Toward a Quantitative Theory of Ultrasmall Liquid Droplets and VaporsLiquid Nucleation

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1 4988 Ind. Eng. Chem. Res. 2008, 47, Toward a Quantitative Theory of Ultrasmall Liquid Droplets and VaporsLiquid Nucleation Zhidong Li and Jianzhong Wu* Department of Chemical and EnVironmental Engineering, UniVersity of California, RiVerside, California Thermodynamic properties of small systems can be drastically different from those corresponding to their macroscopic counterparts due to the surface and fluctuation effects. While the equations of state for macroscopic systems are well advanced, quantitative predictions of the structural and thermodynamic properties of small systems from a molecular perspective remain a daunting challenge. This article illustrates applications of a nonmean-field density functional theory to two types of ultrasmall liquid droplets: one is stabilized in a container of finite size and the other is unstable as appeared during vaporsliquid nucleation. For small systems of simple fluids represented by the Lennard-Jones model, theoretical predictions are compared with results from molecular simulations for the microscopic structure, the droplet size, and the free energy of formation over a broad range of conditions. The numerical agreement of theory with simulation data is comparable to that for the corresponding macroscopic systems. While the Tolman length, a correlation of curvature on surface tension, is negligible at least for small droplets of simple fluids, the vaporsliquid interfacial tension declines with the droplet size approximately proportional to the Gaussian curvature. Surprisingly, the Laplace equation for pressure change across a curved surface remains accurate even for a liquid droplet with the radius only a few times the molecular diameter. I. Introduction Small systems are ubiquitous in nature and in modern technology. 1,2 Common examples include cellular compartments, molecular motors, and viral nucleocapsids in biology, various nanoassemblies of amphiphilic molecules such as micelles, vesicles, and microemulsions, and solidlike metal or semiconductor clusters or quantum dots. Small systems may also emerge as unstable or metastable transition states during various stages of phase transitions in macroscopic systems such as vaporsliquid or liquidssolid nucleation. Whereas it has been well documented that the thermodynamic properties of small systems can be drastically different from those corresponding to their macroscopic counterparts and they may vary at different environments (open, closed, isothermal, isobaric, etc.), it is not until recently that the properties of small systems have become amenable to direct experimental measurements. In contrast to statistical thermodynamics for macroscopic systems, few theoretical tools are available for quantitative analysis of the properties of small systems in terms of their chemical constituents, the intermolecular forces, and interactions with the surrounding environment. In this work, we intend to apply a density functional theory (DFT) to represent the properties of ultrasmall liquid droplets quantitatively. For direct comparison with simulation results, we are concerned here only with a simple liquid model where the intermolecular force can be represented by the Lennard-Jones (LJ) potential. It has been long recognized that for macroscopic systems specified by constant temperature and chemical potential, a liquid droplet is thermodynamically unstable relative to a bulk state. Because of the fractionation effect, however, the same liquid droplet can be stabilized in a closed system of finite size. Toward a better understanding of vaporsliquid transition, formation of liquid droplets in a supersaturated vapor has been scrutinized by extensive molecular simulations. 3 8 While the properties of the critical nucleus, a tiny unstable liquid droplet, * To whom all correspondence should be addressed. jwu@ engr.ucr.edu. Tel.: Fax: have been examined by molecular simulations in great details, theoretical interpretation of the simulation data often relies on the classical nucleation theory (CNT), 9 14 proposed by J. W. Gibbs over 100 years ago. Without any specific knowledge on the microscopic structure of the critical nucleus, CNT presumes that the thermodynamic properties of a tiny liquid droplet are identical to those corresponding to a macroscopic liquid system. Although CNT captures qualitatively the free-energy barrier of a vaporsliquid phase transition far from the spinodal line, its numerical performance is often unsatisfactory and deteriorates