New pathways for nanoparticle formation in acetylene

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formation of larger (linear and branched) hydrocarbons C2nH2 (n = 3, 4, 5), which contribute ... showed that higher-mass hydrocarbon cations and anions up to.

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JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 41 (2008) 225201 (14pp)

doi:10.1088/0022-3727/41/22/225201

New pathways for nanoparticle formation in acetylene dusty plasmas: a modelling investigation and comparison with experiments Ming Mao1,3 , Jan Benedikt2 , Angelo Consoli2 and Annemie Bogaerts1 1

Research group PLASMANT, Department of Chemistry, University of Antwerp, Universiteitsplein 1, B-2610 Antwerp, Belgium 2 Arbeitsgruppe Reaktive Plasmen, Fakult¨at f¨ur Physik und Astronomie, Ruhr-Universit¨at Bochum, Universit¨atsstr. 150, 44780 Bochum, Germany E-mail: [email protected]

Received 27 June 2008, in final form 8 September 2008 Published 23 October 2008 Online at stacks.iop.org/JPhysD/41/225201 Abstract In this paper, the initial mechanisms of nanoparticle formation and growth in radiofrequency acetylene (C2 H2 ) plasmas are investigated by means of a comprehensive self-consistent one-dimensional (1D) fluid model. This model is an extension of the 1D fluid model, developed earlier by De Bleecker et al. Based on the comparison of our previous results with available experimental data for acetylene plasmas in the literature, some new mechanisms for negative ion formation and growth are proposed. Possible routes are considered for the formation of larger (linear and branched) hydrocarbons C2n H2 (n = 3, 4, 5), which contribute to the generation of C2n H− anions (n = 3, 4, 5) due to dissociative electron attachment. Moreover, the vinylidene anion (H2 CC− ) and higher C2n H− 2 anions (n = 2–4) are found to be important plasma species.

growth of nano-size particles is a critical issue. In order to control the nanoparticle growth and physical properties, a fundamental understanding of particle growth kinetics and plasma chemistry is indispensable. In the past decade, several attempts to describe the chemical kinetics of acetylene discharges have been made both by experimental measurements and by computer simulations. Deschenaux et al [13] measured the positive and negative ion mass spectra in a dusty RF acetylene plasma by means of FTIR absorption spectroscopy and mass spectrometry. Their results showed that higher-mass hydrocarbon cations and anions up to nearly 200 amu are formed in the plasma, which indicated that several pathways exist leading to effective particle production. Using molecular beam mass spectrometry, Benedikt et al [17, 18] investigated the temporal evolution of the neutral plasma chemistry products in a RF Ar/He/C2 H2 capacitively coupled plasma. They found that Cn H4 (n = 4, 5, 6) compounds appeared at a very early stage of the discharge which indicated that the polymerization reactions with the C2 H

1. Introduction Nanoparticle (or dust) formation is a well-known phenomenon occurring in reactive gas plasmas, such as silane [1–10], methane [11, 12] or acetylene [13–19]. Under some conditions, the nanoparticle formation is considered to be harmful, whereas for some other applications, it turns out to be beneficial. RF acetylene discharges under low pressure serve nowadays as a source in the plasma-enhanced chemical vapour deposition (PECVD) processing to deposit hydrogenated amorphous carbon (α-C : H) thin films. Due to the extraordinary material properties such as a combination of high hardness, chemical inertness, low abrasive coefficient and infrared transparency, these carbon-based coatings are widely explored for many industrial applications, including tribological materials [20, 21], passivation layers [22] and field emission cold cathodes for future flat-screen displays [23–25]. Among these applications the formation and 3

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radical would not be responsible for their formation. They also proposed C2n H2 (n > 1) molecules as important precursors for negative ion formation via dissociative electron attachment (DEA) reactions. To investigate in more detail the plasma chemistry and the important nanoparticle formation mechanisms, computer simulations are also helpful. A 1D fluid model was developed by Herrebout et al [19] to describe the basic reaction mechanisms of small species in a typical RF acetylene plasma, but neither dust growth nor negative ion formation were taken into account in the model. Stoykov et al [14] developed a zero-dimensional chemical kinetic model containing neutral chain and cyclic hydrocarbons, positive and negative ions and electrons in a low-pressure acetylene RF discharge. However in their model, a constant positive ion concentration was assumed and the role of positively charged hydrocarbon clusters was not investigated. In addition, a Maxwellian distribution was also assumed in the model. Recently, a detailed chemical kinetic scheme, containing electron impact, ion–neutral, and neutral–neutral reactions, has been developed by De Bleecker et al [15, 16] and described in a 1D fluid model to reveal the underlying dust growth mechanisms and the most important dust precursors in RF acetylene plasma. Although the model contains a vast number of chemical reactions, when comparing the results with experimental measurement by Deschenaux et al [13] and Benedikt et al [17, 18], the agreement was not yet satisfying for negative ions, as will be discussed in more detail in section 2.2.1. Hence the growth mechanisms of negative ions should be reevaluated. In this study, we extend our previously developed 1D fluid model [15, 16] by adding new species and related reactions to obtain a more realistic picture of the important pathways for nanoparticle formation in acetylene plasmas. The paper is organized as follows: in section 2, an overview of the model is given, including a new proposed mechanism for negative ion formation. The numerical results are discussed and compared with experimental data in section 3. Finally, the conclusion is given in section 4.

0.00020

2tB 2.5 eV Maxwell 2.5eV 2tB 4.5 eV Maxwell 4.5 eV

Normalized EEDF

0.00015

0.00010

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Figure 1. Comparison of the Maxwellian EEDF and two-term Boltzmann (2tB) EEDF for two different electron mean energies.

2.1. Fluid equations The fluid model employed in this work is based on the iterative and time-dependent solution of the continuity and momentum equations for the electrons, various positive and negative ions, radicals and molecules. The particle balance for each species j (electrons, ions, radicals and neutral molecules) is described by ∂nj ∂j + = Sj , ∂t ∂x

(1)

where nj is the particle’s density and Sj represents the net production rate, determined by different source and sink terms of species j . This includes all chemical volume reactions, as well as the gas inlet and pumping, which are incorporated by introduction of additional source and sink terms. The flux term j is estimated by the drift–diffusion approximation which means that each particle flux consists of two separate terms, a drift and a diffusion term j = µj nj E − Dj

2. Model description

∂nj ∂x

(2)

in which µj and Dj are the mobility and diffusion coefficient of species j and E represents the instantaneous electric field. Because the ions cannot follow the actual electric field E due to their lower momentum transfer frequency, an effective electric field Eeff,j is taken into account which is also adopted in the drift–diffusion approximation. The effect of gas flow on the nanoparticles is negligible compared with the diffusion and drift velocity of the nanoparticles based on the following two reasons: first, the focus in this study is put on the initial stage of nanoparticles formation, i.e. the nucleation. During this stage, the particles grow to a diameter of 2–3 nm. As argued in [26], the electrostatic force remains then the most dominant force in the entire discharge. Second, under the condition of this study (i.e. the same as that of the experiment done by Deschenaux et al [13]), the gas flow velocity is very small in the plasma region for a small gas flow rate of 8 sccm, due to the large

Following our previous work, a capacitively coupled RF discharge (13.56 MHz) in a parallel plate reactor is simulated. Because the electron energy distribution function (EEDF) can often deviate significantly from a Maxwellian distribution and is influenced by the composition of the background gas [27], the EEDF was obtained by solving the Boltzmann equation in two-term approximation to calculate the electron collision rates and electron transport coefficients as a function of the average electron energy. Indeed, as is illustrated in figure 1, the EEDF deviates from a Maxwellian distribution; more specifically a Maxwellian EEDF underestimates the EEDF at lower energies and overestimates it at higher energies. In this section, a general overview of the fluid model is provided at first, then the new mechanism for negative ion growth is introduced by comparing the calculated results with experimental data. Afterwards, a comprehensive description of the complete plasma chemistry is presented. 2

