A density-functional theory for bulk and inhomogeneous Lennard-Jones fluids from the energy route

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 14 8 OCTOBER 2003 A density-functional theory for bulk and inhomogeneous Lennard-Jones fluids from the energy route Yiping Tang a) Honeywell Hi-Spec Solutions, 343 Dundas Street, London, ON N6B1V5, Canada Jianzhong Wu b) Department of Chemical and Environmental Engineering, University of California, Riverside, California Received 22 April 2003; accepted 18 July 2003 A new density-functional theory is developed for representing the structural and thermodynamic properties of Lennard-Jones fluids by unifying the modified fundamental measure theory for the short-range repulsion and the first-order mean-spherical approximation FMSA via the energy route for the attractive part of the intermolecular potential. This theory significantly improves the conventional mean-field approximation for the attractive forces and is applicable to both bulk and inhomogeneous systems using a single set of molecular parameters. The new theory is computationally very affordable and self-consistent with FMSA for bulk systems. It provides accurate radial distribution function, phase diagram, saturation pressure, and chemical potential of bulk Lennard-Jones fluids except very close to the critical point. In addition, it agrees well with simulation results for inhomogeneous systems including the adsorption isotherms and the density profiles of Lennard-Jones molecules near hard walls as well as in attractive slit pores American Institute of Physics. DOI: / I. INTRODUCTION Since van der Waals invented his famous equation of state of simple fluids and the square-gradient theory for inhomogeneous systems, 1 there has been a continuous effort to develop more accurate statistical-mechanical theories that provide faithful representation of both structural and thermodynamic properties of liquid systems. 2 Major progress was made in the late 1960s and early 1970s when analytical solutions to the Ornstein Zernike OZ equation were established for hard spheres, 3,4 Yukawa and electrolyte solutions, 5,6 and based on which perturbation theories were formulated to take into consideration short-ranged intermolecular attractions. 7 9 Except near critical points where correlations are long ranged, liquid-state theories are satisfactory for the phase behavior as well as the microstructures of a broad variety of bulk fluids. Regrettably, liquid-state theories find little application for inhomogeneous systems not only because of the numerical challenges in solving the more complicated integral equations but also because of the inaccuracy of conventional closure relations such as hypernetted chain and Percus Yevick approximations. 2 The most promising approach to represent the structure and the phase behavior of confined fluids is probably from the density-functional theory DFT. 10,11 Early applications of DFT were based on van der Waals square-gradient theory where a bulk fluid is used as the reference and the inhomogeneity is taken into account using a density gradient. 12,13 As in van der Waals original work, these methods are most useful for weakly inhomogeneous systems such as that at the interface of a vapor-liquid coexistence. Later applications of DFT are primarily based on the perturbation approach where a weighted density approximation is applied for the shortrange repulsion and a mean-field theory MFT is used for long-range attractions As for the perturbation theory of bulk systems, the short-ranged repulsion is often represented by hard spheres. Among various versions of weighted density approximations for inhomogeneous hard spheres, the fundamental measure theory FMT by Rosenfeld holds a number of key advantages. In contrast to alternative density-functional approaches, FMT does not require the direct correlation function of bulk hard spheres as an input, which in fact can be directly derived from FMT. Because the weight functions of FMT are independent of density distributions, the weighted densities are more convenient to calculate than those in most other nonlocal density-functional theories. Further, the same formulism is directly applicable to one-component systems as well as to mixtures. 24,25 Recently, a modified FMT MFMT has been proposed by incorporating the Boublik Mansoori Carnahan Starling Leland BMCSL equation of state for bulk hard spheres. 26 In contrast to its great success for hard spheres, DFT progresses only slowly for systems with attractive forces, even for simple Lennard-Jones LJ fluids which often serve as a benchmark for realistic systems. Present applications of DFT relies primarily on MFT, which is computationally convenient and captures some qualitative features of inhomogeneous phenomena such as wetting and drying transitions at a wall fluid interface and the density distributions in slit pores. 27 However, MFT is quantitatively problematic and sometimes qualitatively incorrect because it completely iga Electronic mail: yiping.tang@honeywell.com b Electronic mail: Jwu@engr.ucr.edu /2003/119(14)/7388/10/$ American Institute of Physics

