Application of density functional theory to the structure of hard sphere fluid around different hard and soft spheres
Mohammad Kamalvand * and Ezat Keshavarzi*
Chemistry Department, Isfahan University of Technology, Isfahan, Iran [email protected] ; [email protected]
Introduction
Fluids near solid surfaces are commonplace in nature and engineering, for example, in gas storage, oil recovery, engine lubrication, zeolites, clay swell, in the stability of colloidal systems, heterogeneous catalyst reactions and removal of various pollutants.
The molecular structure of the confined fluid can differ dramatically from that in the bulk.
It is therefore that we cannot use our knowledge of bulk fluids, when we are dealing with fluids in confined geometries.1 Despite the important advances achieved with the combination of modern theories and computer molecular simulations, inhomogeneous situations are not yet completely understood. Even simple spherical fluids show a complex behavior when confined in well defined geometries. When the pore size is of the order of the correlation length, the presence of walls causes a dramatic change in the behavior of these confined systems compared to that exhibited in the bulk phase. In particular, energetic interactions and geometrical confinement modify the character of phase transitions, shift critical points, and new observable metastable states and hysteresis phenomena appear.1
The theory of the structure and phase behavior of simple fluids relies heavily on our knowledge of the hard sphere system, which serves as a standard reference system for determining the properties of more realistic models. The hard sphere model provides excellent testing ground for any theory of the liquid state. This model is also of practical importance because it can be considered as a zeroth-order approximation in the statistical thermodynamics of an extensive variety of more realistic physical systems with soft repulsive cores and attractive interactions, both of which can be handled perturbatively or using more specific approximations. Due to this importance, in this research work, we study the structure of hard sphere fluid around different hard and soft spheres.
The Density Functional Theory
A very successful and general method for determining the thermodynamic and structural properties of inhomogeneous fluids is density functional theory (DFT). This method is based on the formulation of the free energy for an inhomogeneous fluid as a functional of the density profile ρ(r) . At equilibrium condition, grand potential, Ω[ρ] , is minimal with respect to variation in the density distribution. The grand potential is related to the intrinsic Helmholtz free energy functional, F[ρ] as follow2
Ω[ρ(r)] = Fex[ρ(r)] + kT
∫
drρ(r){ln Λ3 ρ(r) −1} +∫
drρ(r)[Vext(r) − µ] (1)where µ is the chemical potential of the system, Vext (r) is the external field and Λ is
thermal de- Broglie wavelength. With minimization of grand canonical potential respect to ρ(r) we have,
ρ(r) = ρ0 exp − βVext (r) + βµex − δFex [ρ(r)] (2) δρ(r)
According to recently developed theory for DFT3, we use fundamental measure weight functions of Rosenfeld 2 for construction of excess Helmholtz free energy density:
Fex [ρ(r)] = kT
∫
drΦ(r) (3)where the excess Helmholtz energy density Φ(r) is a function of density distribution ρ(r) . It consists of scalar and vector contributions2,3
Φ(r) = ΦS (r) + ΦV (r) (4)
Results and Discussion
We solved the integral equation (2) iteratively to find the density profile of a hard sphere fluid outside of a sphere with diameterσ . Figure (1-A) shows very good agreement between this data and simulation results. Then, we obtained density profile of
profile at contact point increases when diameter of the original hard sphere increases.
This behavior is due to tendency of the inhomogeneous system to maximize its entropy.
Additionally, using this approach, we obtained the density profile of a hard sphere fluid around various soft spheres with different force fields. These results summarized in Fig.
(1-C). according to this figure, the height of first peak increases with the hardness of the original sphere. This behavior is because of reducing the effective diameter of the original soft sphere and is due to entropy effects. The entropy effect in this situation and also in confined fluids is may be a version of the effect of the depletion potential.
3.5
simulation (A)
4.0
3.5 D = 1 3.0
(B) T*=2 (C)
3.0 DFT D = 1.5 2.5 T*=1
3.0 D = 2.5 T*=0.5
2.5
2.0
1.5
1.0
0.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0
r/σ
2.5 2.0 1.5 1.0 0.5
D = 3.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
r/σ
2.0
1.5
1.0
0.5
0.0
0.5 1.0 1.5 2.0 2.5 3.0
r/σ
Figure 1 A) g(r) of a hard sphere fluid at ρσ 3 = 0.7 ; b) Density profile of a hard sphere fluid around
various hard spheres with different diameters at around different soft spheres at ρσ 3
= 0.7 .
ρσ 3
= 0.7 ; C) Density profile of a hard sphere fluid
g(r) ρ(r)σ3 ρ(r)
1. M. Dijkstra, J. Chem. Phys. 107, 3277, (1997).
2. Y. Rosenfeld, Phys. Rev. Lett., 63, 980, (1989).
3. Y-X Yu and J. Wu, J. Chem. Phys. 116, 7094, (2002).
