A hybrid method for predicting the microstructure of polymers with ...

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California 92521 and Laboratory of Molecular and Materials Simulation, College of Chemical Engineering,. Beijing University of Chemical Technology, Beijing ...
THE JOURNAL OF CHEMICAL PHYSICS 124, 164904 共2006兲

A hybrid method for predicting the microstructure of polymers with complex architecture: Combination of single-chain simulation with density functional theory Dapeng Caoa兲 Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521 and Laboratory of Molecular and Materials Simulation, College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China

Tao Jiang and Jianzhong Wub兲 Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521

共Received 13 January 2006; accepted 20 February 2006; published online 26 April 2006兲 A hybrid method is proposed to investigate the microstructure of various polymeric fluids confined between two parallel surfaces. The hybrid method combines a single-chain Monte Carlo 共MC兲 simulation for the ideal-gas part of the Helmholtz energy and a density functional theory 共DFT兲 for the excess part that arises from nonbonded intersegment interactions. The latter consists of a modified fundamental measure theory for excluded-volume effect, the first-order thermodynamics perturbation theory for chain connectivity, and a mean-field approximation for the van der Waals attraction. In comparison with a conventional DFT, the hybrid method avoids calculation of the time-consuming recursive functions and is directly applicable to polymers with arbitrary molecular architecture. Its numerical performance has been validated by extensive comparisons with MC data for the density distributions of totally flexible, semiflexible, or rigid polymers and those with starlike architecture. Special attention is also given to the formation of a nematic monolayer by rigid molecules laying perpendicular to a planar surface. The hybrid method predicts the surface pressure versus surface coverage in good agreement with experiment. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2186323兴 I. INTRODUCTION

In the past several decades, much simulation and theoretical effort has been devoted to predicting the segmentlevel structure of polymers, flexible chains in particular, in the bulk or near a surface.1–5 Previous studies indicate that the distribution of polymers near a surface can be strongly affected by the surface-polymer interactions, polymer concentration, surface curvature, and polymer chain length.6 In a good solvent, polymers are depleted from a neutral surface on the length scale comparable to the polymer radius of gyration at low polymer volume fractions but accumulate at the surface, akin to the oscillatory distribution of a high-density hard-sphere fluid, at high polymer volume fractions. Near a strongly attractive surface, on the other hand, long-chain polymers are preferentially adsorbed over short ones due to the intramolecular correlations in the direction parallel to the surface. The structure of polymers near a surface is also sensitive to the surface curvature, in particular, when the surface radii are smaller than the polymer correlation length.7,8 Whereas the microscopic structures of totally flexible polymers at a surface or an interface are relatively well understood, much less is known for semiflexible polymers and those with more complicated chain architecture. It has been shown that the microstructure of rigid molecules can be drasa兲

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tically different from those of flexible polymers.9 In contrast to the flexible chains, the conformation of rigid, or semiflexible molecules is hardly influenced by confinement or solution conditions. In principle, computer simulation can be used to obtain the microscopic structures and interfacial properties of polymers with arbitrary architecture.10 While simulation with atomistic details is limited to relatively short chains, more realistic polymers have been routinely used in various coarsegrained models, in particular, in lattice models where polymer configurations are expressed in a three-dimensional grid.11 However, simulation of long-chain polymers with complex architecture remains computationally challenging even within the framework of the coarse-grained models. On the other hand, analytical methods, including the polymer density functional theory4 共PDFT兲 and self-consistent-field theory 共SCFT兲,12 are not limited by polymer size. In comparison with SCFT, PDFT takes into account the segmentlevel details and therefore its numerical implementation is often more complicated, in particular, for polymers with complex architecture, viz. star or branched polymers.13 For example, introduction of bond-angle potentials into a coarsegrained polymer model leads to an exponential increase of the computational time with the chain length, which makes direct application of DFT difficult for systems with chain length longer than M = 6.14 To avoid the numerical difficulties affiliated with the computation of the recurrence rela-

