1. "War is a matter of vital importance to the State;
the province of life or death; the road to survival or ruin.
It is mandatory that it be thoroughly studied."
Sun Tzu,"The Art of War“ ( approximately 250 BC)
Diffusion
Solid

2. Diffusion
Part1. Constitutional effects
Diffusion is the phenomenon of spontaneous material transport by atomic motion.
Diffusion is classified according to
a) conditions: self-diffusion, diffusion from surface, interdiffusion, fast path diffusion etc.
b) mechanism: interstitial, substitutional;
Part 2.
Non-constitutional effects. Kirkendall effect. Einstein equation.

3. General Note.
Diffusion is a flux of matter in which the atoms or molecules of a certain type move differently (rate, amount etc) with respect to the atoms/molecules of other type.
 Please note the difference from gas or liquid flow, in this case ALL components move in the same way.
The definition: “Diffusion is the movement of molecules from a high concentration to a low concentration” is wrong because there are cases when diffusion process does just opposite.
Flux of matter can be caused not only by the difference in concentration of the atom that diffuses, but also the difference in concentration of other atoms and or gradient of physical parameters ( temperature, pressure, electric or magnetic field).

4. INHOMOGENEOUS DISTRIBUTION OF ONE COMPONENT MAY CAUSE DIFFUSION OF OTHER COMPONENTS.
Example: If two pieces of steel with 0.4% carbon, one containing 3.8% silicon and the other with no silicon, are joined, it might be expected that the carbon would remain homogeneously distributed. However, the chemical potential of carbon is higher in the silicon-containing steel, and having a high mobility, the carbon diffuses into the silicon-free steel.

5. C is the concentration of the diffusion species. x is a coordinate
General thermodynamic considerations lead to a formula for the flux ( amount of matter through a unit area per unit of time): Fick’s law
D- is a diffusion coefficient
C is the concentration of the diffusion species.
x is a coordinate
The sign of the diffusion coefficient (direction of the flux) depends on the conditions of the system. IT CAN BE BOTH POSITIVE AND NEGATIVE ( uphill diffusion, for instance during phase separation)
In the systems, in which D is independent on concentration

6. Examples of solutions:
1. A fixed quantity of solute (B) is plated onto a semi-infinite bar
Boundary conditions:
Solution:
This case is realized if a thin film of diffusant is deposited on a surface.

7. Example 2. Interdiffusion of ONE component
and diffusion from constant source.
Boundary conditions:
Solution:
Notice that the surface concentration remains fixed.
In the case of interdiffusion of TWO components concentration profiles may be very different!

8. In ideal case the point of constant concentration propagates with a rate of (4Dt)-½
If there is a way to trace a point of constant concentration then diffusion coefficient can be determined explicitly. x2 t
The slope is 4D x
This method can be used to measure diffusion coefficient by measuring experimentally :

9. Microscopic Mechanisms of Diffusion
Phenomenological description does not give dependence of the diffusion coefficient on any physical parameters.
Consider two adjacent planes in the crystal one can get that
N sites with n1 atoms
N sites with n2 atoms
v is the number of jumps per second
U is the energy barrier separating two sites
N is the number of atoms per plane 1 2 U a
Energy profile
In ideal case diffusion coefficient exponentially depends on temperature.

10. Examples
Plot of the logarithm of the diffusion coefficient versus the reciprocal of absolute temperature for several metals.

11. Mechanisms of Diffusion
Diffusion is the stepwise migration of atoms from lattice site to lattice site.
For an atom to make such a move, two conditions must be met:
1. there must be an empty adjacent site, and
2. the atom must have sufficient energy to break bonds with its neighbor atoms and then cause some lattice distortion during the displacement.
1. Self-diffusion: there is no gradient of chemical potential:
a) Interstitial
b) Vacancy
b) Kick-out

12. Important.
 During self-diffusion there is no change of chemical potential.
 Realization of each of the mechanisms depends on
 Type of intrinsic defects that prevails in the solid
 Activation energy for each of the mechanisms, if more than one may be realized.
 Presence of other defects (vacancies).
Realization of vacancy or kick-out diffusion is possible only at the temperatures with sufficient concentration of vacancies. Therefore, prevailing mechanism may change with temperature.
In general, EVERY component in solid undergoes self-diffusion, however, if a solid contains more than one component, the ratio between self-diffusion coefficient depends on the type of bonding:
Solids with covalent bonding typically have very low self-diffusion coefficients.
Solids with ionic bonding may have very different self-diffusion coefficients for anion and cation.
Metals and metal alloys usually show fast self-diffusion.