at conditions where the properties of the droplet deviate from those of the bulk phase. Applications of DFT to vaporsliquid nucleation have been well documented. In particular, DFT has been used to describe the interfacial structure, nucleus size, surface tension, free energy of nucleation, and kinetic and mechanical properties In comparison to CNT, DFT surpasses a number of unrealistic approximations including the assumption of a sharp vapors liquid interface and utilization of the bulk interfacial tension. In addition, DFT accounts for the microscopic structure of the critical nucleus explicitly. Regrettably, most previous applications of DFT rely either on the local density approximation (LDA) for the free energy due to the short-range intermolecular repulsion and/or on the van der Waals-like mean-field approximation (MFA) for longer-range intermolecular attractions. Because the MFA is not accurate for the thermodynamic properties of bulk phases, the performance of an MFA-based DFT is at most semiquantitative. In this work, we intend to improve the numerical performance by using a modified fundamental measure theory (MFMT) 22,23 for the short-range repulsion and a quadratic density expansion (QDE) 24,25 for longrange attractions. In our previous publications, 26,27 we have demonstrated that the nonmean-field DFT provides accurate predictions of the properties of bulk phases and planar interfacial tensions. Because DFT is directly applicable to both bulk and inhomogeneous systems on the basis of the same set of /ie070578i CCC: $ American Chemical Society Published on Web 07/20/2007

2 molecular parameters, we expect that it remains reliable for representing the properties of small systems. A second purpose of this work is to examine the interfacial properties of a liquid droplet in terms of its size. The size or curvature-dependence of surface tension is conventionally described by the Tolman equation, 28 which provides a firstorder correction of the surface tension of a liquid droplet with respect to that in the planar limit. The Tolman length can be defined as the linear coefficient in a Taylor expansion of the droplet surface tension as a function of the curvature; it can also be related to the separation between locations of the equimolar dividing surface and of the thermodynamically defined surface in the planar limit. Whereas the Tolman length has been used extensively in surface chemistry, its magnitude and sign remain controversial even for simple fluids. For example, DFT predicts that the Tolman length is negative for LJ fluids and the magnitude is about 15% to 20% of the LJ diameter. 16,20 However, alternative theories for LJ fluids indicate that the Tolman length can be positive at similar conditions. 29 On the basis of DFT calculations, McGraw and Laaksonen 30 asserted that the Tolman length for the LJ fluid is zero and the size dependence of surface tension is mainly attributed to the second-order correction of the curvature. A small Tolman length appears consistent with results from recent molecular simulations. 7,31 Vaporsliquid transition is a dynamic process that involves growing of a liquid droplet that cannot be directly addressed by equilibrium DFT. To represent the free-energy landscape of nucleation, Talanquer and Oxtoby extended DFT to unstable systems. 19,20 The finite size effect is accounted for by assuming that the liquid droplet in contact with a vapor has a density equal to that at the boundary of the container. The density profile and subsequently thermodynamic properties of the critical nucleus can be obtained by using a pseudo grand canonical ensemble. With this method, the properties of an arbitrary liquid droplet can be obtained by inserting different stable clusters into the same supersaturated vapor and by assuming that the internal structures of these stable clusters are fixed. While the pseudo grand canonical ensemble is efficient for calculation of the free-energy profile of vaporsliquid nucleation, the hypothetical boundary condition is sometimes difficult to justify, i.e., the stable cluster may exist in a very small container where the surrounding vapor density is not equal to that at the boundary of the container. Besides, it remains to be validated whether the internal structure of stable clusters depends on the properties of the coexisting vapor. In this work, we propose a new method to obtain the nucleation free-energy landscape directly from numerical iteration. This method has been tested by predicting the density profiles of nonequilibrium nuclei. II. Density Functional Theory We consider ultrasmall liquid droplets of a