Monte Carlo Simulation of Equilibrium Reactions at VaporâLiquid

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J. Phys. Chem. B 2005, 109, 23588-23595

Monte Carlo Simulation of Equilibrium Reactions at Vapor-Liquid Interfaces C. Heath Turner UniVersity of Alabama, Box 870203, Tuscaloosa, Alabama 35487-0203 ReceiVed: May 27, 2005; In Final Form: September 26, 2005

Chemical reactions are known to behave differently, depending upon their local environment. While the interactions with neighboring molecules may alter both the kinetics of chemical reactions and the overall equilibrium conversion, we have performed simulations of the latter. The particular environment that we address is the vapor-liquid interface, since only a few, limited studies have explored the influence of an interface on equilibrium reaction behavior. Simple dimerization reactions are modeled, as well as more complex multicomponent reactions, using the reactive Monte Carlo (RxMC) simulation technique. We find that the conversion of a reaction can be markedly different at an interface as compared to the bulk vapor and liquid phases, and these trends are analyzed with respect to specific intermolecular interactions. In conjunction, we calculate the surface tension of the reacting fluids at the interface, which is found to have unusual scaling behavior, with respect to the system temperature.

1. Introduction The ability to predict the surface tensions of fluids is valuable for understanding a broad range of important chemical and biological systems. Due to the fundamental importance of surface tension, a great deal of experimental1-6 and modeling7-22 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 been modeled at interfaces, showing unique kinetic and transport behavior, depending on the particular system. However, these previous studies have been mainly focused on kinetic analyses, with little attention paid to the equilibrium properties. Previous investigations of reaction equilibrium at interfaces have been limited to simple association behavior10,12,14,20-23 and isomerization reactions.24-27 In this work, we extend the characterization and understanding of reaction behavior at interfaces by modeling three equilibriumlimited chemical reactions at their vapor-liquid (V-L) interfaces. The first reaction is a simple Lennard-Jones model which is used to understand the effect of the interaction parameters on both conversion and surface tension across the vapor-liquid interface. The second reaction is modeled as a more realistic dimerization, with parameters chosen to accurately model nitric oxide dimerization: NO + NO T (NO)2. Finally, we incorporate more complex reaction behavior by modeling the equilibrium of Br2 + Cl2 T 2BrCl. In all three studies, the interfacial composition profiles, surface tension measurements, and reaction conversions are analyzed. To the best of our knowledge, this work represents the first application of the reactive Monte Carlo method to study fluid interfaces as well as the first simulation study to predict the vapor-liquid surface tension of reacting fluids. 2. Previous Work Several experimental studies have been performed to study reactions occurring at interfaces. However, experimental inves-

tigations provide only limited insight into the true molecular nature of these systems and the complex interactions contributing to the reactivity. In retrospect, these experimental studies have helped develop a better understanding of some of the general characteristics of interfacial reactions. For instance, Sitzmann and Eisenthal have shown28 that the isomerization dynamics of 3,3′-diethyloxadicarbocyanine iodine (DODCI) are approximately twice as fast at an air-water interface as compared to the same process in bulk water. However, a definitive reason for this rate acceleration could not be deduced. More recently, Nathanson29 used molecular beam studies to examine interfacial reactions. This work attempted to disentangle the elementary steps involved in chemical reactions at interfaces in order to gain a clearer picture of interfacial phenomena. While some others have also experimentally measured the effects of interfaces on reaction behavior, the molecular details and the reasons for the behavior are typically nonexistent. This limits our true understanding of these systems and our ability to predict behavior for new systems. There are a few examples in the literature of molecular modeling applied to chemical reactions at interfaces.24-26,30,31 Unfortunately, these studies are rather sparse, and the reactions that are modeled are typically limited to isomerizations. This is due to the inherent limitations of the simulation methods previously used. For instance, in 1991, Benjamin reported25 molecular dynamics simulations of a simple Lennard-Jones (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 that 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 1993,26 except that the system was made more interesting by simulating a real molecule (1,2-dichloroethane) at the vapor-liquid interface of water. Again, the reaction was studied at infinite dilution and

