According to the concept of perfect lattice, any given atom of the lattice has a particular lattice site assigned to it, and except from thermal vibration about its mean position on the crystal lattice, it is assumed not to move.
In actual crystals, however, the atoms have more freedom i.e., they wonder from one lattice site to another. This motion is called diffusion. This motion arises due to the non-uniform concentrations of diffusion species (e.g. atoms or vacancies) in crystals and is analogous to the motion of gas molecules in response of pressure gradients.
Classification of Diffusion Processes:
1. Self-Diffusion:
This is due to the jumping of atoms in pure metals.
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2. Inter-Diffusion:
This is observed in binary metal alloy such as Cu-Ni system, in which diffusion of alloying atoms takes place in the lattice of base metal and vice versa.
3. Volume Diffusion:
This is due to the atomic movement in bulk in materials.
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4. Grain Boundary Diffusion:
This is due to the atomic movement along the grain boundaries alone.
5. Surface Diffusion:
This is due to the atomic movement along the surface of a phase.
Mechanism of Atomic Diffusion:
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Let us now describe the mechanism by which the atoms migrate from one lattice site to another. Of course, one cannot look into a crystal and see the individual atomic motions which result in a long-range movement. Nevertheless, it is evident from the consideration of crystal geometry that the atoms move in discrete jumps from one lattice site to the adjacent ones.
These elementary jumps, when added together, permit the atoms to travel over long distance. The detailed mechanisms by which the individual atomic jumps occur are- Vacancy motion, interstitial motion, or some sort of atom interchange mechanism.
The sketches of the motion of atoms necessary to allow diffusion by these methods are presented in Fig. 3.3. Diffusion by vacancy motion occurs when vacancy make interchanges with adjacent atoms.
The sequence of events whereby a vacancy moves through the lattice is shown in Fig. 3.3 (a); clearly, as the vacancy move from site 1 to 2 to 3 to 4. Diffusion by interstitial motion occurs when the interstitial atoms move from one interstitial site to another. Fig. 3.3 (b) shows the motion of the interstitial atom at site 1 to 2 to 3 to 4 to 5 etc.
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An activation energy is associated with interstitial diffusion because, to arrive at vacant site, it must squeeze past neighbouring atoms with energy supplied by the vibrational energy of the moving atoms. Consequently, interstitial diffusion is a thermally activated process. This process is simpler because the presence of vacancies is not required for interstitial atom to move.
This mechanism is important in two cases:
(i) The presence of very small atoms in the interstices of the lattice greatly affects the mechanical properties of metals
(ii) Oxygen, nitrogen and hydrogen can be diffused in metals easily at low temperature.
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Diffusion by interchange mechanism may occur in one of several ways. Simultaneous interchange of two, four, or more atoms may occur; the interchanges of two and four atoms are sketched in Fig. 3.3 (c).
Usually all the three possibilities, occur in all the crystals to lesser or more extent. Their relative importance in a given crystal at a given temperature depends upon the respective energy requirements. Calculations of this energy show that diffusion by vacancy motion should predominate in pure metals and substitutional alloys.
Many experiments support this conclusion. For one group of alloys, however, interstitial diffusion certainly predominates. Such alloys are called interstitial alloys. Usually the motion of small atoms dissolved in the interstices of the structure occurs in such alloys. We now calculate the probabilities of these possibilities to occur. For the sake of simplicity, we first consider the interstitial motion.
Interstitial Motion of Atom:
With reference to Fig. 3.4, consider the motion of an atom from the interstitial site ‘a’ to the interstitial site ‘b’ keeping in mind the potential energy requirements of the atom as a function of position. Since the energy at the mid-point ‘c’ is higher than the energy at the points ‘a’ and ‘b’ (the equilibrium positions) by an amount Em, it is evident that the atom at the site ‘a’ must acquire the energy Em in order to move through across ‘c’.
Usually this magnitude of energy is not available to the atom at any reasonable temperature (Em ≈ 1 eV, whereas the average thermal energy of an atom is less than 0.1 eV) and the atom therefore does not move to the site ‘b’. But when the sufficient energy is available, it certainly moves to the site ‘b’.
The probability of occurrence of interstitial motion refers to the ratio of the number of times an atom actually moves to the site ‘b’ to the total number of times (in the same time) it attempts to do so. Then, if fm is the number of times/sec when it actually moves to the site ‘b’ and v is the total number of times/sec during which it attempts to do so, one can immediately write, using the Maxwell-Boltzmann distribution.
fm is usually called the frequency of jumping and v is the natural frequency of vibration about the equilibrium site.