LJ fluid surrounded by a supersaturated vapor. The system is thermodynamically stable in the canonical ensemble where the total number of molecules is fixed, but it is unstable in the grand canonical ensemble that allows fluctuation of the overall number density. As in Monte Carlo simulations, 7,8 the intermolecular potential is represented by a truncated-and-shifted LJ potential u(r) ) { ulj (r) - u LJ (r c ), r e r c 0, r > r c } (1) where u LJ (r) is u LJ (r) ) 4ɛ [( σ r ) 12 - ( σ r ) 6 ] (2) Ind. Eng. Chem. Res., Vol. 47, No. 15, and σ and ɛ are the size and energy parameters, respectively. Throughout this work, the cutoff distance r c is fixed at 2.5σ. To apply DFT to the LJ fluid, we divide u LJ (r) into a shortrange repulsion and a long-range attraction as introduced originally in the BakersHenderson (BH) perturbation theory 32 u rep (r) ) { ulj (r), r e σ 0, r > σ } (3) u att (r) ) { 0, r e σ u LJ (r), r > σ } (4) For the truncated-and-shifted LJ potential, the intermolecular energy also includes a residual interaction potential u res (r) ) { -ulj (r c ), r e r c -u LJ (r), r > r c } (5) The repulsive part of the LJ potential can be approximated by a hard-sphere (HS) potential with an effective diameter T / d ) T / T /2σ (6) where T* ) k B T/ɛ with k B being the Boltzmann constant and T the absolute temperature. Alternatively, the LJ potential can be divided into a repulsive part and an attractive part following the WeekssChandlersAnderson (WCA) theory. 34 The nonmean-field DFT (NMF-DFT) for the LJ fluids has been reported in our previous publications. 27,35 37 Briefly, the intrinsic Helmholtz functional F is decomposed into four components F ) F id + F ex hs + F ex ex att + F res (7) where F id corresponds to that for a system of ideal-gas molecules, F ex hs represents contribution to the Helmholtz energy due to the HS repulsion, F ex att is that due to van der Waals attraction, and F ex res arises from the residual part of the intermolecular potential. The ideal-gas term is known exactly F id ) k B T dr F(r)[ln F(r) - 1] (8) The excess Helmholtz energy due to the HS repulsion F ex hs can be described accurately by a modified fundamental measure theory (MFMT) 22,23 F ex hs ) k B T Φ hs [n R (r)] dr (9) in which the reduced excess energy density Φ hs depends on six weighted densities n R (r) ) F(r )ω (R) ( r - r ) dr derived on the basis of hard-sphere geometry 38 Φ hs )-n 0 ln(1 - n 3 ) + n 1 n 2 - n V1 n V n π[ 2] n 3 ln(1 - n 3 ) + n 2 3 (n 3 2-3n 2 n V2 n V2 ) (10) 3 (1 - n 3 ) n 3 For the LJ potential, the effective HS diameter d instead of the LJ diameter σ should be used to calculate the weighted densities. 35,36 The excess Helmholtz energy due to the van der Waals attraction F ex att is represented by a quadratic density expansion 15,27,39 F ex att ) dr F(r)f att [n 3 (r)] + k B T 4 dr dr C att ( r - r, F)[F(r) -F(r )] 2 (11)

3 4990 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 with Fj )[F(r) +F(r )]/2. The attractive Helmholtz energy per molecule in the bulk f att b depends on the packing density η ) πf b d 3 /6, which is replaced by n 3 (r) for inhomogeneous systems. att Analytical expressions for f b and for the direct correlation function (DCF) C att (r) are derived from the first-order meanspherical approximation (MSA). 26,35 37 Finally, the Helmholtz energy due to the weak but long-ranged residual potential is accounted for simply by using the mean-field approximation F ex res ) 1 dr dr u res ( r - r )F(r)F(r ) (12) 2 A typical mean-field theory for LJ fluids applies eq 12 to the entire attractive part of the intermolecular potential defined by the WCA perturbation theory. 34 In this case, the attractive part of the Helmholtz energy is given by F ex att ) 1 dr dr u att ( r - r )F(r)F(r ) (13) 2 with u rep (r) ) { ulj (r) + ɛ, r e σ 0, r > σ }, u att (r) ) { -ɛ, r e σ u LJ (r), r > σ } (14) To facilitate a comparison between mean-field and nonmeanfield approximations, we also perform calculations by using the DFT with mean-field approximation (MFA-DFT) for the attractive potential. For consistency, F ex hs is represented by MFMT for both cases. In MFA-DFT, the effective HS diameter is calculated from d ) dr{1 - exp[-u rep (r)/k B T]}. At equilibrium, the density profile satisfies the Eulers Lagrange equation F(r) ) exp{βµ - δf(r)} δβfex (15) where β ) 1/k B T. In eq 15, the chemical potential µ is specified by the coexisting bulk phase for an open system or for a closed system with N particles, it is replaced by the normalization condition F(r) ) N exp{- δf(r)} δβfex dr exp{- δf(r)} δβfex (16) Equations 15 and 16 are solved iteratively by