10.1021/jp0528156 CCC: $30.25 © 2005 American Chemical Society Published on Web 11/15/2005

Equilibrium Reactions at Vapor-Liquid Interfaces

J. Phys. Chem. B, Vol. 109, No. 49, 2005 23589

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 (solvent-solute coupling) decreased the reaction rate. Unfortunately, the study of Benjamin and Pohorille was limited to simple isomerization reactions at infinite dilution. Closely related to our work presented here is a series of papers by Kofke and Singh,20-22 who simulate dimerizing and chainforming fluids at vapor-liquid interfaces. Dimerizing systems are modeled with one-site square-well fluids, while chainforming systems are modeled with two-site square-well fluids. Both systems are studied as a function of temperature and association strength, and it is 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 find 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 our results presented here. In somewhat different calculations, Kuo and Mundy have used ab initio molecular dynamics to study the aqueous liquidvapor interface.32 Although a reaction was not studied, the authors found that the interfacial water molecules tended to be more reactive than bulk water, due to shifts in the electronic structure of the molecules. While these results are notable, we intend to focus on the molecular nature of interfacial reactions, instead of the effects arising from changes in the electronic structure of the individual molecules. In a study related to interfacial reactions, Hayoun and co-workers33 have used molecular dynamics simulations to study solute transfer through a liquid-liquid interface. They found that this is an activated process, similar to a chemical reaction, and that there is a significant activation barrier associated with this process, which has direct implications for interfacial reactions. A recent review has been published27 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. While the work that was reviewed has revealed important information about interfacial reactions, our proposed simulations will explore an area that no one has previously attempted to study. Most notably, we will handle reactions that change in mole number upon reaction, and we will use simulation methods that are capable of dealing with simultaneous reaction and phase behavior. Here, we apply reactive Monte Carlo simulations, to understand and predict simultaneous reaction equilibria and surface tensions at fluid interfaces. 3. Simulations Details 3.1. Methods. The equilibrium conversion of a given reaction can be calculated in virtually any environment using the reactive Monte Carlo (RxMC) simulation method,34,35 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

Pacc ) exp(-βδUrxn)

Ni!

C

∏ ∏ i)1 i)1 (N + ν )! qiνi

i

i

(1)

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. This simulation approach has been shown to accurately model reaction equilibrium within a supercritical CO2 solvent,36 in chemically reacting plasmas (involving multiple simultaneous reactions),37 adsorbed within nanoporous materials,38-42 and in shock waves.43,44 In our application, 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, accurate intermolecular potentials are needed to describe the interaction between all species in a given simulation. These interactions are described with the Lennard-Jones (LJ) potential, with the BrCl model supplemented by electrostatic charges (q), as shown in eq 2:

uij )

[( ) ( ) ]

∑ ∑4iR,jβ iR jβ

σiR,jβ riR,jβ

12

-

σiR,jβ

6

+

riR,jβ

qiRqjβ riR,jβ

(2)

The LJ parameters ( and σ) for the species participating in the reactions are shown in Table 1. The second part of the molecular models consists of the intramolecular partition functions, 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. The free energy data (or partition functions) for both of the realistic reactions have been compiled previously,34,45 and we have used these values in the current models without modification. 3.3. Simulation Analysis. The surface tensions (γ) of these interfaces were calculated using the mechanical virial expression:

γ)

{

LZ 1 〈PZZ〉 - [〈PXX〉 + 〈PYY〉] 2 2

}

(3)

In the above equation, LZ is the length of the simulation cell normal to the interface and 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,46 as well as to both single-site12,16 and molecular fluids,13,17,47-50 including both gas-liquid12,13,16,17,47-51 and liquid-liquid interfaces.52-55 Other more elaborate MC approaches have also been developed, such as the ghost interface method, transition matrix MC, and density of states methods.11,56,57