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distance between the position of the gas inlet and the plasma region (see below). Beside these particle balance and flux equations, the electric field E and the potential V are calculated from Poisson’s equation   ∂ 2V ∂V e  n+ − E=− (3) =− n− − ne 2 ∂x 0 ∂x where 0 is the permittivity of free space, n+ , n− , ne are the positive ion, negative ion and electron densities, respectively. Finally, the electron energy density ωe = ne  (i.e. the product of the electron density and the average electron energy ) is calculated self-consistently from the second moment of the Boltzmann equation ∂ωe ∂ω + = −ee E + Sω ∂t ∂x

2.2.1. New mechanisms of negative ion formation (a) DEA on larger C2n H2 molecules. Particle nucleation in low-pressure plasmas is most probably initiated by negative ions, because they remain confined in the plasma bulk due to the ambipolar potential and have a longer residence time in the discharge compared with the other species. After undergoing successive ion–molecule reactions at a considerable rate, very large negative clusters are generated. In previous work [15, 16], only C2n H− (n = 1–6) anions and a relatively simple growth mechanism referred to as the Winchester mechanism [30] were considered: the primary C2 H− anions, generated through the electron impact dissociative attachment on C2 H2 , trigger a consecutive chain of polymerization reactions with C2 H2 insertion to form bigger anions C2n H− (n = 2–6). However, when comparing the calculated negative ion mass spectra obtained in [15] with experimental data of Deschenaux et al [13], the agreement was not satisfactory. Indeed, the model calculation could not reproduce the high densities of C6 H− , C8 H− , and other negative ions. Therefore, some new mechanisms should be investigated. One possible formation path could be the DEA to C2n H2 (n = 2–5) molecules. However, Consoli et al measured the absolute partial pressures in an Ar/He/C2 H2 plasma [18], and they observed that C4 H2 (maximal partial pressure about 0.1 Pa) was the dominant product, followed by C6 H2 (0.013 Pa), C8 H2 (0.0018 Pa), etc. Up to recently [31] only the DEA cross section of C2 H2 has been reported in the literature [32]. If the DEA cross sections of the C2n H2 molecules were the same as for C2 H2 , we would expect the same decreasing trend for C4 H− , C6 H− , C8 H− , etc. This is clearly not the case in the experimental results from Deschenaux [13]. Recent data on C4 H2 [31] shows that the DEA cross section for this molecule has different energy dependences and is approximately 20 times larger (cf later in figure 2) than that of C2 H2 . This, together with 10× smaller C4 H2 density, can explain the higher signal at mass 49 (C4 H− ) than at mass 25 and 26 amu in the negative ion mass spectrum of Deschenaux et al [13] (cf later in figure 5(b)), corresponding to C2 H− and H2 CC− anions. However, the signal at mass 73 (C6 H− ) is again more than a factor 10 higher than the signal at mass 49. It seems that some mechanism must exist, which promotes the DEA to C6 H2 resulting in a very high C6 H− density. We propose that this mechanism is DEA to branched (maybe even vibrationally excited) C2n H∗2 (n > 2) molecules. Indeed, C2n H2 (n > 2) can exist in a branched configuration in contrast to C2n H2 (n  2), which only exist in a linear form. Moreover, it is well known that vibrational excitation can enhance the DEA cross section by several orders of magnitude [33]. These branched C2n H∗2 (n > 2) molecules are suggested to be formed in the following reactions (see figure 3): during the polymerization process of C2n H2 growth, the radical C2 H is not only attached to the end carbon atoms (ECA) of C2n H2 (n = 2–4) to form C2n+2 H2 with linear structure, as is shown here for C4 H2 → C6 H2 :

(4)

with ω the electron energy density flux ω =

5 5 ∂ωe µ e ωe E − D e 3 3 ∂x

(5)

and µe and De are the electron mobility and diffusion coefficient, respectively. The term Sω in equation (4) represents the loss of electron energy due to electron impact collisions. No energy balance is included for ions and neutrals, since they are assumed to be at the local gas temperature. The system of nonlinear coupled differential equations is solved numerically on an equidistant mesh containing 64 grid points. The spatial discretization of the balance equations is based on an implicit finite-difference technique using the Scharfetter–Gummel exponential scheme [27]. Convergence of the fluid model is defined when the relative changes of the discharge parameters between two succeeding RF cycles are less than 10−6 . The time step within an RF cycle is set to 9.2 × 10−10 s. 2.2. Plasma chemical kinetics Dust formation in reactive gas discharges normally comprises three different stages: nucleation, coagulation and surface growth [8, 9, 13, 28, 29]. In this work, the focus is put on the nucleation process of nanoparticles in reactive acetylene plasmas. As stated in our previous work [15, 16], the smallest species (i.e. radicals, ions, atoms) are generated by electron impact decomposition of the C2 H2 feed gas at the beginning. Then larger clusters can be formed through consecutive polymerization reactions. To describe such a complex process, a detailed chemical kinetics scheme has been developed. In the present model, 78 species containing up to a maximum of 12 carbon atoms, and around 400 volume reactions are taken into account. Due to the lack of exact data on the molecular level of the C2 H2 discharge chemistry, some approximations had to be made. These approximations are validated by comparing our calculated results with experimentally obtained mass spectra by Deschenaux et al [13] for charged particles and by Benedikt et al [17, 18] for neutrals. Some new mechanisms of negative ion formation are proposed at first in the following paragraphs, then a comprehensive description of the complete plasma chemistry is given.

C4 H2 + C2 H → HC≡C–C≡C–C≡CH + H 3

(6)

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and DEA cross sections mentioned in the next paragraph is important for the formation of the C2n H− anions. The lower reactivity can then be compensated by a higher ‘scaling’ factor (see later) used for estimation of the DEA cross sections. Subsequently the branched C2n H∗2 (n = 3–5) radicals are proposed as the main source to form C2n H− anions via enhanced DEA reactions such as

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C2n H2∗ + e− → C2n H− + H (n = 3–5)

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11,24

Because to our knowledge only cross sections for the DEA to C2 H2 and C4 H2 have been measured by May et al [31], the cross sections for DEA to linear C2n H2 (n = 3–5) are proposed to have the same profile as that of C4 H2 , but the DEA cross sections for the branched C2n H∗2 (n = 3–5) molecules are assumed to be a factor of 20 higher. This factor is determined based on the fact that the branched structure gives enhanced reactivity and the latter might be able to enhance the DEA cross section. Moreover, it is based on comparison with the experiment of Deschenaux et al [13], because no other data were found in the literature.

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electron energy (eV) -18

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cross section (m )

(b)

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(8)

23 22

(b) Chain of polymerization reactions starting from H2 CC− . Moreover, the measurements of Deschenaux et al [13] reveal the presence of H2 CC− and higher C2n H− 2 anions (n = 2–4). The anion at mass 26 amu is indeed most probably the vinylidene anion H2 CC− , since it is more stable compared with the vinylidene neutral and stable with respect to autodetachment [31]. Based on the observation from experiment of Deschenaux et al [13], we have assumed that 5% of the production from DEA on C2 H2 is C2 H− , and 95% leads to H2 CC− . Then, similarly to anion–neutral reactions in silane discharges [9], after the formation of H2 CC− via electron attachment to acetylene, a consecutive chain of polymerization reactions could be triggered to form C2n H− 2 (n = 2–4) and (n = 4, 5) as follows: C2n H− 4

34

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12 2 3

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Figure 2. Cross sections of electron impact collisions of C2 H2 , C2 H4 , C2n H2 (n = 2–6) and H2 : (a) vibrational excitation, dissociation and attachment; (b) electron impact ionization. The number on each curve refers to the reaction number specified in table 2. Curves 10 and 25 have also been used for the unknown cross section of reactions (17)–(21) and (26)–(28), respectively. For reactions (29)–(31), the cross sections of (26)–(28) have been multiplied by a factor of 20.

C2 H2 + e− → H2 CC− , −

H2 CC + C2 H2 →

but also to the middle carbon atoms (MCA) to form the branched C2n+2 H∗2 species, i.e. C4 H2 + C2 H → HC≡C–C =C· H + H. | C ||| C

C4 H2−

C2n H2− + C2 H2 → C2n+2 H2− + H2

+ H2 ,

(10)

n = 2, 3, 4

(11)

C8 H4− ,

(12)

H2 CC− + C8 H2∗ → C10 H4− .

