a

-

-

-

-357.314a

exp

Table 3.21: Calculated ∆r H 0 (in kJmol−1 ), ∆r S 0 (in Jmol−1 K −1 ), ∆r G0 (in kJmol−1 ) and Kc (in m3 kmol−1 ) for reactions (d), (e) and (f) of Figure 3.14


-274.549 1049

-264.804

1047

∆r G

-230.788 1041

-253.186

-122.399

-216.692

1039

∆r H

∆r S

∆r G

-28.139

106

∆r G

a

Marsal et al. (1999)

2.09

109

-209.962

∆r S

Krecomb

-48.080

-90.739

∆r H

6.55

-209.799

-110.631

2.35

Krecomb

6.96

-121.014

-266.868

6.38

Krecomb

3.26

-112.878

-111.084

∆r S

-308.203

-297.924

B3P86

∆r H

B3LYP

1.89

108

-39.295

-211.191

-102.261

5.47

1038

-213.078

-121.361

-249.261

2.61

1046

-256.887

-112.347

-290.384

MPW1PW91

7.61

1039

-219.599

-133.554

-259.418

-

-

-

-

1.91

1013

-67.852

-215.602

-132.134

reaction (i)

1040

-225.796

-118.830

-261.225

9.28

1.45

1049

-272.537

-111.439

-305.763

reaction (h)

1047

-265.513

-112.870

-299.165

8.50

BMK

reaction (g)

BB1K

8.78

106

-31.692

-211.836

-94.851

5.48

1039

-218.784

-121.382

-254.974

3.85

1047

-263.553

-111.159

-296.695

BHandHLYP

1.04

-

-

-

-

-

-

-

-

1053

-294.531

-107.837

-326.683

G3B3

2.95

-

-

-

-

-

-

-

-

1052

-291.412

-111.050

-324.522

CBS–QB3

-

-

-

-127.418a

-

-

-

-291.206a

-

-

-

-

exp

Table 3.22: Calculated ∆r H 0 (in kJmol−1 ), ∆r S 0 (in Jmol−1 K −1 ), ∆r G0 (in kJmol−1 ) and Krecomb (in m3 kmol−1 ) for reactions (g), (h) and (i) of Figure 3.14


3.7 Addition/β–scission reactions

(a)

CH3

102

+ CN

O O

(b)

+

CN

N

N

+

(c ) CN CN

O O

O

N

O

O

+

(e)

O

+

(d)

O

N O

O O

O O

(f)

+

N

N

(g)

+

Figure 3.16: The addition/β–scission reactions studied in this level of theory study

3.7

Addition/β–scission reactions

The addition/β–scission reactions presented in Figure 3.16 were studied. As already concluded in previous sections, the 6–311g** provides the optimal cost/ performance ratio from the basis sets tested here. Therefore, this part of the LOTS will be performed using only this basis set. The major problem evaluating these reactions is that there is no experimental data available, with exception of reaction (a). Another problem is opposed by the difficulty of finding a transition state. For instance, for reactions involving a methylcarboxyl radical (see Figure 3.16 (d) and (e)), transition states were not found easily. Searching for the transition state with the B3LYP/6–311g** method yields a transition state for a cycloaddition reaction, as that reaction is favored over the regular addition reaction as shown in Figure 3.16 (d) and (e). Coote et al. (2006) suggested that a transition state for this type of reactions was easily found using the HF method. A transition state was indeed found using the HF method, which is probably because HF does not account for electron correlation. Further investigation to overcome this problem is being dome. For the other

3.7 Addition/β–scission reactions

103

reactions transition states were found, yet only with some methods. An overview of the results for the addition of the methyl radical to propene (see Figure 3.16 (a)) is given in Table 3.23. The equilibrium reaction coefficient Kc was calculated based on the thermodyv −vp

namic calculated data using the relation Kc = Vmr

exp(−∆r G/RT ) (see Equation 3.2).

For this reaction experimental values for ∆r H, k + , k − and Kc are provided. For all the properties, composite methods provide the best agreement with experimental values. But as these methods cannot be used for the other reactions in the test set, the performance of composite methods will not be further discussed. The optimal DFT alternative has to be searched for. BHandHLYP method predicts a value for ∆r H which is the closest to the experimental findings. The k + is best calculated by the BB1K and BMK methods, whereas the B3LYP method yields the best results for k − and Kc . In Table 3.24 the results for reactions (b) and (c) of Figure 3.16 are given. For these reactions no experimental values were found. Therefore, drawing conclusions about the performance of different methods for addition reactions in the basis set would be unappropriate. However, for ∆r H and k + of reaction 1 the BMK and BB1K functionals yields values close to the experimental ones. Based on this small amount of data the BB1K and BMK methods might be considered for further calculations of this type of reactions.

exp

8.930

-

8.692

3.662 103

CBS-QB3 102

8.767

4.021

G3B3

8.722

102

101

4.147

8.826

5.611 102

BMK

BHandHLYP

8.638

5.155 103

MPW1PW91 5.596

8.731

2.412

B3P86

BB1K

8.923

104 102

8.837

1.441 103

-132.516

B3LYP

-98.068

CBS-QB3

-133.962

m3 kmol−1 s−1

-93.236

G3B3

-134.184

m3 kmol−1 s−1

-100.199

BHandHLYP

-130.092

logA+

-108.731

BMK

-133.191

k+

-111.351

BB1K

-133.511

-132.612

-132.889

-

-113.063

MPW1PW91

(J

mol−1 K −1 )

∆r S

-97.8

-112.572

B3P86

exp

-91.990

(kJ

mol−1 )

∆r H

B3LYP

reaction(a)

-

29.332

35.270

40.644

34.772

33.715

28.695

25.899

32.442

(kJ mol−1 )

E+ a

-

-58.560

-53.296

-60.193

-69.944

-71.640

-73.257

-73.033

-52.369

(kJ

mol−1 )

∆r G

13.873 13.764 13.793 13.752 13.782 13.931

4.071 10−11 10−12 1.673 10−11 10−11 10−9 1.043 10−8 4.890

9.486

5.553

8.427

-

13.874

10−10

10−9

13.801

4.456 10−8 1.881

s−1

logA− s−1

k−

1.836 1011

5.315 1010

4.444 1011

8.588 1011

4.387 1013

8.696 1013

1.670 1014

1.526 1014

3.657 1010

Kc 3 (m kmol−1 )

-

125.965

124.880

137.771

141.001

142.538

139.230

135.409

121.366

(kJmol−1 )

E−

Table 3.23: Calculated thermodynamic and kinetic parameters for reaction (a) given in Figure 3.16

3.7 Addition/β–scission reactions 104

reaction(c)

reaction(b)

5.741 5.947 5.672

5.949 10−1 3.704 101 101 101

B3LYP

B3P86

5.549

7.673 10−3 1.037

B3LYP

B3P86

1.931

(m3 kmol−1 s−1 )

(m3 kmol−1 s−1 )

MPW1PW91

logA+

k+

10−1

-200.397

-77.874

MPW1PW91

5.303

5.590

-195.371

-75.843

B3P86

-194.142

-50.983

(Jmol−1 k −1 )

(kJmol−1 )

B3LYP

∆r S

-197.398

∆r H

8.699

5.797

(m3 kmol−1 s−1 )

BMK

(m3 kmol−1 s−1 )

k+

1.888

logA+

-131.685

BMK

MPW1PW91

-205.375

-106.836

MPW1PW91

-197.333

-102.960

B3P86

-196.225

-80.798

(Jmol−1 k −1 )

(kJmol−1 )

B3LYP

∆r S

∆r H

34.177

31.647

43.574

(kJ mol−1 )

E+ a

-18.126

-17.593

6.900

(kJmol−1 )

∆r G

21.133

26.479

23.631

34.194

(kJ mol−1 )

E+ a

-70.453

-47.982

-44.126

-22.294

(kJmol−1 )

∆r G

14.459 14.590

10−9 10−12

5.268

10−6

13.967

13.991

3.516

13.886 10−5

(s−1 ) 5.079 10−3

(s−1 )

k−

3.664 104

2.956 104

1.512

(m3 kmol−1 )

Kc

1.614

logA−

14.249

2.823 10−8 3.025

14.248

(s−1 )

logA−

3.021 10−6

(s−1 )

k−

5.386 1013

6.231 109

1.315 109

1.969 105

Kc 3 (m kmol−1 )

E− a

109.769

105.204

92.279

(kJmol−1 )

E− a

150.493

131.070

124.339

112.753

(kJmol−1 )

Table 3.24: Calculated thermodynamic and kinetic parameters for reactions (b) and (c) given in Figure 3.16


reaction(g)

reaction(f )

6.931

9.437 107 5.927 108

B3LYP

B3P86

log A+ (m3 kmol−1 s−1 ) 6.982 7.327

-195.015 k+ (m3 kmol−1 s−1 ) 2.305 106 2.383 107

BMK

B3LYP

B3P86

BMK

MPW1PW91

-186.711

-195.84

MPW1PW91

6.601

1.447 106

-196.452

-191.054

-195.7683

B3P86

-155.499

mol−1 K −1 )

∆r S

6.429

-171.468

(kJ

mol−1 )

∆r H

5.515

-

-170.624

B3LYP

BMK

(J

6.698

(m3 kmol−1 s−1 )

108

(m3 kmol−1 s−1 )

k+

-

log A+

-246.5362

BMK

MPW1PW91

-163.548

-234.9011

MPW1PW91

-171.594

-232.785

B3P86

-171.256

-215.1974

(J mol−1 K −1 )

(kJ mol−1 )

B3LYP

∆r S

∆r H

2.513

-

-0.288

3.536

(kJ mol−1 )

Ea+

-139.3471154

-137.2678362

-138.8055499

-125.1059732

(kJ

mol−1 )

∆r G

-13.201

-

-10.512

-7.288

(kJ mol−1 )

Ea+

-197.7743638

-184.0295544

-181.6242489

-164.1374236

(kJ mol−1 )

∆r G

14.234 14.079 13.425

3.682 10−25 10−26 10−28 kmol−1 )

Kc

13.535 15.310 15.086 14.404

1.140 10−17 4.699 10−19 10−19 2.290 10−20

1.225

(s−1 )

(s−1 )

k−

6.32374 1025

2.73339 1025

5.08272 1025

2.02324 1023

(m3

5.063

log A−

13.986

6.774 10−23 4.845

(s−1 )

log A−

(s−1 )

k−

1.08875 1036

4.25554 1033

1.61274 1033

1.39323 1030

(m3 kmol−1 )

Kc

Ea−

194.324

194.064

192.006

173.972

(kJ mol−1 )

Ea−

232.437

224.860

220.719

206.376

(kJ mol−1 )

Table 3.25: Calculated thermodynamic and kinetic parameters for reactions (f) and (g) given in Figure 3.16


3.8 Conclusion

3.8

107

Conclusion

In this section a level of theory study was performed on a set of molecules and reactions, typical for reactions involved in the in situ NMP process. Different DFT methods and basis sets were evaluated on their ability to predict molecular geometries and thermodynamic properties, such as heats of formation, molecular entropies and reaction enthalpies, as well as equilibrium coefficients. Additionally, kinetic properties were also investigated, where possible. It was found that the 6-311g** basis set yields the optimal ratio for computational cost vs. quality from the tested basis sets (6–311g, 6–311g**, 6–311++g**). The B3LYP method was found to perform best for geometry calculations. The thermodynamic and kinetic properties were best predicted by the BMK method. Because thermodynamic and kinetic calculations are of major importance in this work, the BMK/6-311g** method is chosen as the best performing DFT method for the in situ NMP reaction network calculations. Additionally, another option could be to perform geometry optimization on a B3LYP level and consequently single point energy calculations with the BMK method.