23590 J. Phys. Chem. B, Vol. 109, No. 49, 2005

Turner

TABLE 1: LJ Interaction Parameters for the Molecular Modelsa molecule

site, bl (nm)

/kB (K)

σ (nm)

charge (e)

A-type B-type NO (NO)2

1 1 1 1, 0.111 85 2, 0.111 85 1, 0.111 45 2, 0.111 45 1, 0.104 95 2, 0.104 95 1, 0.109 28 2, 0.109 28

(1.0) × LJ,A (1.0, 1.25, 1.50, 1.75) × LJ,A 125.0 125.0 125.0 257.2 257.2 178.3 178.3 257.2 178.3

(1.0) × σLJ,A (1.0, 1.25, 1.50, 1.75) × σLJ,A 0.3172 0.3172 0.3172 0.3538 0.3538 0.3332 0.3332 0.3538 0.3332

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0275 0.0275

Br2 Cl2 BrCl a

ref this work this work 34 34 45 45 45, this work

The bond lengths (bl’s) correspond to the site distance from the geometric molecular centers.

TABLE 2: Simulated Surface Tension and Coexistence Densities of a Pure LJ System temp liquid density (T*) (Fl*)

Figure 1. Simulation snapshot during the A/B dimerization reaction at its vapor-liquid interface, corresponding to σB/σA ) 1.50 and B/A ) 1.00. A-type molecules are shown in red, and B-type molecules are shown in blue.

Figure 2. Surface tension of a pure LJ fluid at a reduced temperature of 0.92 as a function of the potential cutoff distance.

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; and (e) collect equilibrium system averages and measure system properties as the simulation continues. As an illustrative example, a simulation snapshot from the A + A T B system during stage e is shown in Figure 1. In our simulations, the systems initially contained approximately 2000 reactant molecules, equilibration lasted ∼50 × 106 MC steps, and averages were taken for ∼200 × 106 MC steps. It is important to recognize the long-range nature of the surface tension calculations. For example, Figure 2 shows our calculation of the surface tension in reduced units58 of a LJ fluid as a function of the potential energy 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 6.5σ. Our

0.92 0.92 0.70 0.70

0.7065 0.707 ( 0.01 0.812 0.815 ( 0.01

vapor density (Fv*)

surface tension (γ*)

ref

0.0294 0.0295 ( 0.002 0.0029 0.0036 ( 0.0004

0.310 0.300 ( 0.019 0.640 0.684 ( 0.020

59 this work 59 this work

reacting systems also exhibit this long-range sensitivity, which suggests the need for a long cutoff distance. Accordingly, our potential cutoff distance for both LJ and electrostatic interactions was set to a fixed value of 2.16 nm in all simulations (without long-range corrections), and our interfacial area was set to 18.66 nm2 in order to accommodate this cutoff. In addition, our surface tension calculations were checked for consistency by comparing them with previous simulations (performed at a cutoff of 2.5σ).59 The results of these simulations are summarized in Table 2 (in reduced units),58 along with standard deviations that were approximated by performing four separate simulation runs. Our simulation results agree closely with the literature data. Slight differences at a temperature of 0.70 could be attributed to standard deviation in the reference values, which was not reported. Our simulations were run for a total of 3 × 108 MC steps, whereas the reference simulations were run for approximately 2 × 108 MC steps. Although not investigated here, one may potentially extrapolate the infinite size surface tension of these systems by following the formalism suggested by Binder.60 This approach has been applied previously to extrapolate the surface tension of associating fluids20-22 and is therefore expected to be reliable when applied to reacting fluids, such as the systems modeled here. 3.4. Reaction Systems. The first systems studied involved model dimerization reactions of the form A + A T B. These reaction systems were intended to explore the effects of simple reactions on interfacial characteristics. The goal was to identify the relationships between surface tension, conversions in the interfacial region, the size and interaction strength of the reactants and products, and the molecular distributions across the interface. The model dimerization reactions were described by singlesite LJ interactions. For example, two A-type molecules react to form a larger B-type molecule, with an arbitrary free energy of reaction assigned to this equilibrium.

