Binary Nucleation Kinetics. 6. Partially Miscible Systems - American

nucleation pathways that avoid the saddles and cross over a ridge in the free energy surface. More recently, Talanquer and Oxtoby3 used density functi...
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11566

J. Phys. Chem. B 2001, 105, 11566-11573

Binary Nucleation Kinetics. 6. Partially Miscible Systems† Barbara E. Wyslouzil* and Shuyu Chen Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 ReceiVed: May 1, 2001; In Final Form: August 3, 2001

We explore vapor-to-liquid phase transitions by numerically solving the birth-death equations for systems where the components are only partially miscible in the bulk liquid. In this case, multiple critical nuclei can coexist and the flux through two saddles contributes to particle formation. When there is a big difference between the monomer impingement rates, we often observe some of the flux crossing the ridge between the two saddles. Despite these complications, the difference between the numerical nucleation rates and the appropriate analytical rate expressions is usually less than 35%.

I. Introduction Advances in experimental techniques make it possible to explore multicomponent vapor-to-liquid phase transitions over the entire range of intermediate gas-phase compositions, even when the bulk liquid exhibits multiple phases.1 In such partially miscible systems, the task of comparing the measured nucleation rates with theoretical predictions is more complicated than in completely miscible systems, because multiple critical nuclei may form under a wide range of gas-phase activities. In binary nucleation, for example, two critical nuclei can form,2,3 and under some conditions both may contribute significantly to the production of particles. Ray, Chalam, and Peters2 were the first to calculate binary nucleation rates in partially miscible systems. Working within the framework of classical nucleation theory, they confirmed that if the two components are miscible for all compositions in the liquid phase, only one type of critical nucleus is possible. On the other hand, systems that are only partially miscible may produce two types of nuclei, and one of them may have a composition that lies within the miscibility gap. In their treatment, Ray et al.2 assumed that the critical clusters were well mixed and neglected both surface enrichment and phase separation within the critical clusters. Furthermore, their analytical approach focused on determining the particle flux through the saddle regions, so they could not answer the question whether two populations of droplets with distinct composition persist and grow to detectable sizes. Nor did Ray et al. address the possibility that some of the new droplets may form by nucleation pathways that avoid the saddles and cross over a ridge in the free energy surface. More recently, Talanquer and Oxtoby3 used density functional techniques to study nucleation in nonideal binary mixtures, including those with a miscibility gap. They also found conditions where more than one nucleus can form. In some cases the critical nucleus itself underwent phase separation. When this occurred, the spherical symmetry of the cluster was lost, and each end of the cylindrical cluster was enriched in one of the components. Although the authors calculated the work of †

Part of the special issue “Howard Reiss Festschrift”. * Corresponding author. E-mail: [email protected]. Fax: 1-508-8315853.

formation of the critical clusters, including those that had phase separated, they did not explore the kinetics of the nucleation process. In any multicomponent system, the net particle flux depends both on the shape of the free energy surface and the impingement kinetics. In our recent work4-7 we explored these issues in detail by numerically solving the kinetics equations describing binary nucleation. Our results let us stringently test the assumptions inherent in analytical binary nucleation rate expressions for both the transient6 and steady states.5 For systems that displayed both positive and negative deviations from ideality in the liquid phase, and over a wide range of relative impingement rates, we found that it was very difficult to find conditions where the numerical rate differed by more than 10%-20% from a modified version5,8 of Stauffer’s rate expression.9 The most significant failure occurred for a strongly positively deviating system with a large difference in the monomer impingement rates. In this case, the major particle flux bypassed the saddle point and crossed a low ridge on the free energy surface.5 In this paper we extend our previous numerical work to investigate binary nucleation in partially miscible systems under conditions where multiple paths in distinctly different regions of the free energy surface contribute significantly to the overall nucleation rate. Here, visualizing the particle formation process is more difficult than when the two components are fully miscible and a single pathway dominates. We studied two partially miscible systems where the excess free energy of mixing was described by a two-suffix Margules equation and the remaining physical properties were those of the ideal o-xylene-m-xylene or ethanol-hexanol systems.5 We assume that all of the clusters are well mixed regardless of their size or composition. In the system based on o-xylene-m-xylene, the biggest contribution to the nucleation rate comes from the fluxes that follow the thermodynamically favored path across the lowest saddle on the free energy surface. The system based on ethanol-hexanol is not symmetric because the vapor pressures of the two species differ by a factor of 226 at the simulation temperature. Thus, kinetics can force more of the flux over the saddle with the higher free energy. In both systems, however, our numerical nucleation rates agree well with the modified Stauffer rate expression,5,8,9 with at most 35% deviation. The good agreement between our numerical results and the analytical rate expressions confirm the validity of the overall approach

