The entropy of fluids confined within nano pores by Density Functional Theory
Mohammad Kamalvand* and Ezat Keshavarzi*
Chemistry Department, Isfahan University of Technology, Isfahan, Iran [email protected] ; [email protected]
Introduction
The behavior of a fluid in a solid porous medium differs markedly from the behavior of the same fluid in the bulk. The confined fluid is constrained to a pore space, generally of complicated shape, is affected by the external force field due to the solid matrix. The equilibrium spatial distribution of fluid and consequently the thermodynamic properties of the confined fluid, result from a balance of external forces on fluid molecules and internal forces among fluid molecules themselves. Therefore, the introduction of wall forces, and the competition between fluid–wall and fluid–fluid forces, can lead to interesting phase changes, include new kinds of phase transitions not found in the bulk phase, e.g. layering and wetting transitions, as well as shifts in regular bulk transitions (e.g. freezing, gas–liquid, and liquid–liquid).
The same as real fluids, the structure and thermodynamic properties of a hard sphere fluid, confined in a porous media, is different from its bulk fluid properties. For a hard sphere fluid, there are not any long range forces and these differences may not due to energetic effects. The only reason to cause this difference is tendency of the system to maximize the entropy. In this research work we show that the entropy of a confined hard sphere fluid is differed dramatically from the bulk fluid entropy.
Key words: Entropy, Confined fluid, Nanoslit, Yukawa, Density Functional Theory
The Density Functional Theory
Density Functional Theory is one the powerful approaches for studying the confined fluids. In a density functional theory for an inhomogeneous hard sphere fluid, the grand potential functional energy is related to the Helmholtz energy functional as follows1:
2
where µ is the chemical potential of the system and Vext (r) is the external field.
According to the variational principle, the equilibrium density distribution function of the non-uniform fluid corresponds to the minimum of the grand potential1,
∂Ω[ρ(r)]
∂ρ(r) = 0 (2)
In the DFT, the functional F[ρ(r)] is normally split into an ideal and an excess part1,
F[ρ(r)] = Fid [ρ(r)] + Fex [ρ(r)] (3)
Where Fid [ρ(r)] is the ideal intrinsic Helmholtz energy and exactly given by1
Fid [ρ(r)] = kBT
∫
drρ(r){
ln Λ3 ρ(r)−1
}
(4)
Where Λ is the thermal de Broglie wavelength, T is the absolute temperature and kB
is the Boltzmann constant? Eq. (2) follows that the equilibrium density distribution satisfies1
ρ (r)Λ3 = exp βµ − β ∂Fex [ρ (r)]
∂[ρ (r)] − βVext (r) (5)
According to fundamental measure DFT, the excess Helmholtz free energy of a hard sphere fluid obtained via2
Fex [ρ(r)] = kT
∫
drΦ(r) (6)where the excess Helmholtz energy density Φ(r) is a function of density distribution ρ(r) and consists of a scalar and a vector contributions.
Results and Discussion
For a hard sphere fluid at each thermodynamic state, the kinetic energy is assumed to be exactly equal to corresponding equivalent ideal gas. Additionally, in a hard sphere fluid there are not any attractive forces. Therefore, the excess part of internal energy for a hard sphere fluid is exactly equal to zero. In the other hand, from the thermodynamic relations we know that F = E − TS where F is the Helmholtz free energy, E is the
4
(vcon
f -macv)av
internal energy and S is the entropy of the system. According to this thermodynamic relation, for a hard sphere fluid, it can be concluded that F ex = −TS ex . As a result, if we could obtain the F ex of a confined fluid, subsequently we can calculate its excess entropy exactly. In Fig. 1 we show the excess entropy of a hard sphere fluid confined within different nonoslits. As is clear in this figure, this thermodynamic quantity has an oscillatory behavior and in macroscopic limits it is closed to the bulk value. Furthermore, the entropy of the confined system increases with increasing the density. This behavior of the excess entropy of the confined system is related to its average molecular volume.
As it is shown in Fig. (1-B), trend of variations in the free volume is similar to the entropy behavior.
1.2
1.0
0.8
0.6
(A) ρ σ3=0.8
ρ σ3=0.7 ρ σ3=0.6
0.8
0.6
0.4
(B) ρσ3=0.6
ρσ3=0.7 ρσ3=0.8
0.4 0.2
0.2
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
H/σ
0.0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
H/σ
Figure 1. A) Difference between excess entropy of the confined and bulk hard sphere fluids at different pore sizes; B) Difference between the average molecular volume in a confined and bulk hard sphere fluids at different pore sizes.
Refrences
1. Y. X. Yu and J. Wu, J. Chem. Phys. 116, 7094 (2002).
2. Y. Rosenfeld, Phys. Rev. Lett., 63, 980 (1989).
(S ex-Sb ) / k
