Langmuir 2007, 23, 2525-2530
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Monte Carlo Simulation of Equilibrium Reactions at Modified Vapor-Liquid Interfaces C. Heath Turner Department of Chemical and Biological Engineering, UniVersity of Alabama, Box 870203, Tuscaloosa, Alabama 35487-0203 ReceiVed October 10, 2006. In Final Form: December 4, 2006 The equilibrium conversion of a chemical reaction is known to be affected by its local environment. Various factors may alter reaction equilibria, including shifts in pressure or temperature, solvation, adsorption within porous materials, or the presence of an interface. Previously, reactive Monte Carlo simulations have been used to predict the equilibrium behavior of chemical reactions at vapor-liquid interfaces. Here, a route is tested for tuning the interfacial conversion of a Lennard-Jones dimerization reaction by adding surfactants to the vapor-liquid interface. Several temperatures are explored as well as several different surfactant models. Even with the addition of a small concentration of surfactants, the simulations predict significant shifts in the conversion at the interface. In general, the shifts in the conversion tend to be related to the values of the interfacial tension.
1. Introduction As the study of chemical processes, biological phenomena, and material science has reached the nanometer length scale, the physics of an interface has become an increasingly important topic. However, at this length scale, system characteristics are often somewhat unpredictable. For instance, it has previously been shown1 that interfacial systems provide unique environments that result in chemical behavior much different from traditional gas or liquid phase behavior. While the density variation along the interface follows a smooth, monotonic trend, the equilibrium conversion of a reaction may exhibit strong spikes or depletions within this region. The equilibrium shifts at the interface become more prominent at lower temperatures, and the directions of the shifts were shown to depend upon the specific parameters assigned to the intermolecular potential. Overall, these simulation results showed a potential relationship between the interfacial tension and the shifts in the conversion at the interface. In order to explore the relationship between interfacial tension and interfacial reaction conversion, additional simulations are reported here that involve the presence of surfactant species. If there is a strong correlation between interfacial tension and conversion, then the addition of carefully chosen surfactants should provide a means to tune the conversion at the interface. Accordingly, this logic has been tested for tuning the equilibrium conversion of a model Lennard-Jones (LJ) dimerization reaction, A + A T B, using the reactive Monte Carlo (RxMC) simulation method. Using this simulation approach, it is possible to maintain precise control over the system parameters, and the detailed molecular-level information concerning the equilibrium properties of the system can be obtained. This includes values such as the spatial and angular distribution of the surfactant molecules, the conversion along the interface, the total surface tension, and surface tension profiles. As described in the following sections, these simulations show that surfactants provide a viable route for tuning reaction conversion at interfaces. Furthermore, only small surfactant concentrations are necessary for inducing significant shifts in the conversion.
2. Previous Work The ability to predict the surface tensions of reacting fluids and to predict equilibrium conversion is valuable for understand(1) Turner, C. H. J. Phys. Chem. B 2005, 109, 23588-23595.
