# Efficient computational strategies for predicting homogeneous fluid structure

## Abstract

A common challenge in materials science is the "inverse design problem," wherein one seeks to use theoretical models to discover the microscopic characteristics (e.g., interparticle interactions) of a system which, if fabricated or synthesized, would yield a targeted material property. Inverse design problems are commonly addressed by stochastic optimization strategies like simulated annealing. Such approaches have the advantage of being general and easy to apply, and they can be effective as long as material properties required for evaluating the objective function of the optimization are feasible to accurately compute for thousands to millions of different trial interactions.

This requirement typically means that "exact" yet computationally intensive methods for property predictions (e.g., molecular simulations) are impractical for use within such calculations. Approximate theories with analytical or simple numerical solutions are attractive alternatives, provided that they can make sufficiently accurate predictions for a wide range of microscopic interaction types.

We propose a new approach, based on the fine discretization (i.e., terracing) of continuous pair interactions, that allows first-order mean-spherical approximation theory to predict the equilibrium structure and thermodynamics of a wide class of complex fluid pair interactions. We use this approach to predict the radial distribution functions and potential energies for systems with screened electrostatic repulsions, solute-mediated depletion interactions, and ramp-shaped repulsions.

We create a web applet for introductory statistical mechanics courses using this approach to quickly estimate the equilibrium structure and thermodynamics of a fluid from its pair interaction. We use the applet to illustrate two fundamental fluid phenomena: the transition from ideal gas-like behavior to correlated-liquid behavior with increasing density in a system of hard spheres, and the water-like tradeoff between dominant length scales with changing temperature in a system with ramp-shaped repulsions.

Finally, we test the accuracy of our approach and several other integral equation theories by comparing their predictions to simulated data for a series of different pair interactions. We introduce a simple cumulative structural error metric to quantify the comparison to simulation, and find that according to this metric, the reference hypernetted chain closure with a semi-empirical bridge function is the most accurate of the tested approximations.