A classical density functional theory model for fragility in the hard-sphere limit

We study, using the classical density functional theory (DFT), the fragility and short-time elastic constants of a soft-sphere liquid. For the amorphous state, the order parameter is the inhomogeneous density function $\rho({\bf r})$ which is described in terms of Gaussian density profiles centered on a set random lattice points $\{{\bf R}_i\}$. The latter is characterized in terms of the Bernel pair function $g_\mathrm{B}(r)$. Based on the Adam–Gibbs-type relation between the $\alpha$ relaxation time $\tau_\alpha$ and the configurational entropy $\mathcal{S}_{\rm c}$, a thermodynamic fragility $m_\mathrm{T}$ for the liquid is defined. The concentration or average density of the liquid is treated as the control parameter here instead of temperature. The configurational entropy of the liquid is calculated using the DFT model. Variations in the short-range structure of the amorphous state are made with different choices for the value of $g_\mathrm{B}(r)$ at short distances, and its implications on the correlation between fragility $m_\mathrm{T}$ and the softness index $n$ are studied. The dependence of Poisson’s ratio $\nu$ on the softness index $n$ of the interaction potential is also obtained from the density dependence of the metastable state free energy. The correlation between $m_\mathrm{T}$ and $\nu$ follows.