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tions, a hybrid of PDFT with single-chain Monte Carlo 共MC兲 simulation has been used by Sen et al.15 and by Yethiraj and Woodward.16 In the latter case, the Helmholtz energy of an ideal reference system was treated exactly via a single-chain Monte Carlo simulation, and the excess Helmholtz energy was represented by the smoothed density approximation. The hybrid method is not limited by the complex architecture of polymers because the single-chain simulation solves the intramolecular interactions exactly. Recently, Goel et al.17 used the hybrid PDFT method to investigate the microstructures and thermodynamic properties of short hard-sphere-chain polymers at various interfaces. Similarly, SCFT can be combined with single-chain simulations and the hybrid method has been used to study the dynamics of phase separation in binary polymer mixtures.18 As pointed out by Yethiraj and Woodward,16 the hybrid method has a number of advantages in comparison to both simulation and conventional PDFT. It avoids calculation of time-consuming complex integrals in solving the recursive functions in the PDFT and inefficient sampling of polymer configurations as encountered in the multichain Monte Carlo simulations. While the single-chain simulation is virtually exact for ideal chains, the numerical performance of the hybrid method depends critically on the formulation of the excess Helmholtz energy accounting for various nonbonded interactions. Built upon the idea proposed by Yethiraj and Woodward,16 we incorporate the single-chain simulation with a PDFT proposed in our previous work14,19,20 to explore the structure of polymer systems with complex architecture. This particular version of PDFT consists of a modified fundamental measure theory for excluded-volume effect, the first-order thermodynamics perturbation theory for chain connectivity, and a mean-field approximation for the van der Waals attraction.21 Extensive comparison with simulation data indicates that the PDFT can faithfully reproduce the microstructure and correlation functions of linear polymers in the bulk or under confinement.14,22 Because the excess Helmholtz energy can be effectively expressed as a functional of segment densities for both linear and complex polymers, we expect that the hybrid method will be accurate for polymers with complex molecular architectures. In this work, we will attest the numerical performance of the hybrid method by applying the PDFT to athermal polymeric systems containing freely connected tangent chains, rigid chains with a fixed bond length and angle, and star polymers. While the emphasis is placed on the numerical accuracy of the theory in comparison with 共multichain兲 simulation data, we also examine the applicability of the hybrid method to predicting the structure and interfacial properties of thermal systems consisting of semiflexible chains, rod-flexible copolymers, and polymers with rigid structure. Special attention is given to the surface tension of a nematic monolayer formed by rigid molecules laying perpendicular to a planar surface and some comparison is made with a similar system recently investigated by experiment.23 The remainder of the article is organized as follows. Section II describes the theoretical framework of the hybrid method in terms of the PDFT and single-chain MC simula-

tion. Section III discusses the theoretical predictions, along with simulation results for comparison, for the structural and interfacial properties of athermal systems consisting of freely jointed tangent chains, rigid chains with a fixed bond length and bond angle, and star polymers, as well as of thermal systems consisting of semiflexible chains, rigid-flexible copolymers, and rigid polymers. Also discussed in Sec. III are the interfacial tension of rigid polymers and comparison of the predicted results with experiments. Finally, Sec. IV summarizes the main results and conclusions.

II. THEORY AND MOLECULAR MODEL A. Polymer density functional theory

The central idea of a polymer density functional theory is that given a grand potential functional ⍀关␳ M 共R兲兴, the equilibrium molecular density profile, ␳ M 共R兲, can be solved from the stationary condition14

␦⍀关␳M 共R兲兴 = 0. ␦␳M 共R兲

共1兲

Once we have the molecular density profiles, all thermodynamic properties and multibody correlation functions can be subsequently evaluated following standard statistical thermodynamic relations. For a polymeric fluid, the grand potential is related to the Helmholtz energy functional F关␳ M 共R兲兴 via the Legendre transform ⍀关␳ M 共R兲兴 = F关␳ M 共R兲兴 +



关␺ M 共R兲 − ␮ M 兴␳ M 共R兲dR, 共2兲

where ␳ M 共R兲 is a multidimensional density profile, and R is a composite vector 共r1 , r2 , . . . , r M 兲 representing the positions of all segments of the polymeric molecule. For a polymer consisting of M identical segments, the molecular density profile ␳ M 共R兲 can be used to specify the segmental densities M

M

i=1

i=1

␳共r兲 = 兺 ␳si共r兲 = 兺



dR␦共r − ri兲␳ M 共R兲,

共3兲

where ␳共r兲 is the total segmental density, ␳si共r兲 is the local density of segment i, and dR stands for a set of differential volumes 共dr1 , dr2 , . . . , dr M 兲. In Eq. 共2兲, ␺ M 共R兲 is the external potential exerting on individual segments, and ␮ M is the chemical potential of the polymer.24 The Helmholtz energy functional F关␳ M 共R兲兴 is conventionally expressed as an ideal contribution corresponding to that for a system of ideal chains 共i.e., interacting only through bonding potentials兲 and an excess part taking into account the contributions from intramolecular correlations and nonbonded interactions24 F关␳ M 共R兲兴 = Fid关␳ M 共R兲兴 + Fex关␳共r兲兴.