2 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Bulk and inhomogeneous Lennard-Jones fluids 7389 nores the correlations of density distributions. The crudeness of MFT is immediately seen in the homogeneous limit where the DFT is reduced to the well-known van der Waals VDW equation of state. To surmount the numerical deficiencies of MFT, a number of theoretical improvements and ad hoc empirical corrections have been proposed. In particular, Tang et al. 33 extended the Barker Henderson BH theory of uniform LJ fluids 34 to inhomogeneous systems. The DFT preserves the bulk thermodynamic properties as predicted by the original BH theory and yields also satisfactory density distributions of LJ molecules near hard walls. Similar applications have been reported for LJ fluids confined in spherical cages. 35,36 An appealing feature of the extended BH DFT is its computational simplicity because it is free of solving integral equations during the iteration process. However, its quantitative performance is not always satisfactory because BH theory under-predicts the liquid density and gives inaccurate radial distribution function. 37 More sophisticated DFT theories have been proposed recently for inhomogeneous LJ fluids These theories are derived from either weighted density approximation WDA or perturbation expansion that resorts to numerical solutions of the Ornstein Zernike OZ equation for the direct correlation functions. While these theories show considerable improvements over MFT, they often demand masssive computational effort for solving the integral equation. Besides, extra parameters may be required to maintain a consistent application of both bulk and inhomogeneous fluids. 40 A less mentioned but potentially fatal problem associated with these methods is that at certain conditions, the OZ equation may lose the solution, causing these theories to break down. 41 Empirical modifications of MFT have also been proposed in the literature such as by letting the size parameter of the LJ potential adjustable with saturated liquid density 42 or by using inconsistent LJ parameters for bulk and inhomogeneous fluids. 32 These and other empirical corrections improve the performance of MFT in some aspects but are often inadequate for others. 32,43 Recently, Tang and co-workers developed a new theory for bulk LJ fluids 44,45 that utilizes the first-order solution of the Ornstein Zernike equation with the mean spherical approximation FMSA. This fully analytical theory is simpler than the BH theory but provides more accurate radial distribution functions and thermodynamic properties of LJ fluids. Unlike the standard integral-equation theories of liquids such as hypernetted chain theory, Percus Yevick theory, and the original full MSA, the FMSA is free of numerical breakdown and more importantly, it is directly applicable to predicting phase transitions. 37 The aim of this work is to reconcile the MFMT that is very accurate for hard-sphere systems and the FMSA theory for bulk LJ fluids within the framework of density-functional theory. The successfulness of MFMT and FMSA warrants that the reconciliation is reliable for both bulk and inhomogeneous systems. II. THEORY The central task of a density-functional theory DFT is to determine the intrinsic Helmholtz free energy F(r) as a functional of the density distribution r. In general, F(r) can be split into an ideal and an excess part: FrF id rf ex r, where the ideal intrinsic Helmholtz energy F id (r) is exactly known, F id rkt drrlnr 3 1. In Eq. 2, k is the Boltzmann constant, T is absolute temperature, and is the thermal wavelength. The excess free energy F ex (r) takes into account the nonideality due to intermolecular interactions and it is in most cases only known approximately. In the context of the grand canonical ensemble, r can be determined by minimizing the grand potential, rfr drrv ext r, 3 where V ext (r) is the external potential and the chemical potential of the inhomogeneous fluid. From the variational principle r 0, r it follows that the equilibrium density distribution satisfies. 