An alternative procedure for modeling of Knudsen flow and surface diffusion

Abstract

An alternative procedure for the calculation of impingement rate distribution and simultaneously the transmission probability in pores under Knudsen diffusion conditions is introduced. It is based on a combination of the finite difference method and a projection approach. Pore entrance and exit effects, and the influence of the pore length on diffusive fluxes are investigated. Later on, it is applied for a simultaneous Knudsen and surface flow system. In the model, the equation system is built without the independent flow and adsorption-desorption equilibrium assumptions. For the conditions investigated, the results indicate that if the surface flow rate is substantial, the independent flow and adsorption equilibrium assumptions become improper estimates for the behaviour of the system. The surface and gas flow rates, the impingement rate distribution and the surface coverage behave much more complex than the characteristics found with such assumptions. KeywordsKnudsen flow · surface diffusion · equilibrium assumption · cylindrical pore · projection approach Acknowledgement One of the authors would like to thank to Mr. Erkan Aksoy and Mr. Denis Chaykin for their comments concerning mathematical calculations.

Connecting theory and simulation with experiment for the study of diffusion in nanoporous solids

“…well [256,[471][472][473]. Hesse [473] has demonstrated that pore lengths close to the average pore diameters lead to a higher transport resistance than obtained from the usual models, in agreement with Eq.…”

“…For gases, a unique nano-confinement phenomenon occurs that has been studied since the beginning of the 20th Century, and is called Knudsen diffusion [3,[256][257][258][259][260][261][262][263][264][265][266][267][268][269][270][271][272], after Martin Knudsen [273,274], who first described it as "molecular flow" (German: Molekularströmung). When the mean free path of gas molecules, , becomes larger or is of the same order of magnitude as the local channel diameter, d , the frequency of the collisions between the molecules and the walls exceeds the intermolecular collision frequency.…”

“…Knudsen assumed that the collisions with the wall are diffuse, rather than specular; that is, upon colliding with the walls, molecules adsorb for a short time, after which energy is redistributed over their various degrees of freedom, and partially exchanged with the walls, so that, when molecules desorb back into the pore space and continue their trajectory, the angle of reflection is independent of the angle of incidence and follows a cosine distribution-similar to Lambert's law of diffuse light reflection [3,[256][257][258][259][260][261][262][263][264][265][266][267][268][269][270][271][272]. Clausing [467] later showed that the cosine distribution is a direct result of the second Law of Thermodynamics, rather than requiring roughness.…”

“…Such "surface barriers" are much better recognized for configurational diffusion in zeolites, but they can also occur for Knudsen diffusion in mesopores. Argönül and Keil [256] have developed a model that describes surface diffusion and Knudsen diffusion without making the assumption of adsorption-desorption equilibrium, and it incorporates automatically the variation of flow rates with pore length, as well as entrance and exit effects. The model also makes use of the outer surface area of the solid to calculate the boundary conditions for the surface flow.…”

Progress report on high aspect ratio patterning for memory devices

“…20) Knudsen transport and surface diffusion can be viewed as separate, independent fluxes or as synergistic, interdependent modes of neutral transport. 20,21) Another benefit of the leaner chemistry used in lowtemperature etching is an improved mask morphology, which translates into improved circularity and sidewall roughness of the etched holes, as shown in Fig. 4.…”