The value of ∆G° was assigned a constant value of 2.0 × (A/kB) at the simulated temperature so that the simulated


Figure 3. Density profile of the A/B dimerization reaction across the vapor-liquid interface at a reduced temperature of 0.92 as a function of σB/σA. Key: σB ) 1.25 (solid line), σB ) 1.50 (dashed line), σB ) 1.75 (dotted line).

conversion falls within a moderate range under the conditions modeled. With this simple model reaction, the system parameters listed in Table 1 were explored at a fixed reduced temperature (T*) of 0.92 (with respect to the A-type species). The second reaction studied is the dimerization of nitric oxide, NO + NO T (NO)2, which is a species of particular interest to both pollution abatement and metabolic regulation.61 Although this system has been modeled previously in its pure liquid and vapor phases,34 the interfacial behavior of this reaction remains unknown. Since the previously developed models of NO and (NO)2 are able to accurately predict the conversion in the liquid and vapor phases, we expect the conversion at the interface to be reliably predicted as well. Also, an important equilibrium reaction62-65 involved in environmental processes, Br2 + Cl2 T 2BrCl, is studied at its vapor-liquid interface. BrCl has been linked to polar ozone depletion,62,66,67 and thus, a clearer picture of this reaction would be beneficial for various environmental analyses. Unfortunately, experimentally measuring the vapor-liquid equilibrium properties of this system is a difficult task, since the phase behavior and the reaction behavior cannot be isolated. RxMC simulations have been used previously to predict the simultaneous reaction and phase behavior of this system. In the previous work,45 the authors approximated the BrCl bond length as an arithmetic combination of the bond distances of Br2 and Cl2. However, we have refined the BrCl geometry with ab initio calculations performed at the MP4/6-311G++(d) level of theory using Gaussian 03.68 In addition, partial charges (instead of an ideal dipole) have been added to the Br and Cl sites in order to account for the molecular dipole using a Mulliken population analysis.69-76 After these small changes were incorporated, we checked for any deviations from the previous simulation results,45 but all previously reported results were accurately reproduced. In contrast to the dimerization reactions that were simulated, the unique nature of the BrCl system arises from the fact that two dissimilar reactants participate in the reaction. Thus, there is the likelihood that each of the two reactants will favor a particular phase (liquid, vapor, or interface), impacting the equilibrium distributions within each of the separate phases, which ultimately controls the overall conversion of the reaction. 4. Results 4.1. A/B Reaction. The RxMC simulation results for the model dimerization reactions are shown in Figures 3-9. In


Figure 4. Density profile of the A/B dimerization reaction across the vapor-liquid interface at a reduced temperature of 0.92 as a function of B/A. Key: B ) 1.25 (solid line), B ) 1.50 (dashed line), B ) 1.75 (dotted line).

Figure 5. Surface tension of the A/B dimerization reaction at a reduced temperature of 0.92 as a function of the LJ parameter ratios σB/σA and B/A. The lines are drawn as a guide to the eye, and the error bars are generated from four separate simulations.

Figure 6. Comparison of the surface tension of the A/B reaction at a reduced temperature of 0.92 with pure A-type and pure B-type fluids as a function of σB/σA.

Figures 3 and 4, we plot the local density of the system (in reduced units, with respect to species A), as a function of the distance along the z-coordinate, which is defined to be perpendicular to the vapor-liquid interface. Figure 3 shows these density profiles as a function of the ratio σB/σA, while Figure 4 shows the density profiles as a function of B/A. With all simulation sets, it can be concluded that the vapor and liquid


Figure 7. Comparison of the surface tension of the A/B reaction at a reduced temperature of 0.92 with pure A-type and pure B-type fluids as a function of B/A.

Figure 8. Mole fraction of B (xB) across the vapor-liquid interface of the A/B dimerization reaction at a reduced temperature of 0.92. Key: σB ) 1.25 (solid line), σB ) 1.50 (dashed line), σB ) 1.75 (dotted line).