APPLYING THE SELECTED DFT METHOD TO REACTIONS INVOLVED IN THE IN SITU NMP PROCESS 108

Chapter 4

Applying the selected DFT method to reactions involved in the in situ NMP process 4.1

Introduction

In order to gain deeper insight in the factors having an influence on the in situ NMP of styrene, a DFT calculation method, BMK/6-311g** has been chosen to model this polymerization process (see Chapter 3). The other method suggested in the conclusion of Chapter 3 (optimization with B3LYP/6–311g** followed by a single point energy calculation with BMK/6–311g**) has also been tested, but does not yield qualitatively deviating results. Experimental studies by the CERM at ULg were taken as a reference (see Chapter 1).

4.2

Influence of the nitrone structure

The main reactions of the in situ NMP process, as proposed by Sciannamea et al. (2005), have been studied for various AIBN/nitrone/Styrene systems. The different types of nitrones are shown in Figure 4.1. An overview of the reactions that will be investigated is given in Figure 4.2, for the case of an AIBN/PBN/Styrene system, yet the analogue reactions are studied for other nitrones. First, the initiator (AIBN) dissociates, yielding two cyanoisopropyl radicals (reaction a). The addition of this initiator radical to the

4.2 Influence of the nitrone structure

109

nitrone (PBN in this case) yields a nitroxide (reaction b), which can recombine with another initiating radical with the formation of an alkoxyamine (reaction c). In a next step in the process, the alkoxyamine releases cyanoisopropyl radicals by increasing the temperature. These radicals initiate the monomer (styrene), yielding a styryl derivative (reaction d). This latter then recombines with the nitroxide, with the formation of a dormant chain, denoted by PNA in Figure 4.2 (reaction e). Remark that – experimentally – the prereaction step involving reactions a, b and c is conducted at 85◦ C, whereas the polymerization involving reactions d and e take place at 110◦ C. Hence, calculated data will be reported for these two reaction temperatures. Reactions a and d are independent of the kind of nitrone used. Calculated data for these reactions is reported in Table 4.1. Large values for the equilibrium coefficient (Kc ) in both reactions suggest that these reactions are strongly shifted toward the products’ side. Hence, from a thermodynamic point of view, generating cyanoisopropyl radicals by AIBN and the formation of a styryl derivative is strongly favored. -

-

O

-

O

+

O

+

N

+

N

N

(c ) PBN

(b) nitrone 2

(a) nitrone1 -

O +

N

N+ -

O

(d) DPN

(e) DMPO

Figure 4.1: Nitrones for in situ NMP: (a) N-tert-butyl-α-isopropylnitrone (nitrone 1), (b) N-tertbutyl-α-tert-butylnitrone (nitrone 2), (c) C-phenyl-N-tert-butylnitrone (PBN), (d) N-αdiphenylnitrone (DPN), (e) 5,5-dimethyl-1-pyrroline-N-oxide (DMPO)

In Tables 4.2–4.4 and 4.6–4.7 the calculated thermodynamic data is given for the reactions b, c and e (see Figure 4.2) for the five nitrones under consideration (see Figure 4.1). In what follows the obtained data will be interpreted and compared with the experimental findings of Sciannamea et al. (2005).


110

N

reaction a

2

N

CN

+ N

CN

CN cyanoisopropyl

AIBN

-

O

O reaction b

2

+

N

N +

CN CN

PBN

CN

nitroxide

CN O

O reaction c

N

N +

CN

CN

CN alkoxyamine

CN

reaction d

+

CN

styrene

styryl derivative

CN

O reaction e

O N

CN

N

+

CN

CN

PNA

Figure 4.2: Reactions involved in the in situ NMP process for which thermodynamic data was obtained using the BMK/6-311g** method


111

Table 4.1: Calculated thermodynamic data for reactions a and d given in Figure 4.2

np -nr a

reaction a reaction d a

∆r H

∆r S

∆r G

Kc

kJmol−1

Jmol−1 K −1

kJmol−1

(kmol m−3 )np −nr

85◦ C

2

33.013

365.355

-97.839

5.460 1015

110◦ C

2

32.879

364.987

-106.966

1.202 1016

85◦ C

-1

-88.042

521.569

-274.842

3.563 1041

110◦ C

-1

-87.869

539.184

-294.457

4.364 1041

number of products minus number of reactants

4.2.1

Substituents in α-position

In this section, the N-tert-butyl-α-isopropylnitrone (nitrone 1), the N-tert-butyl-α-tertbutylnitrone (nitrone 2) and the C-phenyl-N-tert-butylnitrone (PBN) will be compared. These structures differ in the substituent on the carbon of N=C: nitrone 1 bears an isopropyl group, nitrone 2 has a more voluminous tert-butyl group, whereas in PBN the substituent is a phenyl group. It was experimentally observed that the initiation efficiency of the alkoxyamines based on these nitrones increases in the following order : nitrone 2 < nitrone 1 < PBN. The calculated thermodynamic data for the reactions of in situ NMP involving nitrone 1, nitrone 2 and PBN, is provided in Tables 4.2, 4.3 and 4.4, respectively. The initiation efficiency is on the one hand dependent on the amount of alkoxyamine formed during the prereaction. On the other hand the ability of the formed alkoxyamine to release cyanoisopropyl radicals at elevated temperatures will play an important role in the overall polymerization process. Formation of nitroxides (reaction b)

In order to explain the difference in the experimentally observed initiation efficiency, first the affinity of nitroxide formation must be evaluated for the three nitrones under consideration. The ease of carbon centered radical addition to a double bond is dependent on various factors, the most important being the steric effects, exothermicity of the reaction and polar effects (Fischer, 2001). These factors will be investigated in more detail in what follows.


( a)

112

( b)

( c)

Figure 4.3: Spatial representation of nitrone 1 (a), nitrone 2 (b) and PBN (c)

Steric effects

Analysis of the geometries of the three nitrones shows that the steric

hindrance is the highest for nitrone 2 with its tert-butyl group, followed by nitrone 1, with its relatively modest isopropyl substituent. PBN has the lowest steric hindrance. It might seem surprising that a phenyl substituent is considered to induce less steric hindrance than a isopropyl or a tert-butyl group. However, this becomes more clear when the spatial orientation is analyzed (see Figure 4.3): the phenyl group in PBN is oriented in the plane formed by ON=C, whereas the substituents in nitrone 1 and nitrone 2 are oriented out of the plane. This way, the isopropyl group in nitrone 1 acts as a shield over the double bond, and this effect is even more pronounced in case of nitrone 2. Reaction enthalpy

Further, it is theoretically and experimentally verified that the ease

of a carbon centered radical addition to an unsaturated bond in a molecule is related to the reaction enthalpy of this process (Fischer, 2001). In a reaction family, the activation energy of the addition reaction decreases with increasing exothermicity of the reaction (Evans and Polanyi, 1936). The exothermic character of the addition reactions for the nitrones under consideration increases in the order nitrone 2 < nitrone 1 < PBN (see Tables 4.2–4.4), suggesting that more nitroxide will be formed with PBN than with the other nitrones. The same is indicated by the K value obtained for PBN, which is about 104 larger than the K values for the other nitrones. According to this line of reasoning, nitroxide formation is slightly more favored for nitrone 1 than for nitrone 2.


113

Table 4.2: Calculated thermodynamic data for reactions b, c and e given in Figure 4.2 for N-tert-butyl-α-isopropylnitrone (nitrone 1)

np -nr a

nitrone 1

reaction b reaction c reaction e a

∆r H

∆r S

∆r G

Kc

kJmol−1

Jmol−1 K −1

kJmol−1


85◦ C

-1

-96.767

-220.133

-17.927

1.210 104

110◦ C

-1

-96.673

-219.882

-12.425

1.553 103

85◦ C

-1

-68.673

-253.224

22.019

1.806 10−2

110◦ C

-1

-68.604

-253.040

28.348

4.295 10−3

85◦ C

1

-95.862

-70.030

-70.781

1.618 10−12

110◦ C

1

-95.980

-59.269

-73.271

3.263 10−12


Table 4.3: Calculated thermodynamic data for reactions b, c and e given in Figure 4.2 for N-tert-butyl-α-tert-butylnitrone (nitrone 2)

np -nr a

nitrone 2 reaction b reaction c reaction e a

∆r H

∆r S

∆r G

Kc

kJmol−1

Jmol−1 K −1

kJmol−1


85◦ C

-1

-90.190

-231.308

-7.347

3.465 102

110◦ C

-1

-90.103

-231.087

-1.562

5.134 101

85◦ C

-1

-61.681

-233.860

22.076

1.772 10−2

110◦ C

-1

-61.563

-233.542

27.919

4.914 10−3

85◦ C

1

-52.419

-72.264

-26.537

4.587 10−6

110◦ C

1

-52.364

-61.026

-28.982

3.561 10−6


Table 4.4: Calculated thermodynamic data for reactions b, c and e given in Figure 4.2 for PBN

np -nr a

PBN

reaction b reaction c reaction e

b

∆r S

∆r G

Kc

kJmol−1

Jmol−1 K −1

kJmol−1

(kmol

Kc,exp −3 n m ) p −nr

85◦ C

-1

-111.947

-206.497

-37.990

1.020 107

-

110◦ C

-1

-111.799

-206.100

-32.832

9.404 105

-

85◦ C

-1

-84.893

-250.944

4.983

5.515

-

110◦ C

-1

-84.773

-250.621

85◦ C

1

-67.433

-77.164

110◦ C a

∆r H

1

-67.445

-66.113

11.252

9.193

10−1

-

-39.797

5.343 10−8

-

-42.114

10−8

number of products minus number of reactants Experimental result reported by Sciannamea et al. (2007)

5.771

8.7 10−10

b


114

Table 4.5: Calculated Mulliken charges for the atoms of the ON=C double bond in the considered nitrones

Polar effects

nitrone 1

nitrone 2

PBN

DPN

DMPO

C

-0.028

0.121

0.161

0.070

-0.045

N

-0.010

-0.026

-0.078

-0.152

0.003

O

-0.350

-0.432

-0.439

-0.418

-0.351

Finally, the nitroxide formation is evaluated in terms of polar effects.

