Binary Hard-Sphere Mixtures Within Spherical Pores

Journal of the Korean Physical Society, Vol. 35, No. 4, October 1999, pp. 350 354 Binary Hard-Sphere Mixtures Within Spherical Pores Soon-Chul Kim Department of Physics, Andong National University, Andong 760-749 Soong-Hyuck Suh Department of Chemical Engineering, Keimyung University, Taegu 704-701 Chang Hee Lee Hanaro Center, Korea Atomic Energy Research Institute, Taejon 302-353 (Received 16 June 1999) The free-energy model, which is based on the fundamental geometric measures of the particles, has been employed to investigate the structural properties of nonuniform hard-sphere mixtures within spherical pores. Monte Carlo simulation has been performed to calculate the density profiles of hard-sphere mixtures confined in spherical pores, and the simulation has been compared with the calculated results. Comparisons between the theoretical results and the simulation data have shown that the free-energy model demonstrates reliable accuracy and reproduces the simulation data accurately even for larger size ratios of hard spheres. I. INTRODUCTION The physical properties of solid-fluid interfaces are both of fundamental theoretical interest and also of considerable practical importance in connection with phenomena such as adsorption, wetting, catalysis, and filteration [1]. Several theoretical methods have been proposed to describe the structural properties of model fluids confined in special symmetrical systems. Over the past decade, the density functional approaches have evolved into an efficient tool for dealing with these problems. It is well-known that for hard-sphere fluids the density functional approaches based on the free-energy functional approximations simulate the structural properties of inhomogeneous model fluids reasonably well compared with the standard integral equations. The progress made in this field has been described in several review articles [1]. Several different versions for the density functional approaches have been suggested. Among them, a free-energy model, which is based on the fundamental geometric measures of particles developed by Rosenfeld [2], seems to be the most suitable approximation for describing nonuniform hard-sphere mixtures. Many applications have shown that the fundamental-measures freeenergy model generally describes the thermodynamic and the structural properties of confined hard-sphere fluids well compared with computer simulations. E-mail: sckim@andong.ac.kr, Fax: 0571-841-1628 More recently, Gonzalez et al. [3] have applied the fundamental-measures free-energy models, including quasi-0d functional approximations, to investigate the density profiles of hard-sphere fluids confined in spherical pores. They have shown that even for a small spherical pore the free-energy models describe well the density behavior of a confined hard-sphere fluid and that for a large spherical pore the overall performances of the free-energy models are quite similar. However, their applications for confined fluids have been restricted to the structural properties of a hard-sphere fluid in a spherical pore or to a nonuniform binary hard-sphere mixture near a structureless hard wall, and have not be able to handle a nonuniform binary hard-sphere mixture in a spherical pore [4]. An important difference between the spherical pore problem and that with a hard wall is that the fluid (the number of particles) is finite. The structural properties of confined hard-sphere mixtures within a spherical pore have been studied little until recently compared with those of confined hard-sphere mixtures near a hard wall or a semipermeable membrane [5,6]. In this work, we apply the fundamental-measures freeenergy model, which was suggested by Rosenfeld [2], to investigate the density profiles of binary hard-sphere mixtures in hard spherical pores. For comparative purposes, we perform Monte Carlo simulations to calculate the density profiles of binary hard-sphere mixtures. Finally, the calculated results are compared with the computer simulation and those of other approximations. -350-

