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J. Phys. Chem. B 2001, 105, 11656-11661
Two-Pathway Homogeneous Nucleation in Supersaturated Water-n-Nonane Vapor Mixtures† P. E. Wagner Institut fu¨ r Experimentalphysik, UniVersita¨ t Wien, Boltzmanngasse 5, A-1090 Wien, Austria
R. Strey* Institut fu¨ r Physikalische Chemie, UniVersita¨ t zu Ko¨ ln, Luxemburger Strasse 116, D-50939 Ko¨ ln, Germany ReceiVed: April 18, 2001; In Final Form: August 2, 2001
We report an experimental study of homogeneous nucleation in supersaturated vapor mixtures of n-nonane and water exhibiting a pronounced miscibility gap in the liquid state. Using the expansion pulse method, we measured homogeneous nucleation rates J for unary water and n-nonane vapors as well as for vapor mixtures with six different ratios of the partial vapor phase activities a1 and a2. All measurements were performed at the same constant nucleation temperature of 230 K. The measured nucleation rates shape a nucleation rate surface in the (ln J)-a1-a2 space which is found to be qualitatively different from previously observed nucleation rate surfaces in that it exhibits two rather flat parts separated by a limited region with comparatively strong curvature. There is strong indication that the nucleation in water-n-nonane vapor mixtures can be viewed as a superposition of two simultaneous unary nucleation processes. In the flat parts of the nucleation rate surface the nucleation rates are dominated by the nucleation of one vapor component and are thus practically independent of the vapor-phase activity of the other component. On the other hand, in the region of strong curvature it appears that we have observed simultaneous homogeneous nucleation via two kinds of critical nuclei and thus via two separate pathways in the free energy surface. For comparatively low n-nonane vapor content, water vapor was found to nucleate, while the Mie-scattering pattern indicates mainly n-nonane growth at later stages. Thus, the water droplets apparently serve as heterogeneous nuclei and “catalyze” the subsequent condensation of n-nonane onto the droplets.
Introduction Homogeneous nucleation processes are generally viewed as occurring via the formation of critical clusters of a particular size and composition being in unstable equilibrium with the mother phase. A number of experimental studies on homogeneous nucleation in various unary and binary systems have been performed [see refs 1-3 for review]. In some cases nucleation rates as a function of supersaturation were determined from which the size and composition of the critical clusters were obtained by applying the nucleation theorem.4-7 Recently, the first quantitative investigation of the homogeneous nucleation process in ternary water-nonane-1-butanol vapor mixtures was reported.4 In that paper the binary water-nonane was treated as one of the limiting binary cases. Here we will study this system in more detail. The classical description of homogeneous nucleation8,9 is based on the capillarity approximation; i.e., the critical clusters are assumed to be spherical droplets with macroscopic properties. Despite this idealizing assumption, the classical nucleation theory appears to be fairly successful in predicting the unary nucleation rates in supersaturated one-component vapors. Also the behavior of binary systems forming nearly ideal mixtures in the liquid state is reasonably well described by the classical model.6 For the case of nonideal, surface-enriching mixtures, †
Part of the special issue “Howard Reiss Festschrift”. * To whom correspondence should be addressed. E-mail: rstrey@ uni-koeln.de. Fax: 49 221 470 5104.
