A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid

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1 Commun. Theor. Phys. (Beijing, China) 39 (2003) pp c International Academic Publishers Vol. 39, No. 2, February 15, 2003 A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid ZHOU Shi-Qi, CHEN Hong, and ZHANG Xiao-Qi Research Institute of Modern Statistical Mechanics, Zhuzhou Institute of Technology, Zhuzhou , China (Received April 15, 2002; Revised July 2, 2002) Abstract A new bridge functional as a function of indirect correlation function was proposed, which was based on analysis on the asymptotic behavior of the Ornstein Zernike (OZ) equation system and a series expansion whose renormalization resulted in an adjustable parameter determined by the thermodynamics consistency condition. The proposed bridge functional was tested by applying it to bulk hard sphere and hard core Yukawa fluid for the prediction of structure and thermodynamics properties based on the OZ equation. As an application, the present bridge functional was employed for non-uniform fluid of the above two kinds by means of the density functional theory methodology, the resulting density distribution profiles were in good agreement with the available computer simulation data. PACS numbers: Gy Key words: bridge functional, density functional theory, direct correlation function 1 Introduction Recently bridge functional concept combined with the universality principle and test particle trick was made to construct free energy density functional or its first order functional derivative. [1 13] Validity of the new procedure was ascertained by the fact that good predictions for the density distribution profiles of non-uniform fluid were made. The biggest advantage of the new methodology is its computational simplicity and accuracy, which is due to its including all orders of the first order direct correlation function (DCF) functional series expansion beyond the second order by the form of the bridge functional, and thus avoiding the calculation task of weighted density and weighting function as required by the weighted density approximation (WDA) [14] and the fundamental measure theory by Rosenfeld, [15] or the high order DCFs as required by the functional perturbative expansion approximation (PEA). [16] The new methodology uses as input the bulk bridge functional, which is required to be a function of indirect correlation function, such kinds of bridge functionals include the PY bridge functionals, Verlet-modified (VM) bridge functional, and Martynov Sarkisov (MS) bridge functional. [1,17] But all of these bridge functionals forms are only suitable for hard sphere fluid. To apply the methodology to non-uniform non-hard sphere fluid, the mixed bridge functional has to be employed. [2] To go beyond the bridge functional concept, the universality principle and the bulk radial distribution function were combined to make very good predictions for the density distribution profile, [9 13] but these formalisms were a little more computationally complicated. The present article is still based on the bridge functional concept, but a new analytical bridge functional is proposed and tested by applying it to uniform hard sphere fluid in Sec. 2. Then the new bridge functional is employed to study the structure and thermodynamics properties of uniform hard core Yukawa fluid and the density profile of non-uniform hard sphere fluid and hard core Yukawa fluid subjected to an external field in Sec. 3. Finally in Sec. 4, some remarks are made. 2 Formulation of Analytical Bridge Functional and Test by Applying It to Hard Sphere Fluid We denote the bridge functional as a function of indirect correlation function by B(γ), expanding B(γ) around γ = 0 B(γ) = B(0) + B (0)γ + B (0) 2! + B(4) (0) 4! γ 2 + B (0) γ 3 3! γ B(n) (0) γ n +, (1) n! the bulk Ornstein Zernike equation [17] reads γ = h(r) C (2) 0 (r; ρ b ) = ρ b dr h(r )C (2) 0 (r, r ; ρ b ), (2) the closure equation reads g(r) = exp[ β u(r) + γ(r) + B(γ)], (3) The project supported by National Natural Science Foundation of China (Grant No ), Scientific Research Fund of Education Department of Hunan Province of China (Grant No. 02B033) chixiayzsq@yahoo.com

