Preliminary test of time-convolutionless mode-coupling theory based on the Percus-Yevick static structure factor for hard spheres

TL;DR

This study compares the time-convolutionless mode-coupling theory (TMCT) with the ideal mode-coupling theory (MCT) for hard spheres, showing TMCT predicts higher critical volume fractions and similar relaxation behaviors.

Contribution

It provides a preliminary numerical comparison of TMCT and MCT using the Percus-Yevick static structure factor for hard spheres, highlighting differences in predicted critical points.

Findings

TMCT predicts higher critical volume fractions than MCT.

Both theories show similar two-step relaxation processes.

TMCT's critical point depends on wave vector cutoff.

Abstract

In order to investigate how the time-convolutionless mode-coupling theory (TMCT) recently proposed by Tokuyama can improve the critical point predicted by the ideal mode-coupling theory (MCT), the TMCT equations are numerically solved based on the Percus-Yevick static structure factor for hard spheres as a preliminary test. Then, the full numerical solutions are compared with those of MCT for different physical quantities, such as intermediate scattering functions and diffusion coefficients. Thus, the ergodic to nonergodic transition predicted by MCT is also found at the critical volume fraction $\phi_c$ which is higher than that of MCT. Here $\phi_c$ is given by $\phi_c\simeq 0.5817$ at $q_c\sigma_d=40$ and 0.5856 at $q_c\sigma_d=20$ for TMCT, while $\phi_c\simeq 0.5159$ at $q_c\sigma_d=40$ and 0.5214 at $q_c\sigma_d=20$ for MCT, where $q_c$ is a cutoff of wave vector and $\sigma_d$ a particle diameter. The same two-step relaxation process as that predicted by MCT is also discussed.