however, severe discrepancies were observed,10-12 and the classical theory is apparently not applicable in this case. These difficulties seem to be connected with nonuniformities of the cluster composition occurring for surface-enriching mixtures.13,14 The situation can be improved by accounting for the internal structure of the critical clusters in so-called explicit cluster models.15-17 In recent years the classical nucleation theory has been reconsidered by several authors. By adjusting the expression for the cluster formation free energy, a self-consistent version of the classical theory can be obtained,18-20 leading to a somewhat improved prediction of the temperature dependence of the nucleation rates in some cases.21 In the diffuse interface theory22 the clusters are described by assuming a characteristic interface thickness. On the basis of Fisher’s drop model,23,24 various semiphenomenological approaches were introduced25-29 accounting for a curvature dependence of the surface tension in different ways. Recently, scaling relations were introduced,30,31 which constrain the departure from the classical nucleation theory and are useful in connection with the development of phenomenological versions of the nucleation theory. The limitations connected with the capillarity approximation can be avoided by using a nonclassical nucleation theory based on density functional methods.32 This approach has been applied to nonpolar Lennard-Jones fluids33,34 and dipolar Stockmayer fluids.35 The study of binary nucleation using density functional methods36,37 includes a description of the cluster structure and
10.1021/jp011460x CCC: $20.00 © 2001 American Chemical Society Published on Web 11/03/2001
Nucleation in Water-n-Nonane Mixtures thus accounts for surface enrichment. For quantitative comparisons to experimental data, detailed information on the intermolecular potentials is required. A semiempirical fit procedure has recently been suggested38 to obtain realistic expressions for the intermolecular potentials for various compounds. Most nucleation studies reported in the literature so far deal with systems completely miscible in the liquid state, and the critical nucleus is viewed as a cluster of particular size containing a mixture of the condensing liquids with certain composition. Within the framework of the classical nucleation theory, the properties of the critical nucleus can be described using macroscopic quantities. A different and more complex situation is encountered for systems exhibiting a miscibility gap in the liquid state, and therefore a qualitatively different nucleation behavior might be expected.39 However, the results of experimental studies of nucleation in water-1-alcohol vapor mixtures40,41 show a homologous trend. The miscibility gaps occurring for the systems water-1-butanol, water-1-pentanol, and water-1-hexanol do not result in any qualitative difference in the nucleation behavior. Apparently, a macroscopic miscibility gap does not necessarily influence the microscopic behavior. To further investigate possible influences of the macroscopic miscibility of the condensing liquids on the nucleation process, we have considered the binary water-n-nonane system, which exhibits a particularly pronounced miscibility gap in the liquid state. The unary nucleation of both water and n-nonane has already been studied by various authors.42-47 In the present paper we report experiments on the nucleation in supersaturated water-n-nonane vapor mixtures. The results of the present study indicate a significant influence of the miscibility gap on the nucleation behavior and suggest the simultaneous presence of two kinds of critical nuclei. Experimental conditions are identified where significant homogeneous nucleation appears to occur simultaneously via both kinds of critical nuclei and thus via two separate pathways. The applicability of the nucleation theorem in this case is discussed. Experiment The homogeneous nucleation rates presented in this paper were measured using the nucleation pulse method. Supersaturated vapor was obtained by adiabatic expansion of a vaporcarrier gas mixture in an expansion chamber. Significant nucleation actually occurred during a short nucleation pulse. The number concentration of the subsequently growing droplets was determined from light scattering, and thus the nucleation rate was obtained. The experimental system as well as the nucleation pulse method are described in some detail elsewhere.41,48 Here the main features of the experiment are presented emphasizing those aspects which are particularly relevant for this study. Water and n-nonane vapors were generated by evaporation from liquid surfaces. Binary vapor mixtures have been frequently obtained from liquid mixtures with properly chosen compositions. This procedure, however, is inapplicable in the case of partially immiscible liquids as considered in the present study. Accordingly, as indicated in Figure 1, the two vapor components were generated by evaporation from the respective pure liquids in separate vaporizers, V1 and V2. Furthermore, high-purity carrier gas (argon) was obtained from a gas container, G. Selectable quantities of water vapor, n-nonane vapor, and argon were then sequentially passed into the mixing receptacle, R, and thus a binary vapor mixture with arbitrarily selectable composition was prepared in R. Before each single measurement the measuring chamber, M, was flushed at a selectable constant
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11657
Figure 1. Schematic diagram showing the vapor generator and expansion chamber: V1, V2, vaporizers; G, carrier gas container; R, mixing receptacle; RV, electronically controlled regulation valve; M, measuring chamber; E, expansion volume; C, compression volume; VAC, vacuum system.