10.1021/jp011647o CCC: $20.00 © 2001 American Chemical Society Published on Web 10/09/2001

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developed by Ray et al.2 By examining the growth of particles beyond the critical region, we find that although two particle populations with distinctly different compositions exist initially, it is unlikely that these can be distinguished once the droplets have grown to a detectable size. We also examined the analytical timelags associated with each critical nucleus and found that, in some cases, the major steady-state particle flux is associated with the critical nucleus that has the longer timelag. In section II we present the binary kinetics equations, the analytical nucleation rate expressions, and the two partially miscible systems. The steady-state and transient results are presented in section III. Section IV summarizes our work and presents the conclusions. In the Appendix we demonstrate that for partially miscible mixtures where the partial molar volumes are independent of composition, the composition of one of the critical nuclei is always inside the miscibility gap, while the other is outside. II. Basic Equations and Computational Methodology A. Solving the Kinetics Equations. As in our previous papers,4-7 we assume that clusters grow and decay by the addition or loss of a monomer between adjacent cluster sizes. The net flux Jν between clusters of composition (i, j) and (i + 1, j) or (i, j + 1) is determined by the forward and reverse rates as

JA(i,j,t) ) ΓA(i,j) NAf(i,j,t) - EA(i+1,j) f(i+1,j,t)

(1)

JB(i,j,t) ) ΓB(i,j) NBf(i,j,t) - EB(i,j+1) f(i,j+1,t)

(2)

where Γν(i,j) is the forward rate coefficient for adding a monomer of type ν to a cluster containing i molecules of species A and j molecules of species B, f(i,j,t) is the nonequilibrium cluster concentration at time t, Eν(i,j) is the reverse rate coefficient for losing a monomer of species ν from a cluster, and Nν is the number density of monomers of species ν in the vapor. The monomer concentrations are defined as NA ) f(1,0,t) and NB ) f(0,1,t). The forward rate coefficient is the collision frequency between two particles of unequal mass taken from the kinetic theory of gases. A mass accommodation coefficient of unity is assumed for each species. We use the principle of detail balance, together with the forward rate coefficients and the self-consistent equilibrium size distribution, to define the evaporation rate coefficients. The first two papers4,5 in this series contain the explicit expressions for the rate coefficients5 and the equilibrium size distribution.4 The change in cluster number density with time is given by

df(i,j,t) ) JA(i-1,j,t) - JA(i,j,t) + JB(i,j-1,t) - JB(i,j,t) (3) dt for all clusters of composition (i, j) except the dimer. In the latter case, either JA(0,1) or JB(1,0) is set to zero to avoid double counting mixed dimer formation. The maximum number of molecules in any cluster is imax - 1 and jmax - 1. This results in (imax × jmax - 3) differential equations because we fix the concentrations of the monomers and (0, 0) does not correspond to a physical cluster. The initial concentration of clusters containing more than one monomer is zero, f(i,j,0) ) 0 for i + j > 1. The boundary conditions for the differential equations are as follows. (1) The concentrations of the two monomers, NA and NB, are constant. (2) The values of f(imax,j,t) and f(i,jmax,t) are calculated on the basis of the values of f(imax-1,j,t-1), f(imax-2,j,t-1) and f(i,jmax-1,t-1), f(i,jmax-2,t-1), respectively. This second boundary condition is different from the Szilard

