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.
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:

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

Phase diagram for polybutadiene $\tau _{d}\,\!$ 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