total pressure via an electronically controlled regulation valve, RV. Thereby, a binary water-n-nonane vapor mixture of selected composition with well-defined partial vapor pressures was fed into the measuring chamber. A nucleation pulse with a duration of about 1 ms was then achieved by a fast adiabatic expansion into a low-pressure expansion volume, E, followed in a well-defined time sequence by a comparatively small recompression from an elevatedpressure compression volume, C. During this pulse, nucleation occurred and subsequent droplet growth was observed in the measuring chamber. The temperature during the nucleation pulse was chosen to be 230 K for all nucleation experiments reported in this study. Constant nucleation temperature was achieved by keeping the temperature of the measuring chamber fixed and performing each expansion with virtually the same pressure drop. During each measurement series a fixed composition of the binary vapor mixture was considered as prepared in the mixing plenum, R. The vapor-phase activities of the vapors considered were varied during each measurement series by varying the total pressure at which the measuring chamber, M, was flushed before each single measurement. Thereby, each measurement series refers to a fixed nucleation temperature and a fixed ratio of vapor-phase activities. The concentration of the droplets growing in the measuring chamber was determined using the constant-angle Mie-scattering (CAMS) method.49 The droplets were illuminated by a laser beam, and the light flux transmitted through the cylindrical measuring chamber and the light flux scattered by the droplets at a constant forward scattering angle θ of 15° were simultaneously monitored during the droplet growth process. Comparison of the normalized experimental scattered light flux to Lorenz-Mie theory allows an absolute and independent determination of the number concentration and diameter of the growing droplets at various times during the growth process. From the droplet number concentration and nucleation pulse duration, the nucleation rates were directly determined. From time-resolved droplet growth measurements, it has been shown that at the nucleation temperature of 230 K nucleation and condensational droplet growth are nearly decoupled for the vapor-phase activities considered. As far as the applicability of the constant-angle Mie-scattering method is concerned, it should be emphasized that the light scattering behavior of the growing particles depends on their shape and refractive index. The CAMS method is applicable to approximately spherical particles with known refractive index. Parts a and b of Figure 2 show the Mie-scattering intensities at
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Figure 2. Mie-scattering intensities at a forward scattering angle θ of 15° calculated as functions of the particle radius r for the refractive indices of water (a, top) and n-nonane (b, bottom).
a constant forward scattering angle θ of 15° calculated as functions of the particle radius r for the refractive indices of water and n-nonane, respectively. Comparison of these light scattering curves shows that from the morphology of the light scattering patterns information concerning the composition of the growing particles can be obtained. Condensation of totally miscible liquids will result in spherical droplets with a certain refractive index, and thus the CAMS method is applicable. In the case of partially immiscible liquids, a somewhat different behavior might be expected, as will be discussed in the next section. In connection with the present study, the following experimental features of the nucleation pulse system are particularly important: (1) Isothermal data sets are obtained. (2) Data sets with a fixed ratio of vapor-phase activities are obtained. (3) Vapor mixtures are generated regardless of the miscibility of the condensing liquids. (4) The CAMS method allows a direct determination of the nucleation rates. (5) Nucleation and growth are nearly decoupled for the experimental conditions considered. (6) The CAMS method provides information concerning the composition of the condensing droplets. (7) The CAMS method is restricted to approximately spherical particles. Results Homogeneous nucleation has been studied for unary water and n-nonane vapors and furthermore for binary mixtures of these vapors with six different compositions. Argon was used as the carrier gas. Nucleation rates were measured over ranges of vapor-phase activities at a constant temperature of 230 K. During each measurement series the composition of the binary vapor remains unchanged and can be characterized by the actiVity fraction
x)
a2 a 1 + a2
(1)
of n-nonane, where a1 and a2 are the vapor-phase activities of water and n-nonane, respectively. In parts a and b of Figure 3 the experimental nucleation rates are plotted versus a1 and a2, respectively. Each set of data refers to a particular constant value
Figure 3. Experimental nucleation rate J shown as a function of the vapor-phase activities of water (a, top) and n-nonane (b, bottom). Each data set refers to a particular value of the n-nonane activity fraction x. The dashed lines indicate reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1.
Figure 4. Three-dimensional presentation of the experimental nucleation rate J shown as a function of the vapor phase activities a1 and a2 shaping the nucleation rate surface J ) J(a1,a2). The intersections of this surface with planes at the constant reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1 are indicated by full symbols and yield the corresponding onset activities a10 and a20.
of the n-nonane activity fraction x. In the present study n-nonane activity fractions x ) 0, 0.6111, 0.7873, 0.8540, 0.8642, 0.8763, 0.9217 and 1 have been considered. It can be seen from Figure 3 that below an n-nonane activity fraction around 0.85 the nucleation rates show no significant dependence on the n-nonane activity a2, while above this n-nonane activity fraction no significant dependence of the nucleation rates on the water activity a1 is observed. Reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1 are indicated and will be discussed below. As shown previously,6,50 the individual nucleation rate curves measured at various constant activity fractions x determine a nucleation rate surface. A three-dimensional presentation of the nucleation rate J shown as a function of both a1 and a2 is presented in Figure 4. In this diagram the different data sets are viewed as separate curves indeed shaping a surface in the (ln J)-a1-a2 space. To further illustrate the shape of this nucleation rate surface, the curves corresponding to reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1 are indicated
Nucleation in Water-n-Nonane Mixtures
Figure 5. Mutual dependence of the onset activities a10 and a20 corresponding to the reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1 for the water-n-nonane system. The positions of the areas of strong curvature are indicated by the full straight line.