Experimentally, it is observed that it is favorable for an addition reaction to add electrondonor-substituted radicals to double bonds bearing an electron-acceptor group and vice versa (Fischer, 2001). In other words, the addition of a radical of which the carbon center is partially negatively charged to a partially positively charged carbon atom which makes up the double bond, is favored (and vice versa). The fact that the carbon centered radical under consideration bears electron-donor methyl groups, as well as an electron-acceptor cyano group, complicates the analysis. The same problem is encountered for the nitrones, where the carbon which participates in the reaction, bears an electron-donor alkyl (or an electron-acceptor phenyl) group, as well as an electron-acceptor NO group. Hence, in order to evaluate this feature, the Mulliken charges on the atoms of interest where investigated, and are shown in Table 4.5. The carbon atom of cyanoisopropyl radical is slightly negatively charged (-0.090). The carbon atom of the nitrones is negatively charged for nitrone 1 and positively charged for nitrone 2 and PBN. Hence the addition of the radical to PBN should be conducted most easily, followed by the addition to nitrone 2 and, as last, nitrone 1. It is difficult to say what the relative importance of the factors discussed above is on a radical addition to a double bond. However, it is clear that nitrone 2 has the least favorable characteristics for nitroxide formation (highest steric hindrance and least exothermic reaction), followed by nitrone 1, whereas nitroxide formation is more favorable for PBN. This conclusion is supported by the K values for reaction b (Figure 4.2) for these nitrones, as well as by experimental findings. Moreover, for nitrone 1 and PBN reaction rate coefficients were obtained based on the transition state theory. For nitrone 1 the rate coefficient of the forward step in reaction b was found to be much smaller than for PBN −4 3 −1 −1 at 85◦ C). The backward (k+ m3 kmol−1 s−1 , k+ nitrone 1 =2.031 10 P BN =1.352 m kmol s


115

−8 −1 −7 −1 rate coefficients of reaction b were k− s and k− s at nitrone 1 =1.672 10 P BN =1.292 10

85◦ C. These values are in line with the conclusions drawn based on the thermodynamic calculations. Formation of alkoxyamines (reaction c)

The alkoxyamine formation for the radicals derived in the previous step will be evaluated (reaction c in Figure 4.2). In this reaction a cyanoisopropyl radical recombines with the nitroxide radicals. Here again, the steric factors have an important influence. These effects are similar to those discussed for the nitroxide formation, again the steric hindrance increases in the order PBN < nitrone 1 < nitrone 2. The calculated reaction enthalpies increase in the order nitrone 2 < nitrone 1 < PBN, and the K values for alkoxyamine formation based on PBN is almost 103 higher than those for the other nitrones. The polarization effects do not play a significant role in the alkoxyamine formation, as the partial Mulliken charges for the oxygen atom are similar for the three structures (about -0.332). In conclusion it must be mentioned that based on the thermodynamic data presented here for the prereaction of the three considered nitrones, the most alkoxyamine will be generated with PBN, followed by nitrone 1, whereas nitrone 2 generated the least alkoxyamines. Cleavage of alkoxyamines (backward reaction c)

The ability of the formed alkoxyamines to release cyanoisopropyl radicals must be evaluated at 110◦ C, as this is the temperature at which the polymerization is initiated. The cleavage of alkoxyamine is the backward reaction c in Figure 4.2. Based on the K values for reaction c, for the various nitrones, it is clear that nitrone 1 and nitrone 2 will generate cyanoisopropyl radicals with slightly more ease than PBN will. This may also indicate slow recombination of the propagating radical with the nitroxide in case of nitrone 1 and nitrone 2. This way the higher Mn values for these nitrones after 15 hours of polymerization, which were reported by Sciannamea et al. (2005) could be explainedby the lower initiation ¡ of nitrones 1 and 2 compared to that of PBN. This initiation efficiency can be understood from the following reasoning. In the prereaction step, nitrone 1 and nitrone 2 generate less nitroxides than PBN. Hence, the amount of generated alkoxyamines will


116

also be lower than in case of PBN. This means that the initiation efficiency of PBN will be higher than that of the other two nitrones, what was also observed experimentally. Moreover, in the case of nitrone 1 and nitrone 2 there is a great amount of unreacted cyanoisopropyl radicals and AIBN available. At elevated temperatures, the unreacted AIBN generates even more radicals, which initiate the styrene polymerization. This explains the burst effect encountered experimentally when the polymerization was conducted with nitrone 1 and nitrone 2. On the contrary, a small induction period was experimentally observed for PBN. This is in agreement with the calculated value for K of reaction b for PBN as all nitrone and AIBN present were converted into nitroxides, that have to be trapped by the thermally initiated styryl radicals, before the polymerization reaction can proceed in a controlled way. Cleavage/formation of PNA (reaction e)

The cleavage and formation of PNA corresponds with the activation/deactivation reaction, typical for controlled radical polymerization (NMP in this case). In Table 4.4, the calculated and experimentally obtained value for the equilibrium coefficient of this reaction are listed. There is a difference of a factor 102 . This is not a completely unsatisfactory result, because the influence of the chain length of the styryl derivative radical is not yet taken into account. Taking this into account may influence the value of the calculated equilibrium coefficient. 4.2.2

Substituents in β-position

Now, PBN will be compared with DPN (see Figure 4.1). These structures differ in the substitution of the tert–butyl group in PBN for a phenyl group in DPN on the nitrogen of the C=N. The calculated thermodynamic data for DPN is given in Table 4.6. Formation of nitroxides (reaction b)

First, it must be mentioned that for the nitroxide formation with these structures, steric effects are of minor importance, because it is generally known that only the substituents on the carbon atom which participates in the reaction are of importance (Fischer, 2001). In this case, this substituent is the same (phenyl group) for both nitrones. Also the polar


117

Table 4.6: Calculated thermodynamic data for reactions b, c and e given in Figure 4.2 for DPN

np -nr a reaction b reaction c reaction e a

∆r H

∆r S

∆r G

Kc

kJmol−1

Jmol−1 K −1

kJmol−1


85◦ C

-1

-132.236

-208.435

-57.585

7.354 109

110◦ C

-1

-132.086

-208.033

-52.378

4.345 108

85◦ C

-1

-128.734

-232.554

-45.445

1.247 108

110◦ C

-1

-128.633

-232.287

-39.632

7.951 106

85◦ C

1

-121.017

-52.737

-102.129

4.335 10−17

110◦ C

1

-120.993

-41.591

-105.057

1.515 10−16


Table 4.7: Calculated thermodynamic data for reactions b, c and e given in Figure 4.2 for DMPO

np -nr a

DMPO reaction b reaction c reaction e a

∆r H

∆r S

∆r G

Kc

kJmol−1

Jmol−1 K −1

kJmol−1


85◦ C

-1

-140.977

-209.392

-65.983

1.234 1011

110◦ C

-1

-140.853

-209.062

-60.751

6.018 109

85◦ C

-1

-231.839

-237.605

-33.464

2.232 106

110◦ C

-1

-231.442

-237.208

-27.675

1.863 105

85◦ C

1

-119.658

-52.997

-100.677

7.060 10−17

110◦ C

1

-119.661

-41.918

-103.600

2.394 10−16



118

effects are of less importance, because in both cases the double bond carbon atom is positively polarized (see Table 4.5). Yet, there is a significant difference in the reaction enthalpy, as the nitroxide formation for DPN is about 20 kJ mol−1 more exothermic. This is also expressed in the K values for these reaction, with the K value for DPN being 103 higher than for PBN. Formation of alkoxyamines and PNA (reaction c and backward e)

The difference in K values for alkoxyamine formation of about 106 is probably due to the steric hindrance for recombination that is much larger in the case of PBN than in the case of DPN, because there is respectively a t-butyl substituent and a phenyl substituent in β-position of the oxygen radical center. The influence of this substitution is remarkably higher than in section 4.2.1, because in the latter the changing steric hindering substituents were in δ-position of the oxygen radical center. Based on this analysis some important conclusions may be drawn. First, based on thermodynamics, it seems that significantly more alkoxyamine will be generated in the prereaction step with DPN, than with PBN. However, the formed alkoxyamine based on DPN is so stable, that it will release less cyanoisopropyl radicals to initiate the polymerization process. Hence, the advantage of having more alkoxyamine with DPN rather than with PBN is cancelled out, so the initiation efficiency (defined as Mn,theoretical /Mn,experimental ) of these nitrones should not differ much. This was confirmed experimentally (initiation efficiency of 0.15 for DPN and 0.14 for PBN (Sciannamea et al., 2005)). Moreover, the PNA adduct formed (i.e. backward reaction e) for DPN is much more stable than that formed with PBN (the K values are 1/10−16 and 1/10−8 respectively). This explains the significant difference in conversion after 15 hours of polymerization, which is 56% for PBN and only 26% for DPN (Sciannamea et al., 2005). 4.2.3

Structural effects

Finally, the reactivity of the cyclic DMPO is discussed. This nitrone is the least sterically hindered from all the nitrones discussed so far. Moreover, the enthalpy of reaction for nitroxide formation is the most exothermic from all nitrones under consideration. These effects overshadow the partially negative charge on the doubly bonded carbon atom in


119

DMPO. Hence, the nitroxide formation based on DMPO is strongly favored from the thermodynamic point of view. The formation of alkoxyamine based on DMPO is less exothermic than that based on DPN. This might be an explanation for the highest initiation efficiency observed with DMPO: similarly to the case with DPN, there is a lot of alkoxyamine generated in the prereaction. In contrary to the case of DPN though, this alkoxyamine dissociates easier under polymerization conditions, leading to an increased initiation efficiency. Like with DPN, reaction e in Figure 4.2, is strongly shifted toward the PNA adduct, yielding conversions after 15 hours of polymerization which are even lower than those for DPN by a factor 2.

4.3 Conclusion

4.3

120

Conclusion

In conclusion of the previous discussion, it is made clear that thermodynamic calculations can be used to confirm a part of the experimental observations by Sciannamea et al. (2005). However, for a more profound understanding of the influence of the structures of the nitrone/initiator pair on the polymerization behavior, the thermodynamic and kinetic values obtained via ab initio calculations on elementary reactions should be plugged into a (fundamental) kinetic model, that can describe the polymerization process by monomer conversion, (moments of the) molecular mass distribution, etc. A first step toward the development of such a model is given in the next chapter.

MODELING OF (IN SITU) NMP

121

Chapter 5

Modeling of (in situ) NMP 5.1

Formal kinetic model for NMP

In literature various formal kinetic models for NMP have been proposed. These models describe polymerization rates, moments of the molecular mass distribution (MMD) (Mn , Mw and Mz ) and polydispersities as a function of polymerization time. The most simplistic model, which does not account for termination reactions was derived by Matyjaszewski et al. (1997). Subsequently Fischer (1999) expanded this by including termination reactions. A common characteristic of these models is that diffusion limitations are not accounted for and all rate coefficients are assumed to be independent of the chain length. Transfer to solvent is also not included in these models. For a detailed discussion of these models the reader is directed to the Master thesis of Peeters (2007).

In this section an extended formal kinetic model for NMP will be discussed. All significant steps are included, yet diffusion limitations have not yet been accounted for. The rate coefficients of the elementary reactions are assumed to be independent of the chain length. The reactions included in this extended kinetic model for NMP are presented in Table 5.1. It must be noted that a distinction has been made between the activation of an initiator (R0 X) and the activation of a dormant species (Rn X). A distinguishable notation is also applied to deactivation and propagation reactions of the initiator and the polymer radicals. Two types of termination reactions are accounted for in this model, namely termination by recombination and by disproportionation. Transfer to monomer has also

5.1 Formal kinetic model for NMP

122

Table 5.1: Reactions included in the kinetic model for NMP; R0 X = initiator, Rn X = dormant species, Rn = propagating polymer radical, M = monomer, Pn = terminated polymer chain, R0 R0 = product of initiator radical recombination, n = chain length. k

Activation initiator

a0 R0 X → R0 + X

Deactivation initiator radical

da0 R0 + X → R0 X

Propagation initiator radical

R0 + M → R1

Propagation polymer radical

Rn + M → Rn+1

Activation dormant species

Rn X → Rn + X

Deactivation macro radical

Rn + X → Rn X

Transfer to monomer

Rn + M

k

kp0

kp,n

ka,n

kda,n

ktrm,n

→ Pn + R1

ktrm0

Transfer initiator to monomer

R0 + M → P0 + R1

Termination by recombination

Rn + Rm → Pn+m

Recombination initiator and macro radical

tc0 Rn + R0 → Pn

Recombination initiator radicals

tc00 R0 + R0 → R0 R0

Termination by disproportionation

Rn + Rm → Pn + Pm

Disproportionation of initiator and macro radical

td0 Rn + R0 → Pn + P0

Disproportionation initiator radicals

td00 R0 + R0 → P0 + P0

Thermal (self-)initiation

th 3M → 2R0

ktc,n

k

k

ktd,n

k

k

k

been included. Note that in this model, regular NMP is implemented, i.e. an alkoxyamine (unimolecular) initiator (R0 X, see Table 5.1) and the monomer are brought together to start polymerization. This is in contrast to in situ NMP, where a classic initiator, nitrone and – after prereaction – monomer are brought together. Mass balances corresponding to the reactions under consideration have been derived and are given in Table 5.2. Solving these balances would yield a complete molecular mass distribution for the polymer molecules. However, simultaneous calculation of these equations is not feasible from a computational point of view. Hence, average values for the molecular mass distribution of the polymer, the so–called moments Mn , Mw and Mz will be calculated using the momentum method. These moments are defined in Equations (5.1)–(5.6). A distinction has been made between the moments of the dormant and the terminated polymer chains (Mn,RX , Mw,RX , Mz,RX and Mn,P , Mw,P , Mz,P respectively). In these equations Rn X denotes the concentration of the dormant polymer chains, whereas Pn stands for the concentration of terminated