Figure 9. Mole fraction of B (xB) across the vapor-liquid interface of the A/B dimerization reaction at a reduced temperature of 0.92. Key: B ) 1.25 (solid line), B ) 1.50 (dashed line), B ) 1.75 (dotted line).

phases are well-developed, with a possible exception in the case of B/A ) 1.75 (very small liquid phase fraction). After confirming the presence of stable interfaces in the RxMC simulations, we calculated the reduced surface tension (γ*), where γ* ) γσA2/A, associated with each system, according to eq 3. The results of these calculations are all compiled in Figure 5. It is clearly apparent that the surface

Turner tension of the reactive fluid linearly decreases as the size of the product (σB) increases, while the surface tension tends to increase exponentially as a function of the parameter B. These surface tension calculations appear to be quantitatively consistent when compared with the trend of the surface tension of nonreactive fluids composed of pure A-type and pure B-type molecules, as shown in Figures 6 and 7. As expected, the surface tension of the A/B reactive mixture falls within the bounds of the pure A-type and pure B-type fluids. While the simulated surface tensions seem to fall within reasonable bounds, the conversions of these model reactions are less predictable, as shown in Figures 8 and 9. In these figures, we plot the local mole fraction of B (xB) as a function of the z-coordinate. In the bulk gas or liquid phases, the conversion is only a function of the temperature and density. However, at the interface, the surface tension can cause a dramatic departure from either the liquid or gas phase conversions, with a mole fraction of B as much as 40% higher than the maximum in either phase, when σB ) 1.75. The interfacial effects are magnified as the size ratio of B increases. While this unusual behavior was not anticipated a priori, it is proposed that the inverse relationship between the surface tension and σB (shown in Figure 5) favors the formation of the B-type molecules at the interface as these particles increase in size. As the product size increases, the formation of B at the interface has a greater ability to reduce the interfacial tension, and thus, the local environment is expected to favor the product species. When xB is then plotted as a function of B, as shown in Figure 9, the conversion profiles are rather different. Instead of increasing the local conversion in the interfacial region, the interface tends to suppress the formation of the B-type species. The mole fractions of B can be calculated in separate singlephase simulations corresponding to the same densities as those at the interface, and when compared, the interfacial mole fractions tend to be significantly lower than those in the singlephase bulk simulations (which lack interfacial tension). This simulated trend can be explained with the same logic as previously described. Accordingly, since the surface tension now increases with respect to B, the creation of B molecules is suppressed at the interface in order to minimize the surface tension. The general behavior that we observe with the A/B dimerization reaction can be used to analyze our reactions involving realistic molecular species. In particular, the NO dimerization reaction at its interface can be analyzed with respect to its conversion in a single-phase system, at the same density. The logic that we use to explain the A/B results also seems to be consistent with the NO dimerization reaction. 4.2. NO Dimerization Reaction. The simulation results from the A/B model reaction are useful for understanding the results from the realistic reactions that were modeled. Since the intermolecular parameters are fixed, the NO dimerization reaction was modeled as a function of temperature, starting at 110 K and increasing toward the experimental critical point of 179.2 K.77 In the liquid phase, the NO is primarily in the dimer form, while it is mainly monomeric in the gas phase. Since this is an exothermic reaction, increasing the temperature tends to drive the equilibrium toward the NO monomers. Our current simulations of this reaction at its vapor-liquid interface are shown in Figures 10 and 11, corresponding to temperatures of 110 and 170 K, respectively. These figures show profiles of the molecular density, reported in reduced units (NσNO3/V), and the mole fraction of (NO)2 across the vaporliquid interface. In addition, the mole fraction of (NO)2 in these


Figure 10. Mole fraction of (NO)2 and density profiles (NσNO3/V) of the NO dimerization reaction at a temperature of 110 K. Key: density (solid line), mole fraction of (NO)2 (dashed line), {mole fraction of (NO)2 - mole fraction of (NO)2 [bulk]} (dotted line).

