J. Phys. Chem. B 2006, 110, 3511-3516
3511
Unravelling the Peculiar Nucleation Mechanisms for Non-Ideal Binary Mixtures with Atomistic Simulations† Matthew E. McKenzie and Bin Chen* Department of Chemistry, Louisiana State UniVersity, Baton Rouge, Louisiana 70803-1804 ReceiVed: July 18, 2005; In Final Form: August 19, 2005
Recent experiments reveal unusual nucleation behavior for seemingly simple mixtures that cannot be described by the classical theory. Molecular simulations using a combination of aggregation-volume-bias Monte Carlo, umbrella sampling, and histogram reweighting methods were carried out to study the nucleation events involved in the water/ethanol, water/n-nonane, and n-nonane/ethanol mixtures. These simulations reproduced their different nonideal behaviors observed by the experiments. Furthermore, the finding of their strikingly distinct mechanisms, as implied from the calculated free-energy maps, challenges the current theoretical description of this phenomenon.
1. Introduction Understanding the nucleation phenomenon, in which embryos of a new phase emerge from a metastable supersaturated mother phase, as well as predicting the rate of this process, has both fundamental and practical implications. Multicomponent nucleation is of particular interest as it is often involved in processes of atmospheric, environmental, and technological importance. Therefore, with the recent emergence of sophisticated experimental techniques, considerable efforts have been focused on these systems, which produced an enormous amount of nucleation rate data for a variety of binary and ternary mixtures.1-15 These data have painted a rather complex picture for multicomponent nucleation events. Even for common binary fluid mixtures, such as water/alcohol,1-6 water/alkane,7-9 and alkane/alcohol,9,10 significant deviations from the ideal nucleation behavior were found from the experiments. More puzzling is that these seemingly simple systems exhibit rather distinct nonideal nucleation behavior, as indicated by the onset activity plots constructed from the experimental nucleation rate surface. This implies that different systems may take on their own unique mechanisms (or pathways) during the nucleation process. Unfortunately, very little mechanistic information can be obtained directly from the experimental data for these systems. The nucleation theorem,10,16-18 in principle, provides a route to interpret the content of the critical nuclei from the slopes of the experimentally measured nucleation rate surface. However, the obtained composition is an averaged quantity (i.e., over all possible critical nuclei) and, thus, may not reveal the true saddle point in the nucleation pathway.19-23 On the other hand, current theoretical models24-31 for both unary and multicomponent nucleation cannot provide the mechanistic details either. In fact, they rely on an assumption that nucleation proceeds on a welldefined path via the formation of a critical nucleus with a welldefined structure and composition, despite that the validity of this assumption is questionable.10,19-23 For example, in partially miscible mixtures, both density functional theory (DFT) and simulation studies have shown that nucleation could proceed †
Part of the special issue “Michael L. Klein Festschrift”. * Author to whom correspondence should be addressed. E-mail: binchen@ lsu.edu.
via two types of nuclei.19-23 These theoretical studies were partly inspired by the nonideal experimental results obtained on multicomponent systems. However, the relatively simple models (e.g., single Lennard-Jonesium) employed by these studies did not allow a direct, quantitative comparison with the experiments. In this work, simulations using realistic, atom-based force fields were carried out to examine, in detail, the nucleation mechanism for those binary systems that exhibited different nonideal nucleation behavior as shown experimentally. These include the binary water/ethanol, n-nonane/water, and n-nonane/ ethanol mixtures. For the former two systems, a partial nucleation free-energy (NFE) map has been calculated previously by us.32 Here, a more complete NFE diagram was constructed from the simulation to allow an unambiguous identification of the nucleation mechanisms (or pathways) for these systems. We will see that, in these three selected systems, each proceeds on a different nucleation mechanism (see Figure 1). While the water/ethanol mixture nucleates together via a mixed nucleus, there are two independent nucleation channels available for the n-nonane/water system (one via an n-nonaneenriched nucleus and the other via a water-enriched nucleus). In contrast, the n-nonane/ethanol NFE map displays the most striking finding of this study: coexistence of multiple nucleation pathways all over the map. These NFE results were shown to be consistent with the nonideal nucleation behavior observed experimentally for those systems. 