Non-steady diffusion

Here is an example solving a non-steady diffusion equation using the Laplace transform and the inverse Laplace transform. According to Fick s second law, the diffusion equation can be expressed as... 

The non-steady diffusion of surfactant ions is a problem similar to the non-steady diffusion of non-ionic surfactant, which was described in Chapter 4. There is a specific distinction caused by the electrostatic retardation effect. The non-steady transport of ionic surfactants to the adsorption layer is a two-step process, consisting of the diffusion outside and inside the DL. 

The concentration c(K ,t) in Eq. (7.40) has to be expressed in terms of T(t).The general solution for molecular adsorption kinetics was derived by Ward Tordai (1946) in form of Eq. (4.1). This equation can be used for of ionic adsorption too (Miller et al. 1994). It represents the solution of the non-steady diffusion problem, given by the differential equation... 

Diffusional contributions in protein adsorption are described by equation of non-steady diffusion ... 

The scan rate, u = EIAt, plays a very important role in sweep voltannnetry as it defines the time scale of the experiment and is typically in the range 5 mV s to 100 V s for nonnal macroelectrodes, although sweep rates of 10 V s are possible with microelectrodes (see later). The short time scales in which the experiments are carried out are the cause for the prevalence of non-steady-state diflfiision and the peak-shaped response. Wlien the scan rate is slow enough to maintain steady-state diflfiision, the concentration profiles with time are linear within the Nemst diflfiision layer which is fixed by natural convection, and the current-potential response reaches a plateau steady-state current. On reducing the time scale, the diflfiision layer caimot relax to its equilibrium state, the diffusion layer is thiimer and hence the currents in the non-steady-state will be higher. 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

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)] ... 

Figure S5.2 Common geometries for non-steady-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 the diffusant is maintained at a constant value, c0, by gas or liquid flow.

This problem illustrates the solution approach to a one-dimensional, non-steady-state, diffusional problem, as demonstrated in the simulation examples, DRY and BNZDYN. The system is represented in Fig. 4.2. Water diffuses through a porous solid, to the surface, where it evaporates into the atmosphere. 

Abstract. Auto-accelerated polymerization is known to occur in viscous reaction media ("gel-effect") and also when the polymer precipitates as it forms. It is generally assumed that the cause of auto-acceleration is the arising of non-steady-state kinetics created by a diffusion controlled termination step. Recent work has shown that the polymerization of acrylic acid in bulk and in solution proceeds under steady or auto-accelered conditions irrespective of the precipitation of the polymer. On the other hand, a close correlation is established between auto-acceleration and the type of H-bonded molecular association involving acrylic acid in the system. On the basis of numerous data it is concluded that auto-acceleration is determined by the formation of an oriented monomer-polymer association complex which favors an ultra-fast propagation process. Similar conclusions are derived for the polymerization of methacrylic acid and acrylonitrile based on studies of polymerization kinetics in bulk and in solution and on evidence of molecular associations. In the case of acrylonitrile a dipole-dipole complex involving the nitrile groups is assumed to be responsible for the observed auto-acceleration. 

Equation (5.1) is extremely useful to evaluate the flux whenever the concentration gradient can be considered constant with time (i.e., when we can assume a steady state). When a steady state cannot be assumed, the concentration change with time must also be considered. The non-steady-state diffusion is expressed by Pick s second law of diffusion ... 

When the fast reactions occurring in the system have stoichiometries different from the simple one shown by Eq. (5.78), analytical solutions of the diffusion equations are difficult to obtain. Nevertheless, numerical solutions can be obtained by iterative routines, and the results are conceptually similar to those described. The additional complications introduced by non-steady-state diffusion and nonlinear concentration gradients can be similarly handled. 

This technique is based on a non-steady-state approach to diffusion. The non-steady-state diffusion is not treated in our simplified approach to extraction kinetics the interested reader is referred to Yagodin and Tarasov [24] and Tarasov and Yagodin [25]. 

As the electrolysis proceeds, there is a progressive depletion of the Ox species at the interface of the test electrode (cathode). The depletion extends farther and farther away into the solution as the electrolysis proceeds. Thus, during this non-steady-state electrolysis, the concentration of the reactant Ox is a function of the distance x from the electrode (cathode) and the time f, [Ox] = Concurrently, concentration of the reaction product Red increases with time. For simplicity, the concentrations will be used instead of activities. Weber (19) and Sand (20) solved the differential equation expressing Pick s diffusion law (see Chapter 18) and obtained a function expressing the variation of the concentration of reactant Ox and product Red on switching on a constant current. Figure 6.10 shows this variation for the reactant. 

The solutions of a diffusion equation under the transient case (non-steady state) are often some special functions. The values of these functions, much like the exponential function or the trigonometric functions, cannot be calculated simply with a piece of paper and a pencil, not even with a calculator, but have to be calculated with a simple computer program (such as a spreadsheet program, but see later comments for practical help). Nevertheless, the values of these functions have been tabulated, and are now easily available with a spreadsheet program. The properties of these functions have been studied in great detail, again much like the exponential function and the trigonometric functions. One such function encountered often in one-dimensional diffusion problems is the error function, erf(z). The error function erf(z) is defined by... 

Pick s Second Law of Diffusion relates the change in concentration of a diffusing species with time to the diffusion coefficient and the concentration gradient for non-steady-state diffusion ... 

Law et al. [15] determined the diffusion coefficient for benzyl penicillin in thin films of Palacos, Simplex and CMW cements assuming that antibiotic transport can be described by Fick s law using a finite difference approximation to quantify transient non-steady-state behaviour. These investigators found that the diffusion coefficient was increased in the presence of additives and proposed that the finite difference approach could be applied to determine release of antibiotic from preloaded PMMA beads. Dittgen and Stahlkopf [16] showed that incorporation of amino acids of varying solubilities also affected release of chloramphenicol from polymethacrylic... 

Laplace transformation, 1215 Nemst s equation and. 1217 non-steady, 1254 as rate determining step, 1261 Schlieren method, 1235 semi-infinite linear, 1216, 1234, 1255 in solution and electrodeposition, 1335 spherical. 1216. 1239 time dependence of current under, 1224 Diffusion control, 1248... 

However, the important consequences of this analysis are that the complications of the reduction in the density of B between A reactants only develops during the decay of the non-steady-state density of B towards the steady state. The average concentration of B after reaction has begun is less than the initial (or bulk) value [B]0 usually used in the Smoluchowski theory. The diffusion of B towards A is driven by the larger concentration of B at considerable distances from A than the concentration of B nearer to A. The concentration or density gradient of B at A is decreased slightly by this competition between different A reactants for B. Hence, the current B towards one A remains almost as it was in the Smoluchowski theory [eqn. (18)], viz. 

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