Steady-state diffusion in solids

Your objectives in studying this section are to be able to  

Describe the mechanisms of diffusion through polymeric, crystalline, and porous solids. 

Calculate the mean free path for gases and the Knudsen number for gas flow through porous solids. 

Calculate effective molecular and Knudsen diffusion coefficients in porous solids. 

Calculate fluxes through porous solids when both molecular and Knudsen diffusion are important. 

---

In connection with their work on gas-solid reactions Hills and coworkers [48] and Evans [3] have described novel techniques based on the Wicke-Kallenbach apparatus. The former used hollow spheres instead of the conventional diffusion cell the latter followed the quasi-steady state diffusion in a Wicke-Kallenbach type diffusion cell (but with gas entrance and exit lines closed) by observation of the movement of a mercury plug in a horizontal glass tube connected across the cell. 

FIG. 24 Steady-state diffusion-limited current for the reduction of oxygen in water at an UME approaching a water-DCE (O) and a water-NB (A) interface. The solid lines are the characteristics predicted theoretically for no interfacial kinetic barrier to transfer and for y = 1.2, Aj = 5.5 (top solid curve) or y = 0.58, = 3.8 (bottom solid curve). The lower and upper dashed lines denote the... 

FIG. 28 Normalized steady-state diffusion-limited current vs. UME-interface separation for the reduction of oxygen at an UME approaching an air-water interface with 1-octadecanol monolayer coverage (O)- From top to bottom, the curves correspond to an uncompressed monolayer and surface pressures of 5, 10, 20, 30, 40, and 50 mN m . The solid lines represent the theoretical behavior for reversible transfer in an aerated atmosphere, with zero-order rate constants for oxygen transfer from air to water, h / Q mol cm s of 6.7, 3.7, 3.3, 2.5, 1.8, 1.7, and 1.3. (Reprinted from Ref. 19. Copyright 1998 American Chemical Society.)... 

The normal state of affairs during a diffusion experiment is one in which the concentration at any point in the solid changes over time. This situation is called non-steady-state diffusion, and diffusion coefficients are found by solving the diffusion equation [Eq. (S5.2)] ... 

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

Figure 3-3 Steady-state diffusion profile in (a) one dimension with concentrations at the two ends fixed, (b) a solid sphere with constant concentration on the surface (at 5), and (c) a spherical shell (radius from 1 to 5) with concentrations at the two surfaces fixed.

In unbuffered systems, kinetic limitation may come either from slow transport to the cell surface or from slow dissolution of solid species resulting in depletion of the bulk medium. Simple calculations show (Table 1) that, in the open ocean, steady state diffusion of such elements as zinc of iron to the cell surface may match the uptake rate of fast growing algae. This is in accordance with the postulate that, in a stable environment, all potentially limiting elements should effectively be co-limiting primary production and that the average uptake rates should match the diffusion rates (Morel and Hudson, 1985). 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

If in the neighbourhood of the solid particle the melt flows with such intensity that steady-state diffusion through a boundary layer of constant thickness takes place and the melt is in great excess, the particle radius a will decrease at a constant rate ... 

Following [270], we first consider steady-state diffusion to the surface of a solid spherical particle in a translational Stokes flow (Re - 0) at high Peclet numbers. In the dimensionless variables, the mathematical statement of the corresponding problem for the concentration distribution is given by Eq. (4.4.3) with the boundary conditions (4.4.4) and (4.4.5), where the stream function is determined by (4.4.2). 

This is the simplest model of an electrocatalyst system where the single energy dissipation is caused by the ohmic drop of the electrolyte, with no influence of the charge transfer in the electrochemical reaction. Thus, fast electrochemical reactions occur at current densities that are far from the limiting current density. The partial differential equation governing the potential distribution in the solution can be derived from the Laplace Equation 13.5. This equation also governs the conduction of heat in solids, steady-state diffusion, and electrostatic fields. The electric potential immediately adjacent to the electrocatalyst is modeled as a constant potential surface, and the current density is proportional to its gradient ... 

Steady-state difiusion differs from nonsteady-state diffusion in that the concentration of the diffusing atoms at any point, x, and hence the concentration gradient at x, in the solid, remains constant. Steady-state diffusion may be achieved when air diffuses through plastic food wrapping film. 

For simplification of the treatment, we may adopt the treatment described by Gough and Leypoldt and treat the electrode particles as a very thin equivalent electrode layer covered by an equivalent solid electrolyte membrane with a thickness of 5m, as shown in Figure 2.12(B). At the steady state of the reaction, the concentration of oxidant on the thin electrode layer surface is defined as Cq, the concentration at the interface of the equivalent electrode membrane/electrolyte solution at the membrane side is defined as Cm, and that at the solution side is defined as Co, respectively. The oxidant diffusion coefficient in the electrolyte solution is defined as Do and in the membrane as Dm, respectively. Note that the zero point of the x-axis is at the interface of the equivalent electrode membrane/electrolyte solution. In such an electrode configuration, the obtained steady-state diffusion current density was obtained as Eqn (2.81) ... 

It is also possible that the reaction is so rapid that it takes place at the interface only. In section 5.4.2.1 this was called an instantaneous reaction. However, eq. (5.43) describing similar reactions in fluid phases is not applicable here, since there is no flow and consequently no "diffusion layer". Only a non-steady state condition is conceivable. When the reaction product remains "dissolved" in the solid phases, the concentration profiles of the reactants in both phases can be described by the well known Fourier equations, that apply for non-steady state diffusion (with constant diffusivity) for given boundary conditions. For relatively low degrees of conversion die rate of conversion of reactant A can be approximated by the penetration theory ... 

The charge transport equation in an electrolyte with solid or stationary immobilized liquid electrolyte can be derived on the basis of charge balance and assuming steady-state diffusion of charge particles based on ohm s law as... 

Diffusion zone in the solid allows the conponent F to migrate toward surface. As the grain F is emptied and although the zone of diffusion keeps constant dimensions, we cannot make the assumption of steady state diffusion since the amount of F decreases with time at aity point and inparticnlar inthe heart. 

Diffusion of gases and vapors through solid, nonporous polymers is a three-step process. In the first step, the gas has to dissolve in the polymer at the high-partial-pressure side. Then, it has to diffuse as a solute to the low-partial-pressure side. In the third step, the solute evaporates back to the gas phase. Thus, if we consider steady-state diffusion through a membrane of thickness L exposed to a partial pressure difference Ap, the mass flux through the membrane will be given by Eq. (13.2.1) as follows ... 

Figure Bl.14.9. Imaging pulse sequence including flow and/or diflfiision encoding. Gradient pulses before and after the inversion pulse are supplemented in any of the spatial dimensions of the standard spin-echo imaging sequence. Motion weighting is achieved by switching a strong gradient pulse pair G, (see solid black line). The steady-state distribution of flow (coherent motion) as well as diffusion (spatially...

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

---