as well. It is notable that this nucleation rate surface apparently consists of rather flat portions separated by a region with comparatively strong curvature. Accordingly, the shape of the water-n-nonane nucleation rate surface is found to be qualitatively different as compared to the previously measured nucleation rate surface for the 1-pentanol-water system,41 which also exhibits a large, macroscopic miscibility gap. To further describe the behavior of the water-n-nonane system, we consider the onset activities a10 and a20 of water and n-nonane vapor, respectively, corresponding to the abovementioned reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1. In Figure 4 the intersections of the nucleation rate surface with the J ) J0 planes are indicated. The mutual dependence of the onset activities a10 and a20 corresponding to the different values of J0 is shown in Figure 5. It can be seen that for n-nonane onset activities a20 below certain limiting activity values the water onset activities a10 remain nearly unchanged. Conversely, for water onset activities a10 below certain limiting activity values the n-nonane onset activities a20 stay almost constant. This is an indication that within certain ranges of vapor-phase activities the presence of one of the two vapor components does not significantly affect the nucleation of the other component, and thus the corresponding portions of the curves in Figure 5 are nearly linear. Between these rather linear parts of the curves shown in Figure 5, limited regions with strong curvature are observed. Close inspection reveals that the ratio of activities and thus the n-nonane activity fraction x corresponding to the above-mentioned regions of strong curvature vary with varying reference nucleation rate J0. Accordingly, the areas of strong curvature are situated along a slanted line. This behavior will be further discussed in the next section. In connection with the above-mentioned fact that the presence of one of the two vapor components does not seem to affect the nucleation of the other component, the following observation is notable. As mentioned above, from the morphology of the light scattering patterns observed (see Figure 2), information concerning the composition of the growing droplets can be obtained. In fact, for the case of nucleation of water vapor alone (x ) 0), according to the experimental light scattering curves, clearly the growth of water droplets was observed. However, in the presence of n-nonane vapor, even at the lowest n-nonane activity fraction considered (x ) 0.61), the experimental light scattering curves already indicate the growth of n-nonane droplets. At this activity fraction, however, the onset activity of water vapor has remained practically unchanged and the presence of n-nonane vapor has apparently not affected the
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11659
Figure 6. Experimental nucleation rates J in pure water vapor and in pure n-nonane vapor vs the corresponding vapor-phase activity a. Broken lines refer to the classical nucleation theory; full lines were obtained by applying the appropriate constant multiplicative factors (1/3 for water and 2000 for n-nonane).
nucleation process. These observations seem to indicate that particle formation has been initiated by homogeneous nucleation of water vapor and subsequently heterogeneous nucleation of n-nonane vapor occurs, leading to condensational growth of droplets predominantly consisting of n-nonane. In the cases of comparatively low n-nonane vapor content, water vapor apparently nucleates, thereby catalyzing the subsequent formation of n-nonane droplets. As a consequence, in all measurement series reported in this paper, except for the case of pure water vapor, the growth of nearly pure, spherical n-nonane droplets was observed. Hence, the CAMS method was applicable in all cases considered. In this connection it should be mentioned that experiments on particle growth by simultaneous condensation in supersaturated water-n-nonane vapor mixtures51 under certain conditions indicate the formation of nonspherical particles. Discussion At first we refer to the nucleation of the unary systems considered in the present study. The measured unary nucleation rates in pure water vapor and in pure n-nonane vapor are shown vs the corresponding vapor-phase activities in Figure 6. The broken lines in Figure 6 indicate the corresponding predictions by the classical nucleation theory.8 While the data for water nucleation are in fair agreement with theory, some deviations are observed for the n-nonane system. Satisfactory agreement can be achieved by applying single, constant multiplicative factors to the theoretical values (1/3 for water and 2000 for n-nonane). Accordingly, the slopes of the theoretical curves are found to be in good agreement with experiment, while the actual values of the nucleation rates are somewhat different. This behavior is consistent with previous experience concerning various compounds.43,47,52 The experimental data presented in the previous section indicate that for nucleation in water-n-nonane vapor mixtures within certain ranges of vapor-phase activities the presence of one of the two vapor components does not significantly affect the nucleation of the other component. Accordingly, it seems reasonable to consider the nucleation process in the water-nnonane vapor mixtures as a superposition of independent, simultaneous nucleation of the two unary vapors. In this case the experimentally observable total nucleation rate can be expressed as
J ) J 1 + J2
(2)
where J1 and J2 are the unary nucleation rates for water and n-nonane, respectively. Using eq 2, the total nucleation rate in water-n-nonane vapor mixtures can be calculated by means
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Figure 7. Three-dimensional presentation of the calculated nucleation rate J shown as a function of the vapor-phase activities a1 and a2. Calculations were based on the classical nucleation theory for the respective unary vapors accounting for constant multiplicative factors (1/3 for water and 2000 for n-nonane).