2. Simulation Methods The opportunity of carrying out this direct comparison with the experimental data was partly promoted by the emergence of novel nucleation simulation methods.33-40 In particular, the recently developed AVUS-HR approach,40 which combines histogram reweighting41,42 with aggregation-volume-bias Monte Carlo (AVBMC) nucleation simulations using self-adaptive umbrella sampling,38,39 is a powerful tool for simulation studies of the nucleation phenomenon. As an event that happens in the scale of droplet/cm3/s, nucleation poses serious difficulties for molecular simulation. For example, the experimental nucleation rates measured by Strey and co-workers3,4,8-10,12 are around 107 droplets/cm3/s or 10-14 droplets/nm3/s. A number in units of droplet/nm3/s is more appropriate to represent the rate for a
10.1021/jp0539472 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/04/2005
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Figure 1. Contour of the two-dimensional NFEs (in units of kBT) as a function of the number of molecules for the two components involved in ) 1.6 × 10-8 Å-3 and nwater each binary mixture calculated from the simulation at (a) nethanol ) 1.7 × 10-8 Å-3; (b) nnonane ) 1.7 × 10-7 Å-3 and v v v water nonane ethanol -8 -3 -7 -3 -8 -3 nv ) 4.5 × 10 Å ; and (c) nv ) 1.5 × 10 Å and nv ) 3.8 × 10 Å . The contour levels with NFE values of 45, 50, 55, and 60 kBT were depicted as lines for a clear view of the NFE profiles near the saddle point.
simulation because a much smaller length scale (typically, on the order of nanometers) could be afforded by a simulation compared to the experimental length scale of centimeters. Such low occurrence rates are by no means accessible to any conventional simulation techniques even with the most powerful computing system available. The sampling difficulties for these long-time scale events are caused by the large free-energy barriers (or low probabilities for the occurrence of clusters near the critical nucleus size) and the inherent micro-heterogeneity of the phase space. Although the former problem can now be surmounted by a host of free-energy-based methods (including umbrella sampling43), a separate approach is still required to deal with the latter problem. For example, for vapor-liquid nucleation, the micro-heterogeneity arises from the presence of a spectrum of microphase regions (e.g., monomers and clusters). These microphase regions differ to a great extent on both energetic and entropic factors. In contrast, the random displacements used in the conventional Metropolis Monte Carlo scheme44 and the force-driven diffusion employed by molecular dynamics lack the balance of these two factors, which leads to a slow transfer of particles between the microphase regions. The AVBMC technique45,46 was included in the AVUS-HR approach to overcome this problem. In this technique, the space surrounding a molecule was explicitly divided into the associating and nonassociating regions to allow for direct transfer between microphase regions and, thereby, bypass the time and spatial constraints imposed on molecular dynamics and Metropolis Monte Carlo techniques. The incorporation of the configuration-bias Monte Carlo scheme47-49 further improves its efficiency and, most importantly, allows the extension of this method to molecules with articulated structures. In addition, the probability of observing rare critical nuclei is enhanced via the umbrella sampling technique. To let clusters of all sizes of interest be sampled evenly in the simulation, the umbrella potential is chosen as the negative of the NFE, which is solved iteratively.38,39 For computational efficiency, all simulations were carried out using the grand-canonical version of the nucleation algorithm,38,39 where the interactions between the cluster and the gas phase are neglected. As demonstrated previously, this approximation is acceptable in the low-temperature and lowdensity cases where these interactions are negligible.38,39 Since both n-nonane and ethanol are chain molecules, an energy-based Stillinger-type cluster criterion was employed, in which a cluster is defined as a group of molecules of which every molecule has at least one neighbor in the group with an interaction energy less than Uclsa negative threshold energy close in magnitude