2 232 ZHOU Shi-Qi, CHEN Hong, and ZHANG Xiao-Qi Vol. 39 where h(r) = g(r) 1 is the bulk total correlation function, g(r) is the bulk radial distribution function, β u(r) is the inter-particles interaction potential, β = 1/kT with k the Boltzmann constant and T the absolute temperature, and C (2) 0 (r, r ; ρ b ) is the bulk second order direct correlation function in which ρ b stand for the bulk parameters. When the inter-particle separation is very large, β u(r) 0, h(r) 0, from Eq. (2) we know γ(r) 0 as r, finally from Eq. (3) we know that B(γ) 0 as γ(r) 0, which leads to B(0) = 0 when it is combined with Eq. (1). To ascertain the other coefficients in Eq. (1), we compare Eq. (1) with the expansions of other approximate bridge functionals. [1,17] For example, the PY bridge functional B PY (γ) = ln(1 + γ) γ, (4) the MS bridge functional the VM bridge functional B MS (γ) = (1 + 2γ) 1/2 γ 1, (5) B VM (γ) = γ 2 2(1 + 4γ/5). (6) Now we expand Eqs. (4) (6) around γ = 0, B PY (γ) = 0.5γ γ 3, (7) B MS (γ) = 0.5γ γ 3, (8) B VM (γ) = 0.5γ γ 3. (9) From Eqs. (7) (9), we know that it is reasonable to make B (0) = 0.0 and B (0)/2! = 0.5, thus equation (1) reduces to B(γ) = 0.5γ 2 + B (0) 3! = 0.5γ 2( 1 2B (0) 3! γ 3 + B(4) (0) 4! = 0.5γ 2( 1 + 2B (0)/3! 1! γ 2B(4) (0) 4! γ B(n) (0) γ n + n! γ 2 2B(n) (0) γ n 2 n! γ + 2B(4) (0)2!/4! 2! ) γ 2 2B(n) (0)(n 2)!/n! γ n 2 (n 2)! ). (10) The series in Eq. (10) cannot be truncated because γ is not a very small number. It is more accurate to represent the series in Eq. (10) by a function. Considering the similarity between the series in Eq. (10) and the expansion of exponential function, it is reasonable for one to approximate the former with the latter, thus B(γ) = 0.5γ 2 exp( αγ). (11) Equation (11) is the main result of the paper. However, it should be noted that the exponential function exp( αγ) with a chosen α is not consistent with the series in Eq. (10) with a fixed function form B(γ), because each of the terms of the series in Eq. (10) is not equal to the corresponding term of the expansion of exp( αγ), but the substitution avoids the truncation of the series. By choosing the appropriate parameter α, the approximation on each term of the series in Eq. (10) is smoothed out and cancelled out globally, thus the final error is reduced. So one can call this process as renormalization. In fact, the previous VM bridge functional, Eq. (6), is only a limit of Eq. (11) when one expands the exponential function exp( αγ) and truncates at the lowest order with α = 4/5. Equation (11) has advantage over previous Eqs. (4) (6) in that there does not exist any argument region in which equation (11) is not suitable. But obviously equations (4) and (5) will become senseless if γ is smaller than 1 and 0.5, respectively. As regarding Eq. (6), it will also become senseless if γ = This disadvantage of Eq. (6) gives rise to additional computational difficulty and has to incorporate additional approximation when equation (6) is combined with the re-normalization of the indirect correlation function to study on the Lennard Jones fluid in a recent paper. [30] As usual, the adjustable parameter α can be determined by the thermodynamics consistency condition, which requires the equality between the viral P v and the compressibility pressure P c. To test the validity of the new bridge functional, we apply it to the uniform hard sphere fluid for which the virial and compressibility pressures are of the following form where βp v = 1 + 4ηg(σ + ), (12) ρ b β P c = 1 ρ b drc (2) 0 ρ (r, ρ b ), (13) b η = πρ bσ 3 6 is the volume fraction and σ is the hard core diameter. The hard sphere interaction potential is of the following form, r/σ < 1, β u(r) = (14) 0, r/σ > 1.

3 No. 2 A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid 233 Fig. 1 The adjustable parameter α as a function of volume fraction η for hard sphere fluid. Fig. 2 The compressibility factor Z as a function of volume fraction η for hard sphere fluid. In Fig. 1, we plot the numerical value of α as a function of hard sphere bulk volume fraction η. It was shown that α is a continuous function of η, thus one can employ the interpolation procedure to get the value of α at arbitrary hard sphere bulk η. In Fig. 2 we plot the compressibility factor Z = βp /ρ b as a function of the volume fraction η. Figure 2 shows that the predicted compressibility factor is in very good agreement with the prediction of the Carnahan Starling equation of state, [18] which is a very good fit to the simulation data for the hard sphere compressibility factor. The freezing point of the bulk hard sphere fluid is 0.943, while the numerical