Equation (iv) assumes that there is only one site into which the atom at V can jump. But if there are z equivalent neighbouring sites into which it can jump, Eq. (iv) can be modified as-
The probability for the interstitial motion to occur is-
It is now evident from (v) that the frequency of jumping is controlled by the energy of motion of the interstitial atoms, fm depends also on temperature- for example, for carbon diffusing interstitially in iron taking Em about 0.9 eV, and v about 1013 per second, we find from (v) that at room temperature an atom of carbon makes about 1 jump in 25 seconds, whereas at the melting point of iron (1545°C) it makes about 2 × 1011 jumps per second. The temperature dependence is thus very strong.
Vacancy Motion:
With reference to Fig. 3.3. (a), consider the motion of the vacancy at the site 1 to the atomic position 2. Since the motion of the vacancy from site 1 to site 2 is accompanied by the motion of the atom from site 2 to site 1, we may equivalently consider the motion of the atom at the site 2. If we now treat the motion of this atom similar to the motion of the diffusing interstitial atom, we immediately have (for the number of times per second in which it acquires sufficient energy Em) the expression
; v being the total number of attempts made by the atom to acquire it in the same time.
In this case 
does not represent the number of times in which the atom actually moves from site 2 to site 1. This is because there is no certainty that a vacancy does exist adjacent to the diffusing atom. So, in order to find the number of times per second in which the atom at site 2 actually moves to the site, 
must be multiplied by a factor representing the fraction of times in which the atom have a vacancy adjacent to it. If Ev is the formation energy of a vacancy, then this factor is 
, and hence the frequency of jumping f in this case is-
Where z is the number of equivalent sites into which the atom can be excited. Letting Em + Ev = Q, called the activation energy for the process, we finally have-
The probability for the vacancy motion to occur is then-
We consider from (viii) that f in this case is much smaller than the corresponding jump frequency for a typical impurity interstitial. This is because both Em and Ev are of the same order of magnitudes (~1 eV).
Macroscopic Diffusion: Fick’s Laws of Diffusion:
Let us now treat the bulk mass transport of atoms and determine how fast atoms diffuse through a certain plane in the lattice, i.e., the rate of diffusion. For simplicity, the problem is being considered only for the simple cubic lattices.
Let Fig. 3.5 represents a section (cut along a cube face) of the crystal under consideration. In this figure, it is divided into thin slabs of cross-sectional area L2 and an atomic diameter ‘d’ thick. With this geometry, if N is the number of atoms in a given slab and C is the volume concentration of atoms in that slab, then-
C = N/L2d ………. (x)
Since L2d is the volume of the slab. Also, each slab contains one plane of atoms so that different slabs represent different planes of atoms. Consideration is now given to two adjacent planes of atoms, say those which are indicated by the number 1 and 2 and which are separated by the plane A. It is clear that when diffusion occurs, atoms move through plane A in both directions. Under certain circumstances a net flow of atoms exists in one direction or the other. It is this flow which we wish to calculate.
To do this, we suppose that N1 and N2 are the numbers of atoms in the planes 1 and 2 respectively, and f is the frequency of jumping of atoms in either plane. Each atom then makes one jump in 1/f sec.
If we now assume that the atoms are restricted to jumping either to the right or to the left, i.e., after one jump an atom in plane 2 is either in plane 1 or in plane 3, then in a single jump half the atoms in plane 1 jump in either direction and so, during the time 1/f, 1/2 N1 atoms pass the plane A to the right. Similarly 1/2 N2 atoms pass the plane A to the left in that period of time. The net flow of atoms per second. dN/dt, over the plane A is then just the difference between these two, i.e.
dN/dt = f/2 (N2 – N1) … (xi)
where, N1 > N2 and the net flow of atoms is to the right. Using (x), we have-
dN/dt = f/2 L2d (C2 –C1) … (xii)
where, C1 and C2 are the volume concentrations of atoms in the planes 1 and 2 respectively. It is usual to express (C2 – C1) as-
In equation (xiii), the left-hand side is the net flow of atoms per second per unit area of the plane at position A, or the flux (J) of atoms.
∴ J = D dC/dx
Let us now take into account the actual three dimensional jumping of the diffusing atoms. We shall consider again the simple cubic crystal. In this crystal structure, each atom may exchange places with any one of the six nearest neighbour atoms, but only one of these six jumps of an atom in plane 2 carriers the atom into plane 1. Similarly, only one of the six possible jumps of an atom in plane 1 carries the atom into plane 2. The three dimension jumping of the diffusing atoms is then accounted for if we replace (xiii) by-
where D = fd2/6.
The equation (xv) known as the Fick’s first law. The constant D is called the diffusion constant or diffusivity, and has the units m2/sec. The negative sign means that diffusion occurs away from regions of high concentration.