boundary condition, f(imax,j,t) ) 0 and f(i,jmax,t) ) 0. Applying the Szilard boundary condition10 to the system causes the cluster concentration near the boundary to drop abruptly. Our boundary condition stabilizes the numerical scheme near the edge of the grid. The steady-state nucleation rates are calculated by summing all the fluxes crossing any line joining the i axis and j axis. For convenience, we choose to use the 45° line where every cluster has the same total number of molecules. Temkin and Shevelev11 proposed a similar process to describe the nucleation kinetics transition from binary nucleation to unary nucleation. In the steady state the total nucleation rate across any line must be constant. There are some numerical difficulties with this scheme for lines too close to the origin where small net fluxes are the difference between large forward and backward rates. Those fluxes are therefore ignored. B. Analytic Rate Expression. To directly compare our numerical results with analytical predictions, we used the approach outlined in Ray et al. to determine the number of critical clusters and their composition.2 We solved the thermodynamically inconsistent Kelvin equations12,13 (including the surface tension derivatives), because in our numerical scheme the reversible work for cluster formation W(i,j) is based on the overall composition of the cluster. Thus, we implicitly ignore the possibility of surface enrichment or phase segregation within a cluster. As in our previous work, we use Wilemski’s modified5,8 versions of the Reiss14 and Stauffer9 nucleation rate expressions. The expressions for the modified Reiss rate JWR and the modified Stauffer rate JWS are given as eqs 11 and 13 in paper II of this series5 and are not repeated here. The changes introduced by Wilemski ensure a smooth transition between the binary and unary nucleation rates. In the current work, this is important because the critical nuclei are almost always close to a pure component axis. When there are two critical clusters, the net nucleation rate is the sum of the rates associated with each critical cluster. C. Binary Systems Considered. Wyslouzil and Wilemski5 studied six binary systems including two ideal systems (oxylene-m-xylene and ethanol-hexanol), two negatively deviating systems, and two positively deviating systems. All of them were miscible over the entire composition range. The two positively deviating systems they studied were based on the ideal systems with an excess free energy function GE that placed the systems at the bulk upper critical solution temperature (UCST). In particular,

GE/(xAxB) ) 2RTS

(4)

where xν is the mole fraction of component ν, R is the universal gas constant, and TS is the simulation temperature. Here, we consider two additional systems based on the ideal o-xylene-m-xylene and ethanol-hexanol, but now the systems are only partially miscible. We use the following two-suffix Margules equation to describe the excess free energy of mixing,

GE/(xAxB) ) 3RTS

(5)

We call the partially miscible system based on o-xylene-mxylene PM1 and the system based on ethanol-hexanol PM2. The functional form of eq 5 ensures that we can find two saddles for some range of vapor-phase activities. In the PM1 system, the key physical properties of the two components, such as molar volume, surface tension, and vapor pressure, are quite close. Furthermore, the surface tension derivative is small and

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independent of composition. The simulation temperature for PM1, 293.15 K, is the same as that used in our simulations of o-xylene-m-xylene and the positively deviating system PD1. In the PM2 system, the differences in molar volume and surface tensions are more significant, and the surface tension derivative is a linear function of composition. The large difference in vapor pressure between the two components comprising PM2 may also lead to some interesting competition between kinetics and thermodynamics. The simulation temperature for PM2, 260 K, is the same as that used in our calculations for ethanol-hexanol and the positively deviating system PD2. In partially miscible liquids, two liquid phases with compositions x′ and x′′ coexist in equilibrium when the overall composition lies within the miscibility gap. From the equality of the chemical potentials of the components in the liquid phases, we know that

[γAx]x)x′ ) [γAx]x)x′′

(6)

[γB(1 - x)]x)x′ ) [γB(1 - x)]x)x′′

(7)

and

where γν is the activity coefficients of species ν and x ) i/(i + j) is the mole fraction of component A in the mixture. With GE/RTS ) 3x(1 - x) solving eqs 6 and 7 yields the miscibility limits x′ ) 0.07 and x′′ ) 0.93. Because we use the same functional form for the excess free energy the miscibility limits for PM1 and PM2 are identical. The other physical properties for the two systems are summarized in Appendix B of our earlier paper.5 III. Results and Discussion A. Steady-State Results. In unary nucleation, the birth-death kinetics equations have been solved and the numerical results15 agree with analytical predictions. Binary nucleation, however, is more complicated because many paths contribute to the overall nucleation rate. As discussed in Wyslouzil and Wilemski,5 the numerical binary nucleation rates agree with the Reiss14 and Stauffer9 analytical rate expressions if the flux passes through the saddle, but this is not the only way to nucleate a new phase. Depending on the shape of the free energy surface and the ratio of the monomer impingement rates, it is possible for the flux to bypass the saddle point in some miscible systems. Before we look for similar phenomena in our partially miscible systems, we must first locate the regions where two saddles exist. As discussed by Ray et al.2 and Talanquer and Oxtoby,3 under some gas-phase activities the free energy surface of partially miscible systems like PM1 and PM2 will have two saddles corresponding to critical clusters of different compositions. Figure 1 defines the regions in aA-aB space that correspond to the single and multiple critical nuclei regions for PM1 and PM2 at their respective simulation temperatures. Here aν is the gasphase activity of component ν. Although we showed above that the miscibility gap for both systems is the same, the regions of aA-aB space where two critical clusters is found are distinct, because the other physical properties, for example, molar volume, are quite different. We are primarily interested in simulating nucleation in the two saddle regions and we need to ensure that both saddles are well within our standard 80 × 80 computational grid. For our numerical calculations we therefore set aA ) 12 for the PM1 system, aA ) 3.2 for the PM2 system and varied the value of aB across the two saddle regions. Plots of the cluster fluxes