5 r exp 3 F exr V r ext r Because the chemical potential can be expressed in terms of the density of a bulk fluid b, the density distribution can be alternatively determined from, 6 r b exp ex F exr V r ext r where ex is the excess chemical potential of the bulk fluid. Given an expression for the excess Helmholtz energy functional, the density distribution r can be solved from Eqs. 5 or 6 using a suitable numerical iteration method e.g., Picard iteration and subsequently all thermodynamic properties can be calculated. As in a typical perturbation theory of bulk fluids, the excess intrinsic Helmholtz energy is often further split into contributions from the short-ranged repulsion and longranged attraction, F ex rf rep rf att r. For the LJ potential ur4 12 r 12 6 r 6, where both repulsive and attractive components span the entire range of intermolecular separations, the short-ranged repulsion and longer-ranged attraction are often defined following a split proposed by Barker and Henderson,

3 7390 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Y. Tang and J. Wu u rep r4 12 r 12 6 r 6, 0, r, u att r0, 4 12 r, r 12 6 r 6, r, r As usual, and in Eqs stand for the size and energy parameters of the LJ potential, respectively. The same split has been employed in the FMSA theory for bulk LJ fluids. 44 While the longer-ranged attraction is treated as a perturbation, the short-ranged repulsion can be approximated by a hard-sphere potential with an effective diameter, d 0 1e u rep r dr 11 which can be accurately reproduced by T* d T* T* 2, T* kt. 12 Equation 11 can be obtained by minimizing the free-energy difference between the reference fluid with only the shortranged repulsion and the effective hard-sphere fluid. 47 Other expressions for the effective hard-sphere diameter have been proposed in the literature, ranging from that as simple as d to that as complicated as fitting it to the phase diagram of the bulk LJ fluids. 42,43 Evidently the effective hard-sphere diameter is a sensitive parameter in theoretical modeling and an inappropriate choice of this parameter may spoil the calculated results. 32,47 A. Repulsive free-energy functional We use the MFMT to represent the excess Helmholtz energy functional of the reference hard-sphere fluid. 26 As in the original FMT or in a typical nonlocal-density-functional theory for hard spheres, the excess Helmholtz energy functional is presumably expressed as F rep rkt drr, 13 where the excess Helmholtz energy density r is a function of density distribution r. It consists of scalar and vector contributions: SCS VCS, 14 where according to MFMT, the scalar part is given by the Boublik Mansoori Carnahan Starling equation of state, SCS n 0 ln1n 3 n 1n 2 1n ln1n 36n 3 n n 3 1n 3 2, 15 and the vector part follows from the scalar-particle relations, VCS n V1 n V2 1n n 3 2 ln1n 3 12n 3 1n 3 2 n 2 n V2 n V2. 16 The weighted densities n (r) in Eqs. 15 and 16 are defined as n r r rrdr, where the weight functions () (r) are given by 17 0 r d/2r d 2, 18 1 r d/2r, 19 2d 2 rd/2r, 3 rd/2r, V1 r r r V2 r r r d/2r d/2r, 22 2d 23 In Eqs , (r) is the Dirac delta function and (r) is the Heaviside step function. The weight functions as given by Eqs. 22 and 23 are vectors that have the same orientation of r. For systems with special symmetry, Eq. 17 can be expressed more explicitly according to the nature of the density distribution. These expressions have been documented elsewhere and readers may refer to these references for details. 26,48 B. Attractive free-energy functional The attractive free-energy functional is derived by the integration of the Gibbs Helmholtz equation as discussed in Tang s earlier work for FMSA and for the critical phase behavior of bulk LJ fluid. 49 Near the critical point of vaporliquid transition, the long-range density fluctuations in a bulk LJ fluid resemble the density distribution of an inhomogeneous system. Although the critical long-range correlation is quite different from the inhomogeneity introduced by an external field, the underlying free-energy functional can be formulated in a similar manner. According to the Gibbs Helmholtz equation, the freeenergy functional due to the long-ranged attractions can be expressed as an integral with respect to 1/(kT), F att r 0 duatt r. 24 Equation 24 is the so-called energy route to formulate the Helmholtz free energy from the internal energy U att (r), which is in turn given exactly by U att r 1 2dr 1 dr 2 r 1,r 2 u att r 1 r 2. 25