ing a broad range of important chemical and biological systems. For instance, consider the relevance to atmospheric chemistry. These environments are complicated by the presence of aerosol species, which are known to dramatically enhance certain chemical reactions and to facilitate cloud formation. In fact, a recent simulation study2 has shown that both hydrophilic and hydrophobic atmospheric gases adsorb significantly at aqueous air/water interfaces, which suggests that aqueous aerosols contribute significantly to atmospheric chemistry. While the presence and potential global impact of these species is generally accepted, there is still a dire need to include the interfacial chemistry into predictive atmospheric models, since the surface of these particles (rather than the interior) significantly influences atmospheric chemistry. Due to the fundamental importance of surface tension, a great deal of experimental3-8 and modeling9-24 work has been performed to measure this quantity for various systems. With regards to modeling work, several studies have been performed, ranging from simple Lennard-Jones fluids to complex multicomponent alkane mixtures. In addition to pure and mixed fluids, chemical reactions have also been modeled at interfaces, showing unique kinetic and transport behavior, depending on the particular (2) Vacha, R.; Slavicek, P.; et al. J. Phys. Chem. A 2004, 108, 11573-11579. (3) Birdi, K. S. Colloid. Polym. Sci. 1997, 375, 561-566. (4) Lin, H.; Duan, Y. Y. J. Chem. Eng. Data 2003, 48, 1360-1363. (5) Sun, Y. D.; Shekunov, B. Y. J. Supercrit. Fluids 2003, 27, 73-83. (6) Gasior, W.; Pstrus, J., et al. J. Phase Equilib. 2003, 24, 40-49. (7) Gorbachev, M. Y. Phys. Chem. Liq. 2001, 39, 315-325. (8) Law, G.; Watson, P. R. Langmuir 2001, 17, 6138-6141. (9) Gonza´lez-Melchor, M.; Bresme, F., et al. J. Chem. Phys. 2005, 122, 1047101-104710-8. (10) Potoff, J. J.; Panagiotopoulos, A. Z. J. Chem. Phys. 2000, 112, 64116415. (11) Warshavsky, V. B.; Zeng, X. C. J. Chem. Phys. 2002, 117, 3982-3991. (12) Suresh, S. J. and Naik, V. M. Langmuir 1996, 12, 6151-6163. (13) Errington, J. R. Phys. ReV. E. 2003, 67, 012102-1.-012102-4. (14) Tapia-Medina, C.; Orea, P.; et al. J. Chem. Phys. 2004, 120, 2337-2342. (15) Alejandre, J.; Tildesley, D. J.; et al. J. Chem. Phys. 1995, 102, 45744583. (16) Alejandre, J.; Duda, Y.; et al. J. Chem. Phys. 2003, 118, 329-336. (17) Orea, P.; Duda, Y.; et al. J. Chem. Phys. 2004, 120, 11754-11764. (18) Nijmeijer, M. J. P.; Bakker, A. F.; et al. J. Chem. Phys. 1988, 89, 37893792. (19) Paul, S.; Chandra, A. Chem. Phys. Lett. 2003, 373, 87-93. (20) Hill, A. W.; Benjamin, I. J. Phys. Chem. B 2004, 108, 15443-15445. (21) Bresme, F.; Quirke, N. J. Chem. Phys. 1999, 110, 3536-3547. (22) Kofke, D. A.; Singh, J. K. Abs. Pap. Am. Chem. Soc. 2003, 226, U401U401. (23) Singh, J. K.; Kofke, D. A. J. Chem. Phys. 2004, 121, 9574-9580. (24) Singh, J. K.; Kofke, D. A. Mol. Sim. 2004, 30, 343-351.
10.1021/la062979t CCC: $37.00 © 2007 American Chemical Society Published on Web 01/17/2007
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system. However, these previous studies have been mainly focused on kinetic analyses, with less attention paid to the equilibrium properties. Up to this point, investigations of reaction equilibria at interfaces have mainly been limited to studies of association behavior12,14,16,22-25 and isomerization reactions.26-29 For instance, in 1991 Benjamin reported27 molecular dynamics simulations of a simple LJ isomerization reaction occurring at a LJ vapor-liquid interface. Two LJ spheres interacting through a double-well potential were used to model the isomerization, and the simulations measured the potential of mean force on this isolated “reaction” at different locations in the