共4兲

For an ideal-polymer system, the Helmholtz energy is known exactly

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␤Fid关␳M 共R兲兴 =







dR␳ M 共R兲VB共R兲,

共5兲

where VB共R兲 is the bond potential, and ␤ = 共kT兲−1 with k being the Boltzmann constant and T the absolute temperature. For the systems considered in this work, the excess Helmholtz energy functional consists of contributions from the hard-sphere repulsion, the van der Waals attractions, and correlations due to the chain connectivity att hs chain ␤Fex关␳共r兲兴 = ␤Fex + ␤Fex + ␤Fex .

共6兲

14,24,25

Following our previous work, the hard sphere part of the excess Helmholtz energy functional is represented by a modified fundamental measure theory





dr − n0 ln共1 − n3兲 +

n1n2 − nV1 · nV2 1 − n3

+ 共n32/3 − n2nV2 · nV2兲 ⫻



ln共1 − n3兲 12␲n23

1 + 12␲n3共1 − n3兲2

册冎

j

j



␳ j共r⬘兲␻␣j 共r − r⬘兲dr⬘ ,

共7兲

共8兲

and ␼␣j 共␣ = 0 , 1 , 2 , 3 , V1 , V2兲 are weight functions.27 The excess Helmholtz energy functional due to the chain connectivity is given by a generalized first-order thermodynamics perturbation theory24

␤Fchain =



1−M n0 · ␰ ln y hs共␴,n␣兲, dr M

共9兲

where ␰ = 1 − nV2 · nV2 / n22, and y hs共␴ , n␣兲 is the contact value of the cavity correlation function for hard-sphere segments n22␰␴ 1 n2␰␴ + + . y 共␴,n␣兲 = 1 − n3 4共1 − n3兲2 72共1 − n3兲3 hs

共10兲

The Helmholtz energy given by Eq. 共9兲 is not directly related to the bonding potentials of polymeric molecules. Instead, this term takes into account the effect of chain connectivity on the correlations between polymeric segments. Finally, the excess Helmholtz energy functional due to the van der Waals att , is represented by a mean-field attractions, ␤Fex 14 approximation att ␤Fex = 21

冕冕

drdr⬘

␳i共r兲␳ j共r兲␤␸att 兺 ij 共兩r − r⬘兩兲, i,j=A,B

共12兲

␭i共ri兲 =

␦Fex + ␸i共ri兲. ␦␳共ri兲

共13兲

Combination of Eqs. 共3兲, 共12兲, and 共13兲 yields the segmental densities ␳si共r兲

␳si共r兲 =



dR␦共r − ri兲



M

⫻exp ␤␮ M − ␤VB共R兲 − ␤ 兺 ␭ j共r j兲 j=1



共14兲

and subsequently, the total segmental density

where n␣共r兲, ␣ = 0 , 1 , 2 , 3 , V1 , V2 are scalar and vector weighted densities defined by Rosenfeld26 n␣共r兲 = 兺 n␣ j共r兲 = 兺

i=1

where ␭i共ri兲 includes the external potential ␸i共ri兲 and a selfconsistent potential calculated from the functional derivative of the excess Helmholtz energy with respect to the density distribution

␳共r兲 = exp共␤␮M兲 ,



M

␳M 共R兲 = exp ␤␮M − ␤VB共R兲 − ␤ 兺 ␭i共ri兲 ,

dR␳ M 共R兲关ln ␳ M 共R兲 − 1兴

+␤

␤Fhs =

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共11兲

where ␸att ij 共r兲 is the van der Waals potential between segments described in Eq. 共18兲. Minimization of the grand potential with respect to the molecular density profile, namely, Eqs. 共1兲 and 共2兲, gives the Euler-Lagrange equation





M

dR 兺 ␦共r − ri兲 i=1

M



⫻exp − ␤VB共R兲 − ␤ 兺 ␭ j共r j兲 . j=1

共15兲

Because the exponential term exp关−␤VB共R兲兴 represents the probability density of an ideal chain with configuration R,14 i.e., P共R兲 = exp关−␤VB共R兲兴, Eq. 共14兲 can be reformulated as