14. Diffusion of impurities.
a) Interstitial
b) Vacancy
b) Kick-out
Important.
 The diffusion mechanism of an impurity depends on many factors:
type of the solution: interstitial or substitutional;
size of the diffusant and size of the host sites;
temperature;
presence of other impurities;
electronic structure of the host: metal, dielectric or semiconductor.

15. Important.
Special cases:
There are materials where structural properties allow ultra-fast ion movement: superionics. In these materials ( for example AgRb3I4 one of the ions is much smaller than the available sites and there are far more available sites than ions.
Diffusion in polymers and glasses can be described by “randomly opening path” theory. Temperature dependence of diffusion coefficient in these materials is very complicated and time to time activation energy may become negative=> Diffusion coefficient may decrease with temperature.
Diffusion coefficient in anisotropic solids is a strong function of direction. Example: diffusion coefficient of Li and other alkaline metals in graphite along and across the layers may differ by 4 orders of magnitude.

16. Diffusion coefficient and mobility are linked
Diffusion in presence of electric field: Electromigration.
Ionic and charged in impurities in solids can drift in electric field. As a first approximation one can assume that the flux of ions is proportional to electric field and concentration, i.e., one can use a concept of mobility. In this idealized case the flux of ions is given:
(1)
where gi is mobility of ions, Ci is the concentration of ions and E is electric field.
Diffusion coefficient and mobility are linked
Thus mobility is
(Nernst) Einstein equation
Electromigration
Diffusion

17. Limitations.
 Electrical mobility and diffusion coefficient are linked to each other.
 Nernst-Einstein equation is valid whenever the following conditions are met:
The system is not far from equilibrium, i.e., gradient of potential and concentration are small
The diffusion species follow Boltzmann statistics, i.e., they do not interact with the host and with each other.
Important: Nernst-Einstein equation is applicable to electrons in some semiconductors.
 Nernst-Einstein equation is not valid for systems with strong interactions.

18. Comments
Nernst-Einstein equation is a low electric field approximation! It implies that the energy acquired by ion during one jump is mush smaller than the activation energy. =>
Systems with very low activation energy do not obey Nernst –Einstein equation.
Application of a sufficiently high electric field may significantly increase mobility. This electric field is, in fact, comparable with crystal field, the electric field between ions in crystal.
Materials with fast-path diffusion may have different electric fields for each path at which non-linear dependence between mobility and diffusion coefficient becomes noticeable.

19. Thermal diffusion T1 < T2 1 ≠ 2
If a homogeneous alloy is placed in a temperature gradient, one of the elements will diffuse under the influence of the temperature difference. This is known as the Sorét effect, and again shows an example of diffusion occurring without a composition gradient. In the presence of temperature gradients we cannot use Gibbs free energies to define equilibrium conditions, so chemical potential arguments can not be used.
T < T2
 ≠ 2
Thermal diffusion in ionic solids with only one atom mobile leads to thermo-electric voltage, similar to Seebeck effect in electronic conductors.
In practice thermal diffusion (also called thermomigration) can occur both down and up the temperature gradient.
Carbon in austenite thermomigrates up a temperature gradient, because the activation energy in this case is required mainly for preparing the destination site. As the carbon moves, two Fe atoms have to separate to create room for the C atom. This occurs more easily at a higher temperature, so the carbon moves preferentially to the hotter region.

20. Strain field induced diffusion
The presence of strain in the material can have a significant effect on the chemical potential of a solute. For example, in the case of an interstitial solute such as carbon in iron, a tensile strain will increase the space available for the interstitial and so reduce the chemical potential.
The impurities that expand the lattice drift toward dilated regions and impurities that cause contraction of the lattice drift towards compressed regions.