A new empirical structure factor for real liquids using internal pressure

doi:10.1016/S0378-3812(00)00468-4

Fluid Phase Equilibria

Volume 178, Issues 1–2, 1 March 2001, Pages 33-44

https://doi.org/10.1016/S0378-3812(00)00468-4 Get rights and content

Abstract

Customarily structure factor S(Q) of a liquid is determined experimentally (by X-ray or neutron scattering methods) and is used to derive the radial distribution function g(r) which obviously is a laborious and costly procedure. In the present paper, an equation for S(Q) for real fluids is derived which incorporates directly the internal pressure, π_i, based on the fact that, the radial distribution function and hence the structure factor of a hard-sphere and a real fluid are similar. The positions of the principal maxima (PPM) correlate nicely with those adduced by experimental techniques for condensed liquids and liquid metals. In the (X-ray or neutron) scattering experiment, Q refers to the momentum change of the incident beam. In our study Q continues to have the same meaning in the ‘hypothetical’ experiment.

Introduction

The past few years or so have seen a remarkable growth in the understanding of liquid state. Many of these advantages have not yet resulted in modelling the structure factor and any of the thermodynamic property directly, and the present work is aimed at filling this gap.

Structure factor S(Q) enables us to understand both the equilibrium and transport properties of liquids. As mathematical limitations are formidable and that there is not a single theory which can account for all properties of liquids successfully, it is convenient to employ simple relations which are directly accessible from experiments, to express information about the structure of liquids. Therefore, the assumption of liquid molecules as hard-spheres (without any attractive component), helps one to mathematically formulate equations describing the liquid state with relative ease. This also forms the zero-order approximation for higher order perturbation calculations.

The validity of any theoretical model lies in its experimental confirmation, which obviously is not possible with hard-sphere fluids although the simulation procedures perform frequently this role. It is known that the thermodynamics and the relation between thermodynamics and the structure factor are different for hard-spheres and real fluids; nevertheless the radial distribution function g(r) and hence the structure factor of a hard-sphere and a real fluid are similar. In the present work, starting from Wertheim’s solution of Percus–Yevick equation for hard-spheres, we arrive at the S(Q) of real fluids. The resulting simple equation, incorporates the thermodynamic property, the internal pressure (π_i) directly and is tested successfully in the case of a number of real liquids. The internal pressure allows the omission of the attractive forces and hence a direct comparison with the hard sphere fluid. This S(Q) is bound to be complementary, if not alternative, to the S(Q) determined from the costly and time-consuming scattering techniques.

Section snippets

Theoretical background

The static structure factor is described by the Fourier transform C(Q) of the Ornstein–Zernike direct correlation function as $S(Q)=[1−ρC(Q)]^{−1}$ where C(Q) is given by [1] $C(Q)=− 4π Q(1−η)^{4} ∫ 0 σ r sin Qr α+β r σ +γ r σ^{3} d r$ where $α=(1+2η)^{2}, β=−6η 1+ 12 η^{2}, γ= 12 η (1+2η)^{2}, η=πρσ^{3}$ It is interesting to see that all quantities can be expressed by means of the packing fraction η. Block et al. [2] gave the result of the above integration as $C(Q)=− 24η ρx^{6} (1−η)^{4} [αx^{3} (sin x−x cos x)+βx^{2} {2x sin x−(x^{2} −2) cos x−2}+γ{(4x^{3} −24x) sin x−(x^{4} −12x^{2} +24) cos$

Relationship between S(Q) and π_i

The equation of state is one of the most fundamental characteristics of condensed matter and is determined by atomic as well as by statistical properties. It is not surprising that the theoretical calculation of equation of state is a very difficult task, even for simple systems such as noble gases. As a zero-order perturbation approximation, liquid molecules can be treated as hard-spheres resulting in simpler mathematical formalism. The equation of state for hard-sphere fluid is $P=ρkT+ 23 πρ^{2} kTσ^{3}$

Application to metals

It is well known that S(Q) and the pair distribution are themselves insensitive to details of the potential. It is only when these are combined with the potential to calculate pressure or internal energy that sensitivity of the potential is manifest [17]. S(Q) as given by Eq. (15) is plotted against Q for argon and sodium (Fig. 2). Overlapping of S(Q) versus Q curves of argon and sodium supports the prediction [7] that S(Q) for either hard-sphere or real fluids with any sort of interaction

Results and discussion

The equation for S(Q), Eq. (15), contains ρ,T and π_i together for the first time. As desired, the variation of S(Q) with Q is very smooth. This study involves comparison of the positions of principal maxima (PPM) in the S(Q) versus Q curve of Eq. (15) with those of the reported experimental values for liquid metals (Na, K, Rb and In), atomic liquids (Ar and Kr) (Table 3 and Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9). The general equation for S(Q) is given by [3] $S(Q)−1= 4πρ Q ∫ 0 ∞ r sin Qr[g(r)−1] d$

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A new empirical structure factor for real liquids using internal pressure

Abstract

Introduction

Section snippets

Theoretical background

Relationship between S(Q) and πi

Application to metals

Results and discussion

Structure and resistivity of liquid metals

Phys. Rev.

Structure factor and radial distribution function for argon at 85 K

Phys. Rev.

J. Chem. Phys.

J. Chem. Phys.

Mol. Phys.

Phys. Rev.

Fiz. Tverd. Tela

Phys. Rev. B

Relationship between S(Q) and π_i