Linear Irreversible Transport

There are a number of well known equations describing irreversible transport processes. For instance Ohm s law 

Here Jq i is the charge current component a and E/ is the S-component of the electric field, llie quantities ( q,a/3 are the components of the conductivity tensor. Then there is Fourier s law of thermal conductivity 

Here is the local heat fiux density component a due to the / -component of a temperature gradient. Xap are the components of the thermal conductivity tensor. Another example is Pick s law 

Here the quantities AXj are called generalized forces. The prefix A reminds us that we stay close to the equilibrium state. At thermodynamic equilibrium, we have simultaneously for all irreversible processes A/ = 0 and AX, = 0. In the following we shall study the above linear relations and their coefficients in more detail. 

Example Insulation. An important number for the quality of insulation material is its A-value, i.e. the thermal conductivity. The unit of A is W (mK). Table7.1 lists some typical numbers (see also HCP). Notice that the thermal conductivity does depend on temperature and possibly pressure. The numbers given here correspond to usual ambient conditions. 

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This example illustrates that the scavenging coefficient can be a useful parameter provided that it can be estimated reliably and one is aware of its physical meaning and limitations. Use of a scavenging coefficient in (20.2) implies linear, irreversible transport of a species into droplets. 

Solid state reactions occur mainly by diffusional transport. This transport and other kinetic processes in crystals are always regulated by crystal imperfections. Reaction partners in the crystal are its structure elements (SE) as defined in the list of symbols (see also [W. Schottky (1958)]). Structure elements do not exist outside the crystal lattice and are therefore not independent components of the crystal in a thermodynamic sense. In the framework of linear irreversible thermodynamics, the chemical (electrochemical) potential gradients of the independent components of a non-equilibrium (reacting) system are the driving forces for fluxes and reactions. However, the flux of one independent chemical component always consists of the fluxes of more than one SE in the crystal. In addition, local reactions between SE s may occur. 

In thermodynamic equilibrium, the electrochemical potential of a particle k (juk = Hk + zkeq>, juk = chemical potential,

electrical potential, zk = charge number of the particle, e = elementary charge) is constant. Gradients in jlk lead to a particle flux Jk and from linear irreversible thermodynamics [95] the fundamental transport... 

Formally, it will be even necessary to make corrections already in the starting flux equations. The detailed formulation of linear irreversible thermodynamics also includes coupling terms (cross terms) obeying the Onsager reciprocity relation. They take into account that the flux of a defect k may also depend on the gradient of the electrochemical potential of other defects. This concept has been worked out, in particular, for the case of the ambipolar transport of ions and electrons.230... 

Let us now summarize the underlying principle of the IPMC s actuation and sensing capabilities, which can be described by the standard Onsager formula using linear irreversible thermodynamics. When static conditions are imposed, a simple description of mechanoelectric effect is possible based upon two forms of transport ion transport (with a current density, /, normal to the material) and solvent transport (with a flux, Q, that we can assume is water flux). 

We have studied a regular linear fluid mixture where most of the results for transport phenomena (4.137), (4.138), (4.165), (4.166) (viscosity, diffusion, heat conduction and cross effects) are not in a form useful in practice [76, 104, 180, 181]. In this section we transform them into a more convenient form which is also used in linear irreversible thermodynamics [1, 27, 28, 119, 120, 130, 182]. Onsager relations will be also noted and some applications, like Pick law and the electrical conductivity of electrolytes are discussed. 

Expressions (4.514), (4.515) are known as phenomenological equations of linear irreversible or non-equilibrium thermodynamics [1-5, 120, 130, 185-187], in this case for diffusion and heat fluxes, which represent the linearity postulate of this theory flows (ja, q) are proportional to driving forces (yp,T g) (irreversible thermodynamics studied also other phenomena, like chemical reactions, see, e.g. below (4.489)). Terms with phenomenological coefficients Lgp, Lgq, Lqg, Lqq, correspond to the transport phenomena of diffusion, Soret effect or thermodiffusion, Dtifour effect, heat conduction respectively, discussed more thoroughly below. 

The equation for solvent transport consists of a diffusional term and a term due to osmotic pressure. The osmotic pressure term arises by using linear irreversible diermodynamics arguments (20). The osmotic pressure is relat to the viscoelastic properties of the polymer through a constitutive equation. In our analysis, the Maxwell element has been used as the constitutive model. Thus, the governing equations for solvent transport in the concentrated regime are... 

Such curves cannot be represented by usual enzymatic kinetics. The linear part for low light intensities justifies the description of the growth curve as a hnear relationship between growth and light intensity reflecting the irreversible transport step. The slope of this curve, usually denoted as a, is called the photon efficiency, here named yx.i for consistency. Here the parameter is based on biomass, while chlorophyll as a reference is also a choice depending on the detailing of the model. 

Uphill transport of molecules against increasing values of their concentration or potential can be discussed within the framework of linear irreversible thermodynamics (LIT) as presented in Section 3.9. If we have, for example, a 2-component system with linear phenomenological relations of the type of (4.81),... 

Let us now consider the linear irreversible process, of which the flows and the forces are connected by the phenomenological relations 13 and 19. If the transport coefficients are entirely independent of any of the intensive variables and if they are subject to the reciprocity relations, then we can show that... 

GK = Green-Kubo LIT = linear irreversible thermodynamics LRT = linear response theory NEMD = nonequilibrium molecular dynamics NESS = nonequilibrium steady state TTC = thermal transport coefficient TTCF = transient time correlation function. 

As in the previous chapter, the semi-irrfinite diffusion at a planar electrode is considered, where the adsorption is described by a linear adsorption isotherm. The modeling of reaction (2.173) does not require a particular mathematical procedure. The model comprises equation (1.2) and the boundary conditions (2.148) to (2.152) that describe the mass transport and adsorption of the R form. In addition, the diffusion of the O form, affected by an irreversible follow-up chemical reaction, is described by the following equation ... 

The rotating disc electrode is constructed from a solid material, usually glassy carbon, platinum or gold. It is rotated at constant speed to maintain the hydrodynamic characteristics of the electrode-solution interface. The counter electrode and reference electrode are both stationary. A slow linear potential sweep is applied and the current response registered. Both oxidation and reduction processes can be examined. The curve of current response versus electrode potential is equivalent to a polarographic wave. The plateau current is proportional to substrate concentration and also depends on the rotation speed, which governs the substrate mass transport coefficient. The current-voltage response for a reversible process follows Equation 1.17. For an irreversible process this follows Equation 1.18 where the mass transfer coefficient is proportional to the square root of the disc rotation speed. 

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. 

The subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

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. 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

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