Figure 1. Solid (dashed) lines: regions where multiple critical nuclei exist for the PM1 (PM2) binary systems. To the left of the upper boundary, the critical nucleus is rich in component B. To the right of the lower boundary the critical nucleus is rich in component A.

superimposed on the free energy surface are used to visualize the competition between kinetics and thermodynamics. These plots include all of the fluxes whose magnitude is at least 1% of the maximum flux. We will call the saddle corresponding to the component A-rich nucleus saddle A and the saddle corresponding to the component B-rich nucleus saddle B. We start by examining the PM1 system in the region where two saddles exist and have almost the same free energy W. For this system, we expect that the major nucleation flux will always follow the thermodynamically favored path, because of the high degree of symmetry in this system and because the vapor pressures of the two species are very close. On the basis of previous results5 for o-xylene-m-xylene and the positively deviating system PD1 (o-xylene-m-xylene with GE/(xAxB) ) 2RTS), monomer impingement cannot alter the direction of the nucleation path and the major flux will always follow the path of steepest descent. Figure 2a illustrates the results for gas-phase activities aA ) 12 and aB ) 10.5. Saddle A is located at (31.9, 2.4) and W ) 43.4kT. Thus, saddle A lies about 0.5kT lower than saddle B, which is located at (2.6, 33.7) and where W ) 43.9kT. Under these conditions, there are two distinct nucleation paths and the flux through saddle A is greater than the flux through saddle B. Near the saddles, each path moves in the steepest descent direction. Very little flux crosses the ridge separating the two saddles. If we increase aB slightly to aB ) 11, Figure 2b shows that saddle A is located at (31.6, 2.5) with W ) 43.3kT, saddle B is at (2.3, 32.1) with W ) 42.4kT. Two distinct nucleation paths remain, but the flux through saddle B is now greater than the flux through saddle A. As we increase aB further to aB ) 12, Figure 2c, the free energy difference between two saddles is 3.35kT and almost all of the flux goes through the saddle with the lower free energy. As anticipated, the major nucleation path is mainly determined by the location of the saddle point with the lowest free energy and the shape of the free energy surface in this region. As aB increases, at constant aA, saddle B moves closer to the B-axis while saddle A moves away from the A-axis. Because of the large miscibility gap, the saddles are always close to the pure component axes, and their location is therefore not very sensitive to small changes in aB. Under the conditions present in Figure 2a, saddle B lies in the miscibility gap, while in Figure 2b,c, saddle A lies in the

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Figure 3. Compositions of the critical nuclei denoted by the solid and open squares for the PM1 system (aA ) 12) and by solid and open circles for the PM2 system (aA ) 3.2). The dashed lines indicate the miscibility limits for the bulk mixtures. In both cases the critical nuclei become richer in component B as the gas-phase activity of B is increased at a constant gas-phase activity of A. In the PM1 system, the two critical nuclei have the same free energy when aB ) 10.6. In the PM2 system, the two critical nuclei have the same free energy when aB ) 25.2.