123

chains. Mm represents the molecular mass of the monomer. Mn,RX = Mw,RX = Mz,RX = Mn,P = Mw,P = Mz,P =

P R Xn P n Mm Rn X P R Xn2 P n Mm Rn Xn P R Xn3 P n Mm Rn Xn2 P P n P n Mm P P n2 P n P n Mm P n P n 3 P n P n 2 Mm Pn n

(5.1) (5.2) (5.3) (5.4) (5.5) (5.6)

The averages of the complete MMD of the polymer chains can now be rewritten as follows: Mn Mw Mz

P (Pn + Rn X)n Mm = P (Pn + Rn X) P (Pn + Rn X)n2 Mm = P (Pn + Rn X)n P (Pn + Rn X)n3 = P Mm (Pn + Rn X)n2

(5.7) (5.8) (5.9)

The averages of the MMD of the dormant (Mn,RX , Mw,RX , Mz,RX ) and the terminated polymer chains (Mn,P , Mw,P , Mz,P ) can be seen as a function of the so–called moments of the distribution of the dormant and terminated polymer chains. The sth order moment of the distribution of the dormant (τs ) and the terminated (µs ) polymer chains are defined as: τs =

∞ X

ns Rn X

(5.10)

n s Pn

(5.11)

n=1

µs =

∞ X n=1

Using Equations (5.10) and (5.11) the averages of MMD of the dormant and the terminated polymer chains (Equations (5.1)–(5.6)) can be written as: Mn,RX =

τ1 Mm τ0

(5.12)


124

Mw,RX = Mz,RX = Mn,P = Mw,P = Mz,P =

τ2 Mm τ1 τ3 Mm τ2 µ1 Mm µ0 µ2 Mm µ1 µ3 Mm µ2

(5.13) (5.14) (5.15) (5.16) (5.17)

In addition, the averages of the complete MMD of the polymer molecules can be expressed in terms of moments of the distribution of the dormant and terminated polymer chains (see Equations (5.18)–(5.20)). (µ1 + τ1 ) Mm µ0 + τ 0 (µ2 + τ2 ) Mm = µ1 + τ 1 (µ3 + τ3 ) Mm = µ2 + τ 2

Mn =

(5.18)

Mw

(5.19)

Mz

(5.20)

As done for the dormant and terminated polymer chains, the sth order moment of the distribution of polymer radicals (Rn ), which is denoted by λs , can be defined as follows: λs =

∞ X

ns Rn

(5.21)

n=1

Hence, the corresponding averages of MMD of the polymer radicals can be expressed as: λ1 Mm λ0 λ2 = Mm λ1 λ3 = Mm λ2

Mn,R =

(5.22)

Mw,R

(5.23)

Mz,R

(5.24)

The averages of the complete MMD of the polymer molecules are thus completely determined by the moments λs , µs and τs , with s = 0, 1, 2, 3. As mentioned before, the rate coefficients of the different elementary reaction steps are assumed to be independent of the chain length. Hence, differential equations for these moment can be derived by sum-


125

Table 5.2: Momentum equations for the moments of distribution of polymer radicals (λs , s = 0, ..., 3), dormant species (τs , s = 0, ..., 3) and terminated polymer chains (µs , s = 0, ..., 3). Rate coefficients assumed to be independent of chain length. (v = reaction volume)

dλ0 λ0 dv = −kda λ0 X − ktc λ20 − ktc0 λ0 R0 − ktd λ20 − ktd0 λ0 R0 + kp0 R0 M + ka τ0 + ktrm0 M R0 − dt v dt dλ1 = −kda λ1 X − ktc λ0 λ1 − ktc0 λ1 R0 − ktd λ0 λ1 − ktd0 λ1 R0 + kp0 R0 M + ka τ1 dt λ1 dv + ktrm M (λ0 − λ1 ) + kp λ0 M + ktrm0 M R0 − v dt dλ2 = −kda λ2 X2 − ktc λ0 λ2 − ktc0 λ2 R0 − ktd λ0 λ2 − ktd0 λ2 R0 + kp0 R0 M + ka τ2 dt λ2 dv + ktrm M (λ0 − λ2 ) + kp λ0 M + 2kp λ1 M + ktrm0 M R0 − v dt dλ3 = −kda λ3 X − ktc λ0 λ3 − ktc0 λ3 R0 − ktd λ0 λ3 − ktd0 λ3 R0 + kp0 R0 M + ka τ3 dt λ3 dv + ktrm M (λ0 − λ3 ) + kp λ0 M + 3kp λ1 M + 3kp λ2 M + ktrm0 M R0 − v dt dµ0 1 µ dv 0 = ktc λ20 + ktrm λ0 M + ktc0 λ0 R0 + ktd λ0 λ0 + ktd0 λ0 R0 − dt 2 v dt dµ1 µ1 dv = ktc λ0 λ1 + ktrm λ1 M + ktc0 λ1 R0 + ktd λ0 λ1 + ktd0 λ1 R0 − dt v dt dµ2 µ2 dv = ktc λ0 λ2 + 2ktc λ21 + ktrm λ2 M + ktc0 λ2 R0 + ktd λ0 λ2 + ktd0 λ2 R0 − dt v dt dµ3 µ3 dv = ktc λ0 λ3 + 3ktc λ1 λ2 + ktrm λ3 M + ktc0 λ3 R0 + ktd λ0 λ3 + ktd0 λ3 R0 − dt v dt dτ0 τ0 dv = kda λ0 X − ka τ0 − dt v dt τ1 dv dτ1 = kda λ1 X − ka τ1 − dt v dt dτ2 τ2 dv = kda λ2 X − ka τ2 − dt v dt dτ3 τ3 dv = kda λ3 X − ka τ3 − dt v dt

mation of the individual mass balances. These moment equations are presented in Table 5.2. Moreover, the conversion (x) can be determined once the expressions for the moments are known. The conversion, at each time step, is given by the following expression: x=1−

MV M0 V 0

(5.25)

In Equation (5.25) and in the equations presented in Table 5.2 some variables are encountered, such as the concentrations of the monomer and initiator. For each of these


126

Table 5.3: Mass balances for the initiator (R0 X), the initiator radicals (R0 ), the monomer (M ), products of termination of the initiator radicals by disproportionation (P0 ) and recombination (R0 R0 ) (v = reaction volume).

dR0 X R0 X dv = kda0 R0 X − ka0 R0 X − dt v dt dR0 = −kda0 R0 X + ka0 R0 X − 2ktc00 R02 − ktc0 λ0 R0 − kp0 R0 M − ktd0 λ0 R0 dt R0 dv − ktd00 R02 + 2kth M 3 − v dt dX X dv = ka0 R0 X − kda0 R0 X + ka τ0 − kda λ0 R0 − dt v dt dM M dv = −kp M λ0 − kp M R0 − ktrm M λ0 − ktrm0 R0 M − 3kth M 3 − dt v dt dR0 R0 R0 R0 dv 2 = ktc00 R0 − dt v dt dP0 P0 dv = ktd00 R02 + ktd0 λ0 + ktrm0 R0 M − dt v dt

variables, a mass balance has been written down, and the corresponding differential equations are given in Table 5.3. Every rate coefficient in the equations of Tables 5.2–5.3 equals the intrinsic rate coefficient (as diffusion effects are not accounted for), except for ktd00 = 2.ktd00,intrinsic . The formal model discussed in this section has been used as a basis for computational kinetic modeling of the NMP process of styrene, which is the topic of the next section. Remark that possibly important reactions have not been included in this simple formal model. Reactions such as transfer to dimer (herewith also a more detailed description of the self–initiation of styrene) and formation of hydroxylamines (by e.g. thermal decomposition of dormant polymer chains) could be important in the case of a NMP process. This will be the subject of future work. Also, diffusional effects will then be taken into account.

5.2 Simulation results for NMP of styrene

127

Table 5.4: Range of the kinetic parameters investigated in the simulations of the NMP of styrene

parameter

notation

range

unit

Equilibrium coefficient

Kc

10−13 − 10−8

kmol m−3

by recombination

ktc

106 − 109

m3 kmol−1 s−1

by disproportionation

ktd

106 − 109

m3 kmol−1 s−1

Rate coefficient of thermal initiation

kth

100 − 10−9

m6 kmol−2 s−1

Rate coefficient of transfer to monomer

ktrm

10−1 − 103

m3 kmol−1 s−1

Rate coefficient of termination:

5.2

Simulation results for NMP of styrene

As mentioned before, some models describing the NMP of styrene are available in literature (Fischer, 1999). In this section, NMP of styrene will be simulated, using a more extended model than those reported in literature. The main difference would be that the decrease of the free reaction volume in the course of the polymerization process is being accounted for in this model, unlike the model proposed by Fischer (1999). In order to clearly illustrate the influence of this feature on the polymerization rates and control, all simulations have been conducted twice: in the first run the corrections for the decreasing reaction volume have been accounted for (dashed lines in Figures 5.1–5.8), whereas in the second run the reaction volume is assumed to be constant (full lines in Figures 5.1–5.8). Literature has been skimmed for both experimental and calculated kinetic parameters concerning the NMP of styrene with different nitroxides. The most important kinetic parameters, as well as their ranges have been determined (see Table 5.4). The influence of the variation of these kinetic parameters on polymerization rate and control was studied. Herefore some characteristic properties which might indicate a controlled process were analyzed, such as the polydispersity index (PDI), the evolution of the number average molecular mass (Mn ) and ln([M ]0 /[M ]). Also the concentrations of the propagating and the dormant species, as well as the nitroxide concentration were simulated and plotted. For all simulations TEMPO was used as the mediating agent, so the polymerization temperature was chosen to be 120◦ C.