fluid. The model was simple, but some interesting results were found: mainly, that the reaction rate at the interface can be a factor of 2 times larger than in the bulk fluid, and this phenomenon was thought to be primarily due to the density variation. This same type of reaction was studied again by Benjamin and Pohorille in 1993,28 except that the system was intended to simulate a real molecule (1,2-dichloroethane) at the vapor-liquid interface of water. Again the reaction was studied at infinite dilution and the equilibrium conversions at different locations across the interface were not reported. However, the most notable conclusion was that the reaction kinetics at the interface were found to be a combination of two opposing forces: (1) the equilibrium solvation effect increased the rate; while (2) the dynamic solvent effects (solventsolute coupling) decreased the reaction rate. Unfortunately, the study of Benjamin and Pohorille was limited to simple isomerization reactions at infinite dilution. Closely related to the work presented here, is a series of papers by Kofke and Singh,22-24 who simulate dimerizing and chainforming fluids at vapor-liquid interfaces. Dimerizing systems were modeled with one-site square-well fluids while chainforming systems were modeled with two-site square-well fluids. Both systems were studied as a function of temperature and association strength, and it was found that the surface tension goes through a maximum with respect to the association strength, when compared at a fixed reduced temperature (T/Tc). In addition, the authors found evidence of a slight increase in the dimer composition at the interface, depending on the system parameters, and this same behavior is also found in certain cases of the results presented here. While others have also experimentally measured the effects of interfaces on reaction behavior, the molecular details and the reasons for the behavior are challenging to obtain. This limits our true understanding of these systems and our ability to predict behavior for new systems. A recent review has been published29 discussing reaction dynamics at interfaces, covering both experimental and computational investigations, which highlights some of the fundamental aspects believed to affect interfacial reactions: variations in density, viscosity, polarity, and surface roughness. In the following sections the details of the current simulation approach are described, followed by a discussion of the equilibrium reaction behavior in these systems. The simulations demonstrate that surfactant species can significantly alter reactions at vapor-liquid interfaces, and there appears to be strong correlations between the interfacial reaction conversion and the interfacial tension. (25) Lu, J. F.; Fu, D.; et al. Fluid Phase Equilib. 2002, 194-197., 755-769. (26) Rose, D. A.; Benjamin, I. J. Chem. Phys. 1995, 102, 5292-5300. (27) Benjamin, I. J. Chem. Phys. 1991, 94, 662-669. (28) Benjamin, I.; Pohorille, A. J. Chem. Phys. 1993, 98, 236-242. (29) Benjamin, I. Prog. React. Kinet. Mech. 2002, 27, 87-126.
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3. Simulation Details 3.1. Methods. The equilibrium conversion of a given reaction can be calculated in virtually any environment using the RxMC simulation method,30,31 given appropriate intermolecular interaction parameters. The implementation is fairly simple. In addition to the traditional moves in a Monte Carlo simulation (particle displacement and reorientation), forward and reverse reaction steps are attempted (with an equal probability), with the acceptance probability (Pacc) given as C
N i!
C
∏q ∏(N + ν )!
Pacc ) exp(-βδUrxn)
νi
i
i)1
i)1
i
(1)
i