algorithm employed here, which is due to Labik et al., [19] is stable even at bulk density ρ b = 0.93, [20] where the percent relative deviation error of the predicted Z = from the Carnahan Starling equation of state value, Z = , is only 6.75%. By solving the OZ equation from the low density to high density, in which the outcome of the solution at low density is used as inputs of the solution of the equation at high density, we can solve the equation numerically at bulk density very near to the freezing point, i.e. ρ b = Figures 3 5 displayed the hard sphere radial distribution function at several bulk density. It can be seen that the agreement with the simulation data is very good. Figure 6 displayed the structure factor for the hard sphere fluid S(k) = 1 1 ρ b C(k) (15) for two bulk densities. In Fig. 7, the compressibility of the system χ T as a function of the volume fraction η is plotted, χ T = S(0, η), (16) where χ 0 T χ 0 T = 1 βρ b is the compressibility of the ideal system. Again particularly excellent agreement with the simulation data is obtained. Fig. 3 The radial distribution function g(r) for hard sphere fluid of bulk density ρ b = 0.8. The line is from the prediction of the present bridge functional, while the symbol is for the Monte Carlo simulation data. [23] Fig. 4 The same as in Fig. 3, but ρ b = 0.7. From these predictions one can conclude that the present bridge functional, Eq. (11), can close the OZ equation for hard sphere fluid very well. By using the interpolation procedure for the adjustable parameter α, one can obtain the structure and thermodynamics of the bulk

4 234 ZHOU Shi-Qi, CHEN Hong, and ZHANG Xiao-Qi Vol. 39 hard sphere fluid at any density below the freezing point density ρ b σ 3 = Application to Non-uniform Hard Sphere and Hard Core Yukawa Fluid The merit of the present bridge functional lies in that the bridge functional is expressed as a function of the indirect correlation function, and this is exactly the required condition by the DFT methodology in Ref. [1]. So in this section we will firstly apply the bridge functional, Eq. (11), to non-uniform hard sphere fluid based on the DFT in Ref. [1]. In the formalism of DFT, the density profile equation of a non-uniform single component fluid is ρ(r) = ρ b exp βϕ ext (r) + C (1) (r, [ρ]) C (1) 0 (ρ b)}, (17) Fig. 5 The same as in Fig. 3, but ρ b = 0.6. Fig. 6 The structure factor S(k) for hard sphere fluid of two values of volume fraction η. The line is from the prediction of the present bridge functional, while the symbol is for the molecular dynamic simulation data. [24] Fig. 7 Ratio of the isothermal compressibility for hard sphere fluid as a function of the volume fraction η. The line is from the prediction of the present bridge functional, while the symbol is for the molecular dynamic simulation data. [24] where ϕ ext (r) is the external potential responsible for the generation of the density distribution ρ(r), and C (1) and C (1) 0 are the non-uniform and uniform first order DCF, respectively. According to Ref. [1] and the present bridge functional Eq. (11), equation (17) reduces to ρ(r) = ρ b exp βϕ ext (r) + dr (ρ(r ) ρ b )C (2) 0 (r, r, ρ b ) [ ]} + B dr (ρ(r ) ρ b )C (2) 0 (r, r, ρ b ). (18) We apply Eq. (18) to the case of a hard sphere fluid near a hard wall for which the external potential ϕ ext (r) has the form, z/σ < 0.5, βϕ ext (z) = 0, z/σ > 0.5. (19) The second order DCF C (2) 0 (r; ρ b ) from the Percus Yevick approximation [21] is employed in Eq. (18). In Figs. 8 and 9 we plot the density distribution profile for the case of Eq. (19) at two bulk densities, also displayed are the predictions from the VM bridge functional which was shown to be the most accurate predictions for nonuniform hard sphere fluid in Ref. [1]. From Figs. 8 and 9, one can see that the present predictions are even more accurate than that based on the VM bridge functional. It should be noted that the adjustable parameter α in Eq. (18) is exactly the value determined by solving the OZ equation and forcing the thermodynamics consistency to be satisfied at the same bulk density. This point is different from the following case for non-uniform hard core Yukawa fluid. Now we will apply the present bridge functional to a more complicated non-uniform fluid, the hard core Yukawa fluid, for which the interaction potential is of the following form, r/σ < 1, β u(r) = (20) βεσ exp[ λ(r σ)/σ]/r, r/σ > 1.