128

Table 5.5: Input values for the simulations involving variable equilibrium constant Kc

parameter

notation

value

unit

Rate coefficient of polymerization

kp

2 103

m3 kmol−1 s−1

Rate coefficient of termination by recombination

ktc

108

m3 kmol−1 s−1

Rate coefficient of activation

ka

10−3

s−1

Rate coefficient of deactivation

kda

ka K

m3 kmol−1 s−1

The equilibrium constant

The influence of the variation of the equilibrium coefficient Kc

ka kda

on various polymer-

ization process properties is examined and the results are presented in Figures 5.1 and 5.2. The input values different from zero for these simulations are given in Table 5.5. The studied conditions compared to those applied by Fischer (2001) and the simulations confirmed the validity of our model. The polymerization time needed to reach a conversion of 90% decreases as Kc increases (see Figure 5.1 top left). This could be expected, as with increasing Kc , the equilibrium is shifted toward the propagating species. Hence, more chains propagate per time unit as Kc increases. The evolution of the polydispersity with conversion is similar for Kc values smaller than 10−10 m−3 kmol (see Figure 5.1 top right): the PDI reaches its maximal value early in the polymerization process (at less than 5% conversion) then decreases fast, toward a value of 1.0–1.1. With increasing Kc values, however, the maximum for PDI is shifted toward higher conversions (about 25%) and the final PDI value increases as well (more than 1.5 for Kc = 10−8 kmol m−3 ). The increasing polydispersity index with increasing Kc is logical, because an increasing amount of propagating chains leads to more termination reactions and will thus broaden the molecular mass distribution. The evolution of the number average molecular mass with conversion shows a fast decrease at low conversions, followed by a linear increase for higher conversions (see Figure 5.1 bottom left). The minimum value for Mn is reached later (on conversion scale) as Kc increases (for Kc = 10−11 kmol m−3 at 2% conversion). When Kc is larger than 10−10 kmol m−3 , the evolution of the number average molecular mass with conversion does not converge toward the linear increase, which would be expected ideally. This means that the total number of polymer chains (dead and living) is not constant in the course of the polymer-


129

ization process, hence, this is an indication of poor control, as was already suggested by the larger PDI. When ln([M ]0 /[M ]) is plotted as a function of polymerization time (see Figure 5.1 bottom right), it can be seen that the amount of monomer in the system decreases faster as Kc is higher, which is in line with the increasing conversion with increasing Kc values. All the curves have the tendency to deaccelerate and this might be an indication for the occurance of termination reactions. This is not surprising, as termination by recombination is accounted for in this model. Next, the concentrations of the nitroxide, as well as the dormant and the active species will be discussed (see Figure 5.2). A stable amount of dormant and active species is reached earlier with decreasing Kc (at about 10% conversion for Kc = 10−11 kmol m−3 ). For Kc values higher than 10−10 kmol m−3 no steady state is reached for the dormant and active species. This means that the control is too slowly established, or not at all, with the deviation being the most explicit in the case of Kc = 10−8 kmol m−3 . Morever, the amount of nitroxide increases linearly with conversion for Kc lower than 10−10 kmol m−3 , indicating that the amount of termination is independent of the conversion. However, for high Kc values an upward curvature is observed in the nitroxide vs. conversion plot, indicating that termination reactions become more pronounced with increasing conversion. This is confirmed by the pronounced decrease in the amount of dormant species at high conversion for Kc = 10−8 kmol m−3 , while the amount of active species also decreases. This can only be explained by an increasing amount of terminated polymers. Finally, the influence of accounting for the volume (concentration) variation is investigated. The concentration of the monomer ([M ]), will be higher when this variation is accounted for (this case is called ’model A’ from now on, when this is not accounted for, it will be regarded to as ’model B’). Hence, the value for ln([M ]0 /[M ]) will be lower in model A than in model B. Remark that the difference between the two models becomes more pronounced with increasing conversion, which is obvious as the volume decrease becomes more significant at high conversions. Analogously, the higher concentrations of the dormant species and the nitroxide for model A can be explained. Surprisingly, no significant difference between the two models is found for the active species. However, this can be rationalized by remarking that the concentration of the active species is at all times 4–5 orders of magnitude lower than those of the dormant species and the nitroxides,

conversion [%]

M n [g/mol]

0

K=10e-9

K=10e-8

30

0

0

10

20

K=10e-11 K=10e-13

4000 K=10e-10

8000

12000

16000

20000

0

10

20

30

40

50

60

70

80

30

40 50 60 conversion [%]

60 90 120 polymerization time [h]

K=10e-10

70

150

80

90

180

K=10e-13

K=10e-11

PDI [-] ln([M] 0/[M]) [-]

K=10e-8 K=10e-9 90

0

0.5

1

1.5

2

2.5

1

1.5

2

2.5

3

10

20

0

30

K=10e-8 K=10e-9

0

40 50 conversion [%]

60

70

K=10e-10


K=10e-10

30

K=10e-11 K=10e-13

K given in m-3 kmol no volume effect volume effect

90

150

180

K=10e-13

K=10e-11

80

K=10e-9

K=10e-8

5.2 Simulation results for NMP of styrene 130

Figure 5.1: Influence of the equilibrium coefficient Kc on the evolutions of the conversion, polydispersity index (PDI), number average molecular mass (Mn ) and ln([M ]0 /[M ]) as a function of time or conversion.

3 dormant species [mol/m ]

nitroxide [mol/m3]

0

2e-006

4e-006

6e-006

8e-006

1e-005

0

1e-005

2e-005

3e-005

0

0

10

10

20

20

K=10e-13 K=10e-11 4e-005 K=10e-10

5e-005

30

30



K=10e-9

80

70

90

3 active species [mol/m ]

K=10e-9 1e-010 K=10e-10

2e-010

3e-010

4e-010

30

K=10e-8

K=10e-13 K=10e-11 0 0 10 20

K=10e-11 K=10e-13 80 90

K=10e-10

K=10e-9

K=10e-8

70

K=10e-8

5e-010

6e-010


70

K given in m-3 kmol no volume effect volume effect

80

90


Figure 5.2: Influence of the equilibrium coefficient Kc on the concentrations of the dormant species, of the active species and of the nitroxide


132

hence, volume effects are of less importance in this case. Based on the observations discussed above, it is made clear that in order to have a controlled polymerization, the Kc value must be lower than 10−10 kmol m−3 for a good control over the NMP of styrene. Accounting for the reaction volume decrease does not have a significant influence on the process properties which do not involve volumes (inherent to concentrations).


133

The rate coefficient of termination by recombination and by disproportionation Table 5.6: Input values for the simulations involving a varying rate coefficient of termination

parameter

notation

value

unit


Kc

10−11

kmol m−3


ka

10−3

s−1

Rate coefficient of deactivation

kda

108

m3 kmol−1 s−1


kp

2 103

m3 kmol−1 s−1

The influence of the variation of the rate coefficient of termination by recombination (ktc ) and the rate coefficient of termination by disproportionation (ktd ) were investigated independently. However, the resulting effects on the polymerization process characteristics were identical. Hence only the variation of ktc will be discussed here. The simulation results are given in Figures 5.3 and 5.4. The input values different from zero for these simulations are given in Table 5.6. Remark that the same input values were used for the simulations of ktd . The conversion as a function of the polymerization time decreases with increasing rate of termination. This is logical, because when the rate of termination increases, there are less living chains left to propagate, hence, the consumption of monomer decreases too. The PDI decreases with increasing ktc values. The PDI decreases with conversion and ultimately levels off at a values of about 1.05 for high ktc values, whereas for the termination rate coefficient of 106 m3 kmol−1 s−1 the PDI at 90% conversion reaches a value of about 1.2. For Mn and ln([M ]0 /[M ]) the same shapes of the plots are observed as they were in the case when Kc was varied. The ideal, linear increase of Mn is best approximated by a ktc of 109 m3 kmol−1 s−1 . Also here ln([M ]0 /[M ]) shows deacceleration in the first 30 hours of the polymerization process, indicating the occurance of termination reactions. The downward curvature is more explicit when a higher ktc is applied. This is not surprising, as a straight line will be only obtained when no termination reactions occur at all. A constant level is reached after a certain degree of conversion for the concentration of the dormant and the active species as a function of conversion. This level is reached earlier when the rate of termination increases (see Figure 5.4). This could be expected,

conversion [%]

M n [g/mol]

0

0

0

10

20

ktc=10e8 ktc=10e9 30



ktc=10e7

ktc=10e6

30

4000 ktc=10e7

8000

12000

16000

20000

0

10

20

30

40

50

60

70

80

ktc=10e6

70

150

80

ktc=10e9

ktc=10e8

90

180

PDI [-] ln([M] 0/[M]) [-]

90

0

0.5

1

1.5

2

2.5

1

1.5

2

2.5

3

0

0

30

ktc=10e6

10

ktc=10e9

30


60

ktc=10e7


ktc=10e7

20

ktc=10e8

70

150

ktc=10e9

ktc=10e8

80

ktc=10e6

-1 -1

ktc given in m 3kmol s no volume effect volume effect

180

90


Figure 5.3: Influence of the rate coefficient of termination by recombination (ktc ) on the evolutions of the conversion, the polydispersity index (PDI), number average molecular mass (Mn ) and ln([M ]0 /[M ]) as a function of conversion or polymerization time.


nitroxide [mol/m3]

20

30


70

80

90

0

2e-007

4e-007

6e-007

8e-007

1e-006

0

10

20

30


70

80

90

ktc=10e6

ktc=10e7

ktc=10e8

ktc=10e9

0

10

ktc=10e9 0

1.2e-010

1.8e-010

2.4e-010

6e-011

0

ktc=10e7 ktc=10e6

ktc=10e8

ktc=10e9

3e-010

1e-005

2e-005

3e-005

4e-005

5e-005


0

10

ktc=10e8 20

30

70

80

ktc=10e6

-1 -1

ktc given in m 3kmol s no volume effect no volume effect


ktc=10e7

-2 -1

ktc given in m 6kmol s

90


Figure 5.4: Influence of the rate coefficient of termination by recombination (ktc ) on the concentrations of the dormant species, of the active species and of the nitroxide


136

as it is easier to control a small amount of propagating chains. The amount of nitroxide increases with conversion and this increase accelerates toward higher conversions. This acceleration is more pronounced for high values of ktc , confirming the persistent radical effect (for more explanation, see Section 1.1). The effects of accounting for reaction volume decrease with conversion is identical as in the former case of varying Kc (see previous section). Concluding, in order to obtain a low PDI and a high conversion, the ktc should vary between 107 and 108 m3 kmol−1 s−1 , when the other parameters are as listed in Table 5.6.


137

The rate coefficient of thermal initiation Table 5.7: Input values for the simulations involving a varying rate coefficient of thermal initiation

parameter

notation

value

unit


Kc

10−11

kmol m−3


ka

10−3

s−1


kp

2 103

m3 kmol−1 s−1

Rate coefficient of termination by recombination

ktc

108

m3 kmol−1 s−1

The influence of the variation of the rate coefficient of thermal initiation (kth ) was studied. The simulation results are given in Figures 5.5 and 5.6. The input values different from zero for these simulations are given in Table 5.7. The conversion as a function of the polymerization time increases with an increasing rate of thermal initiation, what could be expected, as there are more initiating radicals in the polymerization system and, hence, the consumption of monomer increases too. Remark that this increase becomes significant when kth is about 10−9 m6 kmol−2 s−1 . The PDI increases with increasing kth values, yet decreases slightly in the course of the polymerization process and reaches a values of about 1.2 for a thermal initiation rate of 10−9 m6 kmol−2 s−1 . The increase in PDI with increasing kth is due to the fact that no mediating radicals are available for the additionally generated initiating radicals by thermal initiation, hence, the amount of termination reactions will increase, thus leveling out of this effect by the persistent radical effect (slight decrease in PDI). The number average molecular mass evolution with conversion shows lower values, than predicted for the ideal case, as could be expected, as there are more chains initiated, hence, the average length of each chain decreases. For kth of 10−9 m6 kmol−2 s−1 Mn reaches a value of about 12000 g/mol, whereas the ideal value would be about 18000 g/mol. This is clearly a significant effect. The concentration of active species for kth of 10−9 m6 kmol−2 s−1 is higher than in other cases, which is in line with the expectations, as the additionally generated polymer chains by thermal initiation can not be deactivated, as there is no excess nitroxide available. The amount of nitroxide increases in the course of the polymerization and this might be an indication for termination reactions. The concentration of dormant species for kth of


138

10−9 m6 kmol−2 s−1 shows a different behavior from other values: instead of leveling off at about 10% conversion toward a steady value, the amount of dormant species first increases vastly (till about 10% conversion) and continues that increase until high conversions, yet more slowly. This is surprising however, as the amount of dormant species should decrease if termination reactions become more important. Nevertheless, new (extra) chains are continuously initiated by thermal (self-) initiation. Due to termination reactions, an excess amount of capturing nitroxide radicals is present (the so–called ’persistent radical effect’), so newly initiated radicals are immediately captured, explaining the continuos increase in dormant species.

conversion [%]

M n [g/mol]

