The structure factor of a liquid - part IV

Sum rule for impenetrable systems

The hard sphere liquid is an idealized model, but some of its properties hold for a very large class of systems, those that have an impenetrable core of size \(R_c\) (\(g(r< 2 R_c = 0\)). Let us write the Fourier relation between \(g(r) -1\) and \(S(q) -1\) (the inverse of Eq. (4) in post II):

\begin{equation}
\label{eq:grfinal}
g(r) -1 = \frac{1}{\rho} \frac{1}{(2\pi)^3} \int {\text{d}}^3 \mathbf{q}\, [S(q) - 1] \mathrm{e}^{-i \mathbf{q} \mathbf{r}}
\end{equation}
Setting \(g(0) =0\) yields the sum rule:
\begin{equation}
\label{eq:sumrule}
\frac{1}{(2\pi)^3} \int {\text{d}}^3 \mathbf{q}\, [S(q) - 1] = \frac{1}{2\pi^2} \int_{0}^{\infty} \text{d} q\, q^2 [S(q) - 1]=  - \rho
\end{equation}

Relation \eqref{eq:sumrule} holds for any interaction potential \(u(r)\) as long as it has an impenetrable core: \( u(r) = \infty, \, r \leq 2 R_c \), which is the case of almost all colloidal particles. The exceptions are so-called "soft-core" particles such as polymer coils¹.

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^{1. See e.g. A. A. Louis & al., Phys. Rev. Lett. 85, p. 2522 (2000). ↩}