(13)



H2 CC +

(7)

(9)

C6 H2∗



Because of the lack of precise rate coefficients for these reactions, a theoretical upper limit can be calculated using the Langevin collision rate coefficient [34]. Based on the experimental suggestion in [13], the rate coefficient for the anion reactions is set to 10−18 m3 s−1 .

The total rate constant for the polymerization process (i.e. reaction (6) + (7)) is k = 1.3 × 10−16 m3 s−1 [33], and the branching ratio is determined by the ratio of ECA to MCA in the precursors C2n H2 . For instance, in the case of C4 H2 , 50% leads to the formation of linear C6 H2 and 50% to the branched one. In the case of C6 H2 , 33% and 67% result in the formation of linear and branched C8 H2 , and finally, the branching ratio of linear and branched C10 H2 created out of C8 H2 is equal to 25% versus 75%. This is a very simplified approximation for the reaction of C2 H with C2n H2 molecules, because the reactivities of C2 H with ECA and MCA might be different. However, the product between formation of branched C2n H∗2 species

2.2.2. Complete plasma chemistry described in the model. The above mentioned new anion formation mechanisms are added to the model developed previously in our group [15, 16]. This results in a model where 77 different species besides the electrons are considered (see table 1). It should be noted that both linear, branched and cyclic hydrocarbon species are included. In addition to the species previously considered in the model, such as C2n H+ and C2n H− [15, 16], and the anions 4

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Figure 3. Schematic diagram of the proposed new mechanism for negative ion C2n H− formation. Table 1. Different species considered in the model, besides electrons. Molecules

Ions

C2 H 2 C4 H2 , C6 H2 , C8 H2 , C10 H2 , C12 H2 , C6 H2∗ , C8 H2∗ , C10 H2∗

C2 H+2 , C2 H+ , CH+ , C+2 , C4 H+ , C6 H+ , C8 H+ , C4 H2+ , C6 H2+ , C8 H2+ , C6 H4+ , C8 H4+ , C8 H6+ , C10 H6+ , C12 H6+ H+2 ,H+ C2 H3+ , C2 H4+ , C2 H5+ C4 H3+ , C4 H5+

H2 C, C2 , C2 H4 , C4 H4 , C6 H4 l-C6 H4 , 1-C6 H6 , A1 A2 , PAHs

Radicals C

C2 H− , C4 H− , C6 H− C8 H− , C10 H− , C12 H− H2 CC− , C4 H2− , C6 H2− , C8 H2− , C8 H4− , C10 H4−

+

CH, CH2 C8 H6 , C10 H6 , C12 H6

H C2 H3 , C2 H5 , C4 H3 , C6 H3 C4 H5 , n-C6 H5 , n-C6 H7 A1− , A2− , A1 C2 H, C2 H, C4 H, C6 H C8 H, C10 H, C12 H

Note: The species indicated in bold are newly added, compared with the previous model [15]. Ai denotes an aromatic ring molecule with i fused rings, Ai− represents its radical and PAHs refers to polycyclic aromatic hydrocarbons larger than A2

of the molecule–molecule reactions, the general notation is given in the third column of table 4. The rate coefficients in table 4 reflect the values at a gas temperature of 400 K and a gas pressure of 27 Pa. Most routes to form large clusters by ion–molecule and molecule–molecule reactions have been discussed in detail in our previous work [15, 16] and will not be repeated here. In the following paragraphs, we focus only on the new updates and modifications made in the present model, such as pathways for C2n H4 molecules (n = 1, 2, 3), and charge-transfer reactions. Indeed, C2n H4 molecules were observed by Benedikt et al in the experiment [17], and therefore need to be included in our model. The molecule C2 H4 is proposed to be formed by two hydrogen insertion reactions as follows:

discussed above, some other new species are added to the model, such as C2n H4 (n = 1, 2, 3), which were observed in the experiment of Benedikt et al [17], and related ions (like C2 H+3 , C2 H+4 ) and radicals (like C2 H5 , C4 H5 ). Upon ignition of the discharge, small radicals, ions and atoms are produced by electron impact on acetylene molecules via excitation, dissociation and ionization processes. To calculate the rate constants of these electron-induced reactions, the electron energy distribution function is first calculated by solving the Boltzmann equation with two-term approximation. The 35 electron–neutral reactions included in the model, as well as their threshold energies, are outlined in table 2. Compared with our previous model, 12 new reactions including eight DEA reactions on C2n H2 (n = 1–5) and four reactions on C2 H4 are included. Figure 2 shows the various cross sections for electron impact on C2 H2 , C2 H4 , C2n H2 (n = 2–6) and H2 . The important ion–molecule and molecule–molecule reactions are summarized in tables 3 and 4, respectively. For the temperature- or pressure-dependent rate coefficients

C2 H2 + H → C2 H3 ,

(14)

C2 H3 + H → C2 H4 .

(15)

The rate coefficients for these reactions are taken as k0 = 3.5 × 10−21 m3 s−1 [62] and k0 = 2.14 × 10−16 m3 s−1 [63], 5

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Table 2. Electron impact collisions included in the model and their corresponding threshold energies.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Reaction

Threshold energy (eV)

Reaction type

References

C2 H2 + e− → C2 H2+ + 2e− C2 H2 + e− → C2 H+ + H + 2e− C2 H2 + e− → C+2 + H2 + 2e− C2 H2 + e− → CH+ + CH + 2e− C2 H2 + e− → C+ + CH2 + 2e− C2 H2 + e− → H+ + C2 H + 2e− C2 H2 (0) + e− → C2 H2(v=1) + e− C2 H2 (0) + e− → C2 H2(v=2) + e− C2 H2 (0) + e− → C2 H2(v=3) + e− C2 H2 + e− → C2 H + H + e− C2 H2 + e− → C2 H− + H H2 + e− → H2+ + 2e− H2 (0) + e− → H2(v=1) + e− H2 (0) + e− → H2(v=2) + e− H2 (0) + e− → H2(v=3) + e− H2 + e − → H + H + e − C4 H2 + e− → C4 H + H + e− C6 H2 + e− → C6 H + H + e− C8 H2 + e− → C8 H + H + e− C10 H2 + e− → C10 H + H + e− C12 H2 + e− → C12 H + H + e− C4 H2 + e− → C4 H2+ + 2e− C6 H2 + e− → C6 H2+ + 2e− C2 H2 + e− → H2 CC− C4 H2 + e− → C4 H− + H C6 H2 + e− → C6 H− + H C8 H2 + e− → C8 H− + H C10 H2 + e− → C10 H− + H C6 H2∗ + e− → C6 H− + H C8 H2∗ + e− → C8 H− + H C10 H2∗ + e− → C10 H− + H C2 H4 + e− → C2 H4∗ + e− C2 H4 + e− → C2 H4∗ + e− C2 H4 + e− → C2 H4+ + 2e− C2 H4 + e− → C2 H2 + 2H + e−

11.4 16.5 17.5 20.6 20.3 18.4 0.09 0.29 0.41 7.5 2.74 15.4 0.54 1.08 1.62 8.9 7.5 7.5 7.5 7.5 7.5 10.19 9.55 2.74 1.94 2.74 2.74 2.74 2.74 2.74 2.74 0.12 0.39 10.51 5.8

Ionization Dissociative ionization Dissociative ionization Dissociative ionization Dissociative ionization Dissociative ionization Vibrational excitation Vibrational excitation Vibrational excitation Dissociation Dissociative attachment Ionization Vibrational excitation Vibrational excitation Vibrational excitation Dissociation Dissociation Dissociation Dissociation Dissociation Dissociation Ionization Ionization Attachment Dissociative attachment Dissociative attachment Dissociative attachment Dissociative attachment Dissociative attachment Dissociative attachment Dissociative attachment Vibrational excitation Vibrational excitation Ionization Dissociation