of the classical nucleation theory for the respective unary vapors accounting for the multiplicative factors mentioned above. Figure 7 shows a three-dimensional presentation of the calculated nucleation rate J as a function of the vapor-phase activities a1 and a2 of water and n-nonane, respectively. Comparison with the corresponding experimental nucleation rate surface, shown in Figure 4, reveals a quite similar behavior. The surface clearly consists of rather flat portions separated by a region with comparatively strong curvature. To allow a detailed quantitative comparison between experiment and theory, we consider the onset activities a10 and a20 of water and n-nonane vapor, respectively, corresponding to the reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1. From the above-mentioned calculations of the nucleation rate, based on the classical nucleation theory, these onset activities can in principle be determined. However, we refrain from doing so in view of the fact that we had to adjust the classical theory (cf. Figure 6). Rather, on the basis of the nucleation theorem,5-7 a model-free calculation of the mutual dependence of the abovementioned onset activities a10 and a20 can be performed without reference to a specific nucleation theory. Assuming a simultaneous and independent nucleation of unary water vapor and unary n-nonane vapor, the nucleation theorem allows the (excess) number of molecules in a critical cluster of kind i to be calculated as
∆ni* =
( ) ∂ ln Ji ∂ ln ai
(3)
T
where Ji is the unary nucleation rate and ai is the vapor-phase activity of kind i. It is important to note that the total nucleation rate J is experimentally observable, whereas the nucleation rates Ji cannot be measured individually. However, as shown by Oxtoby and Laaksonen,53 integration of eq 3 and using eq 2 results in the relation
( ) ( ) a10
a10unary
∆n1*
+
a20
a20unary
∆n2*
)1
(4)
describing the mutual dependence of the onset activities a10 and a20 referring to a certain onset nucleation rate J0. In eq 4 a10unary and a20unary are the onset activities for nucleation of the unary vapors referring to the reference nucleation rate J0. The quantities a10unary, a20unary, ∆n1* ) 27.8, and ∆n2* ) 17.6 were obtained from the nucleation rates measured for the unary
Figure 8. Mutual dependence of the onset activities a10 and a20 corresponding to the reference nucleation rates J0 ) 105, 107, and 109 cm-3 s-1 for the water-n-nonane system. The solid lines correspond to calculations using the nucleation theorem for independent nucleation of the two components.53 Experimental data are shown for comparison.
