Expression of flux

Diffusion under electric field 5.5.1. Expression of flux 

We now consider the one-way diffusion of charged particles under the simultaneous action of a gradient of concentration and of an electric field. 

Call zF (F is the Faraday) the molar charge of the particle and V the electric potential at the X-coordinate of the top of the energy barrier. In the direction of the electric force, the total energy to cross dVIdx 0) is given as 

If we assume that the diffusing particles ate diluted (the probability so that the site of reception is free is close to 1), we can express the jump rate for each of the two opposite fluxes. Using the volumetric concentrations and assuming the symmetry of each barrier tc = 1/2), we obtain, at x as X-coordinate, in the increasing x direction  

Expressing the gradient of concentration using the mean value theorem, the total flux, that is, the difference of the two preceding fluxes is then 

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This flux is equal to the expression of flux through the fluid film, according to Equation 5.48 ... 

Substituting the explicit expression of fluxes derived so far into conservation laws, we have the linearized equations of hydrodynamic variables ... 

Then, relation [5.30], which gives the expression of flux under the combined action of an electric field and that of a gradient of concentration, is written as... 

We will generally use the assumption of a pseudo-steady state for diffusion. Under the effect of a concentration gradient (or neutral particles), we will be able to apply for the monodirectional diffusions that of relations [5.23], [5.24], or [5.25] according to the considered geometry. Table 7.1 recapitulates the expressions of flux in these various cases. 

Remark.- With such an expression of flux, the rate is not separable because we caimot define, starting from flux, a reactivity and a geometrical function G, given that (f> depends only on the concentration and G depends only on the space. 

The purpose of all flux models is to express the fluxes in a porous medium in terms of gradients in pressure, composition and temperature. Isothermal flux models are therefore all of the general form... 

The concentration boundary layer forms because of the convective transport of solutes toward the membrane due to the viscous drag exerted by the flux. A diffusive back-transport is produced by the concentration gradient between the membranes surface and the bulk. At equiUbrium the two transport mechanisms are equal to each other. Solving the equations leads to an expression of the flux ... 

For the usual case when R = (total retention of the solute), L petm = 0 3.nd combining these equations gives a general expression for flux in a turbulent-flow membrane system. For any given solute concentration ... 

The negative expression of voltage is according to Lenz s law. which states that The direction of the induced e.m.f. is such that it tends to oppose the change in the inducitig flux . In equation (1,4)... 

To derive an explicit expression of the rate of desorption we restrict ourselves to nondissociative adsorption, listing references to other systems— such as multicomponent and multilayer adsorbates with and without precursors—for which such a treatment has been given, later. We look at a situation where the gas phase pressure of a molecular species, P, is different from its value, P, which maintains an adsorbate at coverage 6. There is then an excess flux to re-establish equilibrium between gas phase and adsorbate so that we can write [7-10]... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

Fig. 7-11 Compilation of the most important photochemical processes in the atmosphere, including estimates of flux rates expressed in moles per year between the earth s surface and the atmosphere and within the atmosphere. (Modified with permission from P. J. Crutzen, Atmospheric interactions - homogeneous gas reactions of C, N, and S containing compounds. In B. Bolin and R. Cook (1983). "The Major Biogeochemical Cycles and Their Interactions," pp. 67-112, John Wiley, Chichester.)...

Dunlop [18] proposed a model for sub-lytic effects in plant cells, based on the same principles, but including four properties postulated to be of particular importance in these systems, namely calcium ion flux, osmo-regulation, cell-cell contact/aggregation and stress protein expression. Of these factors, osmo-regulation (and its inter-relationship with the cell wall) and aggregation patterns, in particular, distinguish plant cells from mammalian cell systems. 

To conclude, a strong correlation was found to exist between the net charge of the proteins in solution, the net charge of the SUM surface, and the extent of protein adsorption, which was expressed in terms of flux losses measured after filtration of the different protein solutions. Moreover, in the case of charge-neutral SUMs, flux losses increased with the hydrophobicity of the nucleophiles bound to the S-layer lattice. All proteins caused higher flux losses on SUMs modified with HDA than on those modified with GME or... 