Binary Hard-Sphere Mixtures Within Spherical Pores Soon-Chul Kim et al. -351- II. THEORY In the fundamental-measures free-energy model for binary hard-sphere mixtures [2], the excess free energy functional F ex [ρ 1, ρ 2 ] is assumed to have the form βf ex [ρ 1, ρ 2 ] = dr Φ[n α (r)], (1) where β = 1/K B T is the inverse temperature and Φ[n α (r)] the excess free energy per volume. It is assumed that the excess free-energy, Φ[n α (r)] = n 0 (r) ln[1 n 3 (r)] + n 1(r)n 2 (r) n V 1 (r) n V 1 (r) [1 n 3 (r)] + n 2(r)[n 2 (r)n 2 (r) 3(n V 2 (r) n V 2 (r))] 24π[1 n 3 (r)] 2, (2) is a function of only the system-averaged fundamental geometric measure of the particles n α (r) = 2 i=1 ds ρ i (s)ω (α) i ( r s ). (3) In Eq. (3), the weighting function ω (α) i (r) represents the geometrical properties of the particles and is defined as ω (3) i (r) = θ(r R i ), ω (2) i (r) = δ(r R i ), (V 2) ω i (r) = r/r δ(r R i ), (4) where R i is the radius of a hard sphere, θ(x) the Heaviside step function, and δ(x) the Dirac delta function. The other functions are proportional to these three and are given by ω (0) i (r) = ω (2) i (r)/4πri 2, ω(1) i (r) = ω (2) (V 1) (V 2) i (r)/4πr i, and ω i (r) = ω i (r)/4πr i. In the density functional theory, the grand canonical potential Ω[ρ 1, ρ 2 ] is written as the density functional of the number density ρ i (r) βω[ρ 1, ρ 2 ] = βf ex [ρ 1, ρ 2 ] 2 [ ] + dr[ln(λ 3 i ρ i (r)) 1+βu ext i (r) βµ i ]ρ i (r),(5) i=1 where µ i is the equilibrium chemical potential, u ext i (r) an external potential, and Λ i = h/ 2πm i k B T the thermal de Broglie wavelength. The equilibrium particle density distribution of the inhomogeneous fluid corresponds to the minimum of the grand canonical potential satisfying the Euler-Lagrange relation δβω[ρ 1, ρ 2 ]/ = 0 [1]. Then, Eq. (5) becomes δβω[ρ 1, ρ 2 ] = δβf ex[ρ 1, ρ 2 ] + ln Λ 3 i ρ i (r) +β[u ext i (r) µ i ] = 0. (6) If the inhomogeneous fluid is in contact with the homogeneous bulk fluid, its chemical potential µ i is equal to that of the homogeneous bulk fluid. Then, the densityprofile equation is given as, after some manipulations, ρ i (r) = ρ bi exp[ βu ext i (r) + i (r; [ρ 1, ρ 2 ]) i (ρ b1, ρ b2 )], (7) where ρ bi is the homogeneous bulk densities. In Eq. (7), i (r; [ρ 1, ρ 2 ]) is the one-particle direct correlation function (DCF) of the inhomogeneous hard-sphere mixtures, which is defined as i (r; [ρ 1, ρ 2 ]) = δβf ex[ρ 1, ρ 2 ], (8) and i (ρ b1, ρ b2 ) the one-particle DCF of the homogeneous hard-sphere mixtures. In the fundamental-measures free-energy model, the one-particle DCF, i (r; [ρ 1, ρ 2 ]), becomes, with the aid of Eqs. (1) and (8), i (r; [ρ 1, ρ 2 ])= ds Φ[n γ (s)] n α α (r) ω(α) i ( r s ). (9) For the homogeneous state, Eq. (9) simply becomes, i (ρ b1, ρ b2 ) = ln(1 n 3 ) n 0v i + n 1 s i + n 2 R i 1 n 3 n 1n 2 v i (1 n 3 ) 2 3n 2 1s i 24π(1 n 3 ) 2 2n 3 2v i 24π(1 n 3 ) 3 (10) with n 0 = 2 i=1 ρ bi, n 1 = 2 i=1 R iρ bi, n 2 = 2 i=1 4πR2 i ρ bi, n 3 = 2 i=1 4πR3 i ρ bi/3, s i = 4πRi 2, and v i = 4πRi 3/3. Equations (7), (9), and (10) constitute the density-profile equation for binary hard-sphere mixtures in an open (grand canonical) pore. For the binary hard-sphere mixture near a hard wall, the equilibrium density profile is calculated from Eq. (7) [5,6]. However, for the binary hard-sphere mixtures confined in a spherical pore, the relevance parameter is the averaged number of particles N i, and thus we need the analogue of Eq. (7) in term of N i [3]. Integrating Eq. (7), we have N i = dr ρ i (r) = ρ bi dr exp[ βu ext i (r) + i (r; [ρ 1, ρ 2 ]) i (ρ b1, ρ b2 )]. (11) From Eqs. (7) and (11), the density-profile equation becomes ρ i (r) = N i exp[ βu ext i (r) + i (r; [ρ 1, ρ 2 ])] / dr exp[ βu ext i (r) + i (r; [ρ 1, ρ 2 ])]. (12)

-352- Journal of the Korean Physical Society, Vol. 35, No. 4, October 1999 Fig. 1. Density profiles of a hard-sphere fluid in a spherical pore with R pore = 5.0σ (N = 277). The solid lines and the circles are from the theory and the computer simulation [8], respectively. It is noted that this density ρ i (r) still corresponds to a system in an open (grand canonical) pore, but not to one in a closed (canonical) pore. Actually, for a model fluid confined in a small spherical pore, significant differences arise between the canonical and the grand canonical ensemble profiles [3,7]. III. RESULTS AND DISCUSSION As a simple application of the fundamental-measures free-energy model, we consider a hard-sphere fluid confined in a spherical pore. In this case, the external potential u ext (r) is given as u ext (r) = r > R pore R = 0 r < R pore R, (13) where R pore and r are the radius of the spherical pore and the distance to the center of the spherical