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Materials Science on CD-ROM User Guide

Introduction to Point Defects

Version 2.1

Andrew Green, MATTER
John Humphreys, UMIST/University of Manchester
Ross Mackenzie, Open University

October 1997

Assumed Pre-knowledge

This module has been developed primarily as a background to the module Atomic Diffusion in Metals and Alloys for which a basic understanding of the nature of vacancies and interstitials is essential. Before starting this module, it is assumed that the student is familiar with:

• the basics of (metallurgical) thermodynamics;
• the terms Gibbs’ free energy, G, enthalpy, H and entropy, S;
• the relationship G = H-TS, where T is temperature;
• basic crystallography, including stacking sequences in FCC, BCC and CPH crystal structures.

Module Structure

The module consists of two sections: Vacancies and Interstitials. As previously mentioned, it is not designed as a comprehensive treatment of this subject, but as a background to the MATTER module Atomic Diffusion in Metals and Alloys.

Vacancies

This section begins with a brief description of some of the important types of point defects in metals and alloys. They include:

• Vacancies
• Self-interstitials
• (Foreign) interstitials
• Substitutional atoms

Each type is illustrated by a graphic.

The remainder of the section concentrates on vacancies, and in particular the equilibrium vacancy concentration. Firstly, an overview is given of the variation in free energy of a crystal, DG with vacancy concentration, X_v. From this relationship, the equilibrium vacancy concentration, X_v^e is shown to be that which results in the minimum free energy.

Having looked at the general relationship between X_v and DG, the module goes on to look in more detail at the underlying theory.

The free energy term DG is first rewritten in terms of enthalpy (DH) and entropy (DS) of vacancy formation, using the familiar relationship:

where T is the absolute temperature. The DH and DS terms are each considered in turn.

Enthalpy of vacancy formation, DH

This is the change in enthalpy resulting from the addition of vacancies and arises from the increase in internal energy caused by breaking interatomic bands (i.e. removing atoms). By making the reasonable assumption that the X_v is low, such that vacancy-vacancy interactions can be ignored, it is shown that DH is proportional to X_v, according to the equation:

DH_v is the molar enthalpy of vacancy formation.

Entropy of vacancy formation, DS

This arises from the increased degree of randomness introduced by the addition of vacancies to a crystal. DS itself can be separated into 2 components:

• Thermal entropy,
• Configurational entropy.

Each of these terms is explained in further detail. It is shown that thermal entropy can be given by:

where DS_v is the molar thermal entropy. Special attention is given to the dominant configurational entropy term, and a side-branch takes the user to background information and related exercises. An expression for configurational entropy as a function of vacancy concentration is derived:

The variation of each component, and hence DS with vacancy concentration is illustrated by a graph.

Having seen how both DH and DS vary with vacancy concentration, the relationship between DG and X_v is revisited. An interactive graph plotting routine allows the user to see the effect of some of the important parameters on the overall shape of the DG vs. X_v curve and hence on the equilibrium vacancy concentration.

For many metallurgical processes, such as phase transformations, diffusion, creep, etc., it is important to know how the equilibrium vacancy concentration, X_v^e varies with temperature, T. The final page of the section shows that this relationship can be given by:

Often it is convenient to split DG into enthalpy and entropy components, i.e.:

The relatively temperature-insensitive term exp(DS_v /R) is normally taken as a constant of value ~3, to give:

Interstitials

Interstitial solid solutions occur when the solute atoms occupy the interstices between the solvent atoms. Such materials form the basis of some very important alloy systems, not least Fe-C.

This section concentrates on the interstitial sites in the three most common metal crystal structures, namely face-centred cubic (FCC), body-centred cubic (BCC) and close-packed hexagonal (CPH). In a series of 3 interactive exercises, the unit cell for each structure is shown and the user asked to click the mouse at each of the points corresponding the octahedral and tetrahedral interstices (voids). An example of each type of interstice is available for those requiring guidance. Each individual interstice becomes highlighted when correctly identified.

Additional questions for the FCC and BCC cases ask the student to calculate the void radii as a fraction of the parent atom radii for both octahedral and tetrahedral interstices.

The final page of the section compares the equilibrium concentration of self-interstitials with that of vacancies.

Bibliography

The student is referred to the following resources in this module: