# Law of Nernst-Einstein

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:

if δξ I is the algebraic length (positive or negative according to the direction of reference) of the jump I .

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:

$= T \ cdot \left(\ Gamma_+ - \ Gamma_-\right) \ cdot \ delta x$
What makes it possible to define the average Speed v :
$v = \ frac \left\{\right\} \left\{T\right\} = \left(\ Gamma_+ - \ Gamma_-\right) \ cdot \ delta x$
This movement under the effect of a force creates a gradient of concentration. However, the random diffusion tends to level the concentrations, and thus is opposed to the “forced” migration, one thus has two flows:
• 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
$j_2 = - D \ cdot \ frac \left\{\ partial C\right\} \left\{\ partial X\right\}$ where D is the Coefficient of diffusion of the species.
Total flow is thus worth:
$j = v \ cdot C - D \ cdot \ frac \left\{\ partial C\right\} \left\{\ partial X\right\}$.

## Stationary mode

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:

$v \ cdot c^ \ infty = D \ cdot \ frac \left\{\ partial c^ \ infty\right\} \left\{\ partial X\right\}$

Let us suppose now that the force is conservative (the most frequent case). It thus derives from a Potentiel η:

$F = - \ frac \left\{\ partial \ eta\right\} \left\{\ partial X\right\}$.
With dynamic balance, the particles are distributed according to a Statistique of Maxwell-Boltzmann:
$c^ \ infty \left(X\right) = c_0 \ cdot \ exp \ left \left(- \ frac \left\{\ eta\right\} \left\{kT\right\} \ right\right)$
where K is the Boltzmann constant and T is the absolute Température. By introducing this into the preceding equation, one obtains:
$v \ cdot c_0 \ cdot e^ \left\{- \ frac \left\{\ eta\right\} \left\{kT\right\}\right\} = - \ frac \left\{D\right\} \left\{kT\right\} \ cdot \ frac \left\{\ partial \ eta\right\} \left\{\ partial X\right\} \ cdot c_0 \ cdot e^ \left\{- \ frac \left\{\ eta\right\} \left\{kT\right\}\right\}$
what gives us the law of Nernst-Einstein

## Friction

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
where B is the mobility of the species ( German Beweglichkeit in ).

The law of Nernst-Einstein thus gives us:

$B = \ frac \left\{D\right\} \left\{kT\right\}$
from where one deduces the law from Einstein :

## Applications

### Chemical potential

The force Fc resulting from the chemical Potentiel μ can be written, with a dimension:

$F_c = - \ frac \left\{\ partial \ driven\right\} \left\{\ partial X\right\}$
and thus the equation of Nernst-Einstein becomes:
$v = - \ frac \left\{D\right\} \left\{kT\right\} \ cdot \ frac \left\{\ partial \ driven\right\} \left\{\ partial X\right\}$

### Electric field

If a particle carries Z elementary charges E , then it undergoes the force Fe (electrostatic Force or Force of Coulomb):

$\ vec \left\{F_e\right\} = Z \ cdot E \ cdot \ vec \left\{E\right\}$
The electric field E drift of a potential V , which is written with a dimension:
$E = - \ frac \left\{\ partial V\right\} \left\{\ partial X\right\}$
thus the law of Nernst-Einstein becomes:
$v = - \ frac \left\{partial Dze\right\} \left\{kT\right\} \ cdot \ frac \left\{\ V\right\} \left\{\ partial X\right\}$

Let us consider the flow of loads jel , also called density of electric current. There is

jel = Z · E · J = Z · E · C · v
being C concentration of charge carriers, and then
$j_ \left\{el\right\} = \ frac \left\{partial Dz^2e^2c\right\} \left\{kT\right\} \ cdot \ frac \left\{\ V\right\} \left\{\ partial X\right\}$

One can make a parallel with the Loi of Ohm connecting this density of electric current jel to the gradient of potential:

$j_ \left\{el\right\} = - \ partial sigma \ cdot \ frac \left\{\ V\right\} \left\{\ partial X\right\}$
σ being the electric Conductivité, which gives us
$\ sigma = \ frac \left\{Dz^2e^2c\right\} \left\{kT\right\}$

## See too

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