Diffusion force

The mass flux vector is also the sum of four components j (l), the mass flux due to a concentration gradient (ordinary diffusion) jYp), the mass flux associated with a gradient in the pressure (pressure diffusion) ji(F), the mass flux associated with differences in external forces (forced diffusion) and j,-(r), the mass flux due to a temperature gradient (the thermal diffusion effect or the Soret effect). The mass flux contributions may then be summarized ... 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

In the absence of pressure diffusion, forced diffusion, and thermal... 

The basic expressions for the mass fluxes and the equations of continuity for multi-component mixtures are given in Sec. II,B. For a -component mixture of ideal gases in a system in which there is no pressure diffusion, forced diffusion, or thermal diffusion, the fluxes are given by... 

This expression has been generalized by Curtiss and Hirschfelder (C12) to include thermal diffusion pressure diffusion, and forced diffusion [see reference (Hll, p. 487)]. 

Consider then a mixture of A and B which are being subjected to a uniform gravitational acceleration or centrifugal acceleration g in the — z direction. It should be noted that there is no mass flux due to forced diffusion according to Eq. (35) inasmuch as jt(F) vanishes when we replace t by —g. The effect of the gravitational field is to produce a pressure gradient, the latter being determined by Eq. (25), which for the case under consideration becomes ... 

Mass flux of A with respect to mass-average velocity (see Table III) Mass flux due to forced diffusion (32)... 

Forced Diffusion from Energy Minimization Calculations... 

Bell et al. (81) presented forced diffusion calculations of butene isomers in the zeolite DAF-1. DAF-1 (82) is a MeALPO comprising two different channel systems, both bounded by 12-rings. The first of these is unidimensional with periodic supercages, while the other is three-dimensional and linked by double 10-rings. The two channel systems are linked together by small 8-ring pores. It is a particularly useful catalyst for the isomerization of but-l-ene to isobutylene (S3) its activity and selectivity are greater than those of ferrierite, theta-1, or ZSM-5. 

The balance over the ith species (equation IV. 5) consists of contributions from diffusion, convection, and loss or production of the species in ng gas-phase reactions. The diffusion flux combines ordinary (concentration) and thermal diffusions according to the multicomponent diffusion equation (IV. 6) for an isobaric, ideal gas. Variations in the pressure induced by fluid mechanical forces are negligible in most CVD reactors therefore, pressure diffusion effects need not be considered. Forced diffusion of ions in an electrical field is important in plasma-enhanced CVD, as discussed by Hess and Graves (Chapter 8). 

No pressure diffusion, thermal diffusion, or forced diffusion (e.g., electrophoresis)... 

For pneumatic transport of solids in a dilute suspension, the effects of apparent mass, Basset force, diffusion, and electric charge of the particles may be ignored. Thus, the dynamic equation of a small particle in a gas medium is given by... 

Debye obtained his result by solving a forced diffusion equation Ci.e., with torque of the applied field included) for the distribution of dipole coordinate p - pcosS, with 6 the polar angle between the dipole axis and tSe field, and the same result for the model follows very simply from equation (3) using the time dependent distribution function in the absence of the field (5). The relaxation time is given by td = 1/2D, which for a molecular sphere of volume v rotating in fluid of viscosity n becomes... 

Figure 1.13 Diffusion bonding process chain. Starting top left stacking, diffusion bonding furnace with mechanical pressure force, diffusion bonding and a cut through a microchannel system after diffusion bonding.

Ordinary diffusion depends on the partial Gibbs free energy and the concentration gradient. The pressure diffusion is considerable only for a high-pressure gradient, such as centrifuge separation. The forced diffusion is mainly important in electrolytes and the local electric field strength. Each ionic substance may be under the influence of... 

Continuum diffusion (Kn 1). Hie different species of a mixture move relative to each other under the influence of concentration gradients (ordinary or concentration diffusion), temperature gradients (thermal diffusion) or external forces (forced diffusion). Here molecule-wall collisions are neglected. 

This model assumes the diffusive flows combine by the additivity of momentum transfer, whereas the diffusive and viscous flows combine by the additivity of the fluxes. To the knowledge of the authors there has never been given a sound argument for the latter assumption. It has been shown that the assumption may result in errors for certain situations [22]. Nonetheless, the model is widely used with reasonably satisfactory results for most situations. Temperature gradients (thermal diffusion) and external forces (forced diffusion) are also considered in the general version of the model. The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes can be added to the diffusion fluxes in the gaseous phase. 

A more general approach to the diffusion problem is needed. The essential concepts behind the development of general relationships regarding diffusion were given more than a century ago, by Maxwell [39] and Stefan [40]. The Maxwell-Stefan approach is an approximation of Boltzmann s equation that was developed for dilute gas mixtures. Thermal diffusion, pressure diffusion, and forced diffusion are all easily included in this theory. Krishna et al. [38] discussed the Maxwell-Stefan diffusion formulation and illustrated its superiority over the Pick s formulation with the aid of several examples. The MaxweU-Stefan formulation, which provides a useful tool for solving practical problems in intraparticle diffusion, is described in several textbooks and in numerous publications [7,41-44]. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

Sung, T. Doping Diamond by Forced Diffusion. Ph.D. thesis. University of Missouri, Columbia, May, 1996. 

This relation is referred to as the Maxwell-Stefan model equations, since Maxwell [65] [67] was the first to derive diffusion equations in a form analogous to (2.302) for dilute binary gas mixtures using kinetic theory arguments (i.e., Maxwell s seminal idea was that concentration gradients result from the friction between the molecules of different species, hence the proportionality coefficients, Csk, were interpreted as inverse friction or drag coefficients), and Stefan [92] [93] extended the approach to ternary dilute gas systems. It is emphasized that the original model equations were valid for ordinary diffusion only and did not include thermal, pressure, and forced diffusion. 

If we drop the pressure-, thermal-, and forced-diffusion terms, the binary diffusivities can be expressed as ... 

The basic concept of diffusion refers to the net transport of material within a single phase in the absence of mixing (by mechanical means or by convection). Both experiment and theory have shown that diffusion can result from pressure gradients (pressure diffusion), temperature gradients (thermal diffusion), external force fields (forced diffusion), and concentration gradients. Only the last type is considered in this book that is, the discussion is limited to diffusion caused by the concentration difference between two points in a stagnant solution. This process, called molecular diffusion, is described by Pick s laws. His first law relates the flux of a chemical to the concentration gradient ... 

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