Figure 2. Steady-state nucleation fluxes superimposed on contour plots of the free energy surface for the PM1 system at aA ) 12. The solid contour lines denote free energies above or equal to the lowest saddle point free energy and are spaced at 3kT intervals. The highest contour value is denoted on the figure to the right of that contour. The dashed contours denote free energies below the lowest saddle point value and are spaced at 5kT intervals relative to each other and to the lowest solid contour. (a) When aB ) 10.5, the flux through saddle A is about 2 times the flux through saddle B. (b) When aB ) 11, more of the flux moves through saddle B. (c) When aB ) 12 almost all of the flux moves saddle B.

miscibility gap. In this system, the saddle with the higher free energy is always in the miscibility gap. Figure 3 shows that for either saddle, the mole fraction of component A decreases as aB increases at fixed aA. For PM1 in the region of two nuclei, one critical nucleus always lies in the miscibility gap. For example, when aB ) 11, the A-rich critical nucleus enters the miscibility gap just as the B-rich nucleus leaves. A short thermodynamic proof in the Appendix explains why this occurs for the simple system explored here. The PM2 system does not display the strong symmetry of the PM1 system, because the vapor pressures of the two species comprising PM2 are very different and the molecular volumes differ by a factor of ∼2. In this system, we expect that kinetics will strongly affect the direction of the nucleation path in some regions and that the flux may not follow the thermodynamically favored path any more. Figure 4a illustrates the results for gasphase activities aA ) 3.2 and aB ) 25. Saddle A is located at (44.1, 3.7) and W ) 33.0kT. Saddle B is located at (1.1, 19.8) and W ) 33.1kT. In this case, most of the flux passes through saddle A, the saddle with the lower free energy. Even though saddle A is quite close to the A-axis, the flux in this region already deviates from the steepest descent direction because of the large difference in the impingement rates of A and B. As we increase the activity of B to aB ) 28, Figure 4b shows that saddle A is now located at (41.8, 4.1) with W ) 32.5kT and saddle B is located at (0.95, 17.9) with W ) 30.9kT. For i + j small enough, two distinct nucleation paths are apparent. For larger cluster sizes, most of the flux that passes through saddle B bends in the direction of the species with the higher impingement rate. Furthermore, some of the flux leaks over the ridge between the two saddles. If we increase aB to aB ) 36, Figure 4c shows that now most of the flux passes through saddle B, the saddle with lower free energy. If we examine the composition of the two saddles as aB is varied, Figure 3 shows that in this asymmetric system both nuclei are outside the miscibility gap near aB ) 20. Figure 5 compares our steady-state numerical rates with the modified Reiss and Stauffer rate expressions for the PM1 and PM2 systems. In PM1 the major flux always follows the path

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Figure 5. Numerical nucleation rates compared to the analytical nucleation rate predictions JWR (solid black line) and JWS (solid gray line). The dashed lines correspond to the unary nucleation rates of components A and B. (a) In the PM1 system (squares), the Reiss and Stauffer rates are both shown but cannot be distinguished. Both are in good agreement with the numerical results, and binary rates are 1-2 orders of magnitude above the unary nucleation rates. (b) In the PM2 system (circles), the Reiss and Stauffer rates coincide when particle formation is dominated by nucleation of pure A or B. At intermediate gas-phase activities of B, the Stauffer rate is in good agreement with the numerical results, while the Reiss expression underestimates the nucleation rate by up to a factor of 5. At worst, the numerical rate is about 35% above JWS.

Figure 4. Steady-state nucleation fluxes are superimposed on contour plots of the free energy surface for the PM2 system at aA ) 3.2. The spacing of the contour levels is the same as in Figure 2. (a) When aB ) 25, the flux though saddle A is higher than the flux though saddle B. (b) When aB ) 28, the free energy of saddle B is lower than that of saddle A but more flux still moves through saddle A. (c) When aB ) 36, most of the flux passes through saddle B, but a reasonable fraction also crosses the ridge between the two saddles.

