Phenomenological transport coefficients

In order to investigate the relation between the phenomenological transport coefficients Ljj and Z) we formulate for the isotope (tracer) diffusion of A in A... 

A primary aim of theoretical work in this field is to obtain the phenomenological transport coefficients in terms of molecular scale processes and in terms of molecular parameters such as the intermolecular pair potential energy. Most of the work prior to 1946 rested on rather crude pictures of the liquid state, and various arbitrary assumptions were made as to the nature of the molecular scale transport processes taking place. Born and Green 3 20 21 and... 

Since the phenomenological transport coefficients can be represented in 2 x 2 matrix form for binary systems. 

This inequality in (26-11) stems from the fact that the determinant of the 2 x 2 matrix of phenomenological transport coefficients [, y] mnst be positive to ensure a positive-definite quadratic form for sq- The contribution from thermal Soret diffusion in the final expression for Ja (see equations 25-76 and 25-77) provides a definition of P in terms of the thermal diffusion coefficient kr and the temperature dependence of cpA-... 

Several transport properties can be evaluated from equilibrium simulations with use of linear response theory, which relates correlation fimctions of spontaneously fluctuating molecular properties to phenomenological transport coefficients. These relations can be used to evaluate diffusion coefficients, thermal conductivities, viscosities, IR spectra, and so on. However, most of these properties are evaluated more directly using appropriately devised techniques of nonequilibrium molecular dynamics. Particularly challenging for polymers is the direct... 

There are three new symbols in equations (4) rj, t]t, and A.. These are the phenomenological transport coefficients of the system, the shear viscosity, the bulk viscosity and the thermal conductivity, respectively. Their definitions, called linear constitutive relations, were postulated (with the exception of bulk viscosity) long before the development of LIT. These definitions in the present formalism are as follows ... 

The ratios of the various phenomenological transport coefficients are described by... 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

It is also evident that this phenomenological approach to transport processes leads to the conclusion that fluids should behave in the fashion that we have called Newtonian, which does not account for the occurrence of non-Newtonian behavior, which is quite common. This is because the phenomenological laws inherently assume that the molecular transport coefficients depend only upon the thermodyamic state of the material (i.e., temperature, pressure, and density) but not upon its dynamic state, i.e., the state of stress or deformation. This assumption is not valid for fluids of complex structure, e.g., non-Newtonian fluids, as we shall illustrate in subsequent chapters. 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

This monograph deals with kinetics, not with dynamics. Dynamics, the local (coupled) motion of lattice constituents (or structure elements) due to their thermal energy is the prerequisite of solid state kinetics. Dynamics can explain the nature and magnitude of rate constants and transport coefficients from a fundamental point of view. Kinetics, on the other hand, deal with the course of processes, expressed in terms of concentration and structure, in space and time. The formal treatment of kinetics is basically phenomenological, but it often needs detailed atomistic modeling in order to construct an appropriate formal frame (e.g., the partial differential equations in space and time). 

Figure 5-11 illustrates the results of an oxide interdiffusion experiment. Clearly, the transport coefficients are not single valued functions of composition. From the data, one concludes that for a given composition, the chemical diffusion coefficients depend both on time and location in the sample [G. Kutsche, H. Schmalzried (1990)]. Let us analyze this interdiffusion process in the ternary solid solution Co. O-Nq. O, which contains all the elements necessary for a phenomenological treatment of chemical transport in crystals. The large oxygen ions are almost immobile and so interdiffusion occurs only in the cation sublattice of the fee crystal. When we consider the following set ( ) of structure elements... 

Since the entropy production is positive, the transport coefficients Lik must satisfy the relation TAA-Lhh>LhA-TAh [S.R. de Groot, P. Mazur (1962)]. This restricts the range for the charges of transport to aA-Oh< 1, see Eq. (8.56) ff. We should also add that whereas the Ly are phenomenological coefficients appropriate for the description of the experiments on transport, the ly relate directly to the SE s (Eqn. (8.28)) and can be derived from lattice dynamics based theoretical calculations. 

Section IIA summarizes the physical assumptions and the resulting mathematical descriptions of the "concentration-dependent (5) and "dual-mode" ( 13) sorption and transport models which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In Section IIB we elucidate the mechanism of the "non-ideal" diffusion in glassy polymers by correlating the phenomenological diffusion coefficient of CO2 in PVC with the cooperative main-chain motions of the polymer in the presence of the penetrant. We report carbon-13 relaxation measurements which demonstrate that CO2 alters the cooperative main-chain motions of PVC. These changes correlate with changes in the diffusion coefficient of CO2 in the polymer, thus providing experimental evidence that the diffusion coefficient is concentration dependent. 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

Irreversible processes are driven by generalized forces, X, and are characterized by transport (or Onsager) phenomenological coefficients, L [21,22], where these transport coefficients, Lip are defined by linear relations between the generalized flux densities,./, which are the rates of change with time of state variables, and the corresponding generalized forces X . 

The kinetic equations serve as a bridge between the microscopic domain and the behavior of macroscopic irreversible processes through the description of hydrodynamics in terms of intermolecular collisions. Hydrodynamics