# Phase diagrams and viscoelasticity

## Introduction

Phase behavior and viscoelasticity are closely related.

Phase diagrams follow from the interplay of forces (energies) and interparticle distances.

Viscoelastic properties follow from the interplay of forces (energies), interparticle distances, and time.

The dimensionless internal energy versus volume fraction, indicating empirically defined zones of liquid-like and solid-like behavior. Goodwin and Hughes, Fig. 5.14.

## Weakly attractive systems

 500 nm polystyrene particles; 0.5 M electrolyte; 3.8 nm surfactant chain. Goodwin and Hughes, Fig. 5.9. Reference
The reduced total energy is:

$E=\frac{\bar{E}a^{3}}{kT}=\frac{9\varphi }{8\pi }+\frac{3}{2}\varphi \int\limits_{0}^{\infty }{r^{2}}g\left( r \right)\frac{V\left( r \right)}{kT}dr$

\begin{align}  & g(r)\text{ is the radial distribution function;} \\ & \varphi \text{ is volume fraction} \\ & \text{and }V\left( r \right)\text{ is the pair potential} \\  \end{align}\,\!

The distance derivative gives force:

$\frac{\Pi a^{3}}{kT}=\frac{3\varphi }{4\pi }-\frac{3\varphi ^{2}}{8\pi a^{3}}\int\limits_{0}^{\infty }{r^{3}g\left( r \right)}\frac{d}{dr}\left( \frac{V\left( r \right)}{kT} \right)dr\,\!$

The (high frequency) shear modulus is:

$\frac{G\left( \infty \right)a^{3}}{kT}=\frac{3\varphi ^{2}}{40\pi a^{3}}\int\limits_{0}^{\infty }{g\left( r \right)}\frac{d}{dr}\left[ r^{4}\frac{d}{dr}\left( \frac{V\left( r \right)}{kT} \right) \right]dr\,\!$

Goodwin and Hughes, Fig. 5.23.

## Reptation and linear viscoelasticity

Goodwin and Hughes, Fig. 5.26
$\tau _{d}\,\!$ is the tube disengagement time. $\tau _{e}\,\!$ is the polymer escaping time. $G_{N}\,\!$ is the cross-over plateau. Goodwin and Hughes, Fig. 5.27

## Effect of temperature on viscoelastic behavior

The secondary bonds of a polymer constantly break and reform due to thermal motion. Application of a stress favors some conformations over others, so the molecules of the polymer will gradually "flow" into the favored conformations over time. Because thermal motion is one factor contributing to the deformation of polymers, viscoelastic properties change with increasing or decreasing temperature. In most cases, the creep modulus, defined as the ratio of applied stress to the time-dependent strain, decreases with increasing temperature. Generally speaking, an increase in temperature correlates to a logarithmic decrease in the time required to impart equal strain under a constant stress. In other words, it takes less energy to stretch a viscoelastic material an equal distance at a higher temperature than it does at a lower temperature.