of steepest descent, the analytical expressions yield identical rates, and our numerical results agree very well with the analytical predictions. Deviations are at most 25%. The abrupt kink in the nucleation rate near aB ) 11 signals the appearance of the second significant nucleation path and the flux through this path increases rapidly with aB. Although the binary nucleation rates follow the behavior of the individual unary rates, they are, on average, 1-2 orders of magnitude higher. Thus, binary nucleation enhances particle formation even where there is a strong miscibility gap and little difference in the pure component surface tensions. For the PM2 system, the analytical and numerical rates all converge at low values of aB and approach the value for unary nucleation of pure A. At high values of aB, the analytical curves converge again but the numerical rate stays higher, consistent with the continued leakage of flux across the low ridge on the free energy surface that is evident in Figure 4c. At intermediate values of aB, the Reiss rate underestimates the nucleation rate by up to a factor of 5. In contrast, Stauffer’s rate expression does a good job of predicting the numerical rates and, in the worst case, underestimates the true rate by about 35%. Another way to compare the behavior of the nucleation paths is to plot the flux as a function of composition for different values of i + j. For PM1, Figure 6a illustrates the behavior of J(x)/J at aA ) 12 and aB ) 11, where J(x) is the flux at x ) i/(i + j) and J is the steady-state nucleation rate. In the subcritical region, i + j ) 15, the contribution of unary nucleation to the overall rate is significant for either path. As i + j increases, however, the unary pathway gradually decays and approaches zero when i + j ) 70. Our numerical solutions predict that the flux through saddle B is 2.2 times larger than the flux through saddle A. In the analytical predictions, both the Reiss and Stauffer theories predict the ratio of fluxes through these two saddles is 2.3. For PM1 under these conditions, two distinct nucleation paths persist well past the critical region, although

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Figure 6. Distribution of the nucleation flux J(x)/J changes as the number of molecules in the clusters increases. (a) In the PM1 system at aA ) 12 and aB ) 11, both critical nuclei contain 34 molecules. (b) In the PM2 system at aA ) 3.2 and aB ) 28, the critical nucleus rich in B contains 19 molecules, while the critical nucleus rich in A contains 46 molecules. By the time clusters contain 70 molecules, almost none of the flux is associated with molecules whose composition corresponds to that of the critical cluster rich in B.

the difference in composition between the two peak fluxes decreases as i + j increases. For PM2, Figure 6b shows the variation of J(x)/J with cluster size at aA ) 3.2 and aB ) 28. Under these conditions, the flux through saddle B is still lower than that through saddle A even though the free energy of saddle A is 1.6kT higher. When i + j ) 19 (the size of the critical cluster rich in B), there are two main nucleation pathways, the contribution of unary nucleation to both pathways is significant, and some of the flux crosses over the ridge between the two saddles. At i + j ) 46 (the size of the critical cluster rich in A) the flux originally associated with saddle B is more spread out in composition space because component A is incorporated into the clusters much more rapidly than component B. This decrease in flux through clusters rich in component B is more pronounced for the largest cluster size, i + j ) 70. This suggests that for large enough cluster sizes the peak flux should be centered at a composition corresponding to the impingement rate ratio of A to B. Another way to examine whether it is possible to form two distinct particle populations with different compositions, is to examine the cluster concentrations directly. Figure 7a illustrates the cluster concentration contours that correspond to the flux plot shown in Figure 2a. The free energy of saddle A is lower

Figure 7. Contours of cluster concentration are spaced as follows. The first solid line corresponds to a cluster concentration 100 cm-3 s-1. The other solid lines correspond to concentration greater than 100 cm-3 s-1, and the dashed lines correspond to concentration less than 100 cm-3 s-1. The contour spacing is 1 order of magnitude in concentration. The saddle points are marked by “+”. (a) For PM1 at aA ) 12 and aB ) 10.5, the cluster concentration contours are quite symmetric, and two cluster populations extend far beyond the saddle region. (b) For PM2 at aA ) 3.2 and aB ) 28, the clusters rich in B die out quickly by the time clusters contain 60 molecules.

than the free energy of saddle B by about 0.5kT, the cluster compositions at the saddles are comparable, and it is easy to distinguish two distinct particle populations for clusters up to twice the size of the critical cluster. Yet if we were to follow the growth of the clusters to much larger sizes, the overall cluster composition would eventually converge to the composition corresponding to the impingement rate of the two species. For supercritical clusters, evaporation is unimportant and the overall cluster composition will eventually be dominated by the mass added during growth. Most particles that can be detected experimentally contain thousands of molecules, and when critical clusters contain fewer than ∼100 molecules, differences in the initial critical nucleus composition quickly become negligible. Thus, two distinct particle populations can only be