4 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Bulk and inhomogeneous Lennard-Jones fluids 7391 FIG. 1. The radial distribution function of a LJ fluid. The symbols are Monte Carlo simulation data and the solid line is predicted by FMSA. In Eq. 25, (r 1,r 2 ) is the two-body density distribution function at positions r 1, r 2. We assume that this function can be approximated by r 1,r 2 r 1 r 2 gr 1 r 2, 26 in which g(r, ) is the radial distribution function RDF of a homogeneous LJ fluid at an average density such as 49 k rg1 re sr 1 e sd dr 0 sz 1 Q 2 sdq 2 z 1 d k 2 e sd sz 2 Q 2 sdq 2 z 2 d with 31 r 1r Alternative choices for can be r 1r 2 2, 28 or as simple as b. We find that the three options have insignificant impact in the calculated results and the last is computationally the most convenient. In the limit of uniform fluids, Eqs reduce to those corresponding to the bulk fluids. It is noted that the mean-field approximation is fully recovered when g(r, ) in Eq. 25 is replaced by unity. The radial distribution function of a homogeneous LJ fluid can be conveniently calculated using the FMSA developed by Tang and co-workers. 44,45 According to this theory, the radial distribution function consists of that for an effective hard-sphere fluid of diameter d as defined in Eq. 12 and a first-order perturbation due to the long-ranged attraction: grg 0 rg 1 r, 29 where, in terms of the Laplace transform, g 0 (r) and g 1 (r) are rg0 re sr dr Lsdesd Qsds 2, 30 Qt St12Ltet 1 2 t 3, 32 St1 2 t 3 61t t1212, 33 Lt 1 2 t12, 1 6 d3, 34 z /, z /, 35 k e z 1 d, k e z 2 d. 36 Readers may refer to earlier FMSA papers for more explicit expressions of g 0 (r) and g 1 (r). 44,45 Figure 1 gives a glance of the radial distribution function RDF of a bulk LJ fluid from FMSA 29 together with simulation data from this work. More extensive discussions about the RDF can be found elsewhere. 37 From Eqs. 25, 26, and 29, the attractive free-energy functional can be simplified to F att r 1 2dr 1 dr 2 r 1 r 2 g 0 r 1 r 2, u att r 1 r 2 1 4dr 1 dr 2 r 1 r 2 g 1 r 1 r 2, u att r 1 r 2 37

5 7392 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Y. Tang and J. Wu FIG. 2. Phase diagrams of the LJ fluid from computer simulation open and filled symbols Refs. 50 and 51 and from FMSA solid line, BHdotted line, and MFT dashed line theories, respectively. in recognizing that g 1 (r) is proportional to. Despite its integral appearance, at the bulk limit the free-energy functional in Eq. 37 is identical to that of the earlier integralfree FMSA. 44 III. RESULTS AND DISCUSSION A. Uniform fluids A reliable density-functional theory DFT is supposed to be applicable to both uniform as well as inhomogeneous conditions. The performance of DFT in the uniform limit is important because on the one hand, an inhomogeneous fluid is often in direct contact with a bulk phase that defines the chemical potential and on the other hand, Helmholtz energy and correlation functions of the bulk fluid are essential in the implementation of DFT for nonuniform systems. Ideally, the equation of state derived from DFT in the bulk limit should be at least as accurate as that used for homogeneous systems. The density-functional theory formulated in this work is exactly reducible to the FMSA for bulk LJ fluids, which has been proved to be very accurate for both correlation functions and thermodynamic properties. 37,44 As an example, Fig. 2 depicts the phase diagrams of LJ fluids predicted by FMSA, BH theory, and MFT. It is evident that FMSA is most accurate except near the critical point and the accuracy is consistent for both vapor and liquid phases. Conversely, the BH theory is good for vapor phase but underestimates liquid densities; and the simple MFT yields too small critical temperature and is totally unreliable as a quantitative theory. Figure 3 compares the performance of the three theories for predicting the saturation pressure. While MFT predicts too large saturation pressure in comparison with the simulation data, both FMSA and BH theories yield satisfactory results except with slightly more deviations near the critical temperature. The inability of FMSA and BH theory for the critical region is due to not taking into account long-range fluctuations which can be remedied by renormalization group transformations. 49 The performance of FMSA, BH theory, and MFT for bulk fluids provides helpful insights to develop reliable density-functional theory of inhomogeneous systems. For instance, phase coexistence is directly associated with wetting and drying processes around solid-fluid interfaces, similar to the phase transition of bulk fluids. 27 Figure 4 presents the excess chemical potentials predicted by FMSA, BH theory, and MFT. Here the open squares are the results from the modified Benedict Webb FIG. 3. Saturation pressures (P*P 3 /) of the LJ fluid from computer simulation open and filled symbols Refs. 50 and 51 and from FMSA solid line, BH dotted line, and MFT dashed line theories, respectively. These curves all terminate at their critical points.