␳si共r兲 = exp共␤␮M 兲





dR␦共r − ri兲 · P共R兲

M



⫻exp − ␤ 兺 ␭ j共r j兲 . j=1

共16兲

Equivalently, Eq. 共16兲 can be expressed in terms of the ensemble average over the configurations of a single chain,16

冓冋

M

␳si共r兲 = exp共␤␮M 兲 exp − ␤ 兺 ␭ j共r j兲 j=1

册冔

,

共17兲

r=ri

where 具¯典 represents the ensemble average of a single chain. In the hybrid method, the ensemble average over the polymer configurations is calculated by carrying out a singlechain Monte Carlo simulation that is independent of the density profiles and the external fields. To calculate the ensemble average, we use a large number 共⬃105兲 of polymer conformations to provide a good representation of the configurational space. B. Molecular models

To test the numerical performance of the hybrid method, we consider the inhomogeneous distributions of polymers of different architectures confined between two parallel surfaces. The theoretical predictions are compared with simulations for both athermal and thermal systems corresponding to polymers in good and poor solvents, respectively. For ather-

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mal systems, the polymers under consideration include freely connected tangent sphere chains, rigid chains with a fixed bond length and bond angle, and starlike polymers; the thermal systems include semiflexible chains, rigid chains, and rigid-flexible copolymers. The pair interaction between polymer segments in an athermal system is represented by the hard-sphere potential. In the thermal systems, the van der Waals attraction between polymer segments is represented by a square-well potential



⬁,

r⬍␴



␸ij共r兲 = − ␧ij , ␴ 艋 r 艋 ␥␴ , 0, r ⬎ ␥␴

共18兲

where ␴ is the segment diameter, ␥␴ is the square-well width, and ␧ij is the energy parameter. Throughout this work, the attractive width is fixed at ␥ = 1.2.

FIG. 1. Total segmental density profiles of hard-sphere chain of four mers confined between two parallel hard walls separated by H = 5␴. The lines are from this work and the points are simulation data 共Ref. 34兲.

C. Multichain Monte Carlo simulations

Extensive simulation data are available from the literature for the microscopic structures of flexible, semiflexible, and rigid polymers with different molecular architectures. However, to the best of our knowledge, no simulation data have been reported for copolymers consisting of a rigid part and a totally flexible part. The rigid-flexible copolymer chains provide a good representation of mesogenic molecules,28,29 such as 4⬘-alkoxy-4⬘-F-benzyliden-aniline 共nOFBA兲 or 4⬘-alkoxy-4⬘-cyano-biphenyl 共nOCB兲, that are made of a linear aromatic core and flexible ends. The aromatic core can be represented by a rigid rod and the flexible part can be described by a freely jointed tangent sphere chain.30,31 To test the numerical performance of the hybrid method, we carry out the multichain canonical Monte Carlo simulations for relatively short chains with rigid-flexible architecture confined between two parallel surfaces.32 At a given pore width and a polymer packing fraction ␩, the simulation box contains at least 50 copolymer chains. The period boundary conditions are applied to the x and y directions of the simulation box, and the box length in the z direction is fixed by the surface separation. The polymer configurations are updated by translational, rotational 共including rotations both for the entire chain and for rigid block only兲, and for the flexible part, by configurational-biased Monte Carlo moves.32,33 The different methods of Monte Carlo updates are allocated approximately with the same frequency. Each simulation runs 1 ⫻ 107 Monte Carlo cycles, with the first one-third for the system to reach equilibrium and the remaining two-thirds for evaluating the ensemble average. III. RESULTS AND DISCUSSION A. Athermal systems

We first consider the microstructures of freely jointed tangent hard-sphere chains, rigid chains with a fixed bond length and bond angle, and starlike flexible chains confined between two parallel surfaces. The predicted density profiles are compared with simulation results from the literature and with the predictions from an earlier version of DFT proposed by Yethiraj and Hall34–36 and Yethiraj.37