5 No. 2 A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid 235 the compressibility pressure, equation (13), remains the same as that for the hard sphere fluid. Firstly we solve the OZ equation with the present bridge functional Eq. (11) for the bulk hard core Yukawa fluid with λ = 1.8 and reduced temperatures T = k B T /ɛ = 2.0 and 1.5 respectively, at various bulk density. Figure 10 displays the value of α as a function of volume fraction η at two reduced temperatures, i.e. T = 2.0 and 1.5. It was shown that the value of α is a smooth and continuous function of η at fixed λ and T. In Fig. 11, we plot the compressibility factor Z as a function of the reduced bulk density ρ = ρ b σ 3. It is shown that the agreement with the simulation data is very good. However, it should be noted from Fig. 11 that even the MC data and MD data deviate a little from each other. From Eqs. (21) and (13) we can see that the compressibility factor is related closely with the Fourier transform of the second order DCF. The fact that the present bridge functional can predict accurately the compressibility factor indicates the ability of its supplying the accurate second order DCF. So in the following application, we employ the numerically obtained C (2) 0 (r; ρ b ) as input into the DFT in stead of the analytical mean spherical approximation solution. [22] Fig. 8 Density distribution profiles of a hard sphere fluid (ρ b σ 3 = 0.575) near a hard wall. The line corresponds to the predictions of the present theory, while the triangles stand for the corresponding computer simulation data. [25] Fig. 10 The adjustable parameter α as a function of volume fraction η for hard core Yukawa fluid of range parameters λ = 1.8 and two reduced temperature T kt/ɛ = 2.0 and T = 1.5 respectively. Fig. 9 The same as in Fig. 8, but ρ b σ 3 = For the hard core Yukawa fluid, the virial pressure, equation (12), should be changed as follows: βp v ρ b = 1 βρ b 6 0 dr4πr 3 du(r) g(r), (21) dr Fig. 11 The compressibility factor Z as a function of volume fraction η for hard core Yukawa fluid of range parameter λ = 1.8 and two reduced temperatures T kt/ɛ = 2.0 and T = 1.5 respectively. The lines correspond to the predictions of the present bridge functional, the triangles stand for the corresponding computer simulation data from Ref. [26], and the dots for the corresponding computer simulation data from Ref. [27]. Now we consider the hard core Yukawa fluid near (i) a structureless hard wall for which the external potential ϕ ext (r) is of the same form as in Eq. (19); (ii) a structureless hard wall with an attractive tail for which the external potential ϕ ext (r) is of the following form, z/σ < 0.5, βϕ ext (z) = βɛ w exp[ λ(z σ/2)/σ], z/σ > 0.5. (22)

6 236 ZHOU Shi-Qi, CHEN Hong, and ZHANG Xiao-Qi Vol. 39 universality principle which means that the free energy density functional is independent of the external potential responsible for the formation of the density distribution. The first order DCF is the functional derivative of the free energy density functional with respective to the density distribution, thus the first order DCF is also independent of the external potential. Comparing Eq. (18) with Eq. (17), one can see that the term in Eq. (18), dr (ρ(r ) ρ b )C (2) 0 (r, r ; ρ b ) [ ] +B dr (ρ(r ) ρ b )C (2) 0 (r, r ; ρ b ) (25) Fig. 12 Density profiles for the hard core Yukawa fluid of ρ = 0.7, T kt/ɛ = 2.0 and λ = 1.8 near a hard wall (a hard wall with an attractive tail). The line corresponds to the predictions of the present bridge functional, while the triangles stand for the corresponding computer simulation data from Ref. [28] for the case of ɛ w/ɛ F = 0.0. Fig. 13 Density profiles for the hard core Yukawa fluid of ρ = 0.7, T kt/ɛ = 1.25 and λ = 1.8 near a hard wall with an attractive tail. The line corresponds to the predictions of the present bridge functional, while the triangles stand for the corresponding computer simulation data from Ref. [28]. For the case of the hard core Yukawa fluid, the value of α in Eq. (18) is determined by the sum rule which specifies the bulk compressibility factor Z by the hard wall contact density ρ w, Z(ρ b )ρ b = ρ w. (23) The value of Z(ρ b ) in Eq. (23) is obtained from solving the present self-consistent integral equation since it supplies the accurate Z(ρ b ) as indicated in Fig. 11. The value of ρ w can be obtained from ρ(0) in Eq. (18) when the external potential has the following form:, z/σ < 0, ϕ ext (z) = (24) 0, 0 < z/σ. Obviously the parameter α is only related to the bulk parameters ρ b, and is not connected to the external potential parameters. This point can be concluded from the is actually the difference between the non-uniform first order DCF and its uniform counterpart, and the uniform first order DCF is a constant for fixed bulk parameters, so the fact that the first order DCF is independent of the external potential means that the term (25), and thus the parameter α in Eq. (18) is independent of the external potential. Thus so specified α in this special case (the structureless hard wall) can be used for an arbitrary external potential case (for example, the structureless hard wall with an attractive tail as considered in the present paper and denoted by the external potential ϕ ext (r) of Eq. (22)). As an example, we consider the case of the range parameter λ = 1.8, which makes the Yukawa fluid qualitatively similar to argon as the density and temperature of the liquid is in equilibrium with its vapor. In Figs. 12 and 13. we plot the density distribution profiles of the hard core Yukawa fluid subjected to an external potential denoted by Eqs. (19) and (22). It is shown that the agreement with the simulation data for the considered bulk parameters is good. 4 Concluding Remarks The proposed bridge functional Eq. (11) combined with the OZ equation and the thermodynamics consistency condition can predict very accurately the structure and thermodynamics properties of the bulk hard sphere fluid and the hard core Yukawa fluid. Although the bridge functional includes an adjustable parameter, it is a continuous and very smooth function of the bulk parameters, so if one solves the OZ equation for the specified models at various bulk parameters, then stores the obtained results, one can use the interpolation procedure to specify the appropriate value of the adjustable parameter α for arbitrary bulk parameters to make routine and fast predictions for the structure and thermodynamics properties. To apply the present bridge functional to other models and to improve the bridge functional furthermore will be left as a future study.

