Diffusion coefficient relation

Note that the effective diffusion coefficient relating N and dc /dz in... 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

To calculate the mixture-averaged diffusion coefficient relating the mass flux with respect to the mass-average velocity V in terms of the mass fraction gradient, write an expression analogous to Eq. 12.158 ... 

Second, the definition of DM, the model diffusion coefficient, relates Ax, At, and AD ... 

Inasmuch as we have seen that the basic diffusion equations lead to a Gaussian profile, we may in most cases describe Gaussian spreading as an apparent diffusion process, with an effective diffusion coefficient related to [Pg.93]

Powerful methods for the determination of diffusion coefficients relate to the use of tracers, typically radioactive isotopes. A diffusion profile and/or time dependence of the isotope concentration near a gas/solid, liq-uid/solid, or solid/solid interface, can be analyzed using an appropriate solution of - Fick s laws for given boundary conditions [i-iii]. These methods require, however, complex analytic equipment. Also, the calculation of self-diffusion coefficients from the tracer diffusion coefficients makes it necessary to postulate the so-called correlation factors, accounting for nonrandom migration of isotope particles. The correlation factors are known for a limited number of lattices, whilst their calculation requires exact knowledge on the microscopic diffusion mechanisms. 

Even the binary system diffusivities in liquid mixtures are composition dependent. Therefore, in multicomponent liquid mixtures with n components, predictions of the diffusion coefficients relating flows to concentration gradients are empirical. The diffusion coefficient of dilute species i in a multicomponent liquid mixture, Dim, may be estimated by Perkins and Geankoplis equation... 

A fully microscopic theory of chemical diffusion can be constructed, however, it requires a careful distinction between the motions of the observed species and the underlying host, and is made complicated by the fact that, as defined, the diffusion coefficient relates flux to the concentration gradient while the actual force that drives diffusion is gradient of the chemical potential. An alternative useful observable is the so-called conductivity diffusion coefficient, which is defined for the motion of charged particles by the Nemst-Einstein equation (11.69)... 

For multicomponent liquid mixtures, it is usually very difficult to obtain numerical values of the diffusion coefficients relating fluxes to concentration gradients. One important case of multicomponent diffusion for which simplified empirical correlations have been proposed is when a dilute solute diffuses through a homogeneous solution of mixed solvents. Perkins and Geankoplis (1969) evaluated several methods and suggested... 

It was shown in Section 3.4 that if the bulk of a dilute binary electrolyte solution may be assumed electrically neutral, then the distribution of reduced ion concentration is governed by the same convective diffusion equation as for a neutral species with an effective diffusion coefficient related to the difference in charge and diffusion coefficients of the positive and negative ions. Once the concentration distribution has been found, the potential distribution in the solution can be obtained by integrating the equation for current continuity (Eq. 3.4.16) to give... 

Mutual diffusion coefficient related to translational mobility of the constituents... 

Self-diffusion coefficient D eif is the diffusion coefficient of the host atoms or ions in random diffusion, which is one of the important diffusion coefficients related to sintering. For lattice diffusion by the vacancy or interstitial mechanisms, Dseir is given by Eq. (5.50) or (5.51). 

In a dilute dispersion it can be assumed that the instantaneous particle velocity V is separable into two parts a rapidly-fluctuating Brownian component, and an interaction component with a fluctuation time of the order of the time for the particle to move a distance K With this assumption, the initial decay of g (K, t) is characterized by an effective, JiT-dependent diffusion coefficient related to the structure in the system ... 

Zw = Rdo (jS)for semi-infinite diffusion contribution and Zw = Rdo (jS) coth (jS) is related to diffusion through a film of thickness d, formed on the electrode, where Rdo is the diffusion resistance and the parameter S= d2(o/D, where d and D are the diffusion thickness and diffusion coefficient related to the transport process (Fedrizzi et al., 1992). 

Fifty years later, Curtiss and Bird [26] proposed that the multicomponent diffusion coefficient relations can be written in simpler form by using the quantities D r = ujiUjj/XiXj)Vsr, rather than T>sr from (2.305). The modified diffusion coefficients... 

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