6 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Bulk and inhomogeneous Lennard-Jones fluids 7393 FIG. 4. Excess chemical potential ( ex * ex /kt) of the LJ fluid at T*1.35. The symbols are the results of the MBWR equation Ref. 52. The solid, dotted, and dashed lines are the predictions from the FMSA, BH, and MF theories, respectively. Rubin MBWR equation of state, 52 which uses 33 parameters to fit massive computer simulation data for the thermodynamic properties of LJ fluids and its accuracy is thus virtually identical to the simulation. The excess chemical potential is important for calculating not only the phaseequilibrium of bulk fluids but also for density distributions in inhomogeneous systems. In particular, computer simulations of inhomogeneous fluids are typically carried out at given chemical potential instead of bulk density. Figure 4 illustrates that the chemical potential from FMSA agrees best with that from the MBWR equation, while the BH theory shows some deviations at moderate densities, and MFT performs poorly as it systematically overestimates the chemical potential over the whole range of densities. The discrepancy between MFT and the MBWR equation is more prominent at intermediate densities. The poor performance of MFT for both phase diagram and excess chemical potential suggests that a similar approach for inhomogeneous cases is not meaningful at least quantitatively. It is worthwhile to mention that as recently demonstrated, 53 FMSA is nearly as accurate as the full MSA solution, while its algebraic manipulation is much easier and avoids the potential breakdown of the full solution in the phase unstable region. One may wonder whether more sophisticated liquid-state theories such as the Percus Yevick, hypernetted chain, and Martynov Sarkisov integral equations can yield better results for the phase diagram. Whereas these integral equation theories may also be applied to inhomogeneous systems, they are not nearly as accurate as the FMSA and the numerical solutions to these integral equations also break down at conditions when the system is unstable. 41,54 For instance, the hypernetted chain approximation yields a much narrower phase diagram even far way from the critical point, and fully loses the criticality 54 due to the numerical breakdown. B. Inhomogeneous LJ fluids We have applied both MFMTFMSA and MFMT MFT density-functional theories to inhomogeneous LJ fluids. In spite of its poor performance for bulk fluids, MFT is included in our calculations for its popularity in the litera- FIG. 5. Density profile (* 3,z*z/) ofalj fluid in contact with a hard wall at T*1.35. The symbols are the computer simulation data Ref. 55. The solid and dotted lines are the predictions from the FMSA and MF theories, respectively; the dashed and dot-dashed lines are those from the same theories with the hard-sphere diameter d.