[36] [37] [37] [37] [37] [37] [38] [38] [38] [37] [31], 5% [39] [40] [40] [40] [41] [37],est. [37], est. [37], est. [37], est. [37], est. [42] [42] [31], 95% [31] [31], est. [31], est. [31], est. [31], est. [31], est. [31], est. [43] [43] [42] [43]

Note: The abbreviation ‘est.’ stands for estimated. The cross sections for reactions (22) and (23) are updated with data from NIST with respect to the previous model [15]. The reactions from (24) to (35) are newly added. The DEA cross sections for C2n H2 (n = 3–5) are assumed the same as that for C4 H2 , and the cross sections for C2n H∗2 (n = 3–5) are assumed 20 times higher than that for C4 H2 .

sometimes became more favourable. Therefore, these reactions are also included in the present model.

respectively. Benedikt et al [17] show that C4 H4 exhibits a very similar temporal evolution as C4 H2 , which means that C4 H4 does not arise from C4 H2 . Hence, we suggest that C4 H4 molecules are generated from the following reaction: C2 H + C2 H4 → C4 H4 + H,

2.2.3. Diffusion and wall deposition losses. Besides chemical reactions in the discharge, species can also be lost by diffusion to the reactor walls followed by deposition. The overall diffusion coefficient Dj of species j in the gas mixture can be calculated by Blanc’s law [74],  pi ptot = , (17) Dj Dij i

(16)

and the rate coefficient is taken as k0 = 1.13 × 10−16 m3 s−1 [60]. As to C6 H4 , it is expected to be formed mainly through a mutual neutralization reaction of C6 H+4 ions with hydrocarbon anions [58]. Some more ion–molecule reactions, including chargetransfer reactions, are taken into account in the present model(see table 3). Most of them have been adopted from astrochemistry research [44]. The charge-transfer process in collisions of slow H+ ions with hydrocarbon molecules has been studied by Kusakabe et al [73]. Their results indicated that if vibrationally excited molecular ions in the exit channel were assumed, the charge-transfer processes

where Dij is the binary diffusion coefficient of species j in every background gas i [35], 1/2

Dij =

3 kB Tgas 4π kB Tgas /mij , 16 ptot π σij2 D ( )

(18)

kB is the Boltzmann constant, Tgas is the gas temperature in Kelvin, ptot is the total gas pressure in Pascals, mij is the 6

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54 C2 H3 + C4 H2+ → C6 H4+ + H 1.2 × 10−15 [47] + + 55 C2 H3 + C4 H3 → C6 H4 + H2 5.0 × 10−16 [47] 56 C2 H3 + C6 H2+ → C8 H4+ + H 4.0 × 10−16 [47] 57 C2 H3+ + H → C2 H2+ + H2 6.8 × 10−17 [55] 58 C2 H3+ + C6 H → C8 H2+ + H2 5.0 × 10−16 [47] 59 C2 H3+ + C4 H → C6 H2+ + H2 4.0 × 10−16 [47] 60 C2 H3+ + C6 H → C6 H2+ + C2 H2 5.0 × 10−16 [47] Reactions between C2 H4 /C2 H+4 and hydrocarbons 61 C2 H4 + C+ → C2 H3+ + CH 8.5 × 10−17 [56] 62 C2 H4 + C+ → C2 H4+ + C 1.7 × 10−17 [56] 63 C2 H4 + C2 H2+ → C2 H4+ + C2 H2 4.1 × 10−16 [53] 64 C2 H4 + C2 H2+ → C4 H5+ + H 3.2 × 10−16 [53] 65 C2 H4 + C2 H3+ → C2 H5+ + C2 H2 8.9 × 10−16 [53] 66 C2 H4 + C4 H+ → C6 H4+ + H 7.5 × 10−16 [47] [47] 67 C2 H4 + C4 H+ → C4 H3+ + C2 H2 7.5 × 10−16 68 C2 H4 + C4 H2+ → C6 H4+ + H2 2.0 × 10−16 [47] 1.0 × 10−15 [47] 69 C2 H4 + C6 H2+ → C8 H4+ + H2 70 C2 H4+ + C4 H → C6 H4+ + H 2.5 × 10−16 [47] 2.5 × 10−16 [47] 71 C2 H4+ + C4 H → C6 H4+ + H 72 C2 H4+ + C6 H → C8 H4+ + H 2.5 × 10−16 [47] 73 C2 H4+ + C6 H2 → C8 H4+ + H2 5.0 × 10−16 [47] Other reactions among hydrocarbons 74 C+2 + CH → CH+ + C2 3.2 × 10−16 [44] 1.1 × 10−15 [57] 75 C+2 + H2 → C2 H+ + H 76 C4 H+ + C4 H → C8 H+ + H 6.0 × 10−16 [47] 77 C4 H+ + C4 H2 → C8 H2+ + H 1.5 × 10−15 [47] Neutralization reactions of hydrocarbon anions with H+ , H+2 and Cn H+m 78 Cx Hy− + H+ → Cx Hy + H, ∼3.0 × 10−14 [58] (x = 1, 2; y = 1, 2, 4) 79 Cx Hy− + H2+ → Cx Hy + H + H ∼2.0 × 10−13 [58] 80 Cx Hy− + Cm Hn+ → Cx Hy + Cn Hm ∼5.0 × 10−14 [58]

Table 3. Ion–molecule reactions taken into account in the model.

Reaction

Rate coefficient (m3 s−1 )