In eq 1, β is the reciprocal of the Boltzmann constant times the absolute temperature, (kbT)-1, C represents the number of species in the reaction, Ni is the number of molecules of species i currently in the system, qi is the partition function for species i, and δUrxn is the change in configurational energy upon reaction. The symbol νi represents the stoichiometry of each species, which is defined to be negative for reactants and positive for products. Since complete forward and reverse reaction steps are performed in the simulations, the high-energy (rare event) processes of bond breaking and bond formation are avoided, leading to rapid equilibration. Furthermore, a reactive potential is unnecessary if this simulation route is followed. The simulations can be performed at high densities, since during the reaction steps, the newly created molecules are chosen to be inserted into the cavities left by the reactants. If necessary, cavity bias techniques32 can also be incorporated in order to accelerate the simulations. This simulation approach has been shown to accurately model reaction equilibria within a supercritical CO2 solvent,33 at high pressures,34 in chemically reacting plasmas (involving multiple simultaneous reactions),35 adsorbed within nanoporous materials,36-40 and in shock waves.41,42 In the current study, once the free energy (or partition functions) of an isolated reaction is known, the conversion can be predicted at any point along the liquid-vapor interface. The interactions due to neighboring molecules are inherently accounted for, and the resulting shifts in chemical equilibrium can be directly calculated. 3.2. Models. The molecular models for the reactants and products consist of two parts. First, intermolecular potentials are needed to describe the interaction between all species in a given simulation. These interactions are described with the Lennard-Jones potential, as shown in eq 2, and the parameters assigned to each molecule in this study are listed in Tables 1 and 2. uij )
∑∑ iR jβ
[( ) ( ) ] σiR,jβ
4iR,jβ
riR,jβ
12
-
σiR,jβ riR,jβ
6
(2)
In these systems, the A species are the reactants, the B species are the products, and a diatomic C species is used to model the surfactants. The A and B species are both single-site LJ monomers, with a value of σB equal to 1.25σA. The various surfactant models (type C) are (30) Johnson, J. K.; Panagiotopoulos, A. Z.; et al. Mol. Phys. 1994, 81, 717733. (31) Smith, W. R.; Triska, B. J. Chem. Phys. 1994, 100, 3019-3027. (32) Brennan, J. K. Mol. Phys. 2005, 103, 2647-2654. (33) Turner, C. H.; Gubbins, K. E. J. Chem. Phys. 2003, 119, 6057-6067. (34) Carrero-Mantilla, J.; Llano-Restrepo, M. Fluid Phase Equilib. 2006, 242, 189-203. (35) Lisal, M.; Smith, W. R.; et al. J. Chem. Phys. 2000, 113, 4885-4895. (36) Turner, C. H.; Johnson, J. K.; et al. J. Chem. Phys. 2001, 114, 18511859. (37) Turner, C. H.; Pikunic, J.; et al. Mol. Phys. 2001, 99, 1991-2001. (38) Lisal, M.; Brennan, J. K.; et al. J. Chem. Phys. 2004, 121, 4901-4912. (39) Hansen, N.; Jakobtorweihen, S.; et al. J. Chem. Phys. 2005, 122, 1647051.-164705-11. (40) Santiso, E.; George, A. M.; et al. Appl. Surf. Sci. 2005, 252, 766-777. (41) Brennan, J. K.; Rice, B. M. Phys. ReV. E. 2002, 66, 021105-1.-02110511. (42) Brennan, J. K.; Rice, B. M. Mol. Phys. 2003, 101, 3309-3322.
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Table 1. Interaction Parameters for the LJ Modelsa molecule
bl/σA
/A
σ/σA
A B CXY(1) CXY(2)
X.Y
1.00 1.00 1.00 1.50
1.00 1.25 0.80 1.60
a The bond length (bl) corresponds to the intramolecular site-site separation distance between CXY(1) and CXY(2), and this value is used to identify the various surfactant models (C06-C16).
Figure 1. Snapshot from the RxMC simulation corresponding to a reduced temperature of 0.92: red ) A-type molecules and blue ) B-type molecules.
