Linear transport processes

In this chapter we formulate the thermodynamic and stochastic theory of the simple transport phenomena diffusion, thermal conduction and viscous ffow (1) to present results parallel to those listed in points 1-7, Sect. 8.1, for chemical kinetics. We still assume local equilibrium with respect to translational and internal degrees of freedom. We do not assume conditions close to chemical or hydrodynamic equilibrium. For chemical reactions and diffusion the macroscopic equations for a given reaction mechanism provide sufficient detail, the fluxes in the forward and reverse direction, to write a birth-death master equation with a stationary solution given in terms of For thermal conduction and viscous flow we derive the excess work and then find Fokker-Planck equations with stationary solutions given in terms of that excess work. 

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The transport processes that we study in this work are linear transport processes [1]. They arise when one has a linear phenomenological relation. 

Electro-osmotic effects are of intrinsic interest also, since they give rise to a number of steady-state phenomena which can be conveniently studied experimentally. These afford examples where the power of non-equilibrium thermodynamics can be easily demonstrated. This is a unique phenomenon where non-linear transport processes have been experimentally observed and studied. 

In the A sector (lower right), the deposition is controlled by surface-reaction kinetics as the rate-limiting step. In the B sector (upper left), the deposition is controlled by the mass-transport process and the growth rate is related linearly to the partial pressure of the silicon reactant in the carrier gas. Transition from one rate-control regime to the other is not sharp, but involves a transition zone where both are significant. The presence of a maximum in the curves in Area B would indicate the onset of gas-phase precipitation, where the substrate has become starved and the deposition rate decreased. 

This result shows that the most likely rate of change of the moment due to internal processes is linearly proportional to the imposed temperature gradient. This is a particular form of the linear transport law, Eq. (54), with the imposed temperature gradient providing the thermodynamic driving force for the flux. Note that for driven transport x is taken to be positive because it is assumed that the system has been in a steady state for some time already (i.e., the system is not time reversible). 

The transient method characterized by linearly changing potential with time is called potential-sweep (potential-scan) voltammetry (cf. Section 5.5.2). In this case the transport process is described by equations of linear diffusion with the potential function... 

Because the conditional scalar Laplacian is approximated in the FP model by a non-linear diffusion process (6.91), (6.145) will not agree exactly with CMC. Nevertheless, since transported PDF methods can be easily extended to inhomogeneous flows,113 which are problematic for the CMC, the FP model offers distinct advantages. 

Substrate transport through the film may be formally assimilated to membrane diffusion with a diffusion coefficient defined as12 Ds = Dch( 1 — 9)/pjort. In this equation, the effect of film structure on the transport process in taken into account in two ways. The factor 1—0 stands for the fact that in a plane parallel to the electrode surface and to the coating-solution interface, a fraction 9 of the surface area in made unavailable for linear diffusion (diffusion coefficient Dcj,) by the presence of the film. The tortuosity factor,, defined as the ratio between the average length of the channel and the film thickness, accounts for the fact that the substrate... 

The stoichiometric matrix N consists of m rows, corresponding to m metabolic reactants, and r columns, corresponding to r biochemical reactions or transport processes (see Fig. 5 for an example). Within a metabolic network, the number of reactions (columns) is usually of the same order of magnitude as the number of metabolites (rows), typically with slightly more reactions than metabolites [138]. Due to conservation relationships, giving rise to linearly dependent rows in N, the stoichiometric matrix is usually not of full rank, but... 

As a simple example, consider the minimal glycolytic pathway shown in Fig. 5. The stoichiometric matrix N has m = 5 rows (metabolites) and m = 6 columns (reactions and transport processes). The rank of the matrix is rank(A) = 4, corresponding to m — ran l< (TVj = 1 linearly dependent row in N. The left nullspace E can be written as... 

If only linear diffusion is the operating mass transport process, it has been shown that the current (i) in a planar electrode is related to the concentration (c) gradient at the surface of the electrode (x = 0) by [332]... 

A plot of (Ink) vs (1/T) yields a linear relationship with the slope equal to (-EJR) and the intercept equal to (lnAf). Thus, by measuring (k) values at several temperatures, the ( a) value can be determined. Low a values (<42 kj mole) usually indicate diffusion-controlled transport processes, whereas higher Ea values indicate chemical reaction or surface-controlled processes [21,25]. 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

The connection between processing conditions and crystalline perfection is incomplete, because the link is missing between microscopic variations in the structure of the crystal and macroscopic processing variables. For example, studies that attempt to link the temperature field with dislocation generation in the crystal assume that defects are created when the stresses due to linear thermoelastic expansion exceed the critically resolved shear stress for a perfect crystal. The status of these analyses and the unanswered questions that must be resolved for the precise coupling of processing and crystal properties are described in a later subsection on the connection between transport processes and defect formation in the crystal.. 

Transport of water and ions between two solutions separated by a clay membrane was studied in [1], using linear transport relations valid when chemical potential differences across the membrane are small [2], Two relaxation processes were found to control the rate at which the solutions came into equilibrium with each other, and both relaxation rates were observed in experiments. Transport was characterised by three independent coefficients, all of which could be estimated from the experimental results. 

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