16 May 2015

The structure factor of a liquid - part I

This post only summarizes some basic concepts and results that will help understand the discussion in the following posts. For a detailed introduction to liquid theory, see one of the many books and review papers [1].

Definitions

Consider a system of \(N\) identical particles contained in the volume \(V\) (in \(D\) dimensions), at a temperature \(T\). Each particle \(i\) is marked by its vector radius \(\mathbf{r}_i\); the particles interact via the isotropic potential \(u_{ij} = u \left ( \left | \mathbf{r}_i - \mathbf{r}_j \right | \right ) = u(r)\). \(P\) is the pressure. Let us also define:

  • \(\rho = \dfrac{N}{V} \qquad\) the number density.
  • \( f(r) = \mathrm{e}^{-\dfrac{u(r)}{k_B T}} - 1 \qquad\) the Mayer \(f\) function, which has the same range as the potential \(u(r)\) (\(u(r) = 0 \Rightarrow f(r) = 0\)). This function measures the ''volume excluded by the particle''; as an example, for a hard-core interaction with radius \(R\), \(f(r) = -1\) for \( r < 2R\), i.e. for the forbidden configurations.
  • \( \dfrac{P}{\rho k_B T} = 1 + \rho B_2(T) + \rho ^2 B_3(T) + \ldots \quad\) is the equation of state, with \(B_i\) the virial coefficients.

Radial distribution function

In order to describe the structure of the system at the microscopic scale (over distances comparable to the size of the particles or to the range of the potential \(u(r)\)) let us define the radial distribution function \(g(r)\), which represents the probability of finding a particle at a distance \(r\) from a reference particle situated at the origin. In other words, \(\rho g(r)\) is the local density around the reference particle and quantifies its effect on the neighbouring particles. Far away from the origin, the density falls back to its average value \(\rho\), and hence the radial distribution function tends towards 1. This function contains all the microscopic information: [2] one can say that the fundamental problem of liquid theory is finding \(g(r)\), knowing \(u(r)\) and \(\rho\).

1. J. A. Barker and D. Henderson, Reviews of Modern Physics 48, 587 (1976).
D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, New York, 1987).
J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Elsevier Academic Press, London; Burlington, MA, 2006).

2. As a consequence, it also contains the thermodynamics of the system; in particular, the equation of state writes \( \displaystyle \dfrac{P}{\rho k_B T} = 1 - \dfrac{\rho}{6 k_B T} \int_V r \dfrac{\text{d} u (r)}{\text{d} r} g(r) \text{d} ^D r \).

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