References

Cluster growth through hydrocarbon anions with C2 H2 1 C2 H− + C2 H2 → C4 H− + H2 1.0 × 10−18 [6] − − 2 C4 H + C2 H2 → C6 H + H2 1.0 × 10−18 [6] 3 C2n H− + C2 H2 → C2n+2 H− + H2 1.0 × 10−18 [6] (n = 3–5) 4 C4 H2− + C2 H2 → C6 H2− + H2 1.0 × 10−18 [6] − − 5 C6 H2 + C2 H2 → C8 H2 + H2 1.0 × 10−18 [6] − Cluster growth through H2 CC with C2n H2 6 H2 CC− + C2 H2 → C4 H2− + H2 1.0 × 10−18 [6] 7 H2 CC− + C6 H2∗ → C8 H4− 1.0 × 10−17 [6] − ∗ − 8 H2 CC + C8 H2 → C10 H4 1.0 × 10−17 [6] + Reactions between H and hydrocarbons 9 H+ + C2 → C+2 + H 3.1 × 10−15 [44] 10 H+ + C2 H4 → C2 H4+ + H 1.0 × 10−15 [45] 11 H+ + C4 H → C4 H+ + H 2.0 × 10−15 [44] 12 H+ + C6 H → C6 H+ + H 2.0 × 10−15 [44] 13 H+ + C2 H2 → C2 H+ + H2 4.3 × 10−15 [46] 14 H+ + C4 H2 → C4 H+ + H2 2.0 × 10−15 [44] 2.0 × 10−15 [47] 15 H+ + C8 H → C8 H+ + H + + 16 H + C8 H2 → C8 H + H2 2.0 × 10−15 [47] 17 H+ + C2 H4 → C2 H3+ + H2 3.0 × 10−15 [45] + Reactions between H2 /H2 and hydrocarbons 6.4 × 10−16 [48] 18 H2+ + H → H+ + H2 + + 19 H2 + C2 H2 → C2 H2 + H2 5.3 × 10−15 [49] 20 H2+ + C2 H4 → C2 H4+ + H2 2.2 × 10−15 [50] 21 H2+ + C2 H4 → C2 H2+ + H2 + H2 8.8 × 10−16 [50] + + 22 H2 + C2 H4 → C2 H3 + H2 + H 1.8 × 10−15 [50] 23 H2+ + C4 H → C4 H2+ + H 1.7 × 10−16 [47] + + 1.0 × 10−15 [47] 24 H2 + C8 H → C8 H2 + H 25 H2 + C2 H+ → C2 H2+ + H 1.7 × 10−15 [49] Reactions between C2 H/C2 H+ and hydrocarbons 5.0 × 10−16 [47] 26 C2 H + C2 H4+ → C4 H3+ + H2 + + 27 C2 H + C4 H → C6 H + H 6.0 × 10−16 [47] 1.3 × 10−15 [47] 28 C2 H + C4 H2+ → C6 H2+ + H 29 C2 H + C6 H2+ → C8 H2+ + H 1.2 × 10−15 [47] 30 C2 H+ + C2 H2 → C4 H2+ + H 1.2 × 10−15 [49] 31 C2 H+ + C2 H4 → C2 H2+ + C2 H3 1.7 × 10−15 [51] + Reactions between C2 H2 /C2 H2 and hydrocarbons 32 C2 H2 + C+2 → C4 H+ + H 1.7 × 10−15 [47] + + 33 C2 H2 + C2 H → C4 H2 + H 1.2 × 10−15 [49] 34 C2 H2 + C2 H3+ → C4 H3+ + H2 2.4 × 10−16 [52] 35 C2 H2 + C2 H4+ → C4 H5+ + H 1.9 × 10−16 [53] 1.5 × 10−15 [47] 36 C2 H2 + C4 H+ → C6 H2+ + H 37 C2 H2 + C4 H2+ → C6 H4+ 1.4 × 10−15 [47] 38 C2 H2 + C2 H2+ → C4 H3+ + H 9.5 × 10−16 [52] 39 C2 H2 + C2 H2+ → C4 H2+ + H2 1.2 × 10−15 [54] 40 C2 H2 + C6 H4+ → C8 H6+ 1.0 × 10−16 [54] 1.0 × 10−16 [54] 41 C2 H2 + C8 H4+ → C10 H6+ + + 42 C2 H2 + C8 H6 → C10 H6 + H2 1.0 × 10−16 [54] 43 C2 H2 + C10 H6+ → C12 H6+ + H2 1.0 × 10−16 [54] 44 C2 H2 + C6 H2+ → C8 H4+ 1.0 × 10−17 [54] + + 45 C2 H2 + C6 H2 → C8 H2 + H2 1.0 × 10−17 [54] 46 C2 H2+ + H2 → C2 H3+ + H 1.0 × 10−17 [53] + + 47 C2 H2 + C6 H2 → C8 H2 + H2 5.0 × 10−16 [54] 1.2 × 10−15 [47] 48 C2 H2+ + C6 H → C8 H2+ + H + + 49 C2 H2 + C6 H2 → C6 H2 + C2 H2 5.0 × 10−16 [47] 50 C2 H2+ + C6 H → C8 H+ + H2 1.2 × 10−15 [47] + Reactions between C2 H3 /C2 H3 and hydrocarbons 51 C2 H3 + C2 H2+ → C4 H3+ + H2 3.3 × 10−16 [47] + + 5.0 × 10−16 [44] 52 C2 H3 + C2 H4 → C2 H5 + C2 H2 53 C2 H3 + C2 H4+ → C2 H3+ + C2 H4 5.0 × 10−16 [44]

Note: Reactions (1)–(3), (19), (25), (33), (39), (40)–(43) and (78)–(79) were considered in our previous model [15], and all the other reactions are new.

reduced molecular mass in amu, σij is the binary collision diameter in Ångstrom. σi + σj σij = , (19) 2 and D ( ) is the diffusion collision integral given by [35] D =

A C E G + + + B eD eF eH

with = Tgas /ij , ij = (i × j )0.5 , and constants A = 1.060 36, B = 0.156 10, C = 0.193 00, D = 0.476 35, E = 1.035 87, F = 1.529 96, G = 1.764 64 and H = 3.894 11. For ions both a mobility and a diffusion coefficient are considered. The ion mobility of an ionic species j in the background gas i can be calculated from the low-electric-field Langevin mobility expression [74] µij = 0.514m−0.5 ij

Tgas −0.5 α , ptot i

(20)

where αi in Å3 is the polarizability of the background gas i. The overall ion mobility µj of ion j in the gas mixture can again be obtained by Blanc’s law. Finally, the ion diffusion coefficient can be derived from Einstein’s relation kB Tion µj , (21) Dj± = e 7

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Table 4. Molecule–molecule reactions included in the model and their corresponding rate coefficients. The general notation of the temperature- and pressure-dependent rate coefficients is given in column 3. Here T is the gas temperature in Kelvin. Reaction

Rate coefficient (m3 s−1 )

Cluster growth through C2 H insertion 1 C2 H + H2 → C2 H2 + H 4.9 × 10−19 2 C2 H + C2 H2 → C4 H2 + H 1.3 × 10−16 3 C2 H + C4 H2 → C6 H2 + H 6.5 × 10−17 4 C2 H + C4 H2 → C6 H2∗ + H 6.5 × 10−17 5 C2 H + C6 H2 → C8 H2 + H 4.3 × 10−17 6 C2 H + C6 H2 → C8 H2∗ + H 8.7 × 10−17 7 C2 H + C8 H2 → C10 H2 + H 3.2 × 10−17 8 C2 H + C8 H2 → C10 H2∗ + H 9.8 × 10−17 9 C2 H + C10 H2 → C12 H2 + H 1.3 × 10−16 10 C2 H + C2 H4 → C4 H4 + H 1.13 × 10−16 11 C2 H + C4 H4 → C4 H3 + C2 H2 6.6 × 10−17 12 C2 H + C2 H2 → C4 H3 2.2 × 10−18 2.2 × 10−18 13 C2 H + C4 H2 → C6 H3 Hydrogen insertion 14 CH + H2 → CH2 + H 1.0 × 10−18 15 C2n H + H2 → C2n H2 + H 4.9 × 10−19 16 C2 H + H → C2 H2 4.1 × 10−16 17 C2 H2 + H → C2 H3 3.5 × 10−16 18 C2 H3 + H → C2 H4 2.14 × 10−16 19 C4 H2 + H → C4 H3 1.2 × 10−18 1.6 × 10−18 20 C6 H2 + H → C6 H3 21 C2n H + H → C2n H2 4.1 × 10−16 22 C6 H4 + H → n-C6 H5 NM × 1.0 × 10−39 23 n-C6 H5 + H → l-C6 H6

NM × 1.0 × 10−39

24 l-C6 H6 + H → n-C6 H7

NM × 1.0 × 10−44

25 C6 H4 + H → A1−

NM × 1.0 × 10−39

26 A1− + H → A1 27 A2− + H → A2 Hydrogen abstraction 28 H + C2 H3 → C2 H2 + H2 29 H + C4 H3 → C4 H2 + H2 30 H + C6 H3 → C6 H2 + H2 31 H + C2 H5 → C2 H4 + H2 32 H + CH2 → CH + H2 33 H + n-C6 H5 → l-C6 H4 + H2 34 H + n-C6 H7 → l-C6 H6 + H2 35 H + A1 → A1− + H2

NM × 1.0 × 10−39 NM × 1.0 × 10−39 6.6 × 10−17 2.4 × 10−17 6.6 × 10−17 5.0 × 10−17 2.8 × 10−16 2.5 × 10−17 2.5 × 10−17 1.8 × 10−22

36 H + A2 → A2− + H2 2.4 × 10−22 Cluster growth through acetylene insertion 37 C4 H + C2 H2 → C6 H2 + H 6.6 × 10−17 38 C2n H + C2 H2 → C2n+2 H2 + H 6.6 × 10−17 39 A1− + C2 H2 → A1 C2 H 1.6 × 10−17 40 A1 C2 H + C2 H2 → A2 + H 8.3 × 10−20 41 C4 H5 + C2 H2 → A1 + H 1.4 × 10−20 42 C2 H3 + C2 H2 → C4 H5 2.3 × 10−21 43 C4 H3 + C2 H2 → n-C6 H5 NM × 1.0 × 10−39 44 C4 H5 + C2 H2 → n-C6 H7 45 C4 H3 + C2 H2 → A1− 46 C4 H3 + C2 H2 → l-C6 H4 + H Other neutral–neutral reactions 47 CH2 + CH2 → C2 H2 + H2 48 CH2 + CH → C2 H2 + H 49 l-C6 H6 + H → A1 + H