1. Freely jointed tangent hard-sphere chains

Freely jointed hard-sphere chains have been extensively used in off-lattice polymer models and thereby provide a good starting point for the calibration of the hybrid method proposed in this work. Figure 1 presents the density profiles of tangent hard-sphere chains with four segments confined between two parallel walls separated by 5␴. Here the lines are predictions of the hybrid method and the points are from simulation.34 In good agreement with the simulations, the theory predicts that at the low packing fraction 共␩ = 0.1兲, the contact value of the density profile is less than the bulk density, suggesting a slight depletion due to the chain connectivity. At higher packing fraction, the contact value of density profile increases owing to the excluded-volume effect. Figure 2 shows the local density distributions of hardsphere chains of M = 20 confined between two hard walls separated at H = 10␴. As before, the lines are the calculated results and points are simulation data.35 The top panel presents the total density profiles and the bottom panel presents the density profiles of the end and middle segments. With the increase of the chain length, the depletion effect becomes more distinct due to the restriction of polymer configurations near a surface. Because the surface has less restrictive effect on the end segments than the middle segments, the contact value of the end-segment density is greater than that of the middle segments. Both Figs. 1 and 2 show that the hybrid method faithfully reproduces the microstructure of freely jointed tangent hard-sphere chains. We notice that for freely jointed tangent sphere chains, direct calculations of the recurrence relations in PDFT are numerically more efficient than the application of singlechain simulations. However, as discussed in the later sections, the advantage of the hybrid method lies in its applications to polymers with more complex architecture. Indeed, from a theoretical point of view, the single-chain simulation merely provides an alternative way to evaluate the recurrence relations due to the polymer chain connectivity. In recent years, several versions of PDFT have been proposed for freely jointed hard-sphere chain fluids.19,38–42 Whereas the underlying approximations vary significantly among differ-

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FIG. 2. Local densities of middle and end segments of hard-sphere 20 mers near a hard surface. The lines are from this work and the points are simulation data 共Ref. 35兲.

ent versions of PDFT, their numerical performance are quite similar in comparison with simulation data. For example, the density profiles of hard-sphere chains near a hard wall predicted by an extension of the thermodynamic perturbation theory 共TPT1兲,19 which is theoretically equivalent to the hybrid method proposed in this work, are very similar to those calculated by PDFT by Zhou41 that relies on the polymer reference interaction site model43 共PRISM兲 and an approximated bridge function for the density expansion of the polymer Helmholtz energy functional. Nevertheless, the situation is quite different when the different versions of PDFT are considered for more realistic polymeric systems, in particular, for those containing polymers of complex molecular architecture. 2. Rigid chains with a fixed bond potential

Figure 3 shows the density profiles of rigid hard-sphere trimers with different bond angles fixed at ␪ = 60°, 120°, 150°. Again, trimers are confined between two parallel hard walls. The lines are calculated results and the points are simulation data.9 As shown in the top panel of Fig. 3, for ␪ = 60°, the end and middle segments are indistinguishable because of the symmetry. For ␪ = 120° and 150°, however, the contact value of the middle-segment density is smaller than that of the end segment due to the surface restriction of the chain configuration. Figure 3 shows that the density profiles predicted by the hybrid method agree well with simulations. Figure 4共a兲 presents the density profiles of straight rigid chains with M = 4 confined between two hard walls separated by H = 10␴. Here the bond angle is fixed at 180°, i.e., the rigid hard-sphere chain resembles a rigid rod. Figure 4共b兲 presents similar results for rigid molecules of ten segments in the pore of H = 16␴. In both cases, the average polymer pack-

FIG. 3. Local densities of middle and end segments of rigid trimers with different bond angles. The lines are from this work and the points are simulation data 共Ref. 9兲.

ing fraction is ␩ = 0.1, and the simulation data are from this work. Figures 4共a兲 and 4共b兲 show that, for the rigid chains, the contact density of the middle segments is very small because, for rigid rod molecules, a middle segment contacts with the surface only when the entire chain is flatly adsorbed. As presented in the top panel of Fig. 4共a兲, the hybrid method successfully captures the three weak peaks appeared in the total density profile for M = 4. Furthermore, it reproduces the double peaks appeared in the density profiles of the middle segments. Figure 4共b兲 shows that as the chain length increases 共M = 10兲, a more uniform distribution of the polymer segments is observed 共except a weak peak at z = 0.5␴ and a shallow valley at z = ␴兲. For both cases, the predictions of the hybrid method are in good agreement with the Monte Carlo simulations. 3. Star polymers

The hybrid method can be applied not only to freely jointed tangent chains or rigid chains with fixed bond length and bond angle but also to polymers with more complex architecture, such as star polymers. Figure 5 shows the density profile of star polymers with M = 16 segments at the bulk packing fraction ␩b = 0.1 and 0.2. Here the simulation data are from the literature.36 The star polymers contain three arms and each arm consists of five segments. Although the calculated density profiles exhibit some fluctuations near the surface 共z ⬍ 0.5␴兲, overall the theoretical predictions are in good agreement with the simulation data. B. Thermal systems

In the previous section, we focus on athermal systems, i.e., all nonbonded intersegment interactions are represented by the hard-sphere potential. We now investigate thermal

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FIG. 5. Microstructure of star polymers near a hard wall. The lines are from this work and points are simulation data 共Ref. 36兲.