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Figure 8. Line marked with filled squares (circles): timelag for the flux through saddle A to reach steady state in the PM1 (PM2) system. Lines marked with open squares (circles): timelag for the flux through saddle B to reach steady state in the PM1 (PM2) system. Solid vertical lines: activity of B where the free energy of saddle B equals that of saddle A. Dashed vertical line: activity of B where the flux through saddle B equals that through saddle A. For the PM1 system the two vertical lines coincide, and the saddle with the lower free energy and higher flux also has the shorter timelag. For the PM2 system, the vertical lines and dashed lines are distinct, and saddle B always has the longer timelag.

detected if we can observe them when they are close enough to the critical size. We note that when conditions favor two initial particle populations, the overall composition of the detectable particles is most likely within the miscibility gap, and the final particles may exhibit internal phase separation. For PM2, Figure 7b illustrates the contours of constant cluster concentrations when aA ) 3.2 and aB ) 28. In this case the concentration of component B-rich clusters dies out rapidly after the saddle region even when the free energy of saddle B is lower than that of saddle A. Here it is hard to distinguish two droplet populations even near the saddle region. B. Timelags. To complement our steady-state work, we examined the behavior of the analytically predicted timelags required for the saddle fluxes to reach steady state in the PM1 and PM2 systems. All of the timelags in Figure 8 are calculated using eq 8 in Wyslouzil and Wilemski6 and are therefore upper bounds on the numerical timelags. At fixed aA, the timelag associated with either saddle decreases as aB increases. Increasing the gas-phase activity of component B increases the vaporphase concentration and the net impingement rate, thereby decreasing the timelag. In addition, higher gas-phase activities decrease the height of the free energy barriers, further reducing the time required to reach steady state. For PM1, the saddle with the lower free energy and higher nucleation rate is always associated with the shorter timelag. On the other hand, in PM2, the net flux through saddle A always reaches steady state before saddle B. Since the timelags only differ by a factor of 5 once the nucleation rate through saddle B is higher than that through saddle A, the net flux across the i + j ) 80 contour increases smoothly, rather than exhibiting a two-step process. IV. Summary and Conclusions

Wyslouzil and Chen and because the vapor pressures of the two species are very close. Kinetics does not contribute significantly to the direction of the nucleation path, which always follows the path of steepest descent. Our numerical nucleation rates agree well with the modified Reiss and Stauffer rate expressions, and the largest deviations were at most 25%. When two types of nuclei form, one nucleus always lies in the miscibility gap. In the asymmetric PM2 system, kinetics modifies the direction of the nucleation path significantly in some regions, and the flux does not always follow the thermodynamically favored path. Our numerical results show better overall agreement with the modified Stauffer rate expression than with that of Reiss. At high values of aB, the numerical values deviate systematically by ∼35% from the analytical values because some of the flux crosses the low ridge between the two saddles. When two nuclei form, both nuclei may lie outside the miscibility gap. Finally, it is unlikely that two droplet populations with different compositions will survive to sizes large enough to observe experimentally, although both types of clusters persist near the critical region. Appendix. Composition of the Critical Clusters In our calculations we observed that when multiple critical nuclei exist in the PM1 system, the composition of one critical nucleus always lies inside the bulk miscibility gap. In contrast, for the PM2 system there was a limited region where both nuclei were outside the bulk miscibility gap. In this Appendix we show that if the partial molar volumes are not functions of composition and the surface tension derivatives are equal to zero, the behavior observed for the PM1 system can be derived from a simple thermodynamic argument. We start by proving that if the composition of one critical cluster corresponds to the composition on one side of the miscibility gap, the composition of the second critical cluster corresponds to the composition on the other side. Within the framework of classical nucleation theory, the composition of the critical cluster is properly found by solving12,13

VB∆µA ) VA∆µB

where Vν is the partial molar volume of component ν and ∆µν is the change in chemical potential between the gas and liquid phases for component ν. We now consider the case where two saddles are present and one of the saddles, saddle A, has a composition that corresponds to the composition at one side of the bulk miscibility gap. Assuming ideal gas behavior, we rewrite (A-1) in terms of the liquid-phase activities of the two components aL,ν and the gas-phase partial pressures, and show that

( )

pA pB0 aL, A(x') ) 0 pB p A

VA/VB

[aL, B(x')]VA/VB

(A-2)

Here pν is the partial pressures of component ν in the gas phase and pν0 is the equilibrium vapor pressure of component ν. At saddle B, the equivalent expression is given by

aL, A(x**) )

