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Diffusion in a Volume-Fixed Frame (V-Frame)
To find the volume-fixed V-frame, assume that a frame, designated an R-frame, exists that relates all local C-frames.
There is an equation of the velocity of a local C-frame with respect to the V-frame. It is the velocity of any inert marker with respect to the V-frame.
The interdiffusivity is designated by Dtilda and is related to the intrinsic diffusivities of components 1 and 2
Relate the V-frame to a laboratory frame suitable for experimental purposes. This is provided by the laboratory frame. The ends of the specimen are unaffected by the diffusion and are stationary with respect to each other since there is no change in the overall specimen volume.
The L-frame and V-frame are thus identical.
The measurement of vcv and Dtilda at the same concentration in a diffusion experiment thus produces two relationships involving D1 and D2 and allows their determination. In the V-frame, the diffusion flux of each component is given by a simple Fick's-law expression where the factor that multiplies the concentration gradient is the interdiffusivity D.
In the V-frame, chemical interdiffusion is described by a single diffusivity. In a local C-frame fixed with respect to the local bulk material, the material flows locally with the velocity vcv relative to the V-frame and the description of the fluxes of the two components requires two diffusivities.
The Kirkendall effect alters the structure of the diffusion zone in crystalline materials. The small supersaturation of vacancies on the side losing mass by fast diffusion causes the excess vacancies to precipitate out in the form of small voids, and the region becomes porous.
In 1946, the Kirkendall effect was observed with inert markers in polymer-solvent systems where the large polymer molecules diffused more slowly than the small solvent molecules.
If the osmotic membrane allows rapid diffusion of A but not B, the pressure PAleft and PAright will then relax to equilibrium values until there is no difference in chemical potential across the membrane. This results in a difference in total pressure across the membrane.
If the membrane becomes free to move, it would move to the left, compressing the left chamber and expanding the right to equilibrate the pressure difference. However, if the membrane is constrained, the fluid may cavitate in the left chamber to relieve the low pressure. This is analogous to the formation of voids in the Kirkendall effect.
3.1.4 Diffusion of Interstitial Particles in a Chemical Concentration Gradient
Another system obeying Fick's law is one involving the diffusion of small interstitial solut atoms (componen 1) among the interstitces of a host crystal in the presence of an interstitial-atom concentration gradient. The large solvent atoms (component 2) essentially remain in their substitutional sites and diffuse much more slowly than do the highly mobile solute atoms, which diffuse by the interstitial diffusion mechanism. The solvent atoms may therefore by considered to be immobile.
L11 can be evaluated by introducing the interstitial mobility M1, which is the average drift velocity, v1, gained by diffusing interstitials when a unit driving force is applied.
There is prediction of diffusive flux which depends linearly on the gradient concentration.
The Nernst-Einstein equation expresses a link between the mobility and the diffusivity
3.1.5 On the Algebraic Signs of Diffusivities
The rate of entropy production is nonnegative. M1 is also nonnegative and L11 must be nonnegative.
A negative diffusivity leads to an ill-posed diffusion equation; so formulations based on fluxes and their conjugate driving forces are preferred to Fick's law and are more physical
3.1.6 Summary of Diffusivities
Four different types of diffusivities include the self-diffusivity in a pure material, the self-diffusivity of solute i in a binary system, the intrinsic diffusivity of component i in a chemically inhomogeneous system, and the interdiffusivity in a chemically inhomogeneous system