FIG. 4. Local densities of middle and end segments and the total densities for rigid hard-sphere chains with 共a兲 M = 4; 共b兲 M = 10. Lines are obtained from the hybrid method and the points are calculated from multichain simulations.

ties affiliated with the calculation of the recurrence relations by replacing the multidimensional integrals with a singlechain simulation. In the hybrid method, the single-chain configurations are generated under the Boltzmann distribution according to the bonding potential. We consider semiflexible chains with M = 20 segments and with the reduced bond-angle energy ␧* = 5. Figure 6 shows the end-segment and total density profiles of the semiflexible chains confined between two parallel hard walls separated at H = 10␴. The average polymer packing fraction inside the pore is ␩ = 0.1. As before, the lines are predictions from the hybrid method and the symbols are simulation data from the literature.37 For z ⬍ ␴, the semiflexible chains are depleted from the neutral surface and in this region, the total density presents a linear increase to the bulk value. However, the end segment exhibits a nearly uniform distribution, which is quite different form that for freely jointed tangent chains shown in Fig. 2. The difference in the end-segment density profiles between semiflexible and freely jointed chains suggests that the polymer conformation becomes less affected by the surface as the molecular stiffness increases. Overall, the density profiles calculated from the hybrid method are in good agreement with the simulation data. To

systems where the van der Waals attractions between nonbond segments are also included. Moreover, we consider the effect of the backbone flexibility on polymer configurations. 1. Semiflexible chains

In semiflexible chains, the bond length is fixed but the bond angle is allowed to vibrate. The intensity of the vibration can be represented by a bending potential,37

␾␪ = ␧共1 + cos ␪兲,

共19兲

where ␧ is an energy parameter, and ␪ is the bond angle. Due to the presence of bond vibration, it is difficult to calculate the recurrence relations directly in a conventional PDFT, especially for systems containing long semiflexible chains. The difficulty arises from the exponential increase of the computational time with the chain length.14 The hybrid method provides an effectual means to overcome the numerical difficul-

FIG. 6. The end-segment density and total density profile of semiflexible chains with 20 segments. The lines are from this work and the points are simulation data 共Ref. 37兲.

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FIG. 8. Density profile of rigid eight mers near a hard wall that attracts only the end segments.

FIG. 7. 共Color online兲 Density profiles of copolymers near a hard wall. The lines are obtained by the hybrid method and the points are from the multichain simulation. 共a兲 A3B5 copolymer; 共b兲 A7B1 copolymer.

the best of our knowledge, this work represents the first theoretical calculation that faithfully reproduces simulation data for semiflexible chains with the chain length up to M = 20. 2. Rigid-flexible copolymers

In Fig. 7, we present the density profiles for copolymers consisting of rigid and flexible blocks confined within two parallel walls that attract the first segment of the rigid block but repel the flexible segments. The surface attraction is represented by a square-well potential

␺Aatt1W共z兲 =



− ␧A1W , ␴/2 ⬍ z ⬍ ␴ 0,

otherwise



,

共20兲

where subscript A1 denotes the first segment of the rigid block and subscript w indicates the wall. ␧A* W = −1 is used in 1 both theoretical calculations and multichain Monte Carlo simulations. All other segments from the rigid block are neutral to the surface. Similarly, a square-shoulder potential is used to represent the interaction between the surface and all * segments from the flexible block, i.e., ␧BW = 1.0, where subscript B indicates the segments from the flexible block. The unlike-pair interaction is represented by the square-shoulder * = 0.5; and the like-pair interaction is reppotential with ␧AB * = −1.0 and resented by the square-well potential with ␧AA * ␧BB = −0.5.