Table 2. Unlike Interaction Parameters for the LJ Models molecule
A,i
B,i
CXY(1),i
CXY(2),i
A,i B,i CXY(1),i CXY(2),i
1.00 1.00 1.70 0.40
1.00 1.00 1.70 0.40
1.70 1.70 1.00 1.00
0.40 0.40 1.00 1.50
all identical, with the exception of the bond lengths. The bond lengths of C were chosen to vary between 0.60σA and 1.60σA and these models are designated as C06 through C16, as shown in Table 1. The LJ parameters of the surfactant species were formulated somewhat arbitrarily. Different parameter sets were sampled until a strong segregation of the C molecules to the interface was observed. In order to limit the scope and maintain the focus of this investigation, the LJ parameters were not modified any further beyond this point (with the exception of the bond lengths). The LJ parameters for the like-like interactions are show in Table 1, the ij parameters for the unlike interactions are shown in Table 2, and the σij parameters for the unlike interactions are given by σij ) 0.50(σi + σj). The second part of the molecular models consists of the intramolecular partition functions (needed in eq 1), which can be calculated quantum mechanically or taken from experimental data. Equivalently, these partition functions can be grouped and related to an overall free energy for the reaction, as is done here. In order to maintain consistency with the previous study,1 a constant ∆G° value of 2.0A was assigned to this reaction (A + A T B) and any thermal variations in this quantity were neglected. This value would need to be accurately obtained if a realistic reaction is to be modeled. 3.3. Procedure. The simulations were initiated using the standard approach: (a) equilibrate the system at a fixed temperature, volume, and total number of particles, where the temperature is below the critical temperature and the density corresponds to the approximate liquid-phase density; (b) expand the box in one of the coordinate dimensions to allow the vapor phase to form; (c) equilibrate the system until the vapor-liquid interface is well-developed; (d) continue equilibrating the system with RxMC simulation moves now enabled; (e) collect equilibrium system averages and measure system properties as the simulation continues. As an example, a simulation snapshot from the A + A T B system during stage e is shown in Figure 1. In these simulations, the systems initially contained approximately 2000 reactant molecules, equilibration lasted ∼50 × 106 MC steps, and averages were taken for ∼500 × 106 MC steps. In all cases, four or more simulation runs were performed at identical conditions in order to estimate standard deviations of the average values obtained. The potential energy cutoff distance (rc) was set at a value of 5.0σA. Beyond this distance, the interactions may still be important, but due to the variations in the density, standard long-range corrections are not applicable. However, the density profile can be used to approximate the long-range corrections in these systems, as shown in eq 3.43 8π ulrc(zk) ) F(zk)2Vs 3
∑∑
[ ( ) ( )]
ab
a
b
12 1 σab
3 r9 c
-
σab6 rc3
(3)
The method was implemented by dividing the system into nz bins (43) Guo, M.; Lu, B. C. Y. J. Chem. Phys. 1997, 106, 3688-3695.
Figure 2. Interfacial tension of a pure LJ fluid at a reduced temperature of 0.92, as a function of the potential cutoff distance. along the length of the simulation cell, corresponding to distances of zk, and evaluating eq 3 for each individual bin at a corresponding density of F(zk). The total long-range correction was then calculated by summing over each bin along the z-direction. The variable Vs represents the volume of each bin, while the a and b indices in the double summation correspond to the LJ sites on each molecule. This long-range contribution was calculated during each Monte Carlo step and was incorporated into the acceptance criteria for the various moves. Following the equilibration period, the surface tensions (γ) of these interfaces were calculated using the mechanical virial expression, as shown in eq 4, and the reduced surface tensions (γ*) reported here are defined as γ* ) γσA2/A. γ)
{
LZ 1 〈PZZ〉 - [〈PXX〉 + 〈PYY〉] 2 2
}
(4)
In the above equation, LZ is the length of the simulation cell normal to the interface, PXX, PYY, and PZZ represent the tangential and normal components of the pressure tensor, respectively (a factor of 1/2 corrects for the periodic boundary conditions applied to the simulation cell, which creates two identical interfaces). This approach to calculating the surface tension of planar interfaces has been applied to both classical and quantum fluids,44 as well as to both single-site14,18 and molecular fluids,15,19,45-48 including both gas-liquid14,15,18,19,45-49 and liquid-liquid interfaces.50-53 It is important to recognize the long-range nature of the surface tension calculations, as well. For example, Figure 2 shows a calculation of the surface tension in reduced units54 of a LJ fluid as a function of the cutoff distance. There is a significant change in the surface tension value as a function of the cutoff, and this trend continues up to about 6.5σA. The reacting systems studied here also exhibit this long-range sensitivity, which suggests the need for a long cutoff distance. Accordingly, the calculation of the virial was truncated at a distance of 5.0σA in all simulations, and beyond this (44) Zhao, X.; Johnson, J. K. J. Chem. Phys. 2004, 120, 8707-8715. (45) da Rocha, S. R. P.; Johnston, K. P.; et al. J. Phys. Chem. B 2002, 106, 13250-13261. (46) Chen, B.; Siepmann, J. I.; et al. J. Am. Chem. Soc. 2002, 124, 1223212237. (47) Rivera, J. L.; Predota, M.; et al. Chem. Phys. Lett. 2002, 357, 189-194. (48) Rivera, J. L.; Alejandre, J. Colloids Surf. A 2002, 207, 223-228. (49) Orea, P.; Duda, Y.; et al. J. Chem. Phys. 2003, 118, 5635-5639. (50) Simmons, V.; Hubbard, J. B. J. Chem. Phys. 2004, 120, 2893-2900. (51) Faraudo, J.; Bresme, F. J. Chem. Phys. 2003, 118, 6518-6528. (52) Bresme, F.; Quirke, N. Phys. Chem. Chem. Phys. 1999, 1, 2149-2155. (53) Bresme, F.; Quirke, N. J. Chem. Phys. 2000, 112, 5985-5990. (54) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford: New York, 1987.
