**Generalized Coupling for the Near-Equilibrium Case**

**Onsager's Symmetry Relation**

**Seebeck and Peltier Effects**

**Diffusion of an Independent Species**

__One Independent Mobile Species: Highlights__

The gradient in chemical potential is the only driving force and the motion of that species is the only flux:

(04-1) |

The chemical potential can be related to local concentration through the activity coefficient :

(04-2) |

Therefore, can be related to :

(04-3) |

If this is compared to the most simple version of Fick's first law,

(04-4) |

The atomic mobility be defined by the the Einstien relation between the average drift velocity and the driving force,

(04-5) |

The flux is related to the average velocity by:

(04-6) |

Therefore

(04-7) |

and the relation between intrinsic diffusivity and mobility is

(04-8) |

If the solution is ideal--as in the case of mixture of radioisotopes of an otherwise identical atomic species--then the diffusivity is called the self-diffusivity and since the activity coefficient is constant:

(04-9) |

__Diffusion in a Crystal__

When substitutional diffusion takes place inside a crystal, the number of sites is conserved. The site conservation provides an additional constraint. The additional constraint, as will be shown below, leads to the consideration of the chemical potential of the vacancy that is switching places with a diffusing substitutional atom. In other words, change in energy associated with a site after one diffusive step has occured is --the chemical energy associated with the site after the jump minus the chemical energy before the jump.

Consider a system consisting of two mobile species and on a lattice with vacancies:

(04-10) |

The site fractions are given by

The volumes associated with each atom are

(04-12) |

Where which will be considered constant and therefore:

(04-13) |

is constant as well.

The relation between the change in entropy per site with the internal energy and work terms when the only additional work term is the chemical work per site is:

(04-14) |

or in terms of densities of extensive quantities:

(04-15) |

Because the sites are conserved (Eq. 4-11), :

(04-16) |

The expression for the entropy production is, for example including the heat flow term as well is:

(04-17) |

where is the flux of -occupied sites on a lattice.

__Self Diffusion of an Isotope in a Crystal__

The above analysis can be applied to isotopic diffusion which is traditionally used to obtain measurements of the self-difussion coefficient.

Let the isotope of be desginated by and while and may be assumed to be chemically identical, intrinsic mobilities may differ due to their mass differences.

Writing out the linearized flux relationships for this case:

(04-18) |

where the constant term has been absorbed into the Onsager coefficients.

If is can be *assumed* that the vacancies are everywhere in equilibrium,^{1}then
; if, futhermore, the vacancy concentration gradient is constant (which
is consistent with the assumption of equilibrium), so that the concentration
gradients

(04-19) |

or using the relation between Onsager coeffients and mobilities,

(04-20) |

which defines the self diffusivity on a lattice in terms of the mobilities of each isotope.

If the solution is not ideal, then by direct extension, the intrinsic diffusivity is:

(04-21) |