to the simulation temperature.32,38-40 In all calculations, the TraPPE-UA50,51 force field was used for ethanol and n-nonane while the TIP4P52 model was used for water. In both TraPPEUA and TIP4P force fields, the intermolecular interactions were modeled by pairwise-additive potentials and many-body effects were approximated in a mean-field way to the thermodynamic state for which the parameters were derived. Although an explicit inclusion of the many-body polarization effects would be desirable for a quantitative comparison to experiments for nucleation events, it increases the computational expense significantly.40,53 On the other hand, previous simulations using these nonpolarizable force fields have shown that they were able to reproduce the nonideal nucleation behavior observed for such nonideal binary systems.32 All simulations were carried out at T ) 230 K except as noted explicitly. 3. Simulation Results and Discussions The calculated contours of the two-dimensional NFEs shown in Figure 1 reveal one of the most striking findings of this investigation; that is, the nucleation mechanism is entirely different between these three seemingly simple but common binary mixtures. While for both water/ethanol and n-nonane/ water systems, the saddle point can be easily located on the NFE maps to allow for an unambiguous identification of the nucleation pathways, for the n-nonane/ethanol mixture, the saddle point stretched all the way from the n-nonane-rich to the ethanol-rich domain. This implies that multiple nucleation pathways may coexist on the nucleation map, in sharp contrast to the other two systems where either one (for water/ethanol) or two (for n-nonane/water) major nucleation channels are present. However, small modifications of the gas-phase densities result in a rapid loss of this feature. For example, at nnonane ) v -8 Å-3, the saddle point ) 4.3 × 10 1.1 × 10-7 Å-3 and nethanol v region for this binary system is no longer widely open (see NFE contour plots shown in Figure 2a). Instead, it is rather localized toward the ethanol-rich sidesan indication of very few nucleation channels present on this contour map resembling a normal nucleation system. This is presumably due to a large increase of the NFEs for the n-nonane-enriched clusters caused even by a small decrease of the n-nonane gas-phase density. Similarly, a small decrease of the ethanol gas-phase density also leads to a significant rise in the NFEs for the ethanol-enriched clusters. As shown in Figure 2b, the saddle-point region shifts to the n-nonane-enriched region at nnonane ) 1.8 × 10-7 Å-3 and v ethanol -8 -3 ) 2.7 × 10 Å . In fact, it exhibits two independent nv domains separated by an ellipsoidal island, indicating that
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Figure 2. Contour of the two-dimensional NFEs (in units of kBT) as a function of the number of molecules for the two components involved in ) 1.1 × 10-7 Å-3 and nethanol the binary n-nonane/ethanol mixture calculated from the simulation at (a) nnonane ) 4.3 × 10-8 Å-3 and (b) nnonane ) v v v -8 -3 1.8 × 10-7 Å-3 and nethanol ) 2.7 × 10 Å . The contour levels with NFE values of 45, 50, 55, and 60 k T were depicted as lines for a clear view B v of the NFE profiles near the saddle point.
Figure 3. Plots of reduced onset activities. In panel a, the solid (circles) and dotted (squares) lines represent the simulation (experimental8-10) data for the binary n-nonane/ethanol and n-nonane/water mixtures, respectively. In panel b, the solid and dotted lines represent the simulation data for the binary water/ethanol mixture obtained at 260 K32 and 230 K, respectively, whereas the experimental results obtained by Viisanen et al.3 at 260 K were shown as diamonds. The dashed straight line corresponds to the ideal case. The asterisk symbols denote the conditions at which the contours of the two-dimensional NFEs shown in Figures 1 and 2 were computed.
nucleation could either proceed through pure n-nonane or clusters containing around five ethanol molecules. The latter channel is expected to become less important if the ethanol gasphase density further decreases. The NFE contours detail a rather surprising nucleation mechanism for the binary n-nonane/ethanol system. In particular, the appearance of multiple nucleation channels at certain combinations of gas-phase densities is directly at odds with the classical view of the nucleation process,24-31 which is based on an assumption that nucleation proceeds on a well-defined path via the formation of a critical nucleus with a well-defined structure and composition, upon which the theoretical formalism of the nucleation rate can be derived. Considering that this peculiar feature has not been discovered before, nor even speculated from the experimental data obtained by Viisanen et al.10 on the binary n-nonane/ethanol mixtures, it remains to be shown whether these contour plots are consistent with the nonideal nucleation behavior observed experimentally for these systems. In the interest of brevity, we will focus on two important aspects, the onset activities and the content of the critical nuclei, which both exhibit puzzling nonideal behavior, as inferred from the experiments.10 Plotted in Figure 3a are the various combinations of onset activities54 for the vapor-liquid nucleation of the binary n-nonane/ethanol mixture. These onset activities are normalized by the activities of neat n-nonane and neat ethanol vapors and correspond either to a constant nucleation rate of
Figure 4. Average mole fraction of n-nonane in the critical nuclei as a function of the normalized activity fraction of n-nonane obtained for the binary n-nonane/ethanol and n-nonane/water mixtures. Symbols as in Figure 3a.