Figure 7共a兲 shows the local density profile of copolymers with the sequence A3B5 confined between two parallel walls separated by H = 10␴. The average packing fraction of the polymers is ␩ = 0.1. Due to the surface selectivity, the overall density of A segments, namely the segments from the rigid block, is greater than that of the segments from the flexible block 共B兲 inside the attractive well of the surface. Beyond the attractive well, the overall density of the rigid segments shows a secondary peak at z = 1.5␴ and a third peak at z = 2.5␴, which is absent in a similar system containing only fully flexible chains. Although the hybrid method substantially overestimates the density of A segments in the well, overall the predicted density profiles agree well with the simulation data. Similarly, Fig. 7共b兲 shows the density profiles of the confined polymers with the sequence A7B1. In this case, the entire molecule is rigid except the freely jointed end B segment. Except slight underestimation of the segment density for the rigid block inside the attractive well, the calculated results again agree well with the simulations for both A and B segments. In particular, the theory successfully captures all the weak peaks in the density profile of the rigid segments appeared at z = 1.5, 2.5, 3.5, 4.5␴. 3. Rigid chains

Figure 8 shows the local density profiles of confined rigid molecules with M = 8 segments tangentially connected in a rodlike structure. Here the van der Waals attraction appears in both the surface potential and the intersegment in* * = −1 and ␧WA = −1. As before, the agreeteractions with ␧AA ment of the hybrid method with simulations is satisfactory. The segment density exhibits a strong peak near the surface, suggesting that, at this condition, most rigid molecules lie parallel on the surfaces. To make the rigid molecules having an average conformation perpendicular to the surfaces, we change the parameters such that the surfaces attract only the first segment 共head兲 of the rigid molecule. Figure 9 shows the local density profiles of rigid molecules at polymer vol* = −1 共top panel兲 and −5 共bottom ume fraction ␩ = 0.1 for ␧WA 1 panel兲. In both cases, the surface attracts only the head segments of the rigid molecules. The top panel of Fig. 9 shows

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Cao, Jiang, and Wu

FIG. 9. Density profiles of rigid molecules at different attractive surface energies for the end segments.

that the overall segment density profile exhibits a weak depletion similar to that for semiflexible chains. As the sur* = −5兲, the end segments are acface energy increases 共␧WA 1 cumulated in the attractive well. The weak second, third, and fourth peaks outside of the well correspond to the distributions of the nonsticky segments from the rigid molecule. The appearance of multiple peaks near the surface reflects the polymer conformations perpendicular to the surface. Although the density profile predicted by the hybrid method exhibits slightly stronger peaks in comparison with the simulation data, overall the theory agrees well with the simulations. To create a monolayer of rigid molecules perpendicular to the surface, we increase the surface separation to H = 20␴ and change the bulk polymer volume fraction to ␩ = 0.01. Figures 10共a兲 and 10共b兲 show the local density profiles for the individual segments of the rigid molecules at two * = −5 and −15, respectively. Figures surface energies ␧WA 1 11共a兲 and 11共b兲 show the corresponding total segment den* sity profiles. For the system with low surface energy 共␧WA 1 = −5兲, Fig. 10共a兲 shows that essentially all the head segments are adsorbed at the surface but the nonsticky segments are more uniformly distributed, suggesting a random distribution of the polymer orientations. For the system with high surface * = −15兲, however, Fig. 10共b兲 shows that all head energy 共␧WA 1 segments are adsorbed at the surfaces and most tail segments are located around the position of z = 7.5␴, suggesting that the rigid molecules are distributed perpendicular to the surfaces forming a nematic monolayer. Figure 11共a兲 shows that at the weakly attractive surface, the total density profile is similar to that of tethered chains. Conversely, near a strongly attractive surface, Fig. 11共b兲 shows that the total density profile resembles that of a nematic monolayer. In the former

FIG. 10. Local density profiles of individual segments in rigid eight mers at * = −5 and −15. two surface energies, ␧WA 1

case, the surface coverage of the head segments is about 20%, which allows a random distribution of the polymer orientations. In the latter case, however, the surface coverage of the head segments exceeds 95%, which compels the nonsticky segments to form an ordered array perpendicular to the surface. The higher the surface coverage is, the smaller is the degree of freedom for the rigid molecules on the surface. Figure 12 shows the dependence of the surface pressure on the molecular density at the surface, or equivalently, the surface area occupied by each molecule in the attractive well. The surface pressure is calculated from21 FH/共2AkT兲 = −