0

30


70

80

90

kth=10e-9

1

1.5

2

2.5

1

0 20

180

kth=10e-10

kth=0

150

0

10


1.5

2

2.5

3

0.5

0

30

kth=10e-10

kth=0

4000

8000

12000

16000

20000

0

10

20

30

40

50

60

70

80

kth=10e-9

PDI [-] ln([M] 0/[M]) [-]

90

0

0

10

30

20

kth=0

kth=10e-9


60


30

70

80

kth=10e-9

150

kth=10e-10 kth=0

kth=10e-10

-2 -1

kth given in m6 kmol s no volume effect volume effect

180

90


Figure 5.5: Influence of the rate coefficient of thermal initiation (kth ) on the evolution of the conversion, of the polydispercity index (PDI), of the number molecular mass (Mn ) and of ln([M ]0 /[M ]) as a function of polymerization time or conversion


nitroxide [mol/m3]

0

0

8e-008

1.6e-007

2.4e-007

3.2e-007

4e-007

0

1e-005

2e-005

3e-005

4e-005

5e-005

6e-005

0

10

10

20

20



30

30

80

kth=0

70

80

kth=0

90

90

kth=10e-10

kth=10e-9

70

kth=10e-9


kth=10e-10

0

4e-012

8e-012

1.2e-011

1.6e-011

2e-011

0

10

20

30

kth=10e-9

70

80

kth=10e-10 40 50 60 conversion [%]

kth=0

-2 -1

kth given in m6 kmol s no volume effect volume effect

90


Figure 5.6: Influence of the rate coefficient of thermal initiation (kth ) on the concentrations of the dormant species, of the active species and of the nitroxide


141

Transfer to monomer

The effect of transfer to monomer (ktrm ) was taken into account and the results of these simulations are presented in Figures 5.7 and 5.8. The input values where the same as in the case of thermal initiation (see Table 5.7). When the values of ktrm are up to 101 m3 kmol−1 s−1 , no influence on the conversion with polymerization time is observed: the plotted evolution is equal to that of a polymerization process with no transfer to monomer with Kc = 10−11 m3 kmol−1 . However, for a ktrm of 103 m3 kmol−1 s−1 , the conversion rate increases significantly. Remark that the evolution observed in this case is the same as observed for the case of Kc = 10−10 m3 kmol−1 , with no transfer to monomer. The influence of ktrm is huge for the PDI and Mn . Where for a ktrm of 10−1 m3 kmol−1 s−1 , the PDI tends to unity during the polymerization process and Mn evolutes nearly linear up to 18000 g/mol, the PDI for a ktrm of 101 m3 kmol−1 s−1 first decreases until about 20% conversion and subsequently increases until its final value of about 1.5 at 90% conversion and the Mn is merely 10000 g/mol at 90% conversion. The situation is even more drastical in the case of a ktrm of 103 m3 kmol−1 s−1 , where the PDI is nearly constant as a function of conversion, at a value of about 1.65. Moreover, the number average molecular mass is also constant over the polymerization process, at a value of about 200 g/mol. This means that with increasing ktrm the polymerization process converts into an oligomerization process. The analysis of the concentration of the dormant species shows that for ktrm of 10−1 m3 kmol−1 s−1 and 103 m3 kmol−1 s−1 the concentration of dormant species is constant, yet this concentration increases for ktrm = 101 m3 kmol−1 s−1 . This is against the expectations, as there is no solid explanation for this. However, for the varying values of ktrm , the active species and the nitroxide concentrations evolve more logically: higher concentrations of active species and a lower concentration for nitroxide in case of ktrm = 103 m3 kmol−1 s−1 than for 10−1 m3 kmol−1 s−1 , indicating less termination reactions in the case of the highest value for ktrm . This discussion indicates that in order to obtain a polymerization reaction with a low polydispersity index the rate of transfer to monomer has to be diminished, with an upper bound of about 10−1 . Otherwise an oligomerization will occure resulting in high PDI values.

conversion [%]

M n [g/mol]

0

0

4000

8000

12000

16000

20000

0

10

20

30

40

50

60

70

80

0

10

30

20

30


70

180

80

90

ktrm=10e3

ktrm=10e1

ktrm=10e-1

150

ktrm=10e-1

ktrm=10e1


ktrm=10e3

PDI [-] ln([M] 0/[M]) [-]

90

0

0.5

1

1.5

2

2.5

1

1.5

2

2.5

3

0

0

10

30

20


60

70

80

150

ktrm=10e-1

ktrm=10e1


ktrm=10e3

30

180

90

ktrm=10e-1

ktrm=10e1

ktrm=10e3

-1 -1



Figure 5.7: Influence of the rate coefficient of transfer to monomer (ktrm ) on the evolution of the conversion, of the polydispercity index (PDI), of the number average molecular mass (Mn ) and of ln([M ]0 /[M ]) as a function of conversion of polymerization time


nitroxide [mol/m3]

0

0

6e-008

1.2e-007

1.8e-007

2.4e-007

3e-007

0

1e-005

2e-005

3e-005

4e-005

5e-005

0

10

10

20

20


70

80

90

80

90

ktrm=10e3

ktrm=10e1

70

ktrm=10e-1

ktrm=10e-1


30

30

ktrm=10e3

ktrm=10e1

0

1e-011

2e-011

3e-011

0

10

ktrm=10e1

20

30


ktrm=10e-1

ktrm=10e3

70

-1 -1


80

90


Figure 5.8: Influence of the rate of transfer to monomer (ktrm ) on the concentrations of the dormant species, of the active species and of the nitroxide 3 active species [mol/m ]

5.3 First step toward a fundamental model for in situ NMP

5.3 5.3.1

144

First step toward a fundamental model for in situ NMP Prereaction with AIBN and PBN

In the previous section, regular NMP was simulated with a formal model. In future work, this formal model will be expanded in order to be able to describe the in situ NMP process. The next step will then be to develop a fundamental model for (in situ) NMP. A first step in this process is made in this Master thesis, describing elementary reactions important for the development of the fundamental model for the prereaction and the reactions that may occur with the monomer, added after the prereaction. No macroradicals or macromolecules were considered in this work, but they will be part of future work. In order to derive a fundamental model for the prereaction, a system comprising of AIBN and PBN was considered. In the first step AIBN releases two cyanoisopropyl radicals. These radicals can generate new radicals by hydrogen abstraction from PBN or by cyanoisopropyl addition to the double bond in PBN (see Figure 5.9 (1)–(5) and (6)–(7) respectively). Here I• initially stands for the cyanoisopropyl radical, however, in the following steps other radicals will be generated in the system, more precisely structures (1)–(21). These radicals can also participate in H–abstraction reactions and are therefore also denoted by I• . In the second step, the radicals which were generated by H–abstraction or addition reactions can cause another addition reaction to the double bond in unreacted PBN. The addition can thereby occur on the C–atom or on the N–atom (see Figure 5.10 (8)–(14) and (15)–(21) respectively). In the third step, the recombination reactions with these generated radicals are listed (see Figures 5.11–5.15). Subsequently, the generation of hydroxylamines in the prereaction is considered (see Figure 5.16). Here a H–atom is abstracted from cyanoisopropyl by an oxygen centered radical to form a hydroxylamine, or a carbon centered radical. Here the structure (58) stands for the radicals (6), (8)–(14), whereas the structure (61) stands for the radicals (1)–(5), (7), (15)–(21). In a last step, disproportionation reactions which may occur in the prereaction will be discussed. As this is the first step toward a fundamental model for the prereaction of AIBN and PBN, some assumptions have been made, in order to reduce the complexity of the system:


145

Structures with two radical centers have been excluded from the model, because the

formation of such species is highly doubtful. Radical formation on structures (22)–(56) is also possible, however the resulting struc-

tures have been excluded from the model to maintain clarity. Next, the probability of the occurance of the structures (1)–(63) will be discussed, based on radical stability theories, as well as experimental observations. Hydrogen abstractions and addition to nitrone

Not all the structures depicted in Figure 5.9 are likely to be encountered in large amounts in a real system. In general, the probability to form a radical increases with the increasing stability of the radical. Structure (1) is a primary carbon centered radical and structure (5) is a secondary carbon centered radical, hence, it is more probable to encounter structure (5) in a real system. However, when the cyanoisopropyl radical approaches the double bond of PBN, there is a choice of either an addition to the carbon radical with formation of a stable nitroxide (structure (6)), or a H–abstraction, with formation of a less stable radical. Therefore, it is most likely that the addition reaction will dominate. The formation of the structures (2)–(4) imply that a H–atom has to be abstracted from a phenyl ring. The CH bond in phenyl, however, is known to have a much higher bond dissociation energy (BDE) than other CH bonds. Hence the formation of these structures is also unlikely. It is experimentally confirmed that structure (6) is being formed during the prereaction of AIBN and PBN (Sciannamea et al., 2007), and there is also evidence that structure (7) exists (Feuer, 1969). Based on this discussion it can be concluded that only structures (6) and (7) depicted in Figure 5.9 will be encountered in a significant amount in reality. Consequently, only a few addition reactions will occur, namely the formation of (8) and (9) in Figure 5.10, because addition to the carbon atom of the C=N bond was prefered (Patai, 1970). Recombination reactions

Subsequently, only a limited number of structures from the recombination reactions depicted in Figures 5.11–5.15 will occur with a certain probability. Only structures (22),


146

-

-

O

O

+

+

N

N

+

I

+

IH

+

IH

+

IH

+

IH

+

IH

(1) -

-

O

O

+

+

N

+

N

I

(2) -

-

O

O

+

+

N

+

N

I

(3) -

-

O

O

+

+

N

N

+

I (4)

-

-

O

O

+

+

N

+

N

I

(5)

-

O

O +

N

N

+

I I

(6) -

O

O

+

N

+

I

N I

(7)

Figure 5.9: Radicals generated in the prereaction step with dissociated AIBN and PBN by H– abstraction ((1)–(5)) and addition ((6)–(7))I• stands for the cyanoisopropyl radical

(23), (24), (29), (30), (36) will be withdrawn from Figures 5.11–5.15. The formation of (30) is doubtful, however, as nitroxide radicals are sufficiently stable.