system water and n-nonane, and the resulting mutual dependence of the onset activities a10 and a20 is shown in Figure 8 and compared to the corresponding experimental values. These calculations are based on the assumption of simultaneous and independent nucleation of n-nonane and water vapor. As can be seen from Figure 8, at the low reference nucleation rate (J0 ) 105 cm-3 s-1) the experimental onset activities are in satisfactory quantitative agreement with the theoretical prediction. With increasing reference nucleation rates J0, increasing deviations of the experimental onset activities are observed; nevertheless, the qualitative behavior of the data remains in agreement with theory. The reasons for these deviations are not completely clear. A possible explanation might be related to the condensational growth of n-nonane droplets, starting immediately after nucleation of water vapor has occurred, and vice versa. According to the comparatively high vapor-phase activities considered, quite fast drop growth can be expected. This condensational growth process will lead to some increase of the system temperature and thus to a decrease of vapor-phase activities and nucleation rates, particularly at high droplet number concentrations and correspondingly at high nucleation rates. It is particularly notable that both the experimental data and the theoretical models show that the mutual dependence of the onset activities exhibits rather linear portions, which are separated by limited regions with strong curvature. Furthermore, for the experimental data as well as for the model calculations, the ratio of activities and thus the n-nonane activity fraction x corresponding to the above-mentioned regions of strong curvature vary with varying reference nucleation rate J0, and thus the areas of strong curvature are situated along a slanted line (compare to Figure 5). Apparently the experimental data are fully consistent with the corresponding model calculations and thus support the assumption of simultaneous and independent nucleation in the water-n-nonane system. Conclusions We have measured homogeneous nucleation rates J in supersaturated water-n-nonane vapor mixtures for various ratios of the vapor-phase activities a1 and a2. The measured nucleation rates shape a three-dimensional nucleation rate surface in the (ln J)-a1-a2 space which is found to be qualitatively different from previously observed nucleation rate surfaces. This peculiar behavior seems to be related to the quite pronounced miscibility gap of the two condensing liquids considered. The nucleation rate surface obtained exhibits two rather flat parts separated by
Nucleation in Water-n-Nonane Mixtures a limited region with comparatively strong curvature. According to the nucleation theorem, this region of strong curvature corresponds to fast changes of the composition of the critical nucleus.53 Accordingly, the observed region of strong curvature suggests the existence of two saddle points in the free energy surface corresponding to two different kinds of critical nuclei. Furthermore, these saddle points are practically located above the unary axes a1 and a2 for water and n-nonane nucleation, respectively. Quantitative comparison to corresponding model calculations indicates that the nucleation in the water-n-nonane vapor mixtures can be viewed as a superposition of two simultaneous unary nucleation processes. In the flat parts of the nucleation rate surface, just one of these processes dominates and the nucleation theorem can be applied to determine the number of molecules in the respective critical cluster. On the other hand, in the region of strong curvature of the nucleation rate surface, it appears that we have observed simultaneous homogeneous nucleation via two kinds of critical nuclei. Accordingly, care should be exercised in applying the nucleation theorem.53 Calculations based on the nucleation theorem assuming individual nucleation of water and n-nonane explain the observed peculiar shape of the experimental nucleation rate surface. Acknowledgment. We are indebted to Prof. M. Kahlweit and the Max-Planck-Gesellschaft for long-lasting support of this experiment. We thank Y. Viisanen for pleasant cooperation. References and Notes (1) Oxtoby, D. W. J. Phys.: Condens. Matter 1992, 4, 7627. (2) Heist, R. H.; He, H. J. Phys. Chem. Ref. Data 1994, 23, 781. (3) Laaksonen, A.; Talanquer, V.; Oxtoby, D. W. Annu. ReV. Phys. Chem. 1995, 46, 489. (4) Viisanen, Y.; Strey, R. J. Chem. Phys. 1996, 105, 8293. (5) Kashchiev, D. J. Chem. Phys. 1982, 76, 5098. (6) Strey, R.; Viisanen, Y. J. Chem. Phys. 1993, 99, 4693. (7) Oxtoby, D. W.; Kashchiev, D. J. Chem. Phys. 1994, 100, 7665. (8) Becker, R.; Do¨ring, W. Ann. Phys. 1935, 24, 719. (9) Volmer, M. Kinetik der Phasenbildung; Steinkopff: Dresden, Germany, 1939. (10) Schmitt, J. L.; Whitten, J.; Adams, G. W.; Zalabsky, R. A. J. Chem. Phys. 1990, 92, 3693. (11) Strey, R.; Wagner, P. E.; Viisanen, Y. In Nucleation and Atmospheric Aerosols; Fukuta, N., Wagner, P. E., Ed.; Deepak: Hampton, VA, 1992; p 111. (12) Viisanen, Y.; Strey, R.; Laaksonen, A.; Kulmala, M. J. Chem. Phys. 1994, 100, 6062. (13) Wilemski, G. J. Chem. Phys. 1984, 80, 1370. (14) Laaksonen, A. J. Chem. Phys. 1997, 106, 7268.