using the Picard method with a step length 0.02σ. The initial guess for the density profile at the vaporsliquid interface is provided by a step function, F(r) )F L for r e r 0 and F(r) )F V for r > r 0, where r 0 is the radius of the liquid droplet. III. Results and Discussion A. Bulk Properties. While the limitations of a mean-field theory for representing the thermodynamic properties of a bulk fluid are well recognized, much less known is its reliability for inhomogeneous systems. In fact, a theory for inhomogeneous systems including DFT must reduce to an equation of state for bulk systems when the local density profile F(r) is replaced by the average bulk density F b. Therefore, we can test the numerical performance of different versions of DFT by comparison with the simulation results in the bulk limit (even though such comparison does not guarantee that a theory good for bulk fluids is necessarily accurate at inhomogeneous conditions). Figure 1 shows the phase diagrams for truncated-and-shifted LJ fluids predicted by our nonmean-field theory and by the mean-field theory according to BH or WCA separation of the attractive Figure 1. Comparison of phase diagrams and saturation pressures for the Lennard-Jones fluid predicted from the mean-field and nonmean-field theories and from Monte Carlo simulation. 42,44 In part a, the dashed lines represent the spinodal curve predicted from the nonmean-field theory. and repulsive LJ potentials. A comparison with the simulation results indicates that, except near the critical point, the nonmeanfield theory faithfully reproduces the coexisting vapor and liquid densities as well as the saturation pressure while predictions from the mean-field theory are at most semiquantitative. Evidently, the WCA separation of the LJ potential overestimates the intermolecular attraction, which is responsible for the small values of the coexisting vapor/liquid densities and the saturation pressure at a given temperature. On the other hand, the BH separation of the LJ potential underestimates the attraction, which leads to less accurate mean-field properties. The dashed lines in Figure 1a show the spinodal curve predicted by the nonmean-field theory. These lines set the boundaries of the metastable region and are important in study of vaporsliquid nucleation. B. Stable Liquid Droplets. We now consider the properties of a liquid droplet coexisting with its own vapor in a closed spherical container. The system was studied by Monte Carlo simulations. 8 Because of the finite size, the liquid droplet is thermodynamically stable and can be represented by a canonical ensemble (i.e., NVT ensemble) along with the coexisting vapor phase. As specified in Monte Carlo simulations, 8 we assume that the external potential due to the cage surface is represented by V ext (R B - r) ) { ulj (R B - r) + ɛ, R B - r e σ 0, R B - r > σ } (17)

4 Ind. Eng. Chem. Res., Vol. 47, No. 15, Figure 2. Density profiles of a stable liquid droplet coexisting with a vapor in a spherical cavity with N Lennard-Jones molecules. (a) T* ) 0.71 and (b) T* ) Solid and dashed lines are predictions of NVTsNMFsDFT and of NVTsMFAsDFT, respectively, and the symbols are simulation results from ref 8. where R B is the radius of the container, which is about 10 times the molecular diameter. Because the container is in direct contact with the vapor phase, the external potential has negligible effect on the properties of the liquid droplet. Figures 2 and 3 present the liquidsvapor interfacial density profiles predicted by mean-field and nonmean-field DFTs in comparison with the simulation results. 8 Partitioning of the LJ molecules among the coexisting liquid and vapor phases ensures that at a given temperature the liquid droplet grows with the total number of particles within the container or shrinks with temperature if the total number of particles is fixed. Because most of the volume is occupied by the vapor phase, theoretical predictions are extremely sensitive to its density, which must be calculated concomitant with the properties of the small liquid droplet. As expected, the performance of NMF-DFT is superior to the MFA-DFT, and in most cases, the nonmean-field theory is able to reproduce the simulation data quantitatively. In particular, the NMF-DFT works the best at low temperature and large droplet size. The discrepancies between NMF-DFT and simulation at high temperature are possibly due to the fact that a small deviation in the vapor density leads to a magnified error for the droplet density. In general, MFA-DFT overestimates the droplet size and liquid density for small droplets or at high