107 droplets/cm3/s as measured experimentally3,4,8-11 or to a combined nucleation barrier height55 of around 50.6 kBT as calculated from the simulations. The overall agreement with the experiments is very satisfactory. In particular, both the experimental and simulation data show that the nucleation of the n-nonane/ethanol system occurs at much higher activities than expected for an ideal mixture (for which the reduced onset activities simply fall on a straight line and sum to unity). These higher onset activities signal a reluctant conucleation between n-nonane and ethanol, which becomes more apparent if one compares these data to those for the n-nonane/water mixture, where independent nucleation of n-nonane and water occurs. In contrast, for the water/ethanol system, the onset activities curved to the opposite side for both the experiments and the simulation (see Figure 3b), an indication of mutual nucleation enhancement.32 The agreement between the simulation and the experiments also extends to the composition for the critical nuclei (see Figure 4).54 Both indicate that the composition of the critical nuclei evolves in a nonlinear way as a function of the normalized activity fraction, fnonane ) (anonane/a0nonane)/(aethanol/a0ethanol + anonane/a0nonane). The sigmoidal shape of these plots arises mainly from the depletion of n-nonane at low n-nonane activity or the depletion of ethanol at low ethanol activity, providing another evidence of the reluctant conucleation between these two species. This also signals a transition of the nucleation mechanism from the ethanol-dominated nucleation channels at low n-nonane activity to the n-nonane-dominated ones at low ethanol activity, consistent with the NFE maps shown in Figures 1 and 2. While at intermediate compositions (i.e., 0.44 < fnonane < 0.58), multiple nucleation channels are expected since a significant mixing of the two components is observed. Although
3514 J. Phys. Chem. B, Vol. 110, No. 8, 2006
Figure 5. Concentration of the critical nuclei as a function of composition, δ () nnonane - nethanol), obtained at a normalized n-nonane activity fraction of 0.5 (or the conditions specified in Figure 1) for the binary n-nonane/ethanol and n-nonane/water mixtures. Symbols as in Figure 3a.
McKenzie and Chen
Figure 7. (a) Radial density profiles and (b) probability density profiles along the z axis for the n-nonane/ethanol system averaged over clusters containing a roughly equal number (20 ( 2) of n-nonane and ethanol molecules. The z axis is defined to be the one that passes through the two centers of mass, one calculated for ethanol molecules only and the other for n-nonane molecules, with the origin positioned at the center of mass of the whole cluster. The solid, dashed, dotted, and dasheddotted lines represent the number density profiles for the ethanol oxygen atoms, the ethanol hydrogen atoms, the ethanol carbon atoms, and the n-nonane carbon atoms, respectively.
Figure 8. Distributions of ethanol molecules over hydrogen-bonded clusters with aggregation number n found in all the n-nonane/ethanol clusters sampled (solid line), in those clusters near the saddle-point region of Figure 1c, that is, with an NFE value between 55 and 60 kBT (dotted), or in the clusters containing 20 n-nonane molecules and 20 ethanol molecules (dashed). Figure 6. Representative snapshots of clusters consisting of 40 molecules shared equally between the two components obtained for the binary n-nonane/ethanol (top two), water/ethanol (bottom left), and water/n-nonane (bottom right) mixtures. Color notation: hydrogen (white), water oxygen (red), ethanol oxygen (blue), and alkyl tails (green stick).
