Case I transport

Pick s first and second laws were developed to describe the diffusion process in polymers. Fickian or case I transport is obtained when the local rate of change in the concentration of a diffusing species is controlled by the rate of diffusion of the penetrant. For most purposes, diffusion in rubbery polymers typically follows Fickian law. This is because these rubbery polymers adjust very rapidly to the presence of a penetrant. Polymer segments in their glassy states are relatively immobile, and do not respond rapidly to changes in their conditions. These glassy polymers often exhibit anomalous or non-Fickian transport. When the anomalies are due to an extremely slow diffusion rate as compared to the rate of polymer relaxation, the non-Fickian behaviour is called case II transport. Case II sorption is characterized by a discontinuous boundary between the outer layers of the polymer that are at sorption equilibrium with the penetrant, and the inner layers which are unrelaxed and unswollen. 

The rate and type of release can be analyzed by the expression Mt/Moo=ktn (76). In the case of pure Fickian diffusion n = 0.5, whereas n > 0.5 indicates anomalous transport, i.e., in addition to diffusion another process (or processes) also occurs. If n = 1 (zero order release), transport is controlled by polymer relaxation ("Case II transport") (76). The ln(Mt/Mco) versus In t plots, shown in Figure 4, give n = 0.47 and 0.67 for samples A-9.5-49 and A-4-56, respectively. Evidently theophylline release is controlled by Fickian diffusion in the former network whereas the release is... 

The rate of entropy production is always positive in the present case, since transport processes are irreversible in nature, i.e. always connected with irreversible losses (dissipation) of energy. 

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... 

Can the Chemical and Electrical Work Be Determined Separately In the case of transport processes, the total driving force for the flow of a particular species j, i.e., the gradient of electrochemical potential. 3pj./3bc, was considered split up into a chemical (diffusive) driving force dp/dx and an electrical driving force for conduction, ZjFdty/dx,... 

When Fickian diffusion in normal Euclidean space is justified, further verification can be obtained from the analysis of 60% of the release data using the power law in accord with the values of the exponent quoted in Table 4.1. Special attention is given below for the values of b in the range 0.75-1.0, which indicate a combined release mechanism. Simulated pseudodata were used to substantiate this argument assuming that the release obeys exclusively Fickian diffusion up to time t = 90 (arbitrary units), while for I, > 90 a Case II transport starts to operate too this scenario can be modeled using... 

The nice fittings of the previous functions to the release data generated from (4.16) and (4.17), respectively, verify the argument that the power law can describe the entire set of release data following combined release mechanisms. In this context, the experimental data reported in Figures 4.8 to 4.10 and the nice fittings of the power-law equation to the entire set of these data can be reinterpreted as a combined release mechanism, i.e., Fickian diffusion and a Case II transport. 

In the most general case, i.e. when intraparticlc and interphase transport processes have to be included in the analysis, the effectiveness factor depends on five dimensionless numbers, namely the Thiele modulus the Biot numbers for heat and mass transport Bih and Bim, the Prater number / , and the Arrhenius number y. Once external transport effects can be neglected, the number of parameters reduces to three, because the Biot numbers then approach infinity and can thus be discarded. 

The methods described so far for studying self-diffusion are essentially based on an observation of the diffusion paths, i.e. on the application of Einstein s relation (eq 3). Alternatively, molecular self-diffusion may also be studied on the basis of the Fick s laws by using iso-topically labeled molecules. As in the case of transport diffusion, the diffusivities are determined by comparing the measured curves of tracer exchange between the porous medium and the surroundings with the corresponding theoretical expressions. As a basic assumption of the isotopic tracer technique for studying self-diffusion, the isotopic forms are expected to have... 

In the case of inhibitors which adsorb on the metal surface and inhibit the corrosion there are two steps, namely (i) transport of inhibitor to the metal surface and (ii) metal -inhibitor interactions. The process is analogous to drug molecule transport transported in the body to the required site and its interaction with the site to provide relief from the ailments. The most important step involves the interaction of the metal with the inhibitor molecule. These are chemical interactions and will be dealt with later. 

Neither the finite element method nor a discretised integration will "catch" eddy or other motion below a certain scale determined by the choice of mesh. Small-scale motion in many cases may be better described as random, in which case the transport of the quantity A is called diffusion. Diffusion can be described by Pick s law, assuming that the flux density /, i.e., the number of "particles" (here a small parcel of a gas), passing a unit square in a given direction, is proportional to the negative gradient of particle concentration n (Bockris and Despic, 2004),... 

In general, tte diffusivities of penetrants that swell glassy and rubbery polymers increase with concentratioiu The sorption i tterms are normally well-described by the Flory-Hug ns equaticm. Clustering of penetrant can also occur and cause deviations from this behavior In the case of as prdymers and strong swelling solvents, so-called Case II transport can occur . As drown in Fig. 6 an initial linear increase in samjde weight with time characterizes II uptake in film samples. 

The rate expressions derived above describe the dependence of die reaction rate expressions on kinetic parameters related to the chemical reactions. These rate expressions are commonly called the intrinsic rate expressions of the chemical reactions. However, as discussed in Chapter 1, in many instances, the local species concentrations depend also on the rate that the species are transported in the reaction medium. Hence, the actual reaction rates are affected by the transport rates of reactants and products. This is manifested in two general cases (i) gas-solid heterogeneous reactions, where species diffusion through the pore plays an important role, and (ii) gas-hquid reactions, where interfacial species mass-transfer rate as wen as solubility and diffusion play an important role. Considering the effect of transport phenomena on the global rates of the chemical reactions represents a very difficult task in the design of many chemical reactors. These topics are beyond the scope of this text, but the reader should remember to take them into consideration. 

A second limiting transport process finds the weight gain of penetrant a linear function of time over the entire sorption range. This process has been termed Case II Transport, and is mechanistically quite different from Fickian diffusion. The rate controlling phenomena are penetrant induced polymeric relaxations. A combination of Case I and Case II processes has been... 

While the ingress of liquids in elastomers is diffusion controlled and thus Case I, because the polymer chains are well above the glass transition temperature Tg, liquid transport into glassy polymers is more complex [112, 135]. The swelling behavior below Tg is characterized by a sharp boundary between the remaining polymer core and the swollen region which proceeds with constant velocity. This behavior is also called relaxation controlled because the diffusion is much faster than the segmental relaxation of the polymer. 

Examination of Eqn. (8.8) shows that there are two limiting cases (i) For kc 1, p < Po whereby the reaction would be limited by the rate of transport of the deleterious gas to the crack tip. (ii) For kc < 1, the pressure p at the crack tip is approximately equal to the external pressure po whereby the reaction would be limited by the rate of reaction of the deleterious gas with the crack-tip surfaces. 

A high ambient temperature conductivity of 10 Scm was measured with LiTFSI dissolved in the P2S5-PEO matrix, i.e, in salt-in-polymer materials. The decoupling of Li+ motions from polymer segmental motions was demonstrated, suggesting that, in certain cases, the transport number for Li+ ions in the phosphorus sulfide-PEO matrix will be much higher than in a typical PEO-based salt-in-polymer system. 

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