2.3 × 10−21 3.0 × 10−17 1.6 × 10−20 5.3 × 10−17 8.3 × 10−17 5.7 × 10−19

Comment

References

1.82 × 10−17 exp(−1443/T )

[59] [59] [59] [59] [59] [59] [59] [59] [59] [60] [61] [62] [62]

1.82T −6.3 exp(−1404/T ) 1.82T −6.3 exp(−1404/T ) 1.82 × 10−22 T 1.79 exp(−840.4/T ) 1.82 × 10−17 exp(−1443/T ) n = 2–6 1.66 × 10−13 T −1 7.25 × 10−18 exp(−1212.7/T ) 2.02 × 10−16 (T /298)0.2 2.82 × 1019 T −11.67 exp(−6441/T ) 7.1 × 1015 T −10.15 exp(−6667.6/T ) 1.66 × 10−13 T −1 n = 2–6 k0 = 1.0 × 10−39 k∞ = 5.48 × 1014 T −10.04 exp(−9467.6/T ) k0 = 1.0 × 10−39 k∞ = 1.83 × 1012 T −9.65 exp(−3525/T ) k0 = 8.0 × 10 − 43T −0.52 exp(−503.6/T ) k∞ = 2.49 × 10−14 T −1.65 exp(−805.7/T ) k0 = 1.0 × 10−39 k∞ = 5.98 × 1047 T −20.09 exp(−14150/T ) k0 = 6.89 × 10−45 exp(−4778/T ) k0 = kreac26 2.65 × 10−11 T −1.6 exp(−1117/T ) 3.98 × 10−11 T −1.6 exp(−1409/T )

[62] [59] [62] [63] [64] [55] [55] [55] [65] [65] [65] [65] [68] [68] [64] [62] [62] [66] [67] [68] [68] [69]

5.0 × 10−18 (Nc /6)1/2 exp(−4080/T ) (T < 1000 K) 4.15 × 10−16 (Nc /6)1/2 exp(−8060/T ) (T  1000 K) k0 = kreac35 [69] n = 3–5 3.48 × 10−15 T −1.07 exp(−2415/T ) 1.03 × 10−18 exp(−2442/T ) k0 = 1.0 × 10−39 k∞ = 7.08 × 10−26 T 1.97 exp(2820/T ) 1.03 × 10−18 exp(−2442/T ) 5.81 × 10−19 exp(4091/T )P 0.8

1.44 × 10−13 T −1.34 exp(−1762/T )

8

[59] [59] [70] [70] [71] [68] [68] [68] [68] [68] [59] [59] [71]

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Table 4. (Continued).

50 51 52 53 54 55

Reaction

Rate coefficient (m3 s−1 ) Comment

References

C2 H3 + CH → C2 H2 + CH2 C4 H3 + H → C2 H2 + C2 H2 C6 H3 + H → C4 H2 + C2 H2 C4 H5 + C2 H2 → l − C6 H6 + H A2− + C2 H2 → PAHs C2 H + C2 H3 → C2 H2 + C2 H2

8.3 × 10−17 1.1 × 10−16 8.1 × 10−17 4.7 × 10−25 2.5 × 10−21 5.0 × 10−17

[72] [62] [62] [71] [68] [72]

2.65 × 10−11 T −1.6 exp(−1117/T ) 3.98 × 10−11 T −1.6 exp(−1409/T ) 9.63 × 10−22 T 1.02 exp(−5489/T )

Note: Reaction coefficients for three-body reactions were calculated with M = C2 H2 . Units for two-body rate coefficients and high pressure limit rate constants for three-body reactions (k∞ ) are in m3 s−1 . Low pressure limits of the rate coefficients for three-body reactions (k0 ) are in units of m6 s−1 . P is in atm. The factor Nc represents the number of carbon atoms in the aromatic molecule and accounts for the increase of the rate coefficient with molecular size. Reactions (4), (6), (8), (10), (11), (17), (31), (48) are newly added compared to the previous model.

normalized with respect to the C4 H+2 hydrocarbon cation. The experimentally obtained mass spectrum of positive ions reproduced from Deschenaux et al [13] is depicted in figure 4(b) for comparison. When compared with our previous calculation results [15, 16], much better agreement is achieved now on the trend of hydrocarbon cation peak variation. Indeed, in the previous model, the positive ions C2n H+3 (n = 1, 2) were not included. Now they are taken into account, and they agree well with experimental data of Deschenaux et al [13]. The highest signal of C4 H+2 is detected by the present model, which is consistent with the measurement of Deschenaux et al [13], as well as the comparable decreasing trend towards larger hydrocarbon positive ions. Figure 5 illustrates a comparison between the theoretical anion mass spectrum and the experimental one, again obtained from Deschenaux et al [13]. To obtain the negative ions mass spectrum, the discharge should be switched off so that the anions can reach the reactor wall. All peaks have been normalized with respect to the C6 H− anions, which has the highest intensity. Compared with our previous work [15, 16], significantly better agreement is reached now with experiment. Indeed, in our previous model, a monotonic decreasing trend was detected, which did not agree with the measured spectrum. This indicates that the new mechanisms for large anion formation proposed in the present model, i.e. DEA on higher hydrocarbons C2n H2 (n > 2), especially the enhanced DEA on the branched C2n H∗2 (n = 3–5), play a key role in the negative ion formation process. We also note that the peak of C6 H− in the simulated mass spectra is not so pronounced as that in the measurements of Deschenaux et al [13]. In the measurements, the difference between the C4 H− peak and the C6 H− peak is a factor of 20, whereas it is only a factor of 2 in our simulated results. So we may conclude that the DEA cross section for C6 H∗2 must probably be even larger than assumed in the model. On the other hand, for the C2n H− 2 anions, the agreement is not yet satisfying. Indeed, our calculated values are much lower than the C2n H− 2 in comparison with experimental data. This indicates that probably beside the polymerization reactions for C2n H− 2 anion growth, also some other path should be evaluated. Also it should be noted that there are more peaks observed in the experimental mass spectra of [13], for both positive and negative ions. However, the dominance of species

where Tion represents the ion temperature which is assumed to be equal to the gas temperature Tgas here. A sticking coefficient model, where the deposition of species at the wall is taken into account, is adopted to describe the plasma–wall interaction. More details about the sticking coefficients assumed for the different species can be found in [15].

3. Results and discussion 3.1. Operating conditions The simulations are carried out for a parallel-plate, capacitively coupled RF discharge, for a pressure of 27 Pa and a power of 40 W. These conditions are chosen very similar to the experiment by Deschenaux et al [13], to allow detailed comparison. Only the pressure is somewhat higher, to ensure the validity of the fluid approach. The gap between the two electrodes is set at 2.5 cm. One electrode is connected to the power supply with a driving frequency of 13.56 MHz, while the other is electrically grounded. The diameters of powered and grounded electrodes in the simulation are both 130 mm. 8 sccm of pure C2 H2 is fed into the discharge and a uniform gas temperature of 400 K is assumed. As described in [75], the gas inlet is in the side wall of the cubic steel vacuum chamber with sides of 400 mm length for the experimental apparatus used by Deschenaux et al [13]. This is far away from the plasma region. Therefore, the gas flow velocity can be neglected compared with the diffusion and drift velocity of nanoparticles. 3.2. Positive and negative ion and neutral species ratios in comparison with experiment In order to validate our simulation results, a theoretical ion mass spectrum is deduced from the ion fluxes toward the reactor wall and their normalized intensities are compared with the experimental mass spectrum which is also normalized. Indeed, comparison of absolute values is not possible because there is a difference between the theoretical mass spectra (i.e. the peaks of ion fluxes) and the experimental one (i.e. the peaks of channel counts). A theoretical positive ion mass spectrum is depicted in figure 4(a). Note that every peak in the spectrum is 9

J. Phys. D: Appl. Phys. 41 (2008) 225201

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10 1

(a) our simulation

+

normalized intensity

C4H2

+

10 0

C2H2

10 -1

+

C6H2 + C6H4

+

C8H2 C8H6 +

+

C10H6

+

H2

+

C12H6

10 -2

10 -3

10 -4 0

10

1

10

0

10

20

30

40

50

60

70

80

90

100 m/z

110

130

140

150

160

170

180

190

(b) measured by Deschenaux et. al

+

C4H2

+

C6H2

+

C2H2

normalized intensity

120

+

C6H4

10 -1

+

C8H6

+

C10H6

+

C8H2

+

10

-2

10

-3

C12H6

+

10 -4

H2

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

m/z

normalized intensity

Figure 4. Mass spectra of positive ions obtained from (a) our simulation and (b) the measurements by Deschenaux et al (adopted from [13] with kind permission of IOP Publishing and Ch Hollenstein). 10

1

10

0

H 2CC 10

C 4H

-

C 4H 2

-1

C 2H

(a) our simulation

-

C 6H

C 8H

-

-

-

C 10H C 10H 4

C 8H 4

-

C 6H 2

-

-

C 12H

-

10-2

10-3

10-4

0

10

20

30

40

50

60

70

80

90

100 m/z

110

120

130

140

150

160

170

180

190

1

10

(b) measured by Deschenaux et al.