What are called physiologically based pharmacokinetic (PBPK) and pharmacodynamic (PBPD) models are more mechanistically complex and often include more compartments, more parameters, and more detailed expressions of rates and fluxes and contain more mechanistic representation. This type of model is reviewed in more detail in Section 22.5. Here, we merely classify such models and note several characteristics. PBPK models have more parameters, are more mechanistic, can exploit a wider range of data, often represent the whole body, and can be used both to describe and interpolate as well as to predict and extrapolate. Complexity of such models ranges from moderate to high. They typically contain 10 or more compartments, and can range to hundreds. The increase in the number of flux relationships between compartments and the related parameters is often more than proportional to compartment count. 

Metabolic control analysis (MCA) assigns a flux control coefficient (FCC) to each step in the pathway and considers the sum of the coefficients. Competing pathway components may have negative FCCs. To measure FCCs, a variety of experimental techniques including radio isotopomers and pulse chase experiments are necessary in a tissue culture system. Perturbation of the system, for example, with over-expression of various genes can be applied iteratively to understand and optimize product accumulation. 

Over-expression of bacterial phytoene synthase led to only modest increases in pigment accumulation (except in the case of chloroplast-contaiifing tissues). Attention turned to CrtI, one gene that might control flux through the entire four desaturation steps from phytoene to lycopene (discussed in Section 5.3.2.4). Only a modest increase in carotenoid content in tomatoes and a variety of changes in carotenoid composition including more P-carotene, accompanied by an overall decrease in total carotenoid content (no lycopene increase), resulted when CrtI was over-expressed under control of CaMV 35S. Apparently, the bacterial desaturase... 

Fortuitously, the bacterial gene product, CRTI, produces di -trans carotenoids and satisfies the stereo-chemical specificity of LYC B for all-trani substrates while also catalyzing the four desaturation steps from phytoene to lycopene. Nevertheless, over-expression of Crtl has been shown to have only a modest effect (two- to fourfold increases in tomatoes and carrots) in increasing flux through the pathway and some unexpected pleiotropic influences on activities upstream and downstream of the desaturations (reviewed by Fraser and Bramley and Giuliano °). 

Since m is the mass of solid remaining at time t, the quantity m/m0 is the fraction undissolved at time t. The time to total dissolution (m/m0 = 0) of all the particles is easily derived. Equation (49) is the classic cube root law still presented in most pharmaceutics textbooks. The reader should note that the cube root law derivation begins with misapplication of the expression for flux from a slab (Cartesian coordinates) to describe flux from a sphere. The error that results is insignificant as long as r0 8. 

Regardless of the flux mechanism, it is clear upon examination of permeability expressions that flux is proportional to the concentration differential across the total barrier to mass transport. This flux is maximal in a given system for a permeant when the penetrating agent is present in the applied phase in a saturated state. There are many situations of pharmaceutical interest where this solubility... 

TfR is low in these patients, PIT is normal and most iron is stored in hepatocytes. In patients with hypoplastic anaemias and with transfusion iron overload, the BM cannot utilize iron, resulting in low TfR expression and decreased iron absorption. Quantitative analysis of all iron fluxes, which can be deduced from Figure 9.1, can assist in understanding the clinical expression of mutations of proteins involved in iron transport. 

We have noted at the end of Section 2 in II that in and near a reverse-biased p-n junction or diode, the departure from electron-hole equilibrium will invalidate (3) and (4) (3) is replaced by (25), with r+0 given by (24) with eFe for eF, and r0+ given by (23), with eFh (4) is replaced by an analogous equation. Thus while the first equality in each of the three flux equations (30)-(32) remains valid, the expression of J+ and / in terms of n0 needs to be modified. The appropriate equations are easily found to be... 

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