7 7394 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Y. Tang and J. Wu FIG. 6. Same as in Fig. 4 except at a higher bulk density. ture. One should bear in mind that our MFT differs from others in the literature not only in terms of MFMT for the short-ranged repulsion but also in the BH diameter as given in Eq. 9 and its attractive force as given in Eq. 10. For brevity in discussion, we use FMSA and MFT to refer to the MFMTFMSA and MFMTMFT density functionals, respectively. Figures 5 and 6 show the density distributions of the LJ fluids near a hard wall at the reduced bulk density *( 3 ) 0.5 and 0.65, respectively. These two states were also studied by Tang et al. 33 using the BH perturbation theory as well as their MFT. As shown in Fig. 5, the LJ molecules are depleted around the solid-fluid interface and the density rises monotonously with the separation from the wall until it reaches the bulk density. The depletion is attributed to the attraction from the bulk liquid, which counterbalances the accumulation of molecules due to the short-range repulsive interactions. Clearly, all these features are quantitatively captured by the FMSA. The performance of the FMSA is slightly better than that of the BH theory, which somehow overestimates the densities around hard wall. The MFT gives an oscillatory density profile and the contact density is much higher than the simulation value. This reveals that the attraction in the MFT is less accounted, which is understandable due to its structureless nature. It may be of interest to mention that our MFT results are very close to those from the MFT by Tang et al. 33 using the same BH diameter. Figure 6 shows the density profile at a higher bulk density. In this case, the density distribution recovers the bulk value in an oscillatory manner, indicating that the repulsion takes more control. The FMSA predicts slightly lower contact value but overall it is comparable to the BH theory. 33 As in Fig. 5, the MFT overestimates considerably the densities around the wall. Thus it suggests that MFT is qualitatively inappropriate to describe LJ fluids near a hard wall. Figures 5 and 6 further illustrates that using an unjustified hard-sphere diameter such as d worsens the results by severely amplifying density oscillation and giving higher densities at contact. Such an amplification and overestimation was also observed before. 33 Figures 7 and 8 give the density profiles of LJ fluids in attractive slit pores with the width H* (H/)3.5 and H*5.0, respectively. The simulation results are from a recent work by Sweatman, 40 using the Steele potential to represent the wall molecule interaction FIG. 7. Density profile of a LJ fluid inside a slit pore (H*H/). The symbols are the computer simulation data Ref. 40. The solid, dashed, and dotted lines are the predictions from the FMSA, Sweatman s, and MF theories, respectively.

8 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Bulk and inhomogeneous Lennard-Jones fluids 7395 FIG. 8. Same as Fig. 7 except at a different bulk density and slit width. V s z w w z 4 w z 3z0.61 3, 38 where the parameters are related to those for the LJ potential, w 0.903, w 12.96, For a planar slit, the external potential is given by V ext zv s zv s Hz. 40 In addition to the simulation, Sweatman 40 developed a DFT in which the repulsive part is taken from the original fundamental measure theory of Rosenfeld, 21 and the attractive part is formulated by a weighted density approximation WDA. The theory resorts to the bulk Helmholtz free energy from a semiempirical equation of state and direct correlation function DCF from the Percus Yevick/hypernetted chain theory PYHNC integral equations to locate the weight functions. Figure 7 shows that the density profile predicted by FMSA is in rather good agreement with the computer simulation data. In particular, it accurately reproduces the first peak of the density profile, which is underestimated by both MFT and WDA. However, analogous to the WDA the second w 4 peak of the FMSA is slightly below the simulation results, but remains better than the MFT. In general, FMSA is more accurate than MFT for both bulk and inhomogeneous systems. However, we find that due to the cancellation of errors, MFT may outperform FMSA at certain conditions. For instance, Fig. 8 shows that the FMSA overpredicts slightly the first peak and the overprediction is magnified around the second peak, while both MFT and Sweatman s theories give better performance. The error cancellation in MFT can be understood by the fact that the excess chemical potential is poorly accounted for by the MFT, as shown in Fig. 4, and thus a good density calculated from Eq. 6 must be resulted from a counterbalanced poor attractive free-energy functional in the theory. In order to have a more comprehensive evaluation of FMSA, we calculated as well the adsorption isotherms of LJ fluids in slit pores. The adsorption isotherm, defined by 0 Hz b dz, 41 provides a systematic check of the overall behavior of density profiles. Figures 9 and 10 present, respectively, the pre- FIG. 9. Adsorption isotherm (* 2 )foraljfluid inside a slit pore. The filled and open symbols are the simulation data of Sweatman Ref. 40, and of van Megan and Snook Ref. 56, respectively. The solid, dashed, and dotted lines are the predictions from the FMSA, Sweatman s, and MF theories, respectively.

