Concentration-driven diffusion flux

Another important leakage mechanism is a concentration-driven diffusive flux in contrast with the pressure-driven hydrodynamic flux considered previously. The flux of species A relative to the average molar velocity of all components is given by Fick s first law. 

Gas permeation through the porous membranes may be driven by pressure or concentration gradient. Under a pressure or concentration gradient, gas will permeate through the membrane in a convective or a diffusive flow, respectively. In general, the pressure-driven convective fluxes are much higher than the concentration-driven diffusion fluxes. 

Figure 1.12. Diffusion flux jj driven by concentration gradient (Cj) surface area A.

Possible driving forces for solute flux can be enumerated as a linear combination of gradient contributions [Eq. (20)] to solute potential across the membrane barrier (see Part I of this volume). These transbarrier gradients include chemical potential (concentration gradient-driven diffusion), hydrostatic potential (pressure gradient-driven convection), electrical potential (ion gradient-driven cotransport), osmotic potential (osmotic pressure-driven convection), and chemical potential modified by chemical or biochemical reaction. 

According to the original concept [1], the field drives the analytes to the accumulation wall of the channel. This concentrating effect is opposed by diffusion, driven by Brownian motion of the analytes, which causes a steady state when the convective flux is exactly balanced by the diffusive flux. The concentration profile is exponential and the corresponding elution... 

In general, the diffusive mass flux is composed of diffusion due to concentration gradients (chemical potential gradients), diffusion due to thermal effects (Soret diffusion) and diffusion due to pressure and external forces. It is possible to include the full multicomponent model for concentration gradient driven diffusion (Taylor and Krishna, 1993 Bird, 1998). In most cases, in the absence of external forces, it is... 

Figure 1.9. Diffusion flux j driven by concentration gradient (Cjo - Cji)/AZ through surface area A.

Permeation is a general term for the concentration-driven membrane-based mass transport process. Differences in the permeabiUty produce a separation between solutes at constant driving force. Because the diffusion coefficients in liquids are typically orders of magnitude higher than in polymers, a larger flux can be obtained with liquid membranes. Application of a pressure difference, an electric held, or temperature considerably intensifies the process, but these special methods are beyond the scope of this book. 

When the pressure is increased, the flux does increase initially. The increase results in a higher rate of convective transport of solute to surface of the membrane. If the system is not "gel-polarized", the solute concentration at the surface (Cs) increases resulting in an increase in the concentration driven back diffusive transport away from the membrane. In fact, Cs will increase until the back-diffusive transport of solute just equals the forward convective transport. 

Molecules disperse in quiescent aqueous media by random walk migration, which is driven by thermal fluctuations in the solvent. As a result, the diffusive flux is proportional to the concentration gradient. 

Diffusive flux is driven by the concentration gradient of the toxic organic at the sediment-water interface. This process is usually described as a simple Fickian process (Equation 13.8) although the flux involves multiple steps that are best approximated using a two-resistance model (Thibodeaux,... 

To take into account the role of surface-active species the transport equations in the bulk and at surfaces for each of them (i = 1,2,...,AO are studied (Dukhin et al. 1995, Danov et al. 1999). In the bulk the change of concentration, C/, is compensated by the bulk diffusion flux, j bulk convective flux, C/V, and rate of production due to chemical reactions, (see Fig. 1). The bulk diffusion flux includes the flux driven by external forces (e.g. electro-diffusion), the molecular diffusive and thermodiffusion fluxes. The rate of production, r/,... 

In these various equations J and Jbi are the electrically driven fluxes in the membrane and the boundary layer, while Jq is the diffusive flux in the boundary layer. The transport numbers of the cation in the membrane and in the boundary layer are t and tb. z is the valence of the cation (z = 1 for Na ) is the Faraday constant v is the electrical current and dc/dx is the concentration gradient in the boundary layer. 

Theories of mass transport in electrolytes or elec-trolyttic solutions take into account that motion of dissolved species / can be driven by gradients in electric potential O (migration), as well as by gradients in molar concentration c, (diffusion) and by motion of material at the bulk velocity v (convection). The most commonly deployed model for electrolyte transport is the Nemst-Planck theory [1], developed in detail by Levich [2]. Within this theory, one constituent of the solution - typically a neutral species in relative excess - is identified as a solvent . The total molar flux of any remaining solute species i, Ni, is then expressed relative to a stationary coordinate frame as... 

Another problem with constitutive law 1 is the assumption that a diffusive flux of i can only be driven by its own concentration gradient. This model does not account for all the possible sources of diffusional drag. For instance, in a solution containing an electrolyte with an additional neutral solute, as well as a solvent, flux of the neutral solute could lead to drag forces that induce gradients in the electrolyte concentration. The recognition of this sort of phenomenon suggests that in principle additional terms, associated with solute/solute interactions, should appear in Eq. 1. 

Cj, and are bulk concentration and flux, respectively, of the th species—note that includes the molecular diffusive flux, the flux driven by external forces (e.g., electrodiffusion [651,662,663]) and the thermodiffusion flux [662]... 

Eq. (1.12) is the generalization of eq. (1.1) i.e. of the Fick s empirical relationship. It is clear that in a multicomponent system, (r >2), the diffusion flux of component i is driven not only by its concentrations gradients but also by the concentration gradients of the other components. 

There are three modes of mass transport in an electrochemical system diffusion, migration, and convection. Diffusion is driven by the concentration gradient where the material transfer occurs from a high concentration to a low concentration. Diffusion is particularly significant near the electrode surface where conversion reaction occurs. Consequently, electrode has a lower reactant concentration than in bulk solution. Similarly, product concentration is higher near the electrode than further out into solution. The diffusion flux (J j mol/cm s) for species j in steady state is expressed for a constant viscosity solution using Pick s first law... 

Another situation is found for the Na+ ions. When the membrane is permeable to these ions, even if only to a minor extent, they will be driven from the external to the internal solution, not only by diffusion but when the membrane potential is negative, also under the effect of the potential gradient. In the end, the unidirectional flux of these ions should lead to a concentration inside that is substantially higher than that outside. The theoretical value calculated from Eq. (5.15) for the membrane potential of the Na ions is -1-66 mV. Therefore, permeabihty for Na ions should lead to a less negative value of the membrane potential, and this in turn should lead to a larger flux of potassium ions out of the cytoplasm and to a lower concentration difference of these ions. All these conclusions are at variance with experience. 

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