-

N

O

O

O +

+

147

O

N

N

N

OR O I

O

N

N I I

(8) -

O

O

+

N

+

N

O

N

O

N

N I

(9)

-

+

N

O

O +

+

-

O

(16)

-

O

+

N

O

I

N

-

O

N

OR I

+

H- abstr

(15)

O

O

N

N

-

N

-

O +

-

OR

N

O

O

+

N

+

N

(10)

(17)

-

O +

N

-

-

O

+

N

O

O

+

+

O

N

N

N

OR -

O +

N

+

O

O

+

-

-

N

N

-

O

+

N

O

O

+

+

O

N

N

H- ab

(18)

(11) -

N

OR

+

O

O

+

-

-

N

N

(12) -

-

O

+

N

O

O

+

+

(19) O

N

N

N

OR +

O

-

+

(13) -

O

O

+

N

+

O

N

N

OR I

O

O

N

N

(14)

-

(20)

O

N

O

N

N

(21)

Figure 5.10: Addition reactions of radicals generated in the prereaction step to PBN

Hydro CN


148

Recombinatie radicalen CN

+

CN

CN

CN

(22) O CN

+

CN

O

N

N

I

I

(23) O CN

+

O

N

N

I

I

CN

(24) -

-

O CN

O

CN

+

+

N

+

N

(25) -

-

O

O

+

CN

+

N

+

N

CN

(26) -

O

+

CN

+

N

+

CN

-

O

N

(27)

-

CN

O +

N

+

CN

-

O +

N

(28) O CN

+

O

N

N

I

I

(29)

CN

Figure 5.11: Recombination reactions of radicals generated in the prereaction step

Mayo (70)

+


O N

149

O

+

I

O

O

N

N

N I

I

I

(30) O N

O

+

O

N

N

I

I

I

O N

I

(31) -

O

O +

N

+

N

+

N

O I

O-

N

I

(32) -

-

O

O

O

+

+

N

+

N

N

O N

I I

(33) -

-

O

O

O

+

+

N

+

N

O

N N

I

I

(34) -

-

O

O

O

+

+

N

+

N

N

O

N

I

I

(35) O N

O

+

N I

I

I

O

N

O N

I

(36)

Figure 5.12: Recombination reactions of radicals generated in the prereaction step Mayo (70)

+


O

O

+

I

N

O

I

I

N

150

N

O

N

I

(37) -

O N

O

O

+

+

N

N

I

I -

O +

N

(38) -

O N

O

+

-

+

N

O +

N I

O N

I

(39) -

O

O N

+

O +

N

+

N

I

O N

I

(40) -

O

O N

+

+

N

I

O N

I +

N -

O

(41)

O N

O

+

I

N

I

I

N

O

O N

I

(42)


Mayo (70)

+


-

151

-

-

O

O

O

+

N

+

+

N

N

+

-

O +

N

(43) -

-

-

O

O

O

+

+

N

+

N

+

N -

O +

N

(44) -

-

-

O

O

O

+

+

N

-

+

N

+

O

N +

N

-

-

O

+

N

+

N

+

(45)

-

O

O +

N

-

O +

N

(46) -

O

O

+

N

+

O

N

N

I

I -

O +

N

-

-

O

-

O

+

+

N

-

N

-

O

(48) -

+

N

+

+

O

+

N

N

-

O

+

-

O

+

N

+

(47)

O

O

N

N

+

(49) -

-

O +

N

-

O +

O

+

N

+

-

O N

N

+

(50)


Mayo (70)

+

Hy


-

O

152

N

N

N

+

O

I

O

+

I

+

N

-

O

(51) -

O

N -

-

O

-

O

+

O

+

N

+

N

+

N

(52) -

O -

-

O

-

O

+

+

N

+

+

N

+

N

O N

(53) O

I -

O

N

O

+

N

N

+

I

+

N

-

O

(54) -

-

O -

-

O

O

+

O

+

N

N

+

+

N

N

+

(55) O

I -

O +

N

N

O N

+

+

N

I

-

O

(56)


Mayo (70)

+

+

H

H

CN

+

CN

+

I

CN

(63)


H

H

+

CN

I’

+

I

153 H

I’

Formation of hydroxylamine

Hydroxylamines are formed when a hydrogen is abstracted by an oxygen–centered radical, such as structure (6), or more in general, structure (58) (see Figure 5.16). For each radical having a hydrogen atom in α position relatively to the radical center, a double bond will be formed upon hydrogen abstraction. In our case only the cyanoisopropyl radical (57) will undergo such a reaction, with formation of an unsaturated structure (59) and a hydroxylamine (60). In addition, cyanoisopropyl radicals, and more in general all radicals bearing a H–atom can undergo a reaction with a radical (61) with either formation of the unsaturated structure (59) if the H–atom is in α position relatively to the radical center (i.e. disproportionation reaction, see next paragraph), or with formation of a radical in all other cases.

+

CN

O

(57)

N

R

H

+

O

I

H

+

C

N R

R

N

OH

(59)

(58)

I

+

CN

R

(60)

I

+

OH

I

+

HC

N

R

R

Figure 5.16: Formation of hydroxylamine by H–abstraction from cyanoisopropyl radical and related reactions

Disproportionation reactions

In the mechanism under study, disproportionationation reactions may occur with two cyanoisopropyl radicals, when one radical abstracts a hydrogen atom from the other radical, yielding a terminated cyanoisopropane (63) and again an unsaturated structure (59). In general, disproportionation may occur for all combinations of two radicals bearing

+ as depicted in Figure 5.17. H–atoms, by hydrogen transfer,

+ (82)

+

+

(83) I

+

I’

I

+

I’

H

I

H

+

O

I

H

+

C

N

R

I

+

OH

I

+

HC

N

R

H R

R


+

CN

H

C

R

154

+

CN

HC

(62)

(61)

+

CN

CN

R

CN

+

CN

(63) I

+

I’

I

+

H

I’

Figure 5.17: Disproportionation reaction involving cyanoisopropyl radicals and a generalized representation

5.3.2

Addition of styrene

In Section 5.3.1 structures were identified, which will most likely be found in a real in situ NMP prereaction step. In the polymerization step, the monomer (in this case styrene), is added to the polymerization medium. First the possible H–abstraction reactions are discussed. These are given in Figure 5.18. Here again, hydrogen is abstracted either from a phenyl ring, or from a carbon atom with a double bond. Analogously as discussed for

+

+

H–abstractions in Section 5.3.1, these structures most likely will not be encountered in the real polymerization system.

(82)

The addition reaction to the double bond of styrene shown in Figure 5.19, however, will occur, with I• being one of the radicals formed in the prereaction step, such as structures (6), (7) and (57) or the radicals generated by H–abstraction given in Figure 5.16, as well

+

+

the newly generated radicals after styrene addition to the system. Besides styrene initiation by addition to the double bond, thermal polymerization of styrene may also occur. In literature there is disagreement about the exact mechanism (83)

of the thermal initiation. The contenders are the Mayo mechanism and the Flory mechaI

+

I’

I

+

I’

H

nism, depicted in Figure 5.20 (Khuong et al., 2005). According to the Mayo mechanism, radical initiation proceeds by a Diels-Alder dimerization of styrene. According to the Flory mechanism styrene dimerizes with formation of a diradical (structure (73) in Figure

Mayo

5.20). A third styrene abstracts a hydrogen atom from the diradical to generate a monoradical (75) and a styryl radical. An other option is the ring closing (70)reaction yielding 1,2-diphenylcyclobutane (structure (74)), which is inactive toward polymerization. These + mechanisms will not be discussed in detail here, but are important to mention because thermal initiation can have significant impact on the in situ NMP process. Fundamental


155

H

+

I

+

IH

+

IH

+

IH

+

IH

+

IH

H (64)

+

I (65)

H +

I (66)

+

I (67)

+

I

(68)

Figure 5.18: Formation of styryl radicals by H–abstraction by radical formed in the prereaction step I

+

I (69) I

+

I (70)

Figure 5.19: Formation of styryl radicals by addition to the double bond by radicals formed in the prereaction step Mayo (70)

reaserch on the topic of thermal polymerization of styrene is being currently performed + by Evelyn Burrick. Next, the formation of hydroxylamines by hydrogen abstraction from styrene adducts by

+ (71)

Flory


156

Mayo (71)

+

(72)

+ Flory (73)

+

+ (75)

(74)

Figure 5.20: Formation of styryl radicals by following the Mayo mechanism and the Flory mechanism

oxygen centered radicals formed in the prereaction step, is presented in Figure 5.21. It is important to account for hydroxylamine formation, as there are experimental observations mentioning the formation of hydroxylamines in an NMP process of styrene (He et al., 2000). Remark here that the formation of structures (76) and (77) in Figure 5.21 is not likely, because the reactant styryl adducts which are formed by hydrogen abstraction, are not likely to be formed themselves, as mentioned before. Note that hydrogen abstraction is also possible with other radicals, however, no hydroxylamine will be formed in that case. Finally, the disproportionation reactions which might occur after the addition of styrene are given in Figure 5.22. Here the disproportionation by two styryl radicals is likely, with formation of an ethylbenzene (82) and a styrene molecule. In addition, disproportionation may occur with two adducts formed according to the Mayo mechanism,

(67)

+

IH


157

(68)

I

+

O

N

R

+

OH

N

R

+

OH

N

R

+

OH

N

R

+

OH

N

R

+

OH

N

R

+

OH

N

R

(69) (76) I

+

O

N

R

(70) (77)

+

O

N

R

(78)

+

O

N

R

(79) I

I

+

O

N

R

(80) I

I

+

O

N

R

(81)

Figure 5.21: Formation of hydroxylamine species by abstraction of a hydrogen atom of styryl adducts

Mayo

(71)

+

yielding structure (83). It must be mentioned that if, in general, one of the radicals is bearing a hydrogen atom in α− position relatively to the radical center, they can undergo a disproportionation reaction. (72)

+ Flory (73)

+

H + (75)

(74)

+

IH

(65)

5.4 Conclusion +

158

IH

+

+

(66)

+

(82)

IH

(67)

+

+

(83)

I I

+

I’

+

I

I’

H

Figure 5.22: Disproportionation reactions by abstraction of a hydrogen atom of styryl adducts (68) I

5.4

Conclusion

In this chapter, a formal kinetic model was used to investigate the influence of some kinetic (69)

and thermodynamic parameters on the regular NMP of styrene. Qualitative insight on the impact of the individual kinetic parameters has been obtained and the impact of the equilibrium coefficient has been enlightened. A formal kinetic model for the in situ NMP of styrene is yet to be developed, hence, the first step toward this model has been made. The most important reactions taking place in the prereaction step have been listed, as well as the reactions involving a single styrene molecule. However, this model has yet to be expanded before a formal kinetic model for in situ NMP can be derived.

Mayo (70)

+

+ (71)

Flory (72)

+

CONCLUSION

159

Chapter 6

Conclusion The in situ NMP of styrene was studied in this Master thesis. First a literature study was performed concerning the NMP and in situ NMP of styrene. The major factors of influence on the polymerization kinetics and control were identified. The structures of the nitrone/initiator pair, as well as the presence of a prereaction step in the absence of monomer, were found to have a fundamental role in the controllability of the polymerization process. The polymerization temperature, the prereaction time and the nitrone/initiator molar ratios have an impact on the equilibrium coefficient (K), which influences the controllability of the process, however, to a lower extent than the choice of the nitrone/initiator pair. The optimal reaction conditions for in situ NMP of styrene are as follows: a prereaction of the nitrone and the initiator at 85◦ C for 4 hours with a nitrone/initiator molar ratio of 2/1, followed by addition of the monomer and a polymerization at 110◦ C. The K varies in a range of 10−11 − 10−9 m3 kmol−1 , depending on the nitrone and initiator structures. Based on the literature study of the computational methods in chemistry, it was clear that composite methods would not be applicable to the majority of the structures under study, because of their size, hence, DFT methods were used. Based on the study of the optimized geometries, it was concluded that the values obtained using the 6–311g basis set tend to overestimate the bond lengths. The expensive 6–311++g** basis set yields values remarkably close to the values obtained using the 6–311g** basis set. In general, BB1K and BHandHLYP methods are found to underestimate bond lengths, and are therefore not suitable for accurate geometry calculations. B3LYP/6–311g** seems to be the best

CONCLUSION

160

DFT method from the applied test set to describe molecular geometries. For the calculation of standard heats of formation, the MPW1PW91/6–311g** method and the BMK/6–311g** method are most suitable, whereas the standard entropies of formation are best predicted by the B3LYP/6-311g** method. The BMK/6–311g** method seems to be the best method to calculate reaction enthalpies. For the kinetic parameters of addition/β–scission reactions, the BB1K and BMK methods have been found to perform well. In this work the focus lies on energy and kinetic calculations, hence, the BMK/6-311g** method is chosen as the best performing method for the calculations of the in situ NMP process of styrene. Applying the chosen method to reactions involved in the in situ NMP of styrene yielded thermodynamic data, based on which it was possible to confirm a part of the experimental observations by Sciannamea et al. (2005). When the ranges of values obtained with the thermodynamic calculations – and also from literature – were plugged into a formal kinetic model for the NMP of styrene, qualitative conclusions about the influence of various kinetic and thermodynamic parameters could be drawn. This indicates that an extended formal model or a more fundamental kinetic model for the in situ NMP might be able to make accurate predictions of the controllability and polymerization rate of in situ NMP systems. Future work

In this Master thesis a first step is made toward the development of a fundamental kinetic model of the in situ NMP of styrene. In the first place, an extention of the formal kinetic model of NMP will have to be done in order to describe the in situ process. Also diffusional effects on the polymerization and more reactions, such as transfer to rimer and the formation of hydroxylamines, should ne accounted for. Next, a more fundamental model, describing (in situ) NMP should be developed. To obtain sufficient kinetic and thermodynamic data to perform simulations, more ab initio calculations on elementary reactions will have to be performed. In a next step, the kinetic model of the in situ NMP polymerization of styrene could be expanded toward other (vinylic) monomers.