pore, respectively. For the comparison with the simulation results [8], we have fixed the (average) number of particles N in the pore and have calculated the density distribution of a hard-sphere fluid confined in a given pore N = Rpore R 0 dr ρ(r). (14) The resulting density profiles ρ(r)σ 3 for a hard-sphere fluid confined to a pore of radius 5.0σ are displayed in Figs. 1 and 2, where σ is the diameter of a hard sphere. In this case, the maximum radius available to the center of a hard sphere is R pore R (or, R pore σ/2). For such one-component hard-sphere fluids, very excellent agreement between the theoretical results and the Fig. 2. Same as Fig. 1 except that N = 342. computer simulation are obtained even for higher densities (N = 342). These results demonstrate the reliable accuracy of the fundamental-measures free-energy model. As expected from the problems (for a large pore R pore = 5.0σ and low packing fraction η = πρ b σ 3 /6), the finite-size effect is very small [3]. Comparison with other approximations shows that the fundamental-measures free-energy model is better than the simple weighteddensity approximation based on the bulk density [9] and the weight-density approximation of Tarazona [10], even though for clarity we did not display their results in these figures. In the case of the simple weighted-density approximation of Zhou [9], a large discrepancy between the theory and the simulation data was detected at the spherical pore wall. As further applications, we consider binary hardsphere mixtures of two species with radii R 1 and R 2 in a spherical pore. In this case, the external potential u ext i (r) is given as u ext i (r) = r > R pore R i = 0 r < R pore R i. (15) In order to provide the basis for a discussion on the validity of theoretical calculations employed in this work, we have also carried out Monte Carlo simulations, which allow the direct computations of pore-density profiles for binary hard-sphere mixtures confined in spherical pores. Our simulation runs followed the standard canonical Monte Carlo procedures [11]. A large number of ensembles, typically between one to two billion steps, were generated to ensure adequate sampling over the configurational space. The maximum distance for a displacement for each species was adjusted in such a way as to achieve 40 50 percentage acceptance in every halfmillion trials. Radial density profiles were calculated by dividing the effective pore section into the equal radial increments of (R pore R 1 )/64.

Binary Hard-Sphere Mixtures Within Spherical Pores Soon-Chul Kim et al. -353- Fig. 3. Density profiles of a binary hard-sphere mixture in a spherical pore (N 1 = 23, N 2 = 66, and σ 2/σ 1 = 1.5). The solid lines and the circles are from the theory and computer simulation, respectively. Throughout the numerical calculations, we have fixed the (average) number of particles N i in a spherical pore as N i = Rpore R i 0 dr ρ i (r), i = 1, 2 (16) where the radius of the spherical pore is R pore = 5.0σ 1. In this case, the maximum radius available to the center of the hard sphere is R pore R i. Figure 3 shows the results for N 1 = 23 and N 2 = 66 with a size ratio σ 2 /σ 1 = 1.5. As can be seen from Fig. 3, the calculated result yields excellent agreement with the computer simulation. In this case, the packing fraction is Fig. 5. Same as Fig. 3 except that (N 1 = 67, N 2 = 16, and σ 2/σ 1 = 2.0). η = π(ρ b1 σ 3 1 + ρ b2 σ 3 2)/6 = 0.2930. For a similar packing fraction η = 0.3080 with different N 1 = 114 and N 2 = 45, a comparison of the calculated results with the computer simulation is presented in Fig. 4. Comparison shows that the calculated result is in good agreement with the computer simulation, even through a slight difference in the density, ρ 1 (r)σ 3 1, may be seen at the second peak. However, the overall picture shows that the free-energy model describes very well the density profiles of binary hard-sphere mixtures confined in spherical pores. Figure 5 shows the results for the density profiles for N 1 = 67 and N 2 = 16 with the size ratio σ 2 /σ 1 = 2.0. As expected from Figs. 3 and 4, the calculated results are in good agreement with the computer simulation. The calculated results for N 1 = 114 and N 2 = 45 have been Fig. 4. Same as Fig. 3 except that (N 1 = 114, N 2 = 45, and σ 2/σ 1 = 1.5). Fig. 6. Same as Fig. 3 except that (N 1 = 47, N 2 = 26, and σ 2/σ 1 = 2.0).