1 ⳵␤⍀ =兺 A ⳵H j



dz␳ j共z兲

⳵␤␸Wj , ⳵z

共21兲

where FH / A represents the force per unit area on a single wall, a factor of 2 comes from the fact that the force is exerted to both surfaces, and ␸Wj is the surface potential exerted on segments j. As expected, the surface pressure decreases with the surface area per molecule or increases with the surface coverage. When the surface area per molecule is less than ␴2, i.e., at high surface coverage, the surface pressure becomes essentially unchanged. However, when the surface area per molecule is larger than ␴2, the surface pressure falls rapidly. Qualitatively, the dependence of surface pressure on surface coverage agrees well with that observed in recent experiment.23 It was found at the surface pressure of stearic acid on a metal surface varies little with the surface

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Microstructure of polymers with complex architecture

FIG. 12. Surface pressure vs the surface area per molecule for rigid molecules in a nematic monolayer.

FIG. 11. Total density profiles of rigid eight mers at the same conditions as those in Figs. 10共a兲 and 10共b兲.

area in the range of A0 ⬍ 112 Å2 / molecule, very close to the cross-section area of a stearic acid molecule 共118 Å2 / molecule兲, but falls rapidly as the surface density is further reduced.

IV. CONCLUSIONS

chains, semiflexible chains, rigid chains with a fixed bond length and bond angle, rigid-flexible copolymers, and polymers with starlike architecture. Encouraged by the good performance of the hybrid method, we have also investigated the structural transition of rigid molecules near an attractive surface. When the surface energy for the head segments of the rigid molecules is sufficiently strong, a nematic monolayer is observed where most molecules are perpendicular to the surface. The surface pressure of the monolayer reaches a constant at high surface coverage while falls rapidly as the surface area per molecule is larger than approximately the crossection area of a single segment. The hybrid method predicts the variation of the surface pressure with concentration in good agreement with experimental results. ACKNOWLEDGMENTS

This work is supported by the National Science Foundation 共CTS 0406100 and CTS0340948兲. D.C. is thankful to the Beijing University of Chemical Technology for the Excellent Talent Fund and National Natural Science Foundation of China 共No. 20646001兲. A. Aksimentiev and R. Holyst, Prog. Polym. Sci. 24, 1045 共1999兲. J. B. Hooper, J. D. McCoy, and J. G. Curro, J. Chem. Phys. 112, 3090 共2000兲. 3 C. E. Woodward, J. Chem. Phys. 94, 3183 共1991兲. 4 S. K. Nath, P. F. Nealey, and J. J. de Pablo, J. Chem. Phys. 110, 7483 共1999兲. 5 D. Chandler, J. D. McCoy, and S. J. Singer, J. Chem. Phys. 85, 5971 共1986兲. 6 R. Israels, J. Scheutjens, and G. J. Fleer, Macromolecules 26, 5405 共1993兲. 7 J. J. Cerda, T. Sintes, and A. Chakrabarti, Macromolecules 38, 1469 共2005兲. 8 A. J. Spakowitz and Z. G. Wang, Phys. Rev. Lett. 91, 166102 共2003兲. 9 S. Phan, E. Kierlik, M. L. Rosinberg, A. Yethiraj, and R. Dickman, J. Chem. Phys. 102, 2141 共1995兲. 10 M. Muller, L. G. MacDowell, and A. Yethiraj, J. Chem. Phys. 118, 2929 共2003兲. 11 Q. Wang, P. F. Nealey, and J. J. de Pablo, J. Chem. Phys. 118, 11278 共2003兲. 12 S. K. Nath, J. D. McCoy, J. P. Donley, and J. G. Curro, J. Chem. Phys. 103, 1635 共1995兲. 13 A. Malijevsky, P. Bryk, and S. Sokolowski, Phys. Rev. E 72, 032801 共2005兲. 1

We proposed a hybrid method for predicting the microscopic structures of polymers with complex architecture by combining single-chain Monte Carlo simulation and a polymer density functional theory 共DFT兲. In this hybrid method, we used the single-chain simulation to calculate the recurrence relations and the DFT to treat the excess free energy. In comparison with conventional DFT, a key advantage of the hybrid method is that it can be easily applied to polymers with arbitrary molecular architectures. Furthermore, the hybrid method avoids inefficient sampling of polymer configurations as often encountered in multichain simulations; it is free of the calculation of the multidimensional integrals appeared in the recurrence relations. Such calculation is extremely time consuming for semiflexible polymers where the computational time for the recurrence relations is an exponential function of the chain length. The density profiles calculated from the hybrid method are in good agreement with the simulation data for freely jointed tangent hard-sphere

2

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