Let us now discuss the feature of diffusivity in solids. After putting (viii) in D = fd2/6, we have for diffusivity.
Grouping z d2v/6 into D0, often called the frequency factor, it is usually rewritten as-
The following features of diffusivity D are now apparent:
(i) The values of D vary greatly for different solids at a given temperature. This variation is at most entirely a result of greatly different values of f, although the variation of d also contributes towards it, but only very slightly. For the common metals, and Ge and Si, at room temperature it commonly lies in the broad ranges 10-20 to 10-50 m2/sec.
(ii) D depends on temperature exponentially whenever a single diffusion mechanism is operative. This temperature dependence is found experimentally to be true for most solids.
(iii) D0 is constant, independent of temperature and should have a value of about 10-5 to 10-7 m2/sec. experimentally, it does appear to be only nearly independent of temperature, and its value for many metal and semiconductors are somewhat larger than 10-5 m2/sec.
Fick’s first law can be used to describe flow under steady state conditions (diffusion flux does not change with time). It is identical to the heat flow under a constant temperature gradient and current flow under a constant electric field gradient. Under steady state flow the flux remains the same at any cross-sectional plane along the diffusion direction.
One common example of steady state diffusion is the diffusion of atoms of a gas through a plate of metal for which the concentration (or pressures) of the diffusing species on both surface of the plate are held constant. This is represented schematically in Fig. 3.6. (a). When the concentration C is plotted vs position (distance) within the solid x, the resulting curve is termed the concentration profile and the slops at a particular point on this curve is the concentration gradient.
Fig. 3.6 (b) shows the concentration—distance profile under steady state flow. The profile is a straight line, when D is independent of concentration, i.e., D ≠ f(C). When D = f(c), the profile will be such that the product D (dc/dx) is a constant. In neither case, the profile changes with time under conditions of steady state flow.
Most practical diffusion situations are non-steady state one. That is, the diffusion flux and the concentration gradient at some particular point in a solid vary with time, with a net accumulation or depletion of the diffusing species resulting. This is illustrated in Fig. 3.7 which shows concentration profiles at three different diffusion times. Under such conditions, use of Eqn. (xv) is no longer convenient.
Consider an element slab of thickness Δx along the diffusion distance x. Let the slab cross section be perpendicular to x and its area by unity. The volume of the slab is then 1 .Δx. Under non-steady state conditions, the flux into the slab, Jx, is not equal to the flux out of the slab, Jx + Δx. The rate of accumulation or depletion of the diffusing atoms within this elemental volume is (c/t). 1 .Δx. It can be expressed as the difference of fluxes in and out of the slab-
Thus, equation J = -D dc/dx can be rewritten in partial form and from above equation we have –
Above equation is known as Fick’s second law for un-directional flow under non-steady state conditions. If D is independent of concentration, the above relation becomes-
Solution to this expression is possible when physically meaningful boundary condition is specified.
Factors Affecting Diffusion:
(i) Diffusion Species:
The magnitude of the diffusion coefficient, D is indicative of the rate at which atoms diffuse. The diffusing species as well as the host material influence the diffusion coefficient.
For example, there is a significant difference in magnitude between self and carbon inter diffusion in α – Fe at 500°C, the D value being greater for carbon inter diffusion. This comparison also provides a contrast between rates of diffusion vacancy and interstitial modes. Self-diffusion occurs by a vacancy mechanism whereas carbon diffusion in Fe is interstitial.
(ii) Temperature:
Temperature has most profound influence on the coefficients and diffusion rates. For example, for the self-diffusion of Fe in α-Fe the diffusion coefficient increases approximately six orders of magnitude in rising temperature from 500°C to 900°C. The temperature dependence of diffusion coefficients is related to temperature according to-
D = D0 exp (-Q/kBT)
Taking natural logarithms of above equation yields-
Log D = log D0 – Q/kB (1/T)
Since D0, KB and Q are all constants, the above equation takes the form of a straight line y = mx + c. Where y and x are analogous, respectively, to the variables log D and 1/T. Thus, if log D is plotted vs. the reciprocal of the absolute temperature, a straight line should result, having slop and intercept of –Q/kB and log D0, respectively. This is, in fact, the manner in which the values of Q and D0 are determined experimentally.
Applications of Diffusion:
i. Diffusion is fundamental to phase changes e.g. -iron to α-iron.
ii. Joining of materials by diffusion bonding e.g. welding, brazing soldering galvanizing and metal cladding.
iii. Important in heat treatment like homogenizing treatment of casting, recovery, recrystallization and precipitation of phases.
iv. Production of strong bodies by powder metallurgy.
v. Surface treatment of steels e.g. case hardening.
vi. Oxidation of metals
vii. Doping of semiconductors.