Diffusion case I

In sorjDtion experiments, the weight of sorbed molecules scales as tire square root of tire time, K4 t) ai t if diffusion obeys Pick s second law. Such behaviour is called case I diffusion. For some polymer/penetrant systems, M(t) is proportional to t. This situation is named case II diffusion [, ]. In tliese systems, sorjDtion strongly changes tire mechanical properties of tire polymers and a sharjD front of penetrant advances in tire polymer at a constant speed (figure C2.1.18). Intennediate behaviours between case I and case II have also been found. The occurrence of one mode, or tire otlier, is related to tire time tire polymer matrix needs to accommodate tire stmctural changes induced by tire progression of tire penetrant. 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

Fig. 4 Time dependence of solvent absorption inside a water-swollen PNIPAAm-gel. The normalized NMR signal of a 0.5 mm thin layer at a distance of 2.78 mm to the solvent-sample interface measures the solvent concentration. The curves are separated by a stepwise offset. 1 D2O/H2O 2 20 vol-% CH3OD 3 40 vol-% CH3OD 4 60 vol-% CH3OD 5 80 vol-% CH3OD 6 40-vol % CH3OD, more cross-linked gel. The small vertical lines mark the end of time lag and the start of case I diffusion (if recognizable). Reprinted from (Kndrgen et al. 2000), p. 77. Copyright (2000), with permission from Elsevier...

If the polymer is above its glass transition tempetature, Tg, it responds rapidly to changes in its physical condition and we have Fickian or Case I diffusion. This is the simplest case, and for T>T, Henry s law is valid for sorption and the diff ion coefficient is a constant (ideal Fickian diffusion). Its temperature dependence is well approximated by a simple Arrhenius expression with a constant activation energy. 

A limiting case of non-Fickian response in which a = 1.0 is typically referred to as Case II diffusion to differentiate it from normal Fickian (Case I) diffusion. Case I (Fickian) diffusion shows a linear increase in sorption as a function of the square root of time. By contrast. Case II kinetics are characterized by linear mass uptake with time as shown in Figure 38, where uptake of n-pentane in polystyrene at high activity (penetrant partial pressure) shows a Fickian response in small spheres and a non-Fickian response in larger diameter spheres (143). The data strongly suggest that diffusion into the small spheres is so rapid that there is insufficient time to generate a Case II concentration profile. Apparently the diffusional equilibration in the small spheres is essentially complete before the complex step concentration profile associated with Case II sorption can be established. This behavior is similar to other observations (144). 

In the aforementioned step 1, the solvent diffuses into polymer networks (case-I diffusion). At this time, the solvent absorption behavior of the gel can be obtained by solving the diffusion equation (Fick s second law). In... 

It has been suggested by Crank (1975) that the kinetics of sorption of moisture in polymers is governed by two limiting cases. Case I (Fickian or diffusion controlled), in which moisture transport is a stochastic process driven by the presence of a concentration gradient. This case predominates in systems where the penetrant has little hygroelastic effect on the polymer or the rate of diffusion is much less than that of relaxation. In Case I diffusion, the weight uptake is initially linear with respect to the square root of time, that is, if the amount sorbed at time t is JCf , then n is equal to 0.5. 

Figure C2.1.18. Schematic representation of tire time dependence of tire concentration profile of a low-molecular-weight compound sorbed into a polymer for case I and case II diffusion. In botli diagrams, tire concentration profiles are calculated using a constant time increment starting from zero. The solvent concentration at tire surface of tire polymer, x = 0, is constant.

