Kinetic Theory of Soft Matter. The Penetrable-Square-Well Model

TL;DR

This paper derives the temperature-dependent Navier--Stokes transport coefficients for a dilute gas of particles interacting via the penetrable-square-well model, revealing regimes where these coefficients resemble those of square-well or penetrable-sphere models.

Contribution

The study provides an analytical derivation of scattering angles and evaluates transport coefficients for the PSW model, bridging the gap between SW and PS models across temperature regimes.

Findings

Transport coefficients match SW at low temperatures.

Transition regime where coefficients interpolate between SW and PS.

At high temperatures, coefficients resemble those of PS.

Abstract

The penetrable-square-well (PSW) pair interaction potential is defined as $\phi (r)=\epsilon_r$ if the two interacting particles are overlapped ($r<\sigma$), $\phi(r)=-\epsilon_a$ inside a corona ($\sigma <r<\lambda$), and $\phi(r)=0$ otherwise ($r>\lambda$). Thus, the potential reduces to the conventional square-well (SW) one in the limit $\epsilon_r\to\infty$ and to the penetrable-sphere (PS) potential if $\epsilon_a\to0$ or $\lambda\to\sigma$. This paper aims at studying the temperature dependence of the Navier--Stokes transport coefficients of a dilute gas of particles interacting via the PSW model. By exploiting the fact that the PSW scattering process is analogous to that of a light ray passing through two concentric spherical media with different refractive indices, the scattering angle is analytically derived as a function of the impact parameter and the relative velocity of the colliding particles; depending on the values of those two quantities, collisions can be soft, hard, or grazing. Next, by standard application of known general results from the Chapman--Enskog method, the Navier--Stokes transport coefficients in the first-order approximation are numerically evaluated. It is found that the PSW coefficients are practically indistinguishable from the SW ones for temperatures low enough ($k_BT\lesssim 0.2 \epsilon_r$), there exists a transition regime ($0.2 \epsilon_r\lesssim k_BT\lesssim 10\epsilon_r$) where the transport coefficients interpolate between the SW and the PS ones, and finally the PSW coefficients are comparable to the PS ones for high enough temperatures ($k_BT\gtrsim 10\epsilon_r$). The results are applied to the temperature profiles of the planar Fourier flow.