Law of Nernst-Einstein
See also: Einstein (homonymy)
The law of Nernst-Einstein is a law which intervenes in the migration of the species in the solids Cristal flaxes, when the species are subjected to a force. By “species”, one understands “crystalline defects”.
This law makes it possible to calculate the speed of migration of the species according to the intensity of the force and the coefficient of diffusion of the species in the crystal.
In absence of force
Let us consider the movements on an axis X (for example by projection on this axis).
In absence of force, the defects migrate by chance, by jumps of a site to a nearby site. These jumps are possible thanks to thermal agitation.
Per unit of time, a species has a probability Γ I of making a jump towards a site I close. The mean velocity of the particles is null (case similar to the Brownian Movement); the quadratic average of < displacements; X 2> during a time T is not it not null and one a:
See the detailed articles Diffusion of the matter and Law of Fick.
Effect of a force
When the species is subjected to a force, that breaks the symmetry of the jumps, the probabilities of two opposite jumps is not equal any more. To simplify, one considers only one species, and a movement in a given direction. If Γ+ is the probability that the particle moves a length +δ X per unit of time, and Γ- the probability that it moves a length - δ X , then the range < X > after a time T is worth:
- a flow J 1 created by the force
J 1 = v · C , where C is the concentration of the species;
- a flow J 2 opposite which follows the law of Fick
where D is the Coefficient of diffusion of the species.
If one waits “sufficiently a long time”, one reaches a stationary mode: flows J 1 and J 2 are compensated, one has a constant gradient of concentration. There is thus J = 0, that is to say, if C ∞ ( X ) is this constant concentration:
Let us suppose now that the force is conservative (the most frequent case). It thus derives from a Potentiel η:
This law resembles a law of fluid Frottement. At the time of a movement at low speed in a nonturbulent fluid, one can estimate that the force of friction is proportional to the speed, and thus that one reaches a stationary mode where speed is proportional to the force (it is the principle of the Parachute):
- v = B · F
The law of Nernst-Einstein thus gives us:
The force Fc resulting from the chemical Potentiel μ can be written, with a dimension:
If a particle carries Z elementary charges E , then it undergoes the force Fe (electrostatic Force or Force of Coulomb):
Let us consider the flow of loads jel , also called density of electric current. There is
- jel = Z · E · J = Z · E · C · v
One can make a parallel with the Loi of Ohm connecting this density of electric current jel to the gradient of potential:
- Walther Hermann Nernst
- Mobility of the charge carriers
- Theory of the kinetics of oxidation of Wagner
|Random links:||Papyrus Edwin Smith | Robert Moray | Castle of Oberhofen | Cultural diversity | Tulia | Moel_Famau|