temperature, and the trend is opposite for large droplets or at low temperature. Besides, the MFA-DFT yields oscillation of the liquid density at low temperature. Because WCA Figure 3. Same as Figure 2 but at different temperatures for (a) N ) 256 and (b) N ) overestimates the intermolecular attraction, this oscillation is possibly due to the fact that the droplets show some ordered structure. C. Unstable Liquid Droplets. In contrast to a finite system, a liquid droplet is thermodynamically unstable if it is surrounded by a bulk supersaturated vapor phase at constant pressure or chemical potential. In this case, coexistence of the liquid droplet with its surrounding vapor reflects a free-energy maximum. As a result, the liquid droplet may either grow or shrink depending on whether its size is larger or smaller than that of the critical nucleus. To calculate the properties of a unstable droplet, we follow a method originally proposed by Zeng and Oxtoby. 21 At a given temperature, the initial guesses of the densities for the liquid and vapor phases, F L, F V, are obtained from the densities of the equilibrium liquid F 0 L and that of the supersaturated vapor F n V. The saddle point is identified by trying different values of the droplet radius r 0, on the basis of the fact that a tiny change of r 0 will lead to either shrinkage or growth of the droplet size. Figure 4 shows variation of the free-energy barrier during iteration near the saddle point at temperature T* ) and supersaturation S ) Here, the supersaturation ratio is defined as S ) p n V /p 0 with p n V and p 0 being the pressure of the supersaturated vapor and that at equilibrium, respectively. The free-energy barrier is defined as Ω ) Ω[F(r)] + p n V V with V being the system volume. Figure 4 indicates that Ω bifurcates around r 0 /σ ) The droplet eventually shrinks when r 0 /σ ) 4.8 and grows when r 0 /σ ) Both curves exhibit a flat plateau lasting for several hundreds of iteration steps within

5 4992 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 and the corresponding vapor spinodal density is Thus, the densities of the supersaturated vapors under study (F b / ) for S ) ) are much lower than that at the spinodal point. In the DFT calculations, the critical nucleation barrier is represented by Ω* and the critical excess number of particles n* is given by n ) 4π [F(r) -F n V ]r 2 dr (18) Figure 4. Nucleation barrier calculated from µvtsnmfsdft as a function of the iteration steps at T* ) and S ) The thin dashed line marks the critical nucleation barrier. Figure 5. (a) Free-energy barrier and (b) excess number of particles of a critical nucleus as a function of supersaturation at T* ) Solid and dashed lines represent predictions of µvtsnmfsdft and CNT, respectively, and the symbols are simulation data from ref 7. which the droplet structure remains nearly unchanged. This plateau marks the critical nucleation barrier Ω*. Figure 5 compares predictions from NMF-DFT with simulation results for vaporsliquid nucleation of the LJ fluid at T* ) in the NPT ensemble. 7 At this temperature, the coexistence vapor and liquid densities are and , respectively, As indicated by Lee et al., 17 Ω in the µvt ensemble is equal to G in the NPT ensemble. In the µvt ensemble, we consider the interactions between the molecules in the system and those in the reservoir explicitly. However, such interaction is absent in the NPT ensemble, which could be significant as the nucleus size increases. Figure 5 indicates that, far from the spindal point, agreement of the theory with the simulation is good for both Ω* and n*. The nucleation barrier falls monotonically with the degree of supersaturation S, which eventually leads to a spontaneous nucleation. Far away from the spinodal point, the droplet size falls with the degree of supersaturation, which is consistent with predictions of the Kelvin equation F 0 L k B T ln(s) ) 2γ/R where γ is the surface tension and R is the droplet radius. Approaching the spinodal point, however, the droplet size may exhibit a shallow minimum. 