-354- Journal of the Korean Physical Society, Vol. 35, No. 4, October 1999 give an accurate description of the pore-density profiles of binary hard-sphere mixtures compared with computer simulations. The free-energy model employed here can generally be used as a reference system for a perturbative analysis in the study of binary Lennard-Jones mixtures [7,12]. It would be very interesting to apply a free-energy model which incorporates the finite-size effect to study the structural properties of binary hard-sphere mixtures in a small circular cavity as was done for a hard-sphere fluid [3,7]. If the finite-size effect is dominant, it is expected that a large discrepancy will be seen at the center of the cavity. We will investigate these problems in the near future. ACKNOWLEDGMENTS Fig. 7. Same as Fig. 3 except that (N 1 = 22, N 2 = 7, and σ 2/σ 1 = 3.0). presented in Fig. 6. Here, the corresponding packing fraction is η = 0.3095. The interesting thing is that in the case of the high packing fraction the distance between the two peaks is the almost same as the diameter of a large hard sphere [5]. On the other hand, for the density profile ρ 1 (r)σ 3 1, a slight density difference between the theory and the simulation can be seen near the center of pore. Figure 7 presents the calculated results for N 1 = 22 and N 2 = 7 with the size ratio σ 2 /σ 1 = 3.0. The calculated result also compared well with the computer simulation, even though the size ratio σ 2 /σ 1 = 3 is extremely large; however, a disagreement with the computer simulation can be seen near the center. Perhaps, the discrepancy near the center comes from the finite-size effect in a spherical cage, even though the effect is small [3]. Thus, at high packing fraction, a strong finite-size effect is expected. We can consider that the finite-size effect is related with the packing fraction and the size ratio σ 2 /σ 1 ; more detailed studies of the finite-size effect on confined binary hard-sphere mixtures are needed. Once again, comparison also shows that even for a large size ratio, σ 2 /σ 1 = 3, the free-energy model is in good agreement with the computer simulation. These calculated results also reveal that the free-energy model is a promising candidate to study much more complicated systems. In this work, we have considered an application of the free-energy model to describe the structural properties of a fluid within spherical pore. It has been shown that up to the size ratio σ 2 /σ 1 = 3, the free-energy model can This work was supported by the Korea Research Foundation made in the program year of 1998 (Project No. 1998-015-D00132). REFERENCES [1] R. Evans, in Fundamentals of Inhomogeneous Fluids, edited by D. Henderson (Morcel Dekker, New York, 1992); H. Löwen, Phys. Rep. 237, 249 (1994). [2] Y. Rosenfeld, Phys. Rev. Lett. 63, 980 (1989); J. Chem. Phys. 98, 8126 (1993); G. Kahl, B. Bildstein and Y. Rosenfeld, Phys. Rev. E54, 5391 (1996); Y. Rosenfeld, M. Schmidt, H. Löwen and P. Tarazona, Phys. Rev. E55, 4245 (1997). [3] A. Gonzalez, J. A. White, F. L. Román, S. Velasco and R. Evans, Phys. Rev. Lett. 79, 2466 (1997); J. Chem. Phys. 93, 4357 (1998). [4] Soon-Chul Kim and Soong-Hyuck Suh, J. Korean Phys. Soc. 31, 708 (1997); Soon-Chul Kim, J. Korean Phys. Soc. 32, 691 (1998); Young-Wha Kim and Soon-Chul Kim, J. Korean Phys. Soc. 33, 444 (1998). [5] M. Muradi and G. Rickayzen, Mol. Phys. 66, 143 (1989); S. Sokolowski and J. Fischer, Mol. Phys. 70, 1097 (1990); R. Leidl and H. Wagner, J. Chem. Phys. 98, 4142 (1993); J. P. Noworta, D. Henderson, S. Sokolowski and K. Y. Chan, Mol. Phys. 95, 415 (1998). [6] P. Bryk, W. Cyrankiewicz, M. Borowko and S. Sokolowski, Mol. Phys. 93, 111 (1998). [7] Soon-Chul Kim, J. Chem. Phys. 110, 12265 (1999). [8] M. Calleja, A. N. North, J. G. Powels and G. Rickayzen, Mol. Phys. 73, 973 (1991). [9] S. Zhou, J. Chem. Phys. 110, 2140 (1999). [10] Soon-Chul Kim, J. Phys.: Condens. Matter 8, 959 (1993). [11] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987). [12] D. Henderson and S. Sokolwski, Phys. Rev. E52, 758 (1995).