Convergence of Fine-lattice Discretization for Near-critical Fluids

Abstract

In simulating continuum model fluids that undergo phase separation and criticality, significant gains in computational efficiency may be had by confining the particles to the sites of a lattice of sufficiently fine spacing, $a_0$ (relative to the particle size, say $a$). But a cardinal question, investigated here, then arises, namely: How does the choice of the lattice discretization parameter, $\zeta \equiv a/a_0$, affect the values of interesting parameters, specifically, critical temperature and density, $T_c$ and ${\rho }_c$? Indeed, for small $\zeta (\lesssim 4$-$8)$ the underlying lattice can strongly influence the thermodynamic properties. A heuristic argument, essentially exact in $d=1$ and $d=2$ dimensions, indicates that for models with hard-core potentials, both $T_c(\zeta )$ and ${\rho }_c(\zeta )$ should converge to their continuum limits as $1/{\zeta }^{(d+1)/2}$ for $d\le 3$ when $\zeta \to \infty$; but the behavior of the error is highly erratic for $d\ge 2$. For smoother interaction potentials, the convergence is faster. Exact results for $d=1$ models of van der Waals character confirm this; however, an optimal choice of $\zeta$ can improve the rate of convergence by a factor $1/\zeta$. For $d\ge 2$ models, the convergence of the {\em second virial coefficients} to their continuum limits likewise exhibit erratic behavior which is seen to transfer similarly to $T_c$ and ${\rho }_c$; but this can be used in various ways to enhance convergence and improve extrapolation to $\zeta =\infty$ as is illustrated using data for the restricted primitive model electrolyte.

To appear in J. Phys. Chem. in honor of David Chandler