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11661 (15) Flageollet-Daniel, C.; Garnier, J. P.; Mirabel, P. J. Chem. Phys. 1983, 78, 2600. (16) Laaksonen, A.; Kulmala, M. J. Chem. Phys. 1991, 95, 6745. (17) Laaksonen, A. J. Chem. Phys. 1992, 97, 1983. (18) Girshick, S. L.; Chiu, C.-P. J. Chem. Phys. 1990, 93, 1273. (19) Girshick, S. L. J. Chem. Phys. 1991, 94, 826. (20) Wilemski, G. J. Chem. Phys. 1995, 103, 1119. (21) Katz, J. L. Pure Appl. Chem. 1992, 64, 1661. (22) Granasy, L. J. Chem. Phys. 1996, 104, 5188 (23) Fisher, M. E. Physics 1967, 3, 255. (24) Stauffer, D.; Kiang, C. S.; Walker, G. H.; Puri, O. P.; Wise, J. D., Jr.; Patterson, E. M. Phys. Lett. A 1971, 35, 172. (25) Delale. C. F.; Meier, G. E. A. J. Chem. Phys. 1993, 98, 9850. (26) Kalikmanov, V. I.; van Dongen, M. E. H. Phys. ReV. E 1993, 47, 3532. (27) Kalikmanov, V. I. van Dongen, M. E. H. J. Chem. Phys. 1995, 103, 4250. (28) Kalikmanov, V. I.; van Dongen, M. E. H. Phys. ReV. E 1995, 51, 4391. (29) Kalikmanov. V. I.; van Dongen, M. E. H. Phys. ReV. E 1997, 55, 1607. (30) Hale, B. N. Phys. ReV. A 1986, 33, 4156. (31) McGraw, R.; Laaksonen, A. Phys. ReV. Lett. 1996, 76, 2754. (32) Oxtoby, D. W.; Evans, R. J. Chem. Phys. 1988, 89, 7521. (33) Zeng, X. C.; Oxtoby, D. W. J. Chem. Phys. 1991, 94, 4472. (34) Shen, Y. C.; Oxtoby, D. W. J. Chem. Phys. 1996, 105, 6517. (35) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1993, 99, 4670. (36) Zeng, X. C.; Oxtoby, D. W. J. Chem. Phys. 1991, 95, 5940. (37) Laaksonen, A.; Oxtoby, D. W. J. Chem. Phys. 1995, 102, 5803. (38) Nyquist, R. M.; Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1995, 103, 1175. (39) Ray, A. K.; Chalam, M.; Peters, L. K. J. Chem. Phys. 1986, 85, 2161. (40) Strey, R.; Wagner, P. E. J. Aerosol Sci. 1988, 19, 813. (41) Strey, R.; Viisanen, Y.; Wagner, P. E. J. Chem. Phys. 1995, 103, 4333. (42) Adams, G. W.; Schmitt, J. L.; Zalabsky, R. A. J. Chem. Phys. 1984, 81, 5074. (43) Wagner, P. E.; Strey, R. J. Chem. Phys. 1984, 80, 5266. (44) Hung, C.-H.; Krasnopoler, M. J.; Katz, J. L. J. Chem. Phys. 1989, 90, 1856. (45) Zdimal, V.; Smolik, J.; Meijer, I. G. N. Collect. Czech. Chem. Commun. 1994, 59, 253. (46) Miller, R.; Anderson, R. J.; Kassner, J. L.; Hagen, D. E. J. Chem. Phys. 1983, 78, 3204. (47) Viisanen, Y.; Strey, R.; Reiss, H. J. Chem. Phys. 1993, 99, 4680. (48) Strey, R.; Wagner, P. E.; Viisanen, Y. J. Phys. Chem. 1994, 98, 7748. (49) Wagner, P. E. J. Colloid Interface Sci. 1985, 105, 456. (50) Strey, R.; Wagner, P. E. In Atmospheric Aerosols and Nucleation; Wagner, P. E., Vali, G., Eds.; Lecture Notes in Physics, Vol. 309; Springer: Berlin, 1988; p 111. (51) Rudolf, R.; Wagner, P. E. In Nucleation and Atmospheric Aerosols; Fukuta, N., Wagner, P. E., Eds.; Deepak: Hampton, VA, 1992; p 221. (52) Rudek, M. M.; Fisk, J. A.; Chakaro, V. M.; Katz, J. L. J. Chem. Phys. 1996, 105, 4707. (53) Oxtoby, D. W.; Laaksonen, A. J. Chem. Phys. 1995, 102, 6846.
