Diffusion

The study of phase transformations concerns those mechanisms: i)By which a system attempts to reach equilibrium state and ii)How long it takes. The fundamental processes that controls the rate at which many transformations occur Is the diffusion of atoms.

The reason why diffusion occurs ? Is always to produce a decrease in Gibbs free energy. As a simple illustration of this Consider Fig. 2.1. Two blocks of the same A—B solid solution, but with different compositions, are welded together And held at a temperature high enough for long-range diffusion to occur.

If the molar free energy diagram of the alloy is as shown in Fig. 2 If the molar free energy diagram of the alloy is as shown in Fig. 2. ib, The molar free energy of each part of the alloy will be given by G1 and G2, And initially the total free energy of the welded block will be G3

However, if diffusion occurs as indicated in Fig. 2. la To eliminate the concentration differences, The free energy will decrease towards G4,i.e The free energy of a homogeneous alloy.

In this case, a decrease in free energy is produced when By A and B atoms diffusing away from the regions of high concentration to that of low concentration, i.e. Down the concentration gradients. What happen when there is a miscibility gap

In alloy systems that contain a miscibility gap (at low temperatures) The free energy curves can have a negative curvature If the free energy curve and composition for the A—B alloy shown in Fig. 2.la Drawn as in Fig. 2.ld

The A and B atoms would diffuse towards the regions of high concentration, i.e. up the concentration gradients, as shown in Fig. 2.ic. However, this is still the most natural process as it reduces the free energy from G3 towards G4 As can be seen in Fig. 2.le and f

The A and B atoms are diffusing from regions where the chemical potential is high to regions where it is low, i.e. down the chemical potential gradient in both cases.

It is known that chemical potential was related to the activity or effective concentration of component A in the solution. It can be shown that Ua (Chemical Potential) at concentration X=Xa is equal to the intercept at 100 % A of the tangent to the free energy curve at that point. Thus the slope of the energy curve (dG/dx) determines the magnitude of the chemical potential.

In practice the first case (concentration gradient) is far more common than the second case (chemical potential), It is usually assumed that diffusion occurs down concentration gradients However, it can be seen that this is only true under special circumstances

Therefore , It is better to express the driving force for diffusion in terms of a chemical potential gradient. Diffusion ceases when the chemical potentials of all atoms are everywhere the same and the system is in equilibrium.

However, since case 1 above is mainly encountered in practice and Concentration differences are much easier to measure than chemical potential differences, It is therefore more convenient to relate diffusion to concentration gradients. Therefore rest of the discussion will be mainly concerned with this approach to diffusion.

Atomic mechanism of diffusion There are two common mechanisms by which atoms can diffuse through a solid i)Substitution, and ii)Interstitial The operative mechanism depends on the type of site occupied in the lattice. Substitution atoms usually diffuse by a vacancy mechanism The interstitial atoms migrate by forcing their way between the larger atoms, i.e. interstitially. Normally a substitutional atom in a crystal oscillates about a given site and Is surrounded by neighbouring atoms on similar sites.

The mean vibrational energy possessed by each atom is given by 3 kT, Therefore vibration increases in proportion to the absolute temperature Since the mean frequency of vibration is approximately constant The vibrational energy is increased by increasing the amplitude of the oscillations.

Normally the movement of a substitutional atom is limited by its neighbours Therefore, the atom cannot move to another site. However, if an adjacent site is vacant It can happen that a particularly violent oscillation results in the atom jumping over on to the vacancy.

in order for the jump to occur The shaded atoms must move apart To create enough space for the migrating atom to pass between.

Therefore the probability that any atom will be able to jump into a vacant site Depends on the probability that it can acquire sufficient vibrational energy. The rate at which any given atom is able to migrate through the solid will be determined by the frequency with which it encounters a vacancy and This in turn also depends on the concentration of vacancies in the solid. It will be shown that both the probability of jumping and the concentration of vacancies are extremely sensitive to temperature.

Interstitial Diffusion When a solute atom is appreciably smaller in diameter than the solvent, it occupies one of the interstitial sites between the solvent atoms. In fcc materials the interstitial sites are midway along the cube edges or, equivalently, in the middle of the unit cell, Fig. 2.3a. These are known as octahedral sites since the six atoms around the site form an octahedron. In the bcc lattice the interstitial atoms also often occupy the octahedral sites which are now located at edge-centring or face-centring positions as shown in Fig. 2.3b.

Usually the concentration of interstitial atoms is so low that only a small fraction of the available sites is occupied. This means that each interstitial atom is always surrounded by vacant sites Therefore, it can jump to another position as often as its thermal energy permits And to overcome the strain energy barrier to migration, Fig. 2.4.

Jump process for an interstitial atom Effect of Temperature— Thermal Activation Due to the thermal energy of the solid all the atoms will be vibrating about their rest positions Occasionally a particularly violent oscillation of an interstitial atom, or Since the diffusion coefficient is closely related to the frequency of such jumps, F, it is of

The rest positions of the interstitial atoms are positions of minimum potential energy. In order to move an interstitial atom to an adjacent interstice The atoms of the parent lattice must be forced apart into higher energy positions as shown in Fig. 2.6b. The work that must be done to accomplish this process causes an increase in the free energy of the system by Gm (m refers to migration) as shown in Fig. 2.6c

Gm is known as the activation energy for the migration of the interstitial atom. In any system in thermal equilibrium the atoms are constantly colliding with one another and changing their vibrational energy.