Fickian diffusion theory

In order to improve the predictive power of the Fickian diffusion theory, a concentration dependent diffusion coefficient is used in Eqs. (15) and (16). Equation (16) is then rewritten and solved with the appropriate boundary conditions ... 

For diffusion of liquid through rubbery polymer composites, Fickian and non-Fickian diffusion theories are frequently used to describe the mechanism of transport, but for gas or vapour, other models have been developed to fit experimental data of diffusion profiles. The models of gas transport include Maxwell s model," free volume increase mechanism," solubility increase mechanism," nanogap hypothesis," Nielsen model, " " Bharadwaj model, ° Cussler model " " and Gusev and Lusti model, " etc. 

The second adhesion theory to emerge from the Soviet Union in the mid-years of the twentieth century was the diffusion theory. The basic principles were set out by S.S. Voyutskii and his colleagues. Their compatriot, R.M. Vasenin, contributed to its quantitative development using Fickian diffusion theory (Voyutskii 1963 Wake 1982). 

Compared to the molecules themselves, the channels and pores through which the molecules diffuse are typically large, and the main interaction for a particular molecule is collision with other molecules, the driving force for bulk diffusion. In some cases for flow in very small pores and channels, however, the molecular collisions with the solid walls of the channel become a significant component of the overall number of collisions the molecule experiences, and the effective diffusion will deviate from that predicted with Fickian diffusion theory. Consider normal diffusion in a cylindrical pore (Figure 5.14). The molecular interactions with the channel wall are negligible compared to the collisions with other molecules. 

The development of the theory of solute diffusion in soils was largely due to the work of Nye and his coworkers in the late sixties and early seventies, culminating in their essential reference work (5). They adapted the Fickian diffusion equations to describe diffusion in a heterogeneous porous medium. Pick s law describes the relationship between the flux of a solute (mass per unit surface area per unit time, Ji) and the concentration gradient driving the flux. In vector terms. 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

The major pathway of drug transport across buccal mucosa seems to follow simple Fickian diffusion [17]. Passive diffusion occurs in accordance with the pH-partition theory. Considerable evidence also exists in the literature regarding the presence of carrier-mediated transport in the buccal mucosa [18,19]. Examination of Eq. (1) for drug flux,... 

The analysis of transfer mechanisms of drugs across the intestinal epithelial layer has passed a long way since the theory of lipid pore membrane [118] in which the total pore area of the intestinal membranes was calculated (and found to be low compared with the total surface of the mucosal aspect of the gut), through the Fickian diffusion calculations of the transport of unionized moieties of drug molecules (the Henderson-Hasselbach equation), which led to the conclusion that acidic drugs are absorbed in the stomach [119,120]. 

Basic Equations All of the processes described in this section depend to some extent on the following background theory. Substances move through membranes by several mechanisms. For porous membranes, such as are used in microfiltration, viscous flow dominates the process. For electrodialytic membranes, the mass transfer is caused by an electrical potential resulting in ionic conduction. For all membranes, Fickian diffusion is of some importance, and it is of dom-... 

However, the multicomponent Fickian diffusivities, Dgy, do not correspond to the approximately concentration independent binary diffusivities, Dsr, which are available from binary diffusion experiments or kinetic theory determined by the inter-molecular forces between s —r pair of gases. Instead, these multicomponent Fickian diffusion coefficients are strongly composition dependent. 

A multicomponent Fickian diffusion flux on this form was first suggested in irreversible thermodynamics and has no origin in kinetic theory of dilute gases. Hence, basically, these multicomponent flux equations represent a purely empirical generalization of Pick s first law and define a set of empirical multi-component diffusion coefficients. 

Kitchener and coworkers (12, 16) have shown that when colloidal size particles with the same charge sign as that of the planar surface diffuse to and sorb on the latter, the rate of uptake may be considerably less than that predicted by simple Fickian diffusion across the unstirred liquid film, principally because of the double layer repulsive forces, as treated in the Derjagin-Landau-Verwey-Overbeek theory of colloid stability. It was found in some systems, particularly at high ionic strength, that the sorption rate decreased with time, eventually becoming negligible. 

For a thin film gel attached to a rigid substrate, the characteristic wavelength of the surface pattern can be assumed to be proportional to the deformation along the z axis (normal to the substrate), u z,t) and can be explained by the linear diffusion theory, or Fickian-type kinetics [21, 23, 24] ... 