(NσNO3/V)

Figure 11. Mole fraction of (NO)2 and density profiles of the NO dimerization reaction at a temperature of 170 K. Key: density (solid line), mole fraction of (NO)2 (dashed line), {mole fraction of (NO)2 - mole fraction of (NO)2 [bulk]} (dotted line).

simulations is compared against the mole fraction of (NO)2 in a single-phase bulk system, at the same corresponding temperature and density. The difference between these two values more clearly illustrates the effect of the interface on the conversion. As expected, the mole fractions of (NO)2 in the well-developed liquid and gas sections show virtually no difference with the bulk conversions. The slight discrepancy in the liquid section of around 2% is likely due to differences in the potential truncation in the two simulations. The potentials in the vaporliquid simulations were truncated at 2.16 nm (with no longrange corrections), while the single-phase simulations were truncated at 1.66 nm with standard long-range corrections applied. At a temperature of 110 K, the surface tension is calculated to be 28.21 mJ/m2, and the negative shift in conversion within the interfacial region is significant. However, at a temperature of 170 K, the surface tension has fallen to a value of 4.23 mJ/ m2, which greatly reduces the influence of the interface on the reaction equilibrium. The tendency of the reaction conversion to decrease with increasing surface tension seems to follow the general trends observed with the A/B reaction system. This can be seen by performing a simple analysis of the intermolecular potentials used in the simulations. For instance, a rough estimate of the and σ ratios of (NO)2 and NO would yield (although the dimer model is nonspherical) (NO)2/NO ) (125.0 K + 125.0 K)/125.0 K ) 2.0 and σ(NO)2/σNO ) (0.3172 nm + 0.111 85 nm)/0.3172 nm ) 1.35. Although this is not a rigorous


Figure 12. Mole fraction of BrCl and density profiles (NσBr23/V) of the BrCl reaction at a temperature of 270 K. Key: density (dashed line), mole fractions (solid lines).

Figure 13. Mole fraction of BrCl and density profiles (NσBr23/V) of the BrCl reaction at a temperature of 333 K. Key: density (dashed line), mole fractions (solid lines).

comparison, it would suggest that for this system the ratio is greater in magnitude than the σ ratio and that the ratio should dominate the effects arising from the molecular size difference. This is in fact what we observe. The formation of the dimer is suppressed at the interface, and the effect is magnified as the surface tension increases. 4.3. BrCl Reaction. The interfacial behavior of the BrCl reaction is more difficult to analyze, mainly due to the multicomponent nature of the system. The conversions of the A/B reaction and the NO dimerization reaction were functions of only the temperature and the density, since both reactants are identical in these reactions. However, the BrCl equilibrium is a function of temperature, density, and the reactant mole fractions. Thus, even though we begin with an equimolar mixture of Br2 and Cl2, the reacting molecules will unevenly partition between the vapor, liquid, and interfacial regions, making comparisons with a single-phase bulk fluid unreasonable. Regardless of these analytical limitations, we have plotted the interfacial profiles of this system at two different temperatures of 270 and 333 K in Figures 12 and 13, respectively. As with the previous reactions, the density profiles and the local mole fractions appear to be very well-developed. Lacking a direct comparison with the BrCl equilibrium in the bulk, we are not able to make any definitive conclusions regarding the influence of the interface on the equilibrium of this reaction. However, the profiles across the interface show no obvious departure from the adjoining liquid and gas phases, aside from a slight enhancement of Cl2 at the interface in Figure 13. If the intermolecular parameters of the reactants and products are


Turner eq 4. The unusual behavior of the NO system is likely due to the chemical equilibrium shift that occurs at the interface, which was reasoned to be more dramatic than that for the BrCl system. Although somewhat arbitrary, we find that the NO reaction data can be fit to the previous equation, when supplemented by an additional term involving the free energy of the reaction, where the value 10 is merely a fit parameter:

[

γ ) γ0(1 - (T/Tc))2ν exp

Figure 14. Surface tension versus temperature in the BrCl simulations. The symbols represent simulation data, and the solid line is taken from eq 4. Error bars are generated from four separate simulations.