the good agreement between the experiments and the simulations seems to suggest the applicability of using the nucleation theorem10,16-18 in the experimental interpretation of the average content of the critical nuclei (i.e., determined from the slopes of the nucleation rate surface), caution is needed to use this average content of the critical nuclei to identify the true saddle point in the nucleation pathway.19 As shown from the NFE contours, a wide range of compositions could be present for the critical nuclei. A more quantitative picture can be provided by plotting the density distributions for the critical cluster nuclei as a function of the composition (see Figure 5). As should be expected, this distribution is relatively more flat for the n-nonane/ethanol system, in accordance with a saddle-point region that is more open at this condition (see Figure 1c). More interestingly, it exhibits a concave shape with a wide dip centered around a δ () nnonane - nethanol) value of 0, resembling the n-nonane/water mixture. This dip suggests that a full miscibility remains unfavorable despite that the mole fraction of ethanol averaged over all the critical nuclei is very close to 0.5. In fact, similar to that for the n-nonane/water mixture, this
value appears to accidentally arise from the formation of two major critical nuclei, one n-nonane-enriched and the other ethanol-enriched. Such a behavior was also noticed previously for partially miscible mixtures by both simulations and DFT studies employing relatively simple Lennard-Jonesium (LJ) models.19-23 It should be noted that the appearance of the other dip at a high δ value is consistent with the presence of an elevated ellipsoidal island on the NFE map toward the n-nonane dominated region (see Figure 1). This indicates that clusters with a molecular content of about two ethanols are discouraged while clusters with five ethanols tend to be favored, presumably because of the formation of stable hydrogen-bonded aggregates (i.e., cyclic ethanol pentamers).56,57 Visual inspection of the cluster configurations provides further support on the reluctant mixing behavior observed between n-nonane and ethanol (see Figure 6). Even for critical nuclei with equal numbers of n-nonane and ethanol molecules, these two components still tend to separate from each other microscopically, as a result of the aggregation of ethanol molecules through hydrogen-bonding interactions.56,57 In addition, these hydrogen-bonded aggregates attempt to wrap around the nonpolar core formed by the n-nonane molecules, resembling, to some extent, the core-shell structure of the water/ethanol clusters. Occasionally, further clustering of these aggregates or the formation of one large hydrogen-bonded aggregate leads to a fully phase-separated double-layer structure, similar to that observed for the n-nonane/water mixture. Analysis of the density
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Figure 9. Contour of the two-dimensional NFEs (in units of kBT) as a function of the number of molecules for the two components involved in the binary n-nonane/ethanol mixture calculated from the simulation at (a) T ) 300 K, nnonane ) 3.9 × 10-6 Å-3, and nethanol ) 4.2 × 10-6 Å-3 and v v -5 Å-3, and nethanol ) 3.8 × 10-5 Å-3. In panel a, the curves depicted for a clear view of the NFE profiles near (b) T ) 360 K, nnonane ) 1.7 × 10 v v the saddle point are for contour levels with NFE values of 30, 35, 40, and 45 kBT, whereas in panel b, these curves correspond to levels with NFE values of 20, 25, 30, and 35 kBT.
profiles of mixed clusters containing roughly equal number (20 ( 2) of n-nonane and ethanol molecules also supports this picture. As shown in the radial density profile (see Figure 7a), there is a noticeable enrichment of the ethanol oxygen and hydrogen densities approaching to the surface, which is consistent with the core-shell structure pattern shown above. On the other hand, the probability density profiles along the axis that passes through the centers of mass of the two species show the kind of separation expected for a double-layer structure (see Figure 7b). This appears to resemble those found by Talanquer and Oxtoby21 for partially miscible LJ mixtures, whereas in ten Wolde and Frenkel’s simulation work employing similar LJ models, no evidence of phase separation inside the critical nucleus was found. However, neither chainlike molecules of n-nonane nor the orientationally dependent hydrogen-bonding interactions (involved by ethanol) can be modeled through the use of a spherical LJ bead.39 Thus, it is difficult to draw connections between the systems examined by previous theoretical studies employing LJ models and the n-nonane/ethanol mixture studied here with atom-based force fields. A majority of these hydrogen-bonded aggregates58 consist of between four and eight ethanol molecules, as shown by those representative snapshots and also by the distribution of molecules over aggregate size plotted in Figure 8. In particular, cyclic pentamers are extremely stable, which gives rise to a sharp peak in the distribution as well as the aforementioned presence of an ellipsoidal island on the NFE maps. Clearly, a very diverse spectrum of hydrogen-bonded aggregates are present inside these clusters, which seems to already resemble those observed for the bulk alcohol phases.51 It is evident from both the structural analysis and the composition of the critical nuclei that the n-nonane/ethanol clusters (up to the size range sampled by the current simulations) display an inherent microscopic immiscibility. Would this microscopic separation persist even for conditions where these two species are known to be fully miscible macroscopically? Some additional nucleation simulations were carried out for the n-nonane/ethanol mixture at higher temperatures (300 and 360 K). As evident from both the NFE maps (see Figure 9) and the composition plots of the critical cluster nuclei (see Figure 10), the saddle point region remains broad for intermediate compositions even up to T ) 360 K. Despite this feature, at 360 K, the saddle surface already adopts a more normal shape. Similarly, we might expect such a classical picture of the nucleation process (i.e., formation of a mixed critical nuclei with a welldefined composition) for this system at even lower temperatures
Figure 10. Concentration of the critical nuclei as a function of composition, δ () nnonane - nethanol), obtained for the binary n-nonane/ ethanol mixture at the conditions specified in Figure 9. The dashed and solid lines correspond to the results at 300 and 360 K, respectively.