-

C6H

0

normalized intensity

10

-

C6H2

-

10-1

C4H

-

H2CC

-

C8H C8H4

-

C4H2

-

C10H4

-2

10

-

C10H

-

C2H

-

C12H

-3

10

10-4

0

10

20

30

40

50

60

70

80

90

100 m/z

110

120

130

140

150

160

170

180

190

Figure 5. Mass spectra of negative ions obtained from (a) our simulation and (b) the measurements by Deschenaux et al (adopted from [13] with kind permission of IOP Publishing and Ch Hollenstein).

with even carbon atom number is evident and it indicates that the strong carbon triple bond of the initial acetylene molecule (H–C≡C–H) persists upon consecutive integration of acetylene in larger building units. Therefore, mostly carbon

species with an even number of carbon atoms have been incorporated in the model and the number of species is limited in order to reduce the computational effort. However, as is clear from figures 4(b) and 5(b), many more species are present in 10

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our new model calculation and the experimental data from Consoli et al [18] is quite satisfactory.

Normalized density [% of C2H2 density]

100

experiment by Consoli et al. our simulation

3.3. Relative contribution of different chemical reactions to nanoparticle growth

10

To identify the governing processes in the nanoparticle growth, we calculated the relative contributions of the chemical reactions to the total production and destruction rate of every species using [76]

1

0.1

ci,j Ri,j , αi,j =  k ci,k Ri,k

where αi,j is the relative contribution of reaction j to the total production or destruction of species i and Ri,j is the spatially and time-averaged reaction rate of reaction j that is defined by  nl , (23) Ri,j = kj

0.01 C2H2

C2H4

C4H2

C4H4

C6H2

C6H4

C6H6

C8H2

(22)

C8H6

Species

Figure 6. Comparison of the neutral species’ densities obtained from our simulation and from the measurements by Consoli et al [18].

l

where kj is the rate coefficient of reaction j and nl the density of the reacting species. The coefficient ci,j in equation (22) is the associated stoichiometric number that accounts for the number of particles i that are lost or created in reaction j . A detailed overview of the relative contributions of the chemical reactions to the total production and destruction of the most important species, including radicals (C2n H (n = 1–6), C2n H3 (n = 1–2)), cations (C2n H+2 (n = 1–4), C2n H+6 (n = 4–6)) and anions (C2n H− (n = 1–6)), is given in table 5. The percentage is relative to the total production or total destruction. There are two major pathways which contribute to the generation of the C2n H (n = 1–6) radicals, i.e. electron impact dissociation of C2n H2 (n = 1–6) molecules and neutralization reactions of the C2n H− (n = 1–6) anions with hydrocarbon cations. It is clear from table 5 that the electron impact dissociation reactions become less important as a production mechanism when the C2n H radicals become bigger, while the neutralization reactions become predominant. As far as loss mechanisms of the C2n H (n = 1–5) radicals are concerned, the association reactions between C2n H (n = 1–5) and C2 H2 to form larger C2n+2 H2 molecules are the most important, except for C12 H, where this process was not taken into account anymore, as larger (e.g. C14 H2 ) molecules are not considered in the model. Indeed, the C12 H radicals are therefore mainly lost due to H2 insertion reactions to form C12 H2 . With respect to the C2n H3 (n = 1–2) species, they are predominantly formed from insertion of H or C2 H into C2 H2 , and they are lost as precursor to aromatic ring formation. As far as the cations are concerned, for the small positive ions C2n H+2 (n < 4), electron impact ionization of C2n H2 (n < 4) are the principal routes, while the association reactions of smaller cations with C2 H2 play a key role in the formation of the larger cations (C8 H+2 and C2n H+6 (n = 4–6)). With respect to the loss routes, the association reactions with C2 H2 are dominant for C2n H+2 (n < 4) and C2n H+6 (n = 4–5), whereas neutralization reactions with anions are the most important for C8 H+2 and C12 H+6 .

the plasma, but at lower densities. Furthermore, impurities such as acetone (CH3 –O–CH3 , m/z = 46 and 59) were found in the experiment of Deschenaux et al but they were also not taken into account in this study. Finally, a comparison of the neutral species’ densities between our simulations and the measurements of Consoli et al [18] is shown in figure 6. As argued in [18], a qualitative comparison of the neutral mass spectrum of Deschenaux et al [13] and reconstructed spectra of Consoli et al [18] showed good agreement, which indicates that the presence of argon and helium in the acetylene plasma does not change significantly the C2 H2 plasma chemistry. Additionally, a 1% of C2 H4 impurity was found in the experiment [18], which is therefore also taken into account in our calculations for the present comparison study. Indeed, the existence of the C2 H4 impurity affects the formation of C4 H4 (see equation (16) above). In order to investigate the influence of C2 H4 on the formation of C4 H4 , calculations are carried out and compared for the case with and without 1% of C2 H4 . As expected, the density of C2 H4 increases 1 order of magnitude with a 1% of C2 H4 impurity. Since the formation of C4 H4 is assumed to occur from C2 H insertion into C2 H4 in the present model, it is quite logical that the density of C4 H4 also increases 1 order of magnitude. For the C2n H2 (n = 1–4) molecules, an excellent agreement is achieved when compared with the experimental measurement [18]. We should point out here that the densities of C2n H2 (n = 3, 4) are the sums of the densities of linear C2n H2 (n = 3, 4) and branched C2n H∗2 (n = 3, 4). Note that the densities of larger species C2n H2 (n = 3–5) have dropped a lot compared with our previous model due to DEA to these molecules, especially the enhanced DEA to the branched C2n H∗2 (n = 3–5) species. For the other species such as C2n H4 (n = 1–3) and C2n H6 (n = 3, 4), a relatively bigger discrepancy appears but it should be stressed that the agreement is already significantly better than that for our previous model [15]. In general, we can conclude that the agreement between 11

J. Phys. D: Appl. Phys. 41 (2008) 225201

M Mao et al

Table 5. Calculated relative contributions of the most important production and loss processes of the various plasma species. The numbers in parentheses refer to the number of the table and the reaction, respectively. For example, 2-10 means reaction 10 in table 2. Production process C2 H Electron impact dissociation of C2 H2 (2-10) Neutralization reaction between C2 H− and Cm H+n (3-80)

%

C8 H+2 Reaction between C2 H2 and C6 H+2 (3-45) Reaction between C2 H+2 and C6 H2 (3-47) C8 H+6 Reaction between C6 H+4 and C2 H2 (3-40) C10 H+6 Reaction between C8 H+6 and C2 H2 (3-42) Reaction between C8 H+4 and C2 H2 (3-41) C12 H+6 Reaction between C10 H+6 and C2 H2 (3-43)

%

99.9 Reaction between C2 H and C2 H2 to form C4 H2 (4-2) 0.1 Reaction between C2 H and C4 H2 to form C6 H2 (4-3) Reaction between C2 H and C2 H2 to form C4 H3 (4-12) Others

