Knudsen

MetallorganicMBE (MOMBE). tire solid source Knudsen cells in conventional MBE are replaced witli gaseous beams of organometallic precursors, directed toward a heated substrate in UHV. Compared to MOCVD, MOMBE eliminates gas phase reactions tliat may complicate tire deposition surface reactions, and provides lower growtli temperatures. 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

Strictly speaking, this expression is correct for a semi-infinite region bounded by a plane wall and containing a gas at rest. Here it is applied to a bounded region surrounded by a curved wall, and the molecules have a drift velocity parallel to che wall. Knudsen was concerned that this drift velocity might invalidate the treatment, but Pollard and Present [8] showed Chat this is not che case. 

Note the use of a script for the binary pair mutual diffusion coefficient, as distinct from the Roman D already used to represent Knudsen diffusion coefficients. This convention will be adhered to throughout. 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

The complete problem with composition gradients as well as a pressure gradient, may be regarded as a "generalized Poiseuille problem", and its Solution would be valuable for comparison with the limiting form of the dusty gas model for small dust concentrations. Indeed, it is the "large diameter" counterpart of the Knudsen solution in tubes of small diameter. 

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The solute species therefore diffuse independently, rather as in Knudsen diffusion, but with effective diffusion coefficients D, where... 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

The Knudsen diffusion coefficients are given by equations (2.11) in which K. is independent of pressure and proportional to pore diameter a, so that we can write... 

It may seem curious that Knudsen diffusion coefficients still appear in equations (5.18) and (5.19), which supposedly give the flux relations at the limit of bulk diffusion control. However, inspection reveals that only ratios of these coefficients are effectively present, and from equation (2,11) it follows that... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

The first case corresponds to a situation in which all Knudsen diffusion coefficients are equal, and all binary pair bulk diffusion coefficients are equal ... 

It ls not surprising chat such a relation should hold at the Limit of Knudsen diffusion, since Che Knudsen diffusion coefficients are themselves inversely proportional to the square roots of molecular weights, but the pore diameters in Graham s stucco plugs were certainly many times larger chan the gaseous mean free path lengths at the experimental conditions. 

To appreciate the questions raised by Knudsen s results, consider first the relation between molar flow and pressure gradient for a pure gas flowing through a porous plug, rather than a capillary. The form predicted by the dusty gas model can be obtained by setting = 1, grad = 0 in equation... 

Though by no means a complete theory, this is at least a reasonable explanation of the Knudsen minimum, and it then remains to explain why the minimum is not observed for flow through porous media. Pollard and Present attributed this to the limited length/diameter ratio of the channels in a typical porous medium and gave a plausible argument in favor of this view. 

Knudsen diffusion, but the dependence on physical and geometric conditions... 

Che Knudsen diffusion coefficient is independent of composition and pressure,... 

Remick and Geankoplis made flux measurements for both species in the isobaric diffusion of nitrogen and helium through their tube bundle. Pressures spanned the interval from 0.444 nim, Hg to 300,2 ram Hg, which should cover the whole range between the limits of Knudsen streaming and bulk diffusion control. Then, since K and K, are known in this case, the form of the proposed flux relations could be tested immediately by plotting the left hand side of equation (10.15) against... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

Knudsen diffusion coefficient for the test gas in a micropore. represents the total void fraction and c that part of of the void fraction... 

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