(as long as no miscibility gap is present) by decreasing the gasphase densities so that the formed critical clusters are sufficiently large. 4. Conclusions In summary, the AVUS-HR approach allows the use of a realistic, atom-based force field to study nucleation events for systems of practical interest. Using the TraPPE-UA force field, it was found that these simulations were able to reproduce the different nonideal nucleation behaviors observed experimentally for the water/ethanol, n-nonane/water, and n-nonane/ethanol mixtures. In particular, good agreements were found between the experiments and the simulation for both onset activities and the composition of the critical nuclei. Additional analysis on the composition and structure provides further support on the observed nonideal behavior for these systems (i.e., the reluctant mixing/conucleation between n-nonane and ethanol). Moreover, from the calculated NFE contour maps, it was shown that these behaviors arose from rather distinct nucleation mechanisms, with the binary n-nonane/ethanol system displaying the most striking one, where multiple nucleation channels coexist. Thus, those theoretical models based on the assumption that nucleation proceeds through the formation of mixed critical clusters with a certain well-defined composition would have to be modified accordingly for an accurate prediction of the nucleation rates for these systems. Acknowledgment. Financial support from the LSU startup fund, the National Science Foundation (CHE-0448918), and the Petroleum Research Fund administered by the American Chemical Society (Grant 41933-G9) is gratefully acknowledged.
3516 J. Phys. Chem. B, Vol. 110, No. 8, 2006 Part of the computer resources were provided by the Center for Computation and Technology at LSU. References and Notes (1) Zahoransky, R. A.; Peters, F. J. Chem. Phys. 1985, 83, 6425. (2) Schmitt, J. L.; Whitten, J.; Adams, G. W.; Zalabsky, R. A. J. Chem. Phys. 1990, 92, 3693. (3) Viisanen, Y.; Strey, R.; Laaksonen, A.; Kulmala, M. J. Chem. Phys. 1994, 100, 6062. (4) Strey, R.; Viisanen, Y.; Wagner, P. E. J. Chem. Phys. 1995, 103, 4333. (5) Peters, F.; Rodemann, T. Exp. Fluids 1998, 24, 300. (6) Wyslouzil, B. E.; Heath, C. H.; Cheung, J. L.; Wilemski, G. J. Chem. Phys. 2000, 113, 7317. (7) Peeters, P.; Hruby´, J.; van Dongen, M. E. H. J. Phys. Chem. B 2001, 105, 11763. (8) Wagner, P. E.; Strey, R. J. Phys. Chem. B 2001, 105, 11656. (9) Viisanen, Y.; Strey, R. J. Chem. Phys. 1996, 105, 8293. (10) Viisanen, Y.; Wagner, P. E.; Strey, R. J. Chem. Phys. 1998, 108, 4257. (11) Strey, R.; Viisanen, Y. J. Chem. Phys. 1993, 99, 4693. (12) Wyslouzil, B. E.; Seinfeld, J. H.; Flagan, R. C.; Okuyama, K. J. Chem. Phys. 1991, 94, 6827. (13) Looijmans, K. N. H.; Luijten, C. C. M.; van Dongen, M. E. H. J. Chem. Phys. 1995, 103, 1714. (14) Anisimov, M. P.; Koropchak, J. A.; Nasibulin, A. G.; Timoshina, L. V. J. Chem. Phys. 1998, 109, 10004. (15) Heath, C. H.; Streletzky, K. A.; Wyslouzil, B. E.; Wo¨lk, J.; Strey, R. J. Chem. Phys. 2003, 118, 5465. (16) Oxtoby, D. W.; Kashchiev, D. J. Chem. Phys. 1994, 100, 7665. (17) Oxtoby, D. W.; Laaksonen, A. J. Chem. Phys. 1995, 102, 6846. (18) Ford, I. J. J. Chem. Phys. 1996, 105, 8324. (19) ten Wolde, P. R.; Frenkel, D. J. Chem. Phys. 1998, 109, 9919. (20) Oxtoby, D. W.; Laaksonen, A. J. Chem. Phys. 1995, 102, 6846. (21) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1996, 104, 1993. (22) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1997, 106, 3673. (23) Napari, I.; Laaksonen, A. J. Chem. Phys. 1999, 111, 5485. (24) Becker, R.; Do¨ring, W. Ann. Phys. 1935, 24, 719. (25) Volmer, M. Kinetik der Phasenbildung; Steinkopff: Dresden, Germany, 1939. (26) Reiss, H. J. Chem. Phys. 1950, 18, 840. (27) Stauffer, D. Aerosol Sci. 1976, 7, 319. (28) Wilemski, G. J. Phys. Chem. 1987, 91, 2492. (29) Jaecker-Voirol, A.; Mirabel, P.; Reiss, H. J. Chem. Phys. 1987, 87, 4849. (30) Laaksonen, A.; Taalnquer, V.; Oxtoby, D. W. Annu. ReV. Phys. Chem. 1995, 46, 489. (31) Noppel, M. J. Chem. Phys. 1998, 109, 9052.