40,41 In this case, the Kelvin equation becomes inappropriate because at higher density the vapor phase cannot be treated as an ideal gas and the structure of the nucleus becomes very different from that of a macroscopic droplet. We did not calculate the droplet size at high supersaturation in this work because the low free-energy barrier makes the critical nucleus hardly discernible. For comparison, we have also calculated the properties of critical nuclei by using the classical nucleation theory (CNT). According to CNT, the critical nucleation barrier Ω* and droplet size n* are given by, respectively, Ω / ) 16πγ 3 (19) 3 p 2 n / ) 32πγ 3 3 p (F L 3 0 -F V n ) (20) where γ is the surface tension of a planar vaporsliquid interface and p ) p L n - p V n with p L n being the pressure of a hypothetical liquid droplet that has the same chemical potential as that of the vapor. By applying the NMF-DFT to a planar interface, we obtained that γ / ) γ σ 2 /ɛ ) 0.527, which is close to the simulation result In consistency with the previous work, 7,41 CNT predicts the size of critical nuclei accurately. For the free-energy barrier, the performance of CNT is comparable to that of NMF-DFT at low supersaturation. But, contrary to NMF-DFT, the accuracy of CNT deteriorates as the degree of supersaturation increases. Figure 6 shows the density profiles of the critical nuclei at three values of supersaturation (S ) 1.53, 1.79, and 2.04) and T* ) The lines are predicted by the NMF-DFT, and the symbols are the simulation results. 7 Unlike that for closed systems, the performance of the DFT is near-perfect. For sufficiently large droplet (S ) 1.53), the density of the liquid core is close to that of the bulk liquid at the same chemical potential of the surrounding vapor. In that case, both the size of the critical droplet and liquid density falls as the vapor pressure increases. D. Tolman Length. The Tolman length measures the extent by which the surface tension of a small liquid drop deviates from its planar value. It can be formally defined as the leading

6 Ind. Eng. Chem. Res., Vol. 47, No. 15, Figure 6. Density profiles for the critical nuclei at different degrees of supersaturation S at T* ) The thick lines and symbols correspond to predictions from µvtsnmfsdft and from simulation, 7 respectively. The thin horizontal lines represent the corresponding density in the bulk liquid that has the same chemical potential as that of the vapor. coefficient in the Taylor expansion of the surface tension in terms of the curvature: γ ) γ (1-2δ ) TL R + (21) where δ TL is the Tolman length and 1/R is the curvature of a spherical liquid droplet. At T* ) 0.7, the coexistence vapor and liquid densities are and , respectively. The vapor spinodal density is which is much higher than the density of supersaturated vapor used in the work (F / b ) for S ) ). By using the nonmeanfield DFT, we predict γ / ) 0.599, which agrees well with the simulation result To define the droplet size or curvature, we use an equimolar dividing surface to partition the liquid droplet and its surrounding vapor. The radius of the equimolar dividing surface is calculated from R 3 3 e ) V [F(r) -F V n ]r 2 3 n / dr ) F(0) -F n 4π[F(0) -F V n ] (22) In terms of R e and γ e, the free energy of nucleus formation is given by Ω / )- 4 3 πr e3 p + 4πR e 2 γ e (23) On the basis of the droplet radius R e and the free energy of formation Ω* calculated from DFT, we obtain the surface tension γ e. Subsequently, the Tolman length can be estimated by fitting γ e in terms of the Helfrich formula 43 γ e ) γ - 2δ TL γ + k s (24) R 2 e R e where k s is the rigidity coefficient. Equation 24 can be obtained from eq 21 by replacing R and γ by R e and γ e, respectively, and by neglecting the high-order terms. Figure 7a shows γ e calculated from eq 23 as a function of 1/R e at T* ) 0.7 and By fitting the data with eq 24, we estimate that δ TL /σ ) and βk s ) for T* ) 0.7, and δ TL /σ ) and βk s ) for T* ) Figure 7. (a) Surface tension γ e and (b) pressure drop p versus 1/R e for the critical nuclei at T* ) 0.7 and Symbols are from µvtsnmfsdft. The thick lines are obtained by the best fitting (a) of eq 24 and (b) of the Laplace equation p ) 2γ /R e, respectively. The thin horizontal line in part a marks the surface tension at the planar limit. Because the Tolman length δ TL is close to zero, the first-order correction to the surface tension appears insignificant at least for the LJ fluid. Our DFT calculations support an earlier conjecture by McGraw and Laaksonen assuming that the Tolman length is zero. 30 Figure 7a indicates that the surface tension rises with the droplet size and it reaches the planar limit asymptotically. Minimization of Ω* with respect to R e gives the generalized Laplace equation 30,43 p ) 2γ e R e + γ e R e ) 2γ R e - 2δ TL γ R e 2 (25) Because the Tolman length is close to zero, eq 25 suggest that the original Laplace equation is accurate even for a very small liquid droplet. Figure 7b shows an excellent linear relation as predicted by the original Laplace equation p ) 2γ /R e. The good fix also suggests that δ TL can be ignored for the LJ fluids. E. Free-Energy Profile of VaporsLiquid Nucleation. Different from the method of the pseudo grand canonical ensemble, we find that the nucleation free-energy landscape can be directly obtained from numerical iteration in the grand canonical ensemble. To illustrate, Figure 8 shows variation of the free energy for different values of r 0 and plots the freeenergy barrier Ω as a function of the excess number of particles n during the iteration at T* ) and S ) Except the initial iteration steps, the shrinkage (r 0 /σ e 4.8) or

