Currents of diffusing particles: Diffusion, drift and thermodiffusion.

Diffusion processes always aim at uniformity. A diffusion current of particles tends to make an inhomogeneous particle density $n(\vec r)$ uniform. This is expressed by Fick's law for the density of diffusion current

$$\vec j (\vec r)\vert _{\rm {Diff}} = - D \vec\nabla n (\vec r),$$ (14)

where $D$ is the diffusion coefficient. An inhomogeneous distribution of temperature is rendered uniform through a heat current, corresponding to Fourier's law

$$\vec q = -\lambda \vec\nabla T$$ (15)

for the heat current density $\vec q$ in terms of temperature gradient and thermal conductivity $\lambda$. In this simplified description of diffusion processes, however, thermodiffusion does not occur!

The plausible expressions (14) and (15) are incomplete: The first is valid only at constant temperature, the second only at constant density. For the general case ''cross terms'' need to be added: a term proportional to $(-\vec\nabla T)$ in (14), and one proportional to $(-\vec\nabla n)$ in (15). The additional term for the particle current describes the effect of thermodiffusion. A derivation of this term is given in the following section (eqn. (24)). (Mathematically speaking, the diffusion coefficient $D$ in (14) and the thermal conductivity $\lambda$ in (15) are only the diagonal elements of a $2 \times 2$ matrix of transport coefficients. The off-diagonal elements of this matrix provide the ''cross terms'' in the general expressions for particle and heat current. The fact that the off-diagonal transport coefficients are not necessarily positive, in contrast to the diagonal coefficients $D$ and $\lambda$, is an indication that the corresponding terms for particle and heat current are not so immediately obvious as the main terms (14) and (15).)

Yet another complication needs to be taken into account if the diffusing particles are electrically charged like the conduction electrons. For example, accumulated electrons are driven apart by their strong Coulomb repulsion, which is not taken into account in expression (14). Such an effect is described by an additional ''drift current'' proportional to the local electric field $\vec E(\vec r)$, which is given by

$$\vec j (\vec r)\vert _{\rm {Drift}} = l_{11} \vec E (\vec r)\, .$$ (16)

This expression contains the transport coefficient $l_{11}$ related to the electrical conductivity $\sigma$ by

$$l_{11} = - \sigma/e \, .$$ (17)

The total electron current is the sum of the diffusion and the drift current. As follows from the kinetic theory for the conduction electrons (see, e.g., Ref [3]), this total current is given by the expression

$$\vec j (\vec r)\vert _{\rm {Diff} + \rm {Drift}} = -l_{11} \vec\nabla \varphi_{e-ch} (\vec r)$$ (18)

which obtains from eqn. (16) if the gradient of the purely electric potential is replaced by that of the electrochemical potential (8).