C4 H Electron impact dissociation of C4 H2 (2-17) 97.0 Neutralization reaction between C4 H− and Cm H+n (3-80) 3.0 C6 H Electron impact dissociation of C6 H2 (2-18) 37.2 Neutralization reaction between C6 H− and Cm H+n (3-80) 62.8 C8 H Electron impact dissociation of C8 H2 (2-19) 20.8 Neutralization reaction between C8 H− and Cm H+n (3-80) 79.2 C10 H Electron impact dissociation of C10 H2 (2-20) 9.0 Neutralization reaction between C10 H− and Cm H+n (3-80) 91.0 C12 H Electron impact dissociation of C12 H2 (2-20) 0.6 Neutralization reaction between C12 H− and Cm H+n (3-80) 99.4 C2 H3 Reaction between H and C2 H2 (4-18) 100

C4 H3 Reaction between C2 H and C2 H2 (4-12) Neutralization reaction between C4 H+3 and Cx H− y (3-80) Reaction between H and C4 H2 (4-19) C2 H+2 Electron impact ionization of C2 H2 (2-1) Charge transfer between H+2 and C2 H2 (3-19) Reaction between H2 and C2 H+ C4 H+2 Electron impact ionization of C4 H2 (2-22) Reaction between C2 H2 and C2 H+2 (3-39) Others C6 H+2 Electron impact ionization of C6 H2 (2-23) Charge transfer between C2 H+2 and C6 H2 (3-49) Others

Loss process

86.0 10.4 1.5 2.1

Reaction between C4 H and C2 H2 to form C6 H2 (4-37) Reaction between H2 and C4 H to form C4 H2 (4-15)

99.6 0.4

Reaction between C6 H and C2 H2 to form C8 H2 (4-38) Reaction between H2 and C6 H to form C6 H2 (4-15)

99.6 0.4

Reaction between C8 H and C2 H2 to form C10 H2 (4-38) Reaction between H2 and C8 H to form C8 H2 (4-15)

99.6 0.4

Reaction between C10 H and C2 H2 to form C12 H2 (4-38) Reaction between H2 and C10 H to form C10 H2 (4-15)

99.6 0.4

Reaction between H2 and C12 H to form C12 H2 (4-15) Reaction between H and C12 H to form C12 H2 (4-21)

99.4 0.6

Reaction between C2 H2 and C2 H3 to form C4 H5 (4-42) Reaction between H and C2 H3 to form C2 H4 (4-18) Reaction between C2 H3 and C4 H+3 (3-55) Others

56.2 17.5 11.5 14.8

88.8 Reaction between C2 H2 and C4 H3 to form A1− (4-45) 10.4 Reaction between C2 H2 and C4 H3 to form n-C6 H5 (4-43) 8.8

88.2 11.8

93.9 Association with C2 H2 to form C4 H+2 (3-39) 5.3 Association with C2 H2 to form C4 H+3 (3-38) 0.8 Others

55.1 43.6 1.3

74.1 Association with C2 H2 to form C6 H+4 (3-37) 25.5 Association with C2 H to form C6 H+2 (3-28) 0.4

99.6 0.4

98.2 Association with C2 H2 1.0 Association with C2 H2 0.8 Association with C2 H4 Association with C2 H3 Others

to form C8 H+2 to form C8 H+4 to form C8 H+4 to form C8 H+4

(3-44) (3-45) (3-69) (3-56)

85.3 Neutralization reactions with Cx H− y (3-80) 14.7

36.9 36.9 16.5 5.5 4.2 100

Association with C2 H2 to form C10 H+6 (3-42) Neutralization reactions with Cx H− y (3-80)

98.9 1.1

95.9 Association with C2 H2 to form C12 H+6 (3-43) 4.1 Neutralization reactions with Cx H− y (3-80)

98.9 1.1

100

100

Neutralization reactions with Cx H− y (3-80)

100

Neutralization reactions with Cm H+n (3-80) Reaction between C2 H− and C2 H2 (3-1)

69.6 30.4

98.5 Neutralization reactions with Cm H+n (3-80) 1.5 Reaction between C4 H− and C2 H2 (3-2)

62.4 37.6

100



C2 H Dissociative electron attachment to C2 H2 (2-11) C 4 H− Dissociative electron attachment to C4 H2 (2-25) Reaction between C2 H− and C2 H2 (3-1)

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Table 5. (Continued). Production process

%

Loss process

%



C6 H Dissociative electron attachment to C6 H∗2 (2-26) Dissociative electron attachment to C6 H2 (2-29) Reaction between C4 H− and C2 H2 (3-2) C 8 H− Dissociative electron attachment to C8 H∗2 (2-27) Dissociative electron attachment to C8 H2 (2-30) Reaction between C6 H− and C2 H2 (3-3) C10 H− Dissociative electron attachment to C10 H∗2 (2-28) Dissociative electron attachment to C10 H2 (2-31) Reaction between C8 H− and C2 H2 (3-3) C12 H− Reaction between C10 H− and C2 H2 (3-3)

87.3 Neutralization reactions with Cm H+n (3-80) 2.3 Reaction between C6 H− and C2 H2 (3-3) 10.4

58.9 41.1

19.4 Neutralization reactions with Cm H+n (3-80) 1.5 Reaction between C8 H− and C2 H2 (3-3) 79.1

43.1 56.9

14.3 Neutralization reactions with Cm H+n (3-80) 1.1 Reaction between C10 H− and C2 H2 (3-3) 84.6

44.3 55.7

100

Neutralization reactions with Cm H+n (3-80)

Finally, it is clear from table 5 that DEA reactions to C2n H2 and C2n H∗2 (n = 3–5) molecules play an important role in the generation of C2n H− (n = 3–5) anions. The relative contribution decreases for larger anions, because the densities of larger C2n H2 molecules become gradually smaller. The so-called Winchester mechanism [30], i.e. a consecutive chain of polymerization reactions between C2n H− and C2 H2 , which was assumed as the only anion growth path in our previous model [15, 16] appears to be still the most important production mechanism for the larger anions C2n H− (n = 4–6). Finally, neutralization of the C2n H− (n = 1–5) anions with various hydrocarbon cations, and the above mentioned polymeric reactions with C2 H2 are more or less equally important as the loss mechanism for C2n H− (n = 1–5) anions.

100

impact ionization as well as association reactions between small cations and C2 H2 play a key role in the formation of larger cations. Finally, DEA to C2n H2 (n = 2–5) and C2n H∗2 (n = 3–5) molecules appears to be an important production mechanism for the C2n H− anions, besides the polymerization reactions of smaller anions by insertion of C2 H2 .

Acknowledgments This project was financially supported by the Fund for Scientific Research (FWO) Flanders (Project G. 0068.07), the Interuniversity Attraction Poles Programme of the Belgian State (Belgian Science Policy; Project P6/42) and the CALCUA computing facilities of the University of Antwerp. The authors would like to thank T Martens for interesting discussions and suggestions about section 3.3 and Professor Ch Hollenstein for giving us the permission to use figure 6 in [13].

4. Conclusions A detailed chemical kinetics scheme has been developed to investigate the initial nanoparticle formation and growth mechanisms in acetylene dusty plasmas. A reasonable agreement is achieved by comparing our simulations with experimental measurements. Some new mechanisms are proposed for anion formation. The first mechanism is based on the generation of branched (and probably vibrationally excited) molecules C2n H∗2 (n = 3–5) and subsequent enhanced DEA to form C2n H− anions. The second mechanism involves the formation of the vinylidene anion (H2 CC− ) and the consecutive chain of polymerization reactions to form C2n H− 2 (n = 2–4) and C2n H− 4 (n = 4, 5). Excellent agreement between our calculated results and experimental measurement indicates that the C2n H2 molecules are an important precursor for fast formation of negative ions and play an important role in nanoparticle formation because the negative ions are very probably the precursors for dust particle formation, since they are confined in the plasma. The study of the relative contribution of the chemical reactions to the total production and destruction of the various species shows that electron impact dissociation as well as anion–cation neutralization are two predominant routes for C2n H generation. Electron

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