CHAPTER 1

Phase Transformation in Solids 1.1. INTRODUCTION

In the Universe matter exists in three states; solid, liquid and gaseous state. The phase of a solid substance is stable when the thermodynamic variables like volume, pressure, temperature and energy are minimum. If any of the thermodynamic variables is varied, the Gibb's free energy of the system also changes continuously. If the variation in free energy leads to change in structural details of a phase, a "phase transformation or phase transition" is said to occur. The term "Phase transformation" is more common among metallurgists, materials scientists and chemists.

For a different set of

thermodynamic conditions there exists another structure with minimum free energy. hence the system undergoes a phase transformation to that new structure [I-51. The free energy varies continuously if the thermodynamic variables like temperature or pressure is varied and the rate of variation is system and structure dependent. On alteration of the external conditions such as pressure and temperature, the initial state of the system is no more in the equilibrium state.

The Gibb's free energy determines whether a system is at equilibrium or not. The Gibb's free energy 'G' of a system is given by G = H - TS, where T is the temperature, S is the entropy and the Enthalpy H is defined in terms of internal energy U, pressure p, and volume V of the system. Then the Gibb's free energy can be written as G=U+PV-TS

(1.1)

dG=dU+PdV+VdP-TdS-SdT

(1.2)

dG = VdP - SdT, since dU = TdS - PdV

(1.3)

First and second order derivatives of Gibb's free energy can be written as (1%/8p)~ = v

(1.4)

(XICT)~= -s

(1.5)

( $ G / ~ P ' ) ~= - 1N (8V/8P)T = P

(1.6)

( s 2 ~ i a r 2 )=, -T ( a s ~ a r =) -c,

(1.7)

( 2 ~ 1 a ~=n 1N ) (aviar)p= a

(1.8)

where C,, a and

p

are heat capacity, volume thermal expansivity and

compressibility, respectively. During a phase transformation, whereas the free energy of the system remains continuous, thermodpamic quantities like entropy, volume, heat capacity and so on undergo discontinuous change. If a discontinuous change occurs in the first derivatives of the Gibb's free energies such as volume and entropy the transformation is said to be first order phase transformation. Correspondingly if the discontinuous change occurs in the second derivatives 2

of Gibb's free energy, i.e. in heat capacity, thermal expansivity and compressibility, the transformation is said to be a second order phase transformation. The third and higher order transformation involves further differential quantities.

1.2.

THERMODYNAMICS OF PHASE STABILITY AND PHASE TRANSFORMATIONS Thermodynamics can be used to predict the phase stability and phase

transformations. The Gibb's Free energy provides stability criteria that are based only on the properties of a system at constant pressure and temperature. For example, for phase transformations occurring at constant T and P, the relative stability of the phases in a system is determined by their Gibb's free energies AG = A H - T A S = G , ,

-G,,,,

If AG < 0

Process is spontaneous (process allowed)

AG > 0 => Process is not spontaneous (process forbidden) AG = 0 = Process in equilibrium.

Any phase transformation that results in a decrease in Gibb's Free energy is thermodynamically possible. Therefore, as noted above. a necessary criterion for any phase transformation is

-

AG = Grind Gin,id< 0

Understanding of phase stability is very important in materials science since all properties of a material (optical, electronic, thermal, magnetic, mechanical) depend on its phase composition.

1.3. CLASSIFICATION OF PHASE TRANSFORMATIONS TWO microscopic modes of transformation; Homogenous and Heterogeneous phase transformation in materials have been identified [ 6 ] . Homogeneous transformation takes place over the entire volume of the system simultaneously. But heterogeneous transformation occurs at specific sites of the system leaving the remaining system untransformed. The classifications under homogeneous and heterogeneous phase transformations are depicted in Table 1.1. Heterogeneous transformations occur by the nucleation and growrh of the product phase. Heterogeneous transformation can be broadly classified as liquid - solid transformation and solid - solid transformation. Crystallization and melting are typical examples for liquid transformation.

- solid

heterogeneous mode of

Based on mode of growth, solid-solid transformation is

further classified into thermally activated and athennal growth. If the growth rate is strongly temperature dependent and composition of the parent and product phases differ appreciably it is called thermally activated growth. In arhermal mode of growth, composition of parent and product phases remains exactly the same and the growth rate is high and independent of temperature and the transformed region undergoes a change of shape. 4

T s b k 1.1. C W h L i o m of Pbssc T~m.sZorm~tb.s

+

Phase lransrormations 1

I

Ilelerogeneous Phase Transli rmalions

I

Liquid-Solid Transformations

Solid-Soli

I lomogeneous Phase l'ransformalions

Transformations

Thermally Activated Growth

I

ul

I -

Without Atherrnal

Short Range 'I'ransport

Crystallizalic~n -Melting

-l'c>Iyn~~~rpl~ic Transfi~rrnation -Massive 'l.ran,lbrn~a~ion -Order-diu>rder

Athermal Growth

I With Athemal Component

Mcdium and long Range l'ransporc

I

-liulccloid Itcaction -