25 May 2015

The structure factor of a liquid - part III

This is the third part in a series. In part I and part II we defined the basic concepts used in the theory of liquids, in particular the radial distribution function \(g(r)\) and the structure factor \(S(q)\).

The simplest system one can imagine is the ideal gas. There is no interaction between particles: \(u(r) = 0\), leading to \(g(r) = 1\) (the particle at the origin does not affect the position of its neighbors) and \(S(q) = 1\). The ideal gas is a trivial case, but it can be seen as the reference state for other systems. In particular, one could say that the functions \(g(r) - 1\) and \(S(q) - 1\) that appear in Equation (4) of part II quantify the difference with respect to the ideal gas (due to the interaction potential \(u(r) \neq 0\).)


Hard spheres

We will now discuss the case of a hard sphere liquid, where \(u(r) = \infty\) for \(r < 2R\) (with \(R\) the hard sphere radius) and \(0\) elsewhere.

The figure above shows the structure factor of a (three dimesional) hard sphere liquid for different volume concentrations \(\phi = \rho V_{\text{part}}\), where \(V_{\text{part}} = (4\pi/3) R^3\) is the particle volume. There is a single length scale in the problem, the radius \(R\) (or the diameter \(D = 2R\)). We can thus use the normalized scattering vector \(Q = qD\): the structure vector of systems with different \(D\) (but the same \(\phi\)) is the same when plotted against \(Q\).

Dilution law

One could object that another length scale can be obtained from the number density: \(L \sim \rho^{-1/3} = D (\pi/6)^{1/3} \phi^{-1/3}\). In a crystal, \(L\) would define the position of the Bragg peaks (up to a numerical constant). However, until very high values of \(\phi\), the first maximum of \(S(q)\), \(q_{\text{max}}\), is constant and arises close to \(Q = 6\) (see the right panel in the figure). The relevant distance is \(2\pi /q_{\text{max}} \simeq D\), and not \(L\)! The hard sphere liquid does not obey the dilution law one would expect based on the value of \(L\) (solid dots in the figure). The peak position only shifts at high concentrations \(\phi\), close to the random close packing (RCP) limit, which marks the end of the liquid range.

To understand this apparently paradoxical behaviour, remember that \(S(q) - 1\) is the Fourier transform of \(g(r) - 1\) (Eq. (4) in part II). The strongest feature of \(g(r)\) is the discontinuity in \(r=D\) which, on Fourier transforming, causes an oscillation in \(S(q)\) and gives rise to its peaks. The amplitude of the discontinuity increases with \(\phi\), but its position does not change and neither does that of \(q_{\text{max}}\).

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