The two major approaches to equation development for turbulent diffusion are the Fickian diffusion model and the description of the basic turbulent process by the theory of diffusion by continuous motion. Only the Fickian approach will be presented here. Fick proposed that the molecular diffusion of matter could be expressed in the same manner as the conduction of heat or electricity in a conducting body, as used by Fourier and Ohm, respectively, in their work. This can be stated as making the rate of diffusion in any direction directly proportional to the concentration gradient in that direction. Mathematically, this becomes... 

Diffusion is the mass transfer caused by molecular movement, while convection is the mass transfer caused by bulk movement of mass. Large diffusion rates often cause convection. Because mass transfer can become intricate, at least five different analysis techniques have been developed to analyze it. Since they all look at the same phenomena, their ultimate predictions of the mass-transfer rates and the concentration profiles should be similar. However, each of the five has its place they are useful in different situations and for different purposes. We start in Section 15.1 with a nonmathematical molecular picture of mass transfer (the first model) that is useful to understand the basic concepts, and a more detailed model based on the kinetic theory of gases is presented in Section 15.7.1. For robust correlation of mass-transfer rates with different materials, we need a parameter, the diffusivity that is a fundamental measure of the ability of solutes to transfer in different fluids or solids. To define and measure this parameter, we need a model for mass transfer. In Section 15.2. we discuss the second model, the Fickian model, which is the most common diffusion model. This is the diffusivity model usually discussed in chemical engineering courses. Typical values and correlations for the Fickian diffusivity are discussed in Section 15.3. Fickian diffusivity is convenient for binary mass transfer but has limitations for nonideal systems and for multicomponent mass transfer. 

D8. Use the Wilke-Chang theory to estimate the infinite dilution Fickian diffusivity of methanol in liquid water at 293.16 K. Data are available at http //www.engineeringtoo1box.com/ and http //www.cheric.org/research/kdb/ (click on Korean Physical Properties Data Bank). 

Cutress IJ, Dickinson EJF, Compton RG (2011) Electrochemical landom walk theory. Probing voltammetry with small numbers of molecules stochastic versus statistical (Fickian) diffusion. J Electrotmal Chem 655 1-8... 

Mass transport also plays a major role in several other important disciplines. Environmental processes are dominated by the twin topics of mass transfer and phase equilibria, and here again an early and separate introduction to these subject areas can be immensely beneficial. This text provides detailed treatments of both phase equilibria and compartmental models, which are all-pervasive in the environmental sciences. Transport, where it occurs, is almost always based on Fickian diffusion and film theory. The same topics are also dominant in the biological sciences and in biomedical engineering, and the text makes a conscious effort to draw on examples from these disciplines and to highlight the idiosyncrasies of biological processes. 

It is remarked that even though the latter two Fickian diffusivity proposals, i.e., (2.309) and (2.311), are symmetric, the multicomponent Fickian diffusivities Dsr and Djr do not correspond to the approximately concentration independent binary diffusivities, i.e., Dsr, which are available from binary diffusion experiments or kinetic theory determined by the inter-molecular forces between s — r pair of gases. Instead, these generalized multicomponent Fickian diffusion coefficients are strongly composition dependent. For these reasons, the generalized Fickian diffusivities are hardly used in practice because of the large number of costly experimental data sets that are required for parameter fitting. 

A comparison with the results of the kinetic theory of dilute gases with those from the irreversible thermodynamics also shows that dJ = aso- By use of (2.314) it is concluded that the generalized thermal diffusion coefficients Df, have the property (2.304). Moreover, it was shown in the previous subsection that the various generalized Fickian diffusivities T>sr, Dsr, Dsr and derived from kinetic theory can be related to the phenomenological coefficients by the following relations (2.306), (2.307), (2.309) and (2.311). 

All the PI-PDMS data show that the overall Rouse dynamics prevails, although it is affected and accelerated by interface undulations, which contribute mainly at high Q and counteracts in parts the effect of grafting one chain-end to the surface. Secondly, at lower Q, in agreement with other studies, the diffusion is essentially slowed down by a confinement to the interface. This diffusion can be estimated remarkably well based on the Rouse theory and Fickian diffusion in 2D. Hence, while the overall global dynamics slows down, the chain dynamics is still similar to Rouse motion. 

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