7 4994 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Figure 8. Free-energy landscape obtained by using different initial guesses of r 0 at T* ) and S ) 1.53 calculated from µvtsnmfsdft. Figure 10. Density profiles for several precritical nuclei at T* ) and S ) Lines represent results from µvtsnmfsdft, and symbols are simulation data from ref 7. free-energy landscape during liquid droplet formation. Figure 9 shows that the energy landscape from the pseudo µvt ensemble is systematically lower than that from the exact result. The difference is probably due to the assumption that the internal structures of the stable clusters are independent of the supersaturated vapor where they are inserted. Figure 10 presents the density profiles of several precritical nuclei in comparison with the simulation results 7 at T* ) and S ) Except for the two smallest nuclei, the structures of all the other nuclei are faithfully reproduced by the NMF- DFT. The good agreement also supports the rationality of the new method for determining the free-energy landscape. Figure 9. Free-energy landscape β Ω at different degrees of supersaturation S at T* ) Solid and dashed lines are predictions of µvtsnmfsdft and pseudo µvtsnmfsdft, respectively. the growth (r 0 /σ g 4.82) of the nuclei follow a unique route independent of r 0. This unique pathway corresponds to the freeenergy change at different nucleation stages, i.e., the free-energy landscape. Figure 9 shows the free-energy landscape of vaporsliquid nucleation at T* ) and different degrees of supersaturation. Several essential features are captured by these results. First, the energy barrier is zero when small nuclei disappear in the vapor phase and approaches a negative infinity when the large nuclei become the liquid phase. Second, near the spinodal point, the free-energy landscape narrows and the critical barrier becomes smaller. We speculate that the energy barrier disappears at the spinodal point. For comparison, we have also carried out NMF-DFT calculations in a pseudo µvt ensemble as proposed by Talanquer and Oxtoby. 19,20 For these calculations, we encapsulate a fixed number of particles in an imaginary spherical container and the density of surrounding vapor is taken into account by assuming it to be identical to that of the container boundary. A set of stable clusters are obtained by minimization of the Helmholtz energy with respect to the container volume. By assuming that the internal structures of the stable clusters remain unchanged when they are inserted into a given supersaturated vapor, we are able to calculate the IV. Conclusions We have investigated the properties of ultrasmall liquid droplets and vaporsliquid nucleation for the Lennard-Jones fluids by using a nonlocal density functional theory. Unlike a typical mean-field approach, here the excess Helmholtz freeenergy functional is represented by a modified fundamental measure theory for the short-range repulsion and by a quadratic density expansion for the long-range attraction. By extensive comparison with simulation results, we find that the DFT can be used for predicting the microscopic structure of liquid droplets, the size, and the free-energy barriers of critical nuclei. We have also calculated the Tolman length of the Lennard- Jones fluids and studied the influence of the droplet curvature on the surface tension. Our calculation suggests that the Tolman length is at least close to zero but the surface tension is still curvature-dependent. We find that the McGrawsLaaksonen assumption is quite reasonable. As the droplet becomes smaller, the surface tension falls gradually from the planar limit. We are able to obtain the free-energy landscape of vapors liquid nucleation directly from numerical iteration. This new method allows us to capture structural and thermodynamic properties of ultrasmall liquid droplets at either equilibrium or nonequilibrium conditions. Acknowledgment We thank Drs. Dong Fu and Vicente A. Talanquer for fruitful discussions. This research is sponsored by the U.S. Department of Energy (DE-FG02-06ER46296) and uses the computational

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