]

-∆G°(T) 10RT

(5)

This parametrized equation is also included in Figure 15. We refrain from assigning any fundamental significance to eq 5, as we have only applied this equation to a single reaction. Furthermore, a second-order polynomial can fit the data equally well. We anticipate investigating other reactions in the future and plan to consider the generality of eq 5. Notwithstanding, we apply eq 5 to predict a critical temperature of 180.6 K for the NO dimerization reaction, which compares closely to an experimentally reported value of 179.2 K for pure NO.77 At the critical temperature, the reaction equilibrium is expected to strongly favor the NO monomers, and thus, we expect the critical temperature of our reactive fluid to be equivalent to that experimentally measured for the pure nitric oxide. 5. Conclusions

Figure 15. Surface tension versus temperature in the NO dimerization simulations. The symbols represent simulation data, the solid line is calculated from eq 4, and the dashed line is calculated from eq 5.

analyzed, the σ and ratios are found to be essentially identical, suggesting negligible influence of the interface on the reaction conversion. Although this is not a rigorous analysis, these results seem to be consistent (or at least not in conflict with) the logic used to interpret the NO dimerization and A/B reaction results. 4.4. Theoretical Comparisons. As stated previously, the surface tensions of the NO and BrCl reactions were calculated over a range of temperatures, approaching the critical point of the two systems. For pure fluids, surface tension versus temperature data can usually be fit to the following equation:78

γ ) γ0(1 - (T/Tc))2ν

(4)

In eq 4, ν is the critical exponent of the correlation length, γ0 is a fit parameter, and Tc is the critical temperature. Figure 14 shows a least-squares fit of eq 4 to our simulated surface tension data for the BrCl reaction, with the value of γ0 being equal to 91.7 mJ/m2, a predicted critical temperature of 514.2 K, and ν being equal to 1.22. Despite the fact that the interfacial composition of this system significantly changes over this temperature range, eq 4 provides an accurate representation of the data. The estimate of the mixture critical point of the BrCl reaction from eq 4 is the first that we are aware of (from either simulation or experiment). The scaling behavior of the NO dimerization reaction is more unusual, as seen in Figure 15. Most notably, the surface tension data clearly show negative curvature as a function of the temperature, which is opposite of the BrCl system. Figure 15 shows that the data cannot even be qualitatively reproduced with

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. We have taken such an approach in this work, to predict the behavior of equilibrium-limited chemical reactions at V-L interfaces using the RxMC simulation method. We have found that the A/B equilibrium, NO dimerization, and BrCl reaction equilibrium all seem to follow similar trends at their vapor-liquid interfaces, when analyzed with respect to their intermolecular potentials. First, the surface tensions in these systems tend to decrease as product/reactant decreases and as σproduct/σreactant increases. As a consequence, the reaction equilibrium tends to shift in the direction that minimizes the surface tension, depending on the intermolecular parameters of a particular system. In tandem, as the surface tension increases, the magnitude of the equilibrium shift is more dramatic. While our calculations are based on results from only three systems, we plan to check these trends in the future with other reacting systems. Finally, we show that the surface tensions in these systems show unusual behavior as functions of temperature, qualitatively different from pure fluids. A predictive model is suggested for the trends observed, which will be investigated further in future work. Also, by analyzing the surface tension versus temperature data, we accurately predict the critical temperature of nitric oxide and predict the critical temperature of the BrCl reactive mixture for the first time. Acknowledgment. Financial support for this work was provided by a Research Advisory Committee grant from the University of Alabama. References and Notes (1) Birdi, K. S. Colloid Polym. Sci. 1997, 375, 561. (2) Lin, H.; Duan, Y. Y. J. Chem. Eng. Data 2003, 48, 1360. (3) Sun, Y. D.; Shekunov, B. Y. J. Supercrit. Fluids 2003, 27, 73.

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Monte Carlo Simulation of Equilibrium Reactions at VaporâLiquid

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