McKenzie and Chen (32) Chen, B.; Siepmann, J. I.; Klein, M. L. J. Am. Chem. Soc. 2003, 125, 3113. (33) ten Wolde, P. R.; Frenkel, D. J. Chem. Phys. 1998, 109, 9901. (34) Yoo, S.; Oh, K. J.; Zeng, X. C. J. Chem. Phys. 2001, 115, 8518. (35) Oh, K. J.; Gao, G. T.; Zeng, X. C. Phys. ReV. Lett. 2001, 86, 5080. (36) Kusaka, I.; Wang, Z.-G.; Seinfeld, J. H. J. Chem. Phys. 1998, 108, 3416. (37) Kusaka, I.; Oxtoby, D. W. J. Chem. Phys. 1999, 110, 5249. (38) Chen, B.; Siepmann, J. I.; Oh, K. J.; Klein, M. L. J. Chem. Phys. 2001, 115, 10903. (39) Chen, B.; Siepmann, J. I.; Oh, K. J.; Klein, M. L. J. Chem. Phys. 2002, 116, 4317 (40) Chen, B.; Siepmann, J. I.; Klein, M. L. J. Phys. Chem. A 2005, 109, 1137. (41) Wilding, N. B. Phys. ReV. E 1995, 52, 602. (42) Potoff, J. J.; Panagiotopoulos, A. Z. J. Chem. Phys. 1998, 109, 10914. (43) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1974, 28, 578. (44) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (45) Chen, B.; Siepmann, J. I. J. Phys. Chem. B 2000, 104, 8725. (46) Chen, B.; Siepmann, J. I. J. Phys. Chem. B 2001, 105, 11275. (47) Siepmann, J. I.; Frenkel, D. Mol. Phys. 1992, 75, 59. (48) Frenkel, D.; Mooij, G. C. A. M.; Smit, B. J. Phys.: Condens. Matter 1992, 4, 3053. (49) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1999, 103, 4508. (50) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1998, 102, 2569. (51) Chen, B.; Potoff, J. J.; Siepmann, J. I. J. Phys. Chem. B 2001, 105, 3093. (52) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926. (53) Chen, B.; Siepmann, J. I. Theo. Chem. Acc. 1999, 103, 87. (54) In the simulation, this property was calculated through averaging the composition of all clusters with the critical cluster size weighted by their corresponding probability densities, whereas for the experiments, this property was derived from the application of the nucleation theorem10,16-18 to the measured nucleation rate surface. (55) In the calculation of this barrier, the two-dimensional NFE profiles were projected onto a single coordinate, the combined size of the cluster, as follows: exp[-∆Gtot(n)/kBT] ) ∑nn1)0 exp[-∆G(n1, n - n1)/kBT]. A value of around 50.6 kBT for the barrier height corresponds to a concentration of 100 droplets/cm3 for the critical nuclei, from the following relationship between them: Ptot(n) ) exp[-∆Gtot(n)/kBT], where Ptot(n) is the concentration of clusters with a combined size n in units of droplets/Å3. (56) Stubbs, J. M.; Siepmann, J. I. J. Am. Chem. Soc. 2005, 127, 4722. (57) Stubbs, J. M.; Siepmann, J. I. J. Phys. Chem. B 2002, 106, 3968. (58) A cutoff distance of 3.5 Å for oxygen-oxygen separation was used to determine whether two ethanols belong to the same hydrogen-bonded aggregate.51
