1. (a) Macrostate

• Most effect of molecular motion not observed because random motions of molecules tend to average themselves out.
• The residual effects are the quantities that describe the macrostate of the system.
• E.g. molecules constantly bombard the walls of the container. This gives rise to pressure in the container which can be measured and monitored using macroscopic instrument like a pressure gauge.
• Macrostate observes only the mechanical effects of microscopic motion.
• A small number of parameters specify the macrostate of a system, these parameters give the condition of the system at any instant of time. The values of these parameters are such that if the no. of molecules and two other variables are known, all other macroscopic parameters can be calculated and the macrostate is specified.
• Examples of variables that specify the macrostate are:

• Pressure (due to bombardment of walls by atoms/molecules of gas)
• Volume (imposed on the system by fixed boundaries or let the system find its own volume by natural expansion or contraction)
• Amount of substance (system either contains fixed no. of particles, or if particles are allowed to enter or leave the system, then the number and type of the particles that enter or leave must be known)
• Energy (kinetic energy due to atoms in motion, and potential energy due to interaction of atoms with one another – these 2 sources of energy give rise to the internal energy of the system)
• Others like external stresses, electric field strengths, magnetic field strengths, magnetisation, polarisation…Normally these parameters are assumed absent.

1. (b) Microstate

• Consists of a list describing the velocity and position of atoms and molecules at any instant of time.
• A small system contains 2.5 billion paricles.

1. (c) Equilibrium

• If a system is left undisturbed, its macroscopic variables will settle down to definite values. No further change occurs, i.e. the composition, structure and amount of substance do not vary with time at a given temperature and pressure. When this occurs, equilibrium is said to have occurred.
• Equilibrium states are at the lowest value of Gibbs free energy, and there is no net change in the Gibbs free energy, i.e. dG = 0.

1. (d) Activation energy

• The energy barrier that particles have to overcome before a reaction can take place.
• This energy barrier is overcome through thermal activation.
• According to Arrhenius equation, the rate of reaction is related to the activation energy by the equation:

• The higher the temperature, the faster the rate of the process since there is higher probability that atoms have sufficient energy to overcome the activation energy.

1. (a) Types of interfaces in crystals

• Coherent interface

• Arises when two crystals match perfectly at the interface plane so that the two lattices are continuous across the interface.
• The interfacial plane has the same atomic configuration in both phases.
• Energy of this interface is low, typically 0.05 J/m² because although there is a sharp change in chemical composition, there is no structural change.
• This interface is usually only possible with small second phase particles.

• Semi-coherent interface

• Two crystals have slightly different lattice spacings.
• Boundary still coherent, but has more energy associated with it.
• Strain gets bigger as boundary grows sideways.
• As paricle grows, the strain builds up until it is relieved by the injection of dislocations to give a semi-coherent boundary.
• Energy of this boundary varies from 200-500 J/m².

• Incoherent interface

• When phases meet at boundary are large and differ in both chemical composition and crystal structure.
• No possibility of good matching across the interface.
• Phase boundary has a high energy, typically between 500-1000 J/m².

1. (b) ANISOTROPY

• Given the equilibrium surface energy, an object will tend to assume a shape that will lower its free energy.
• Although a spherical shape satisfy this minimum energy condition for liquids, the same is not true for crystalline solids because atoms in a solid cannot move about freely and arrange themselves in the lowest energy configuration.
• The origin of surface free energy is that atoms in the layers nearest to the surface is deprived of some of its neighbours. From this, we can infer that the surface energy of a crystalline surface is strongly dependent on the types of crystallographic planes. In other words, the value of the interfacial tension between a solid phase and another phase is dependent on the indices of the planes of the solids forming the surface – this is anisotropy of surface energies.
• Due to the varying efficiency of packing of different faces, it can be expected that the planes with the lowest values of surface tension is most likely to develop an interface.
• A more closed-packed surface will have the lowest energy; hence this plane is more likely to develop an interface.

• The surface tension acting on the boundary between the a and b phase will attempt to shorten the length of the surface. Likewise, the phase boundary between a and g , and b and g will behave to shorten the grain boundaries between them.
• At equilibrium, the surface tension forces acting between each surface must balance.

• Size of the angles, a , b and g will depend on the surface tension forces.
• For a pure polycrystalline material cooled in equilibrium from the molten state, the three surface tension forces are equal.

s a b /sin g = s b g /sin a _{= s a g} /sin b

so, a = b = g = 120^o

• A grain shape that satisfies this requirement exists. It also packs together to fill space. The shape has 14 faces - a tetrakaidecahedron. Grains in a polycrystalline material tend to be tetrakaidecahedral when the grain boundary energy is the dominating influence.
• The above equation is completely true for a liquid film. In solids however, it is complicated by the rigidity of the material and the factor of preferred crystallographic orientation.

Lens-shaped crystals formed at grain boundary

• Consider a second phase particle which meet the grain boundary at an angle.

Resolving the forces for equilibrium,

Cos q = s a a _/ 2s a b

Equilibrium is only possible if s a a _/ 2s a b _£ 1

• If s a a _/ 2s a b _® 1, i.e. q = 0, then the second phase will spread along the boundary as a thin layer. This is wetting of the grain boundary.

• If energy of interphase boundary is isotropic, then spherical particles (with the least surface area) will be formed to reduce the interphase boundary energy.

• If energy of interphase is anisotropic, then different parts of the particle will experience different surface tensions.

• If the grain boundary energy is equal to the interphase energgy, the second phase will form complete grains. The phases will have tetrakaidecahedral shapes when they come together.

(iii)

(iv)

6. (a) Martensitic transformation

• Diffusionless (steel is quenched sufficiently rapid to prevent any C diffusion).
• Instantaneous (martensite grains nucleate and grow rapidly at the speed of sound).
• Athermal (reaction is not thermally activated; extent of reaction depends on temperature at which reaction takes place and is independent of time).
• Large numbers of atoms experience cooperative movement, only a slight displacement of each atom relative to their neighbours.
• Martensite is a non-equilibrium single-phase structure – will transform if heated and diffusion rates become appreciable.

(b) (i) Martensite

(ii) Lower bainite. Name of heat treatment is tempering.

(iii) Fine pearlite

(iv) Martensite. Name of heat treatment is martempering or marquenching.

(v) Bainite. Name of heat treatment is austempering.

(vi) Spheroidite