9 7396 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Y. Tang and J. Wu FIG. 10. Same as in Fig. 9 except at H*3.50 and with an additional set of simulation data, the filled triangles, by Tan and Gubbins Ref. 29. dicted adsorption isotherms over a wide range of densities in slit pores as those in Figs. 7 and 8. In general, the adsorption isotherms predicted by the FMSA are satisfactory while the MFT consistently underestimates the simulation results for the range of densities considered in this work. These results are understandable based on their snapshots in Figs. 7 and 8. For both isotherms, the DFT reported by Sweatman yield rather good results. It is notable that our FMSA coincides better with the simulation data of Tan and Gubbins 29 in Fig. 10. Despite its better accuracy, Sweatman s theory needs tedious implementation procedures and much more computational work. The theory makes use of two inherently inconsistent methods for completing its functional, i.e., an empirical equation of state for the bulk thermodynamic properties and the HNCPY integral equations for the structure information. To resolve the inconsistency, a number of adjustable or scaling parameters have to be introduced empirically. The theory could break down, as cautioned by the author, if the nonsolution region of the two integral equations is encountered. In contrast, the present theory is computationally much less extensive. Its implementation is nearly as straightforward as that for MFT. Since both bulk properties and structure information are retrieved from the solution of the FMSA, the thermodynamic consistency is warranted. IV. CONCLUSIONS In summary, we have developed a new densityfunctional theory for bulk as well as inhomogeneous Lennard-Jones fluids by combination of MFMT for hard spheres and FMSA for attractive forces. Its applications to homogeneous LJ fluids have been demonstrated by comparing the calculated phase coexistence curve, saturated pressure and chemical potential with simulation data and with results from the BH and MFT theories. For inhomogeneous systems, it has been applied to describing the LJ density profiles around hard wall and within attractive slit pores. The predictions from the new density-functional theory are found rather satisfactory and they are substantially better than those from mean-field theory in general. In addition, the predicted adsorption isotherms for the LJ fluids in slit pores agree well with simulation data, better than the mean-field results. It is also demonstrated that the proposed theory is computationally convenient, thermodynamically consistent, and free of any numerical troubles of breakdown as encountered in more complicated DFT theories. Since FMSA has been extended to the critical region of the LJ fluid, 49 to associating LJ chains 57 and associating LJ chain mixtures, 58 the successful implementation of the MFMTFMSA in this work paves a way for studying systems consisting of more complex molecules, as demonstrated individually for the MFMT 24,25 and for the FMSA. 57,58 ACKNOWLEDGMENT J.W. gratefully acknowledges the financial support from University of California Research and Development Program and from the National Science Foundation Grant No. CTS J. D. van der Waals, Z. Phys. Chem. 13, J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, 2nd ed. Academic, London, M. Wertheim, Phys. Rev. Lett. 10, E. Thiele, J. Chem. Phys. 39, E. Waisman and J. L. Lebowitz, J. Phys. Chem. 52, L. Blum, Mol. Phys. 35, J. A. Barker and D. Henderson, Rev. Mod. Phys. 48, D. Chandler, in The Liquid State of Matter: Fluids, Simple and Complex, edited by E. W. Montroll and J. L. Lebowitz North-Holland, Amsterdam, 1982, p D. Chandler, J. D. Weeks, and H. C. Andersen, Science 220, R. Evans, in Fundamentals of Inhomogeneous Fluids, editedbyd.henderson Dekker, New York, 1992, p H. T. Davis, Statistical Mechanics of Phases, Interfaces, and Thin Films VCH, New York, G. F. Teletzke, L. E. Scriven, and H. T. Davis, J. Chem. Phys. 77, G. F. Teletzke, L. E. Scriven, and H. T. Davis, J. Chem. Phys. 78, C. Ebner, W. F. Saam, and D. Stroud, Phys. Rev. A 14, V. K. Shen and P. G. Debenedetti, J. Chem. Phys. 114, L. Perez, S. Sokolowski, and O. Pizio, J. Chem. Phys. 109,

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