Blanc provided a simple limiting case for dilute component i diffusing in a stagnant medium (i.e., N 0), and the result, Eq. (5-205), is known as Blanc s law. The restriction basically means that the compositions of aU the components, besides component i, are relatively large and uniform. 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

We will first consider the simple case of diffusion of a non-electrolyte. The course of the diffusion (i.e. the dependence of the concentration of the diffusing substance on time and spatial coordinates) cannot be derived directly from Eq. (2.3.18) or Eq. (2.3.19) it is necessary to obtain a differential equation where the dependent variable is the concentration c while the time and the spatial coordinates are independent variables. The derivation is thus based on Eq. (2.2.10) or Eq. (2.2.5), where we set xj> = c and substitute from Eq. (2.3.18) or Eq. (2.3.19) for the fluxes. This yields Fick s second law (in fact, this is only a consequence of Fick s first law respecting the material balance—Eq. 2.2.10), which has the form of a partial differential equation... 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

The rate is independent of particle size. This is an indication of neghgible pore-diffusion resistance, as might be expected for either very porous particles or sufficiently small particles such that the diffusional path-length is very small. In either case, i -> 1, and ( rA)obs = ( rA)inl for the surface reaction. 

If the total current can be assumed to be limited by diffusion to the STM tip, Case III is similar to diffusion to a microdisk electrode (one electrode) thin-layer cell (63). Murray and coworkers (66) have shown that for long electrolysis times, diffusion to a planar microdisk electrode TLC can be treated as purely cylindrical diffusion, provided that the layer thickness is much smaller than the disk diameter (66). In contrast to the reversible case discussed above (Case I), the currents in this scenario should decrease gradually with time at a rate that is dependent on the tip radius and the thickness of the interelectrode gap. Thus, for sufficiently narrow tip/sample spacings, diffusion may be constrained sufficiently (ip decayed) at long electrolysis times to permit the imaging of surfaces with STM. 

For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v = 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator V2, also frequently denoted as A. Thus, for a case of diffusion and flow, equation (10) becomes ... 

As a first example, the transient case with Henry isotherm can be considered. Expressions developed in Section 2.3 apply with D replacing Dm,ct m replacing cM (including the substitution of c v M by < M and cfsM by c f ) and Ku (defined as r/cM(r0,t) in both cases, i.e. with or without the presence of L) by AT i / (1 + Kc ). Other cases with analytical solutions arise from the steady-state situation. The supply flux under semi-infinite steady-state diffusion is [57] ... 

The theory of linear differential equations indicates that long-term evolution depends on the boundary conditions and the determinant of the coefficients preceding the second spatial derivatives (which can actually be considered as effective diffusion coefficients). Such a system is likely to be highly non-linear. One extreme case, however, is particularly interesting in demonstrating how periodic patterns of precipitation can be arrived at. We assume that (i) species i diffuses very fast and dC /dp is large so that P is small and (ii) that species j is much less mobile and P is large. The... 

The kinetic problem for the intramolecular cross-linking reactions in general form was not yet solved. Only some particular cases, i.e. the cvclization of macromolecules, the intramolecular catalysis and diffusion-controlled collision of two reactive groups were studied theoretically bv Xorawetz, Sisido and Fixman... 

Equations 2.26 and 2.27 carmot be solved analytically except for a series of limiting cases considered by Bartlett and Pratt [147,192]. Since fine control of film thickness and organization can be achieved with LbL self-assembled enzyme polyelectrolyte multilayers, these different cases of the kinetic case-diagram for amperometric enzyme electrodes could be tested [147]. For the enzyme multilayer with entrapped mediator in the mediator-limited kinetics (enzyme-mediator reaction rate-determining step), two kinetic cases deserve consideration in this system in both cases I and II, there is no substrate dependence since the kinetics are mediator limited and the current is potential dependent, since the mediator concentration is potential dependent. Since diffusion is fast as compared to enzyme kinetics, mediator and substrate are both approximately at their bulk concentrations throughout the film in case I. The current is first order in both mediator and enzyme concentration and k, the enzyme reoxidation rate. It increases linearly with film thickness since there is no... 

A special case of the thin-film approximation considers that the substrate can diffuse within the film with negligible depletion. This corresponds to both Case V (unsaturated enzyme). Case I (saturated enzyme) and the border between them. A thin film is the only situation where we can fit experimental data to a theoretical calibration curve for the whole range of glucose concentrations with ... 

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