# Homogeneous flow of metallic glasses: A free volume perspective

Started by Lauren Hartle, Fall 2011.

F. Spaepen, "Homogeneous flow in metallic glasses: a free volume perspective", Scripta Materialia: 54, 363-367 (2006).

## Contents

## Keywords

Creep, Deformation, Glass Transition Temperature, Free Volume Theory, Fulcher–Vogel–Tammann Equation, Metallic Glass, Modulus, Plastic Flow, Poisson's Ratio, Shear, Shear Band, Shear Transformation Zone, Strain, Stress, Structural Relaxation, Viscosity

## Introduction

The paper frames "the temperature-dependence of the liquid viscosity, structural relaxation, and deformation-induced softening" of glass in terms of changes in free volume. The ultimate goal is to propose an atomic-scale mechanism that predicts both liquid flow and describes the intermediate mechanism(s) between liquid flow and the appearance of shear bands in metallic glasses. Figure 1 shows the various flow regimes of a metallic glass. Figure 2 demonstrates the dependence of liquid and glass viscosity on temperature and time.

## Fundamentals of the Free Volume Approach to Homogeneous Flow in Metallic Glasses

### Plastic Flow

Plastic flow, or irreversible shear, occurs when a material experiences sufficiently large local shear strain (on the order of 100%) to change the atom's nearest neighbors. These "shear transformation zones" experience a macroscopic strain rate of:

<math>\frac{d\epsilon}{dt} = 2 c_f k_f \frac{\epsilon_0 \nu_0 \sigma}{\Omega} sinh\left( \frac{\epsilon_0 \nu_0}{2 k T}\right)</math>

where <math>\epsilon_0</math> is the strain, <math>c_f</math> is the defect concentration, <math>\nu_0</math> is the defect volume, <math>\sigma</math> is the stress, and <math>\Omega</math> is the atomic volume. At low stresses, the sinh term can be approximated as linear, and the result is the familiar Newtonian fluid flow. Figure 3 overlays a plot of strain rate vs stress with a sinh fit. The predicts an activation volume of 141 cubic Angstroms, reasonable based on data from other lead-based glasses.

### Free Volume Approach to Liquid Flow

For a liquid undergoing a given external stress, the strain rate equation includes only the linear term of the right hand side, and the strain rate and thus the viscosity is proportional to <math>c_f k_f T^{-1}</math>, where the constants are temperature-dependent (the defect concentration, and a rate constant). The Fulcher–Vogel–Tammann (FVT) equation, applicable for liquids in internal equilibrium, requires that the viscosity have an exponential temperature dependence:

<math>\eta = \eta_0 exp \left[\frac{B}{T-T_0}\right]</math>

The free volume model assigns this temperature dependence via the defect concentration:

<math>c_f = exp \left[\frac{-\gamma\nu^{*}}{\nu_f}\right]</math>

The exponential temperature-dependence of liquid viscosity is thus explained by atomic volume expansion:

<math>v_f = \alpha \Omega \left( T-T_0\right)</math>

### Structrural Relaxation

At a low enough stress, the impact of structural relaxation on viscosity overwhelms that of deformation-induced defects. In this regime, it can be empirically demonstrated (see Figure 4) that the viscosity linearly increases with time if the temperature is sufficiently lower than the glass transition temperature, <math> T_g</math>.

By the strain rate equation,

<math> \frac{d\eta}{dt} \propto - \frac{1}{c_f^{2}} \frac{dc_f}{dt}</math>

and in this regime,

<math>\frac{d\eta}{dt} = const</math>

hence, we arrive at the bimolecular equation:

<math>\frac{dc_f}{dt} = -k_r c_f^2</math>

This result might be explained as the physical result of annihilation of some free space when two voids combine. As the system approaches the glass transition temperature, structural relaxation becomes negligible. We add an empirically-supported term (See Figure 5) to the rate of change in the defect concentration:

<math>\frac{dc_f}{dt} = -k_r c_f ^2 + k_c c_f</math>

where <math> c_eq = \frac{k_c}{k_r} </math>

This deformation-induced volume creation process is analogous to dislocation climb--it occurs near existing voids.

### Deformation-induced softening

A series of creep tests on <math> Pd_{41}Ni_{10}Cu_{29}P_{20}</math> glass at 450 MPa, 14 MPa, then 450 MPa again, revealed that the strain rate is a unique function of "instantaneous stress, temperature and defect concentration". No shear bands formed during the experiment. Figure 6 shows the relevant data.

The results support the argument that stress-driven defect creation and annihilation primarily mediate creep in this regime. Free volume theory provides a functional form to estimate the free volume production rate:

<math> \frac{d\nu_f}{dt} = 2 c_f k_f \frac{kT}{S} \left[cosh\left(\frac{\sigma \epsilon_0 \nu_0}{2kT} \right)- 1 \right] </math>

where <math> \nu</math> is the Poisson's Ratio, <math> \mu</math> is the shear modulus, and

<math> S = \frac {2 \mu \left( 1 + \nu \right)}{3 \left( 1 - \nu \right)} </math>

## Conclusions/Soft Matter Connection

In its final form, the free volume theory interpretation predicted an activation volume consistent 2-D bubble raft studies, 3-D colloid studies, and with the assumption of shear strain on the order of unity. As a set of fitting functions, the model appears to improve descriptions of creep while accounting for behavior at important stress and temperature limits. That said, the mechanism assumed by the model only accounts for a single atom at a time--due to the large free volume predicted by the data fit, the model ought to be expanded to account for the coordination of multiple atoms.

This paper attempts to connect atomistic scale mechanisms with macroscopic mechanical behavior. Likewise, when modeling complex phenomena like polymer interactions, surface and wetting interactions, and complex polymer networks, one must balance teasing out most important physical principles while remaining true to the complexity of the system, all while developing a theory with useful implications for experiments.