PHASE FIELD MODELING OF MARTENSITIC MICROSTRUCTUTES

Martensitic transformations are diffusionless military transformations which are temperature

dependent but time independent (in most cases) [1]. Diffusionless transformation occurs when

the cooling rate exceeds a critical value. The transformation begins at a temperature called the

Martensitic start temperature (Ms) and ends at Martensitic finish temperature (Mf). The

characteristic of martensitic transformation is the existence of an undistorted plane or Habit

plane which does not change on transformation [2].

There are mainly two aspects of martensitic transformations to be understood firstly, the

geometric or crystallographic theory and secondly, the thermodynamics behind the

transformation.

1. Geometry

Geometrically martensitic transformations can be explained in simple terms by Bain Model of

fcc > bct transformation. But this model has many drawbacks like

i. It does not explain the orientation relationship between initial and final lattices [3]

ii. Does not involve shear a characteristic of martensitic transformation

iii. It does not well establish the existence of habit planes which makes it difficult to

explain the invariant plane strain associated with martensitic transformations [4]

Later, other crystallographic theories like Wechsler Lieberman Read (WLR) and Bowles and

MacKenxie (BM) were developed which explained the habit plane orientation and multidomain

structure of martensite reasonably well [5].

Both the theories are almost similar to each other but mathematically different. They are

characterized by the following features

i. lattice deformation ( > ' Bain distortion) ii. Lattice invariant shear

iii. Lattice rotation (ii. and iii. essential for invariant plane upon transformation) [3]

2. Thermodynamics

The driving force for any microstructural evolution is the possibility to reduce free energy of the

system. In general, free energy (F) of any system can contain the following terms:

F = Fbulk + Fint + Fel + Ffys

Fbulk = Bulk free energy determines the composition and volume fraction of equilibrium phases

Fint = Interfacial energy

Fel = Elastic strain energy

Ffys = Energy terms due to magnetic or electrostatic interactions of domains

Classical thermodynamics assumes properties are homogeneous throughout the system but in

Phase Field method free energy is formulated as a functional of the set of the phase-field

variables (which are functions of time and spatial co-ordinates) and their gradients [6].

In martensitic transformations, free energy contributions are of two types

i. Chemical free energy (Fbulk) ii. Non-chemical free energy (Fint + Fel) [3]

In general for martensite, the free energy is given in terms of the following:

F = Fch + Fint + Fel

The first term in the above equation is negative change in chemical free energy which provides

the driving force of the reaction, and the next two terms represent the positive free energy due to

the interface and accommodation of elastic strain (latter may be regarded as an effective stress

acting on the interface to restrict the spontaneous growth of the plate) [1].

i. Chemical free energy

The basic equation of chemical free energy evolved from a basic calculation is given below [3]

= (1 x ) + x

+

x = conc. of composition A in atom fractions; M = Mixture of solid solution of two species

The evolution of this simple term to more complex forms has been gradually achieved and terms

such as chemical free energy have been incorporated in one of the recent model [7].

ii. Non-Chemical free energy

In general the contribution of non-chemical free energy is not of much significance in other

reactions but in martensitic reaction its contribution is quite significant. Major contributions to

this free energy are discussed below.

a. Interfacial energy

Interfacial energy between matrix and martensite depends on the coherency of the two phases

i. e., on orientations and indices of the interface of two crystals. Initially, with an assumption of

constant interfacial energy per unit area and lenticular shape with negligible thickness, the

interfacial energy was given by the following expression [3]

Affects the equilibrium compositions and

volume fractions of coexisting phases

Fint = 2r2

r is the radius of the martensitic crystal; is the surface energy per unit area

Later other interfacial energies were calculated using other theories but after the introduction

micromechanical theory of branching domain walls on the interface (by Kohn and Muller)

between martensite and the parent phase, which is caused by a balance between the stress

energy interfacial energy. [7]

b. Energy for plastic deformation

Slip or twinning occurs in martensitic crystals in order to relax the stress due to the shape change

associated with lattice transformation. The energy needed for occurrence of deformation is very

large but is not properly quantified.

c. Energy for elastic deformation

Elastic distortion occurs within and outside martensitic crystal and this contributes to the elastic

energy. This is the most important factor that contributes to free energy, and the one which has

undergone several changes from a simple equation to complex equations by incorporating

various changes in conditions gradually.

Several conditions have been incorporated into models depending on the dimensions of

transformation modeling, materials, stress conditions etc., Examples of few of the them are given

below;

Homogenous and inhomogeneous material

Inclusion of non-linear non-local (strain gradient) term in strain energy of transformation

Effect of free surface in a body

Considering single and multivariant orientation of martensite

Constrained and unconstrained transformation

Multilayer systems

Application of external stress

References:

1. J. W. Christian, The Theory of Transformations in Metals and Alloys, vol. 7, Pergamon

press, Oxford, p 13 & 921;

2. David A. Porter, Kenneth E. Easterling, Phase Transformations in Metals and Alloys, 3rd

edition, CRC Press, New York

3. Z. Nishiyama, Martensitic Transformation, Academic Press, New York, p. 342, 355, 211,

213, 216;

4. T. V. Rajan, Heat Treatment: Principles and Techniques p. 92

5. Y. Wang and A. G. Khachaturyan, Three-Dimensional Field Model and Computer

Simulation of Martensitic Transformation, Acta Metallurgica, vol. 45, 759-773, (1997)

6. Nele Moelans, Bart Blanpain, An Introduction to Phase-field Modeling of Microstructure

Evolution, Computer Coupling of Phase Diagrams and Thermochemistry, 32, 268-294, (2008)

7. A. Artemev, Y. Yin and A. G. Khachaturyan, Three Dimensional Phase Field Model of

Proper Martensitic Transformation, Acta Meteriali, 49, 1165-1177, (2001)