Nonsteady-state diffusion

The normal state of affairs during a diffusion experiment is one in which the concentration at any point in the solid changes over time, as in the example of tracer diffusion described in Section 7.1. This situation is called nonsteady-state diffusion, and diffusion coefficients are found by solving the 

Initial concentration Cq Thin-film planar sandwich 

Open planar thin film Small spherical precipitate 

Initial concentration, cq, maintained constant Open plate 

In these examples, the initial concentration of the tracer is hxed and the amount remaining as the diffusion progresses will diminish over the course of the experiment. An important case of nonsteady-state diffusion occurs when the initial concentration at the surface is maintained as a constant throughout the experiment. This happens when gas molecules diffuse into a solid and the gas supply is constantly replenished. The solution for diffusion into a plate 

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The Laplace transfonnation method of solving nonsteady-state diffusion problems was briefly treated in Chapter 4. Thus, one can study all sorts of problems by using various types of current or potential stimuli (as in researches using transients see Section 7.7) and analyzing how transport in solution influences the response of the system. For example, a sinusoidally varying current, density can be used with... 

The treatment of nonsteady-state diffusion is a question of solving Fick s second law of diffusion. In many cases, however, the equations can be taken from the treatments of the analogous problems in heat flow in solids. The point is that heat flow and diffusion are described by mathematically similar methods. 

In this paper, solutions to three important heterogeneous diffusion problems are presented, and their implications for transport in biological systems are discussed. While the detailed methods of solutions and subtleties are presented in other papers (2-6), the asymptotic solutions are easily described, and they define the important physics of diffusion for most of the ranges of interest. In particular, a) nonsteady-state diffusion through oil-water multilaminates (2,3) b) desorption from oil-water multilaminates (4) and... 

Nonsteady-state diffusion through oil-water multilaminates has been used extensively as a model for the optimal biological response of a series of congeners with respect to partition coefficient (3,7, 8). Actual solution of this model reveals the deficiencies of multilaminates as a model for biological transport, but it does show the extraordinary separation factors of these multilaminates in the nonsteady-state regime. 

Pick s first law represents steady-state diffusion. The concentration profile (the concentration as a function of location) is assumed constant with respect to time. In general, however, concentration profiles do change with time. In order to describe these nonsteady-state diffusion processes use is made of Pick s second law, which is derived from the first law by combining it with the continuity equation (9n,/9t =... 

The first law of Pick tells one how the concentration gradient is related to the flux under steady-state conditions it says nothing about how the system goes from nonequilibrium to steady state when a diffusion source or sink is set up inside or at the boundary of the system. Thus, it says nothing about how the concentration changes with time at different distances from the source or sink. In other words. Pick s first law is inapplicable to nonsteady-state diffusion. For this, one has to go to Pick s second law... 

The kinetics of sorption of a penetrant into a polymer film of thickness i serves to illustrate problems of nonsteady-state diffusion. At t < 0, C = 0 for all x whereas, at t > 0, the surfaces at x = 0 and i assume the value C, which after enough time is the value achieved for all x, that is, equilibrium. If M. denotes the total amount of permeant that has entered the film at time t, and M , denotes the amount at equilibrium = A A Cg, then the solution to Equation 8 for these boundary conditions is... 

Figure 7.5 Common geometries for nonsteady-state diffusion (a) thin-film planar sandwich (b) open planar thin film (c) small spherical precipitate (d) open plate and (e) sandwich plate. In parts (a)-(c) the concentration of the diffusant is unreplenished in parts (d) and (e) the concentration of diffusant is maintained at a constant value, Co, by gas or liquid flow...

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. 

Nonsteady-state diffusion is characterised by the fact that the concentration at ... 

Nonsteady-State Diffusion Process of Reactant when the Electrolyte Solution at the Static State 48... 

In convective systems, the situation is different, because the thickness of the nonsteady-state diffusion layer, is limited by that of the steady-state diffusion layer 5, which depends only on the prevailing hydrodynamic conditions (see Sect. 4.3). As the non-steady-state diffusion layer grows, convection contributes increasingly to the transport of the reacting species. Eventually, when becomes comparable to the steady-state diffusion layer thickness 5 the concentration profile no longer changes with time. 

Tutorial Video Steady-State and Nonsteady-State Diffusion... 

Pick s second law— diffusion equation for nonsteady-state diffusion (in one direction)... 

Figure 5.5 Concentration profiles for nonsteady-state diffusion taken at three different times, fj, t2, and fj.

How do I Decide Which Equation to Use for a Specific Nonsteady-State Diffusion Situation ... 

Becanse this is a nonsteady-state diffusion problem in which the surface composition is held constant, Eqnation 5.5 is nsed. Valnes for all the parameters in this expression except time t are specified in the problem as follows ... 

Because this is a nonsteady-state diffusion situation, let us first employ Equation 5.5, nsing the following values for the concentration parameters ... 

Diffusion in semiconducting materials was discussed in Section 5.6. For both predeposition and drive-in treatments, diffusion is nonsteady-state—solutions to Pick s second law were provided for both. Nonsteady-state diffusion and these treatments are two of the processing components for silicon, as noted in the following concept map ... 

SS For a nonsteady-state diffusion situation (constant surface composition) in which the surface and initial compositions are provided, as well as the value of the diffusion coefficient, develop a spreadsheet that allows the user to determine the diffusion time required to achieve a given composition at some specified distance from the surface of the solid. 

Pick s second law. The time rate of change of concentration is proportional to the second derivative of concentration. This relationship is used in nonsteady-state diffusion situations. 

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