7 No. 2 A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid 237 The present bridge functional is expressed as a function of the indirect correlation function, and this is exactly the requirement by DFT methodology in Ref. [1]. When we apply the bridge functional to non-uniform fluid (hard sphere and hard core Yukawa fluid), good predictions for the density distribution are produced, as one expected. However, we observe that the agreement with the simulation data is better for non-uniform hard sphere fluid than that for the non-uniform hard core Yukawa fluid even we use the sum rule for the latter. We guess that it is due to the insufficiency of the bridge functional for the hard core Yukawa fluid being expressed as a function of the indirect correlation function only. So it is an interesting problem to devise a method to incorporate the renormalization of the indirect correlation function in Eq. (18) for non-uniform fluid as in the bridge functional expression for uniform fluid in Ref. [29]. References [1] S. Zhou and E. Ruckenstein, J. Chem. Phys. 112 (2000) [2] S. Zhou and E. Ruckenstein, J. Chem. Phys. 112 (2000) [3] S. Zhou, J. Chem. Phys. 113 (2000) [4] S. Zhou and X. Zhang, Phys. Rev. E64 (2001) [5] S. Zhou, X. Sun, H. Chen, and H. Li, J. Chem. Phys. 115 (2001) [6] S. Zhou, J. Phys. Chem. B105 (2001) [7] S. Zhou and X. Zhang, J. Colloid Interface Sci. 242 (2001) 152. [8] N. Choudhury and S.K. Ghosh. J. Chem. Phys. 114 (2001) [9] S. Zhou, Phys. Rev. E63 (2001) [10] S. Zhou, Phys. Rev. E63 (2001) [11] S. Zhou, Commun. Theor. Phys. (Beijing, China) 37 (2002) 543; S. Zhou, Commun. Theor. Phys. (Beijing, China) 38 (2002) 355. [12] S. Zhou, J. Chem. Phys. 115 (2001) [13] S. Zhou and X. Zhang, Chin. Phys. 11 (2002) [14] R.L. Davidchack and B.B. Laird, Phys. Rev. E60 (1999) [15] Y. Rosenfeld, Phys. Rev. E50 (1994) R3318. [16] S. Zhou and E. Ruckenstein, Phys. Rev. E61 (2000) [17] J.P. Hansen and I.R. McDonald, Theory of Simple Liquids, 2nd ed., Academic, New York (1986). [18] N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 635. [19] S. Labik, A. Malijevsky, and P. Vonka, Mol. Phys. 56 (1985) 709. [20] W.G. Hoover and F.H. Ree, J. Chem. Phys. 49 (1968) 204. [21] E. Thiele, J. Chem. Phys. 39 (1963) 474; M.S. Wertheim, Phys. Rev. Lett. 19 (1963) 321. [22] E. Waisman, Mol. Phys. 25 (1973) 45; J.S. Hoye and G. Stell, Mol. Phys. 32 (1976) 195. [23] J.A. Barker and D. Henderson, Mol. Phys. 21 (1971) 187. [24] W.E. Alley and B.J. Alder, Phys. Rev. A27 (1983) 3158; W.E. Alley, B.J. Alder, and S. Yip, Phys. Rev. 27 (1983) [25] R.D. Groot, N.M. Faber, and J.P. Van der Eerden, Mol. Phys. 62 (1987) 861. [26] P.T. Cummings and G. Stell, J. Chem. Phys. 78 (1983) [27] C. Rey, L.J. Gallego, and L.E. Gonzalez, J. Chem. Phys. 96 (1992) [28] W. Olivares-Rivas, L. Degreve, D. Henderson, and J. Quintana, J. Chem. Phys. 106 (1997) [29] J.M. Bomont, N. Jakse, and J.L. Bretonnet, J. Chem. Phys. 107 (1997) [30] N. Choudhury and S.K. Ghosh, J. Chem. Phys. 116 (2002) 8517.

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