Linear diffusion chemical potential

The most important driving forces for the motion of ionic defects and electrons in solids are the migration in an electric field and the diffusion under the influence of a chemical potential gradient. Other forces, such as magnetic fields and temperature gradients, are commonly much less important in battery-type applications. It is assumed that the fluxes under the influence of an electric field and a concentration gradient are linearly superimposed, which... 

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. 

Let us assume that the molecular transport is governed only by the differences in the chemical potential (diffusion) and neglect a possible order parameter transport by the hydrodynamic flow [1,144,157]. Then, one can postulate a linear relationship between the local current and the gradient of the local chemical potential difference p(r) [146,147] as... 

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

As described in the introduction, submicrometer disk electrodes are extremely useful to probe local chemical events at the surface of a variety of substrates. However, when an electrode is placed close to a surface, the diffusion layer may extend from the microelectrode to the surface. Under these conditions, the equations developed for semi-infinite linear diffusion are no longer appropriate because the boundary conditions are no longer correct [97]. If the substrate is an insulator, the measured current will be lower than under conditions of semi-infinite linear diffusion, because the microelectrode and substrate both block free diffusion to the electrode. This phenomena is referred to as shielding. On the other hand, if the substrate is a conductor, the current will be enhanced if the couple examined is chemically stable. For example, a species that is reduced at the microelectrode can be oxidized at the conductor and then return to the microelectrode, a process referred to as feedback. This will occur even if the conductor is not electrically connected to a potentiostat, because the potential of the conductor will be the same as that of the solution. Both shielding and feedback are sensitive to the diameter of the insulating material surrounding the microelectrode surface, because this will affect the size and shape of the diffusion layer. When these concepts are taken into account, the use of scanning electrochemical microscopy can provide quantitative results. For example, with the use of a 30-nm conical electrode, diffusion coefficients have been measured inside a polymer film that is itself only 200 nm thick [98]. 

Care has to be taken when considering simple concentrations of the permeant since the driving force for diffusion is really the chemical potential gradient. As stated above the maximum flux should occur for a saturated solution of the permeant. However, if supersaturated solutions are applied to the skin, it is possible to obtain enhanced fluxes [27]. This can only be true if the outer skin lipids are capable of sustaining a supersaturated state of the diffusant. Figure 4.4 shows the linear increase in skin permeation with degree of supersaturation, and Fig. 4.5 demonstrates... 

In the generalized Maxwell-Stefan equations, chemical potential gradients, which are the thermodynamic forces, are linear functions of the diffusion flows... 

Nonisothermal reaction-diffusion systems represent open, nonequilibrium systems with thermodynamic forces of temperature gradient, chemical potential gradient, and affinity. The dissipation function or the rate of entropy production can be used to identify the conjugate forces and flows to establish linear phenomenological equations. For a multicomponent fluid system under mechanical equilibrium with n species and A r number of chemical reactions, the dissipation function 1 is... 

RT(wk/ck)(5/3x)ck where wk denotes the thermodynamic factor d In ak/d In ck and ak the activity—finally determined by the concentration gradient. If, however, the chemical potential is virtually constant, as it is the case for systems with a high carrier concentration (metals, superionic conductors), zkF(3/3x)< remains as driving force. For the linear approximation to be valid A must be sufficiently small. Experimental experience confirms the validity of Fick s and Ohm s law (that immediately follow from Table 3 for diffusion and electrical transport) in usual cases, but questions the validity of the linear relationship Eq. (96) in the case of chemical reactions. For a generalized transport we will use in the following the relation 173,178,181... 

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... 

Diffusion of any nonpolar component i does not depend on the electric field. The effect of other components, temperature and pressure shows up in a change of its chemical potential. If we disregard magnetic and gravity forces and assume the presence of hydrodynamical equilibrium, the only force, which compels nonpolar component to move, is gradient of its chemical potential. Relative to it the diffusions linear equation will assume the format... 

From a physical point of view, it seems that measurable quantities are mixture invariant (cf. end of Sect. 4.4). Such are the properties of mixture like y, T (see (4.94), (4.236), (4.240), (4.225)) but also the chemical potentials ga. Note that also heat flux is transformed as (4.118) (with functions (4.223)) and therefore heat flux is mixture invariant in a non-diffusing mixture (all = o) in accord with its measurability. But heat flux is mixture non-invariant in a diffusing mixture, consistently with our expectation of difficulties in surface exchange (of masses) of different constituents with different velocities together with heat. We note that all formulations of heat flux used in linear irreversible thermodynamics [1, 120] (cf. Rems. 11 in this chapter, 14 in Chap. 2) are contained (by arbitrariness of rjp) in expression (4.118) for heat flux in a diffusing mixture. 

In this section we will deal with the analysis of adsorption kinetics of a multicomponent system. First we will deal with the case of a single zeolite crystal to investigate the effect of the interaction of diffusion of all species inside a zeolite crystal. This interaction of diffusion is characterized by a diffusivitv matrix which is in general a function of the concentrations of all species involved. This concentration dependence will take a special functional form if we assume that the driving force for the diffusion inside the zeolite crystal is the chemical potential gradient and that the mobility coefficients of all species are constant. Only in the limit of low concentration such that the partition between the fluid phase and the adsorbed phase is linear, the diffusivity matrix will become a constant matrix. 

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