Linear Fickian

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

The linear Fickian diffusion equation (9.2-3) has the form (written in nondimensional form)... 

Upon reviewing a large amount of data, it became possible to relate fluid sorption in polymers and polymeric composites by means of six schematic curves relating weight gain to y/i as sketched in Fig. 4.2 below. The scales marked in that figure apply to the linear Fickian (LF) plot alone. 

Fig. 4.2 Schematic curves representing four categories of recorded non-fickian weight-gain sorption. The solid line, designated by LF, corresponds to linear Fickian diffusion...

Unidirectionally reinforced [0°4]AS4/3501-6 graphite/epoxy coupons were immersed in simulated sea water at room temperature within pressure chambers at 13.8 and 20.7 MPa for 2 months and weight gain data recorded periodically up to saturation (Gao and Weitsman 1998). A third set of specimens was immersed at atmospheric pressure 0.1 MPa. Each circumstance involved at least four replicate samples. All weight data could be matched with linear Fickian predictions. Results are summarized in Table 5.1. 

If De crr 1, non-Fickian behavior may appear, ie, a dependence other than linearity with the square root of time may be observed. 

This solution is valid for the initially linear portion of the sorption (or desorption) curve when MtIM is plotted against the square root of time. These equations also demonstrate that for Fickian processes the sorption time scales with the square of the dimension. Thus, to confirm Fickian diffusion rigorously, a plot of MJM vs. Vt/T should be made for samples of different thicknesses a single master curve should be obtained. If the data for samples of different thicknesses do not overlap despite transport exponents of 0.5, the transport is designated pseudo-Fickian. ... 

THE BERENS-HQPFENBERG MODEL. The Berens and Hopfenberg model considers the sorption process in glassy polymers as a linear superposition of independent contributions of a rapid Fickian diffusion into pre-existing holes or vacancies (adsorption) and a slower relaxation of the polymeric network (swelling).(lS) The total amount of sorption per unit weight of polymer may be expressed as... 

Rate equations There are two basic types of kinetic rate expressions. The first and simpler is the case of linear diffusion equations or linear driving forces (LDF) and the second and more rigorous is the case of classic Fickian differential equations. 

The computed mole fractions as a function of distance along the tube are shown in Fig. 12.14. The mole fractions exhibit a nearly linear drop between their equilibrium values at the surface and zero at the top of the tube. This behavior is not unexpected for this simple system, in which the very dilute species diffuse into one dominant species (air) that is present in great excess. In such a case we expect (and observe) Fickian behavior. That is, we could solve this problem using one of the mixture-averaged formulations discussed in Section 12.7.4 with very little error. 

When the fractional drug release from an initially dehydrated hydrogel sheet is plotted as a function of square root of time as shown in Figure 1 for thiamine HC1 release from a poly(2-hydroxyethyl methacrylate) sheet, linearity in the plot is observed only at large times. This illustrates the non-Fickian and time-dependent nature... 

Non-Fickian diffusion, 696 Non-functional structural groups, 129 Non-linear optics, 347 Non-Newtonian viscosity, 554 Non-redox doping, 345 Non-relaxed dielectric constant, 325 Normal... 

Laatikainen and Lindstrom [100] used TSM devices to investigate absorption in cellulose acetate and poly-(hexamethylene adipamide). In addition to measuring absorption isotherms and partition coefficients, they reported on transient responses to changes in methanol concentration for a cellulose-acetate-coated TSM device (Figure 4.10). At low concentrations, the linear response with Vr is consistent with Fickian behavior, and diffusion coefficients can be evaluated (Z> =... 

Many homogeneous reactions occur in the liquid phase, but consume reactants that must be supplied by mass transfer from a gas phase (or occasionally from another liquid phase). This is a typical problem of reaction engineering and is treated in some detail in most modem texts of that field [1,3,4,9,16,17]. Customarily, a power law is assumed for the rate of the chemical reaction and is then combined with a standard linear-driving force or Fickian diffusion treatment of mass transfer. A mass-transfer limitation lowers the rate, which in some extreme situations can become entirely mass transfer-controlled. Certain types of multistep reactions, however, can produce a totally different and very interesting behavior that may involve instability. 

The linear dependence of uptake on time results from the existence of a cleaily dis-cernable propagating solvent-induced swelling front which moves thrmi the pconstant velocity. The polymer ahead of the front is largely free of penetrant, while the polymer behind the front has essentially reached its equilibrium swelling value corresponding to the temperature and pressure of the experiment. For Fickian... 

One criterion of Fickian behavior holds that Mt/Moo vs. ti /2 plot should be linear up to 60% reduced sorption. 

The most rigorous formulation to describe adsorbate transport inside the adsorbent particle is the chemical potential driving force model. A special case of this model for an isothermal adsorption system is the Fickian diffusion (FD), model which is frequently used to estimate an effective diffusivity for adsorption of component i (D,) from experimental uptake data for pure gases.The FD model, however, is not generally used for process design because of mathematical complexity. A simpler analytical model called linear driving force (LDF) model is often used. ° According to this model, the rate of adsorption of component i of a gas mixture... 

Mechanical dispersion is assumed to mathematically follow a Fickian diffusion formulation, i.e., the mechanical dispersive flux is assumed to be linearly proportional to the concentration gradient. As such the hydrodynamic dispersion is the sum of diffusive and mechanical dispersive terms, and the total dispersive flux is written as ... 

Fickian Diffusion and Linear Driving Force models are generally used to describe the transport of water vapour into the alumina particles. For isothermal adsorption of water vapour from a constant partial pressure (P ) batch adsorption system on a spherical adsorbent particle of radius Rp, the uptake profiles are given by [13] ... 

Due to this difficulty it is preferable to transform the Fickian diffusion problem in which the mass-flux vector, js, is expressed in terms of the driving force, ds, into the corresponding Maxwell-Stefan form where is given as a linear function of jg. The key idea behind this procedure is that one intends to rewrite the Fickian diffusion problem in terms of an alternative set of diffusivities (i.e., preferably the known binary diffusivities) which are less concentration dependent than the Fickian diffusivities. 

If the penetrant enters the glassy matrix faster than the polymer can adapt itself by volume relaxation, the solvent front advances linearly with time. This behaviour is called case-II diffusion or relaxation-controlled diffusion. It is a special case of anomalous diffusion, where the mean square particle displacement is proportional to t. It commonly applies to polymers in the glassy state [Wei2]. Here the system 1,4-dioxane/PVC is an example (Fig. 10.1.8(b)). Due to the softening of the material behind the diffusion front, the polymer relaxation in the already swollen matrix is fast enough to adapt to a new situation created by further solvent uptake. Therefore, solvent ingress as well as swelling behind the diffusion front is Fickian. 

To obtain diffusion coefficients, the following equation can be applied to the linear portion of the normalized sorption/desorption data assuming Fickian behavior with constant diffusion coefficient (Crank, 1975) ... 

The sorption/desorption experiments were carried out as a function of a both above and below the Tg of the matrix. The data obtained from both experiments closely resembled Fickian diffusion. Below 0.55a (between 0 < Mt and Mf < 0.5) and above 0.55a (between 0.08 < Mf and Mf < 0.75) initial slope of Mf/Mf vs. curves were linear with respect to abscissa (R > 0.98). Diffusion coefficients were obtained using the linear portion of the normalized moisture sorption (Mt/Moo vs. curves from Equation 46.14. 

Then we see that the shear strain rate e j corresponds to the flux through a plane in Fickian diflusion, and the linear shortening rate e corresponds to the gain or loss term dQ/dt per unit volume. 

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