( )

pA pB0 p 0 pB A

We solved the birth-death equations describing binary nucleation in partially miscible systems. In the PM1 system, the major flux always follows the thermodynamically favored path because of the high degree of symmetry in this system

(A-1)

VA/VB

[aL,B(x**)]VA/VB

(A-3)

where x** is the as yet unknown composition of the second critical cluster. Because the partial molar volumes are not functions of composition, eq A-3 can be simplified to give

Binary Nucleation Kinetics

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11573

aL, A(x**) ) aL, A(x′)

(

)

aL, B(x**) aL, B(x′)

VA/VB

(A-4)

or

aL, A(x**) aL, A(x′′)

)

(

)

aL, B(x**) aL, B(x′′)

VA/VB

(A-5)

In going from eq A-4 to eq A-5 we have made the substitution aL,ν(x′) ) aL,ν(x′′) across the miscibility gap. Equation A-5 clearly holds if x** ) x′′. To show that x** must equal x′′, we rely on the fact that in the regions of stability, the values of a′L,B are always positive, while a′L,A are always negative, where a′L,ν are the derivatives of aL,ν with respect to x ) i/(i + j). Therefore, x** < x′′ implies that aL,A(x**) > aL,A(x′′) and aL,B(x**) < aL,B(x′′). On the other hand, x** > x′′ implies that aL,A(x**) < aL,A(x′′) and aL,B(x**) > aL,B(x′′). When x** > x′′ or x** < x′′, eq A-5 is contradicted for any constant values of VA and VB. Thus, x** ) x′′ and we see that if one saddle is at the bulk miscibility limit, the second saddle must also be at the miscibility limit. The remainder of the proof is simply to recognize that the compositions of the critical clusters do not vary independently. If the gas-phase activities are changed so that one of the critical clusters becomes richer in component ν, so does the second cluster. Thus, if changes in the gas-phase activity push one of the clusters into the bulk miscibility gap, it must simultaneously push the second critical cluster out of the miscibility gap. In our PM1 system, the partial molar volumes are constant and

quite similar, and the surface tension derivative is very small. Thus, the thermodynamically inconsistent Kelvin equations are very close to eq A-1. In the PM2 system, the partial molar volumes are constant but differ by a factor of ∼2, the surface tension derivatives are larger and are functions of composition. Thus, the Kelvin equations differ enough from eq A-1 that the existence of a region where both saddles lie outside the activity gap is not surprising. Acknowledgment. This work was supported by the National Science Foundation, Division of Chemistry under Grants No. CHE-9502604 and CHE-9729274. We thank G. Wilemski for useful discussions. References and Notes (1) Strey, R.; Wagner, P.; Viisanen, Y. J. Phys. Chem. 1994, 98, 7748. (2) Ray, A. K.; Chalam, M.; Peters, L. K. J. Chem. Phys. 1986, 85, 2161. (3) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1996, 104, 1993. (4) Wilemski, G.; Wyslouzil, B. E. J. Chem. Phys. 1995, 103, 1127. (5) Wyslouzil, B. E.; Wilemski, G. J. Chem. Phys. 1995, 103, 1137. (6) Wyslouzil, B. E.; Wilemski, G. J. Chem. Phys. 1996, 105, 1090. (7) Wyslouzil, B. E.; Wilemski, G. J. Chem. Phys. 1999, 110, 1202. (8) Wilemski, G. J. Chem. Phys. 1975, 62, 3763. (9) Stauffer, D. J. Aerosol Sci. 1976, 7, 319. (10) As discussed by W. J. Dunning in Chemistry of the Solid State (Garner, W. E., Ed.; Academic: New York, 1955, p 159) and in Nucleation (Zettlemoyer, A. C., Ed.; Marcel Dekker: New York, 1969; p 1). (11) Temkin, D. E.; Shevelev, V. V. J. Cryst. Growth 1984, 66, 380. (12) M. Kahlweit, M. Z. Phys. Chem. (Frankfurt) 1962, 34, 251. (13) Wilemski, G. J. Chem. Phys. 1988, 88, 5134. (14) Reiss, H. J. Chem. Phys. 1950, 18, 840. (15) Abraham, F. F. J. Chem. Phys. 1969, 51, 1632.