APPENDIX A

Appendix A

161

487.478 355.272 309.014 332.331 234.718 435.404 318.294 459.972 229.484 440.433 344.803 267.671 356.355 381.188 357.678 390.622 287.52 592.287

AIBN

benzoyloxy radical



hydroxylamine

TEMPO-H

dimethylnitrone

PBN

aminoxyl radical

TEMPO

styrene

benzene

ethylbenzene

1-methylethylbenzene

ethylbenzene radical

1-methylethylbenzene radical

phenyl radical

TEMPO-Styryl

Domalski and Hearing (1993)

397.562

diacetylperoxide

a

541.906

BPO

B3LYP

589.111

287.412

390.099

356.991

380.782

356.95

267.584

344.992

439.483

229.024

458.34

317.729

434.123

234.3

334.582

300.666

354.657

486.403

393.487

534.249

B3P86

587.789

287.11

389.823

356.753

380.292

356.196

267.353

347.151

438.337

228.99

457.617

317.051

433.17

234.086

333.595

299.863

354.209

484.453

393.361

531.618

MPW1PW91

-

286.349

389.187

350.711

374.945

366.016

266.558

348.226

429.571

228.543

454.558

311.867

428.471

233.513

336.197

295.503

481.683

387.671

514.651

BB1K

573.714

287.169

392.045

352.105

377.087

355.678

267.186

339.394

435.115

229.003

452.73

310.009

426.316

233.789

332.837

295.294

-

-

393.428

527.224

BMK

578.597

285.457

386.405

353.887

376.376

354.038

265.692

344.552

432.462

228.94

452.01

315.101

427.927

233.572

331.833

234.672

348.318

477.055

391.052

523.877

BHandHLYP

388.57a

360.475

269.250

345.1a

236.18

exp

0 for the entire test set (6-311g basis set) (in Jmol−1 K −1 ) Table 6.1: Calculated values for the standard molecular entropy Sm

APPENDIX A 162

-34.833

5.078E+00

-99.734

-220.466

-34.002

3.632E+00

11.059

-368.866

121.036

2.421E-27

∆r G

Kc

∆r H

∆r S

∆r G

Kc

∆r H

∆r S

∆r G

Kc

b

Patai (1983) Luo (2003)

-168.638

∆r S

a

-85.112

∆r H

B3LYP

4.804E-21

85.109

-374.422

-26.525

3.986E+03

-51.347

-207.845

-113.316

7.695E+02

-47.272

-175.065

-99.468

B3P86

BB1K

-

-

-

-

2.104E+05

-61.174

-203.335

-121.798

1.077E-22

94.518

-374.368

-17.100

1.189E-21

88.569

-382.213

-25.388

reaction (c )

5.704E+02

-46.530

-206.365

-108.058

reaction (b)

8.447E+01

-41.798

-176.800

-94.511

reaction (a)

MPW1PW91

-

-

-

-

3.097E+08

-79.247

-197.160

-138.030

-

-

-

-

BMK

1.253E-27

122.669

-378.071

9.947

3.754E+26

-182.412

-78.292

-205.755

1.442E+14

-111.582

-172.759

-163.090

BHandHLYP

-

-

-

-

-

-

-

-130.733b

-

-

-

-124.700a

exp

Table 6.2: Calculated ∆r H 0 (in kJmol−1 ), ∆r S 0 (in Jmol−1 K −1 ), ∆r G0 (in kJmol−1 ) and Kc (in m3 kmol−1 ) for reactions (a), (b) and (c) of Figure 3.14 (6-311g basis set)

APPENDIX A 163

-313.155

3.104E+49

-462.487

-134.457

-422.399

4.373E+68

-411.460

-124.042

-374.477

1.741E+60

∆r G

Kc

∆r H

∆r S

∆r G

Kc

∆r H

∆r S

∆r G

Kc

Yao et al. (2003)

-115.931

∆r S

a

-347.720

∆r H

B3LYP

1.530E+62

-385.567

-123.925

-422.515

3.509E+70

-433.263

-134.436

-473.345

3.342E+51

-324.748

-114.649

-358.931

B3P86

BB1K

3.195E+51

-324.637

-99.303

-354.244

4.666E+69

-428.264

-134.399

-468.335

9.351E+58

-367.232

-124.139

-404.244

1.207E+61

-379.274

-128.850

-417.691

reaction (f )

2.194E+67

-414.984

-134.365

-455.045

reaction (e)

4.448E+48

-308.342

-115.165

-342.678

reaction (d)

MPW1PW91

2.494E+61

-381.073

-129.566

-419.703

4.459E+69

-428.152

-134.591

-468.280

5.013E+51

-325.753

-111.035

-358.858

BMK

7.859E+58

-366.801

-124.637

-403.962

1.276E+68

-419.347

-134.374

-459.410

3.474E+48

-307.730

-114.457

-341.855

BHandHLYP

-

-

-

-

-

-

-

-471.551a

-

-

-

-357.314a

exp

Table 6.3: Calculated ∆r H 0 (in kJmol−1 ), ∆r S 0 (in Jmol−1 K −1 ), ∆r G0 (in kJmol−1 ) and Kc (in m3 kmol−1 ) for reactions (d), (e) and (f) of Figure 3.14 (6-311g basis set)

APPENDIX A 164

-239.387

3.642E+36

-233.583

-119.637

-197.913

1.957E+29

-86.296

-205.824

-24.930

9.329E-02

∆r G

Kc

∆r H

∆r S

∆r G

Kc

∆r H

∆r S

∆r G

Kc

-249.341

-109.332

-281.938

B3P86

1.147E+02

-42.556

-207.363

-104.381

2.697E+31

-210.118

-119.968

-245.886

2.024E+38

Marsal et al. (1999)

-109.374

∆r S

a

-271.997

∆r H

B3LYP BB1K

3.436E+36

-239.242

-109.638

-271.931

4.083E+30

-205.441

-115.708

-239.939

4.804E+00

-34.695

-207.301

-96.502

reaction (i)

2.225E+28

-192.526

-119.775

-228.237

reaction (h)

1.411E+35

-231.332

-109.512

-263.983

reaction (g)

MPW1PW91

1.323E+05

-60.025

-213.506

-123.682

5.871E+29

-200.635

-123.407

-237.429

1.155E+37

-242.248

-109.822

-274.991

BMK

1.247E-01

-25.649

-207.752

-87.590

3.934E+29

-199.644

-119.143

-235.166

1.783E+36

-237.617

-109.976

-270.406

BHandHLYP

-127.418a

-291.206a

exp

Table 6.4: Calculated ∆r H 0 (in kJmol−1 ), ∆r S 0 (in Jmol−1 K −1 ), ∆r G0 (in kJmol−1 ) and Kc (in m3 kmol−1 ) for reactions (g), (h) and (i) of Figure 3.14 (6-311g basis set)

APPENDIX A 165

APPENDIX B

Appendix B Overview of the ab initio calculations testset structures

– BPO: geometry optimizations and frequency calculations – BPO rad: geometry optimizations and frequency calculations – DAP: geometry optimizations and frequency calculations – DAP rad: geometry optimizations and frequency calculations – AIBN: geometry optimizations and frequency calculations – AIBN rad: geometry optimizations and frequency calculations – H2NOH: geometry optimizations and frequency calculations – H2NO – TEMPOH: geometry optimizations and frequency calculations – TEMPO: geometry optimizations and frequency calculations – DMN: geometry optimizations and frequency calculations – PBN: geometry optimizations and frequency calculations – cumene: geometry optimizations and frequency calculations – cumyl: geometry optimizations and frequency calculations – benzene: geometry optimizations and frequency calculations – phenyl: geometry optimizations and frequency calculations – ethylbenzene: geometry optimizations and frequency calculations – ethylbenzene rad: geometry optimizations and frequency calculations – styrene: geometry optimizations and frequency calculations – TEMPO styryl: geometry optimizations and frequency calculations

166

APPENDIX B

testset additions

– methyl propene * scan: scanning for approximate TS structure * prod: geometry and frequency calculations for the products * TS: transition state geometry and frequency calculations – AIBNrad DMN * scan: scanning for approximate TS structure * prod: geometry and frequency calculations for the products * TS: transition state geometry and frequency calculations – AIBNrad styrene * scan: scanning for approximate TS structure * prod: geometry and frequency calculations for the products * TS: transition state geometry and frequency calculations – DAPrad DMN * scan: scanning for approximate TS structure * prod: geometry and frequency calculations for the products * TS: transition state geometry and frequency calculations – DAPrad styrene * scan: scanning for approximate TS structure * prod: geometry and frequency calculations for the products * TS: transition state geometry and frequency calculations – phenyl DMN * scan: scanning for approximate TS structure * prod: geometry and frequency calculations for the products * TS: transition state geometry and frequency calculations – phenyl styrene * scan: scanning for approximate TS structure * prod: geometry and frequency calculations for the products * TS: transition state geometry and frequency calculations

nitrones

167

APPENDIX B

168

– nitrone1 * thermo: geometry and frequency calculations for nitrone, nitroxide, alkoxyamine, PNA * TS: transition state geometry and frequency calculations – nitrone2 * thermo: geometry and frequency calculations for nitrone, nitroxide, alkoxyamine, PNA * TS: transition state geometry and frequency calculations – PBN * thermo: geometry and frequency calculations for nitrone, nitroxide, alkoxyamine, PNA * TS: transition state geometry and frequency calculations – DPN * thermo: geometry and frequency calculations for nitrone, nitroxide, alkoxyamine, PNA * TS: transition state geometry and frequency calculations – DMPO * thermo: geometry and frequency calculations for nitrone, nitroxide, alkoxyamine, PNA * TS: transition state geometry and frequency calculations

APPENDIX B

169

Overview of the input values for simulations Due to the size of the simulation files, these will not be included on the DVD. In what follows, only the simulation input parameters are given.

kda

(m3 kmol−1 s−1 )

108 108 108 108 108

108 108 108 108

108 108 108

108 108 108

ka

(s−1 )

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

10−3

simulation 1

simulation 2

simulation 3

simulation 4

simulation 5

simulation 6

simulation 7

simulation 8

simulation 9

simulation 10

simulation 11

simulation 12

simulation 13

simulation 14

simulation 15

108

108

108

108

108

108

1010

108

107

106

108

108

108

108

108

(m3 kmol−1 s−1 )

ktc

0

0

0

kth

0

0

0

10−9

10−10

0

0

0

0

0

0

0

0

0

0

(m6 kmol−2 s−1 )

Variation ktrm

0

0

0

Variation kth

0

0

0

0

Variation ktc

0

0

0

0

0

Variation Kc

(m3 kmol−1 s−1 )

ktd

103

101

10−1

0

0

0

0

0

0

0

0

0

0

0

0

(m3 kmol−1 s−1 )

ktrm

10−11

10−11

10−11

10−11

10−11

10−11

10−11

10−11

10−11

10−11

10−13

10−11

10−10

10−9

10−8

(kmol m−3 )

Kc

APPENDIX B 170

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