# Phase diagrams and viscoelasticity

## Contents

## 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.

## Weakly attractive systems

The reduced total energy is: |
<math>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</math> <math>\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}\,\!</math> |
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The distance derivative gives force: |
<math>\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\,\!</math> |

The (high frequency) shear modulus is: |
<math>\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\,\!</math> |

## Phase diagram for polybutadiene

From (http://en.wikipedia.org/wiki/Polybutadiene). Polybutadiene is a synthetic rubber that is a polymer formed from the polymerization of the monomer 1,3-butadiene. It has a high resistance to wear and is used especially in the manufacture of tires. It has also been used to coat or encapsulate electronic assemblies, offering extremely high electrical resistivity. It exhibits a recovery of 80% after stress is applied, a value only exceeded by elastin and resilin. Polybutadiene is a highly resilient synthetic rubber. Due to its outstanding resilience, it can be used for the manufacturing of golf balls. Heat buildup will be less in polybutadiene rubber based products subjected to repeated flexing during service. This property leads to its use in the sidewall of truck tires. Good abrasion resistance of this rubber also leads to its use in the tread portion of truck tires; however, skidding may be a problem in passenger car tires due to low rolling resistance. For high temperature curing, polybutadiene may be blended with natural rubber and other rubbers, due to resistance in reversion of physical properties. Polybutadiene rubber can be used in water seals for dams due to its low water absorption properties. Rubber bullets and road binders can be also produced by polybutadiene rubber.

## Reptation and linear viscoelasticity

## 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.