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Figure 3. Mole fraction of component B along the liquid-vapor interface (no surfactants). distance, the following estimate was used to correct for the truncated interactions: Vs π γlrc(zk) ) F(zk) 2 A
∫ dr ∫ ∞
rc
r
-r
Figure 4. Mole fraction profile across the vapor-liquid interface with the addition of various surfactant models (C06-C16) at a reduced temperature of 0.80. For clarity, the curve corresponding to C14 has been removed.
d∆z ×
nz
∑[F(z ) - F(z i
i)1
i-1)]
dU dr
[r2 - 3(∆z)2] (5)
Equation 5 was implemented by dividing the system into bins (nz ) number of bins between z and zk, ∆z is the difference between z and zk), and numerically evaluating the double integral over the local density in each bin, F(zi). The total contribution of the longrange component of the interfacial tension was then calculated by summing over the local contributions from each bin, γlrc(zk). Using this approach, the long-range corrections to the interfacial tension was estimated by numerically integrating the local interfacial tension along the density profile obtained in the simulations.43,55 A few different implementations of these long-range corrections have been developed, and they have all shown to give consistent results.55 With respect to the current simulations, this tail correction adds an additional 15-20% to the total value of the interfacial tension.
4. Results work,1
In order to maintain consistency with the previous the current simulations of the LJ dimerization reaction began at a reduced temperature of T* ) 0.92. However, the current system is slightly different from the previous system with respect to the long-range interactions. Here, a cutoff distance of 5σA is used (versus 6.5σA used previously) and the long-range corrections described in the previous section are now used, in order to compensate for the shorter cutoff distance. In addition, the direction normal to the interface (the z-dimension) has been extended from a length of 46σA to 100σA, and the x and y box lengths are each 10σA. Simulations were performed at a total of three reduced temperatures: T* ) 0.80, 0.92, and 1.00. The concentration profiles of the B molecules were calculated during the RxMC simulations along the z-dimension, and these profiles are illustrated in Figure 3 for the pure LJ dimerization (without surfactants). At all three temperatures there is a significant spike in the mole fraction of B at the vapor-liquid interface, and this effect becomes more pronounced at the lower temperatures. Compared to the liquid phase, the mole fraction of B at the interface is increased by 58% at T* ) 0.80, 26% at T* ) 0.92, and 11% at T* ) 1.00. The temperature also has a noticeable influence on the width of the liquid phase, and this is the result of several competing factors. At higher temperatures, the molecules have a tendency to evaporate, reducing the size of the liquid phase. Also, at higher temperatures, the density of the (55) Goujon, F.; Malfreyt, P. J. Chem. Phys. 2002, 116, 8106-8117.
Figure 5. Mole fraction profile across the vapor-liquid interface with the addition of various surfactant models (C06-C16) at a reduced temperature of 0.92. For clarity, the curve corresponding to C14 has been removed.
liquid phase decreases, leading to a moderate expansion. Finally, due to the presence of a reaction, the conversion in the liquid phase changes with respect to changes in the temperature. In this particular case, higher temperatures lead to higher conversion, resulting in a higher fraction of B molecules. The B molecules are physically smaller than the two A molecules from which they are formed, so the higher conversions tend to reduce the width of the liquid phase. Due to these competing factors, a monotonic change in the volume of the liquid phase with respect to the temperature is not observed. After characterizing the pure fluids, various surfactant molecules (C06-C16) were added to these systems in an attempt to control the interfacial conversion. The surfactant species were added during the equilibration periods by performing random particle insertions into the simulation cell. Once a predetermined number of surfactants had been inserted into the system, these moves were no longer attempted. In all cases shown, a total of 50 surfactant molecules were added. For comparison, simulations were also performed with only 25 surfactant molecules added, but the effects from such a very low concentration were difficult to distinguish. Also, it should be noted that the mole fraction of B was calculated without counting the surfactant species. Therefore, the deviations that are observed at the interface are not simply dilution effects due to the selective adsorption of the surfactants to the interface. These are true shifts in the conversion. While the LJ parameters were held fixed, several values of the surfactant bond length were investigated, ranging from 0.60σA (model C06) up to 1.60σA (model C16). Figures 4, 5, and 6 illustrate the mole fraction profiles of B at reduced temperatures
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Figure 8. Distribution of surfactant orientations with respect to the xy-plane at a reduced temperature of 0.80. Figure 6. Mole fraction profile across the vapor-liquid interface with the addition of various surfactant models (C06-C16) at a reduced temperature of 1.00. For clarity, the curve corresponding to C12 has been removed.
Figure 9. Distribution of surfactant orientations with respect to the xy-plane at a reduced temperature of 0.92.
Figure 7. Surfactant density across the vapor-liquid interface. The profiles for C06-C14 (red and black) are very similar at a given temperature, whereas the profiles for C16 (green and blue) are somewhat different.
of 0.80, 0.92, and 1.00, respectively. Several observations can be made from these three figures. First, there is definite asymmetry between the two mole fraction peaks at each temperature. This is a direct result of the location of the surfactant molecules. Although the surfactants are inserted randomly in the simulation cell, the surfactants have a tendency to completely populate one of the interfaces before any significant accumulation occurs at the other interface. This is shown in Figure 7, where the average concentration of the surfactants is plotted along the length of the simulation cell at the three different temperatures. Since only the left interface has a significant concentration of surfactant molecules, only the conversion at the left interface is affected. Exceptions are observed at T* ) 0.80 with C16 and at T* ) 1.00 with C16. In these two cases, the concentration of the surfactants at the other interface is no longer negligible, and the reaction at the right interface is affected. The reason for this behavior may be attributed to the steric interactions among the surfactants. As the bond lengths grow, the surfactants become bulkier, crowding the interface, initiating the migration of some of the C16 surfactants to the opposite interface. At all three temperatures, shifts are observed in the mole fraction of B only in the regions where there is a measurable accumulation of surfactant molecules. As expected, there are no other detectible changes in the other regions (bulk gas or bulk liquid) of the simulation cell. Furthermore, the behavior at the modified interfaces is a strong function of the particular surfactant model used, especially at the lowest temperature. The conversion of the reaction is increasingly suppressed at the interface as the bond length of the surfactants grows. This may be due, in part, to a
Figure 10. Distribution of surfactant orientations with respect to the xy-plane at a reduced temperature of 1.00.
variety of steric effects, as the overall size of the surfactants grows and as the orientations of the surfactants change. The orientation of the surfactants at the interface depends upon the bond length, and these distributions become more pronounced at lower temperatures. The orientation profiles are shown in Figures 8, 9, and 10, corresponding to reduced temperatures of 0.80, 0.92, and 1.00, respectively. The orientation is defined as the angle between the surfactant bond and the xy-plane of the simulation cell. The trend in the mole fraction of B may be rationalized, as was done previously,1 by comparing the interfacial tension at each condition. As the surfactant bond length varies, there is an induced shift in the interfacial tension, as shown in Figure 11. This shift tends to scale linearly with respect to the bond length of the surfactant model, with the longer bond lengths resulting in lower interfacial tension values. It is important to recognize that the interfacial tension is changing only at the interface populated by the surfactants. This is illustrated in Figure 12, where the total interfacial tension contribution has been divided along the length of the simulation cell. From this analysis, it is apparent that only the tension at the left interface is changing, as expected. Figure 12 shows only the results from a reduced
2530 Langmuir, Vol. 23, No. 5, 2007
Figure 11. Reduced surface tension corresponding to different surfactant bond lengths (C06-C16) at three different temperatures. Error bars correspond to one standard deviation from the mean.
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C06-C10 models (left) is higher than the unmodified interface (right), but the mole fraction of B is actually higher at the unmodified interface. Obviously, the interfacial tension is not the sole driving force for this reaction. This is a three-component system, and the interactions can be rather complex. There are additional interactions that can affect the conversion, such as the attraction between the reacting molecules and the surfactants. If there is a favorable interaction between the surfactants and a particular reacting species, then the reaction would likely shift in the direction to increase the favorable interactions (or in order to avoid unfavorable interactions). In the current simulation study, the interaction between the A-type species and the surfactant is slightly stronger, and this tends to mitigate the effect of the increased interfacial tension at the left interface. However, at a fixed surfactant concentration, the conversion tends to correlate strongly with the variation of the interfacial tension.
5. Conclusions
Figure 12. Local surface tension profile corresponding to a reduced temperature of 0.80.
temperature of 0.80, since the behavior at the higher temperature is qualitatively similar. In all cases, these simulations predict that the surfactant-induced lowering of the interfacial tension reduces the driving force for the creation of the B molecules. Higher concentrations of B (versus A) molecules results in lower interfacial tension values.1 Therefore, in order to minimize the overall free energy of the system, the conversion at the interface shifts, in order to increase the concentration of B. As more effective surfactant molecules are used, the interfacial tension lowers, resulting in less of a thermodynamic driving force for the reaction at the interface. Likewise, if comparisons are made among the three temperatures, this trend is still followed. As the temperature decreases, the interfacial tension rises, creating a larger driving force for the reaction, and thus, larger spikes in the conversion are observed. An exception to this general explanation can be quickly seen by examining the local interfacial tension at the left and right interfaces in Figure 12. The interfacial tension resulting from the
In contrast to the challenges faced by experimental investigations, simulations provide a complimentary paradigm to study interfacial systems. The relevant dimensions are on the order of nanometers, which is a length scale suitable for current computational analyses. Here, RxMC simulations have been used to predict the behavior of equilibrium-limited chemical reactions at modified vapor-liquid interfaces. It is generally accepted that reaction rates can be significantly altered at vapor-liquid interfaces, and it has previously been shown that this applies to reaction conversions, as well. In this work, it is shown that it is also possible to tune the conversion of a reaction at an interface by the addition of surfactant species. The effects of the surfactants are most prominent at the lower temperatures, and this appears to be due, in part, to the larger values of the interfacial tension. In cases where the vaporliquid interface makes a significant contribution, such as in atmospheric chemistry, the interface (and any modifications to the interface) may cause large deviations from bulk-phase equilibrium behavior. The overall composition shift of the aerosol particles may be negligible, since typically particle diameters fall in the micrometer size range, but the local conversion shift in the “chemically active” region of these aerosol particles may be substantial. In future work, interfacial systems with significant electrostatic interactions will be modeled, such as weak acid dissociation at aqueous interfaces. Acknowledgment. Support for this work was provided by a Research Advisory Committee grant from the University of Alabama and the Simmons Endowed Excellence Fund. Computing resources were provided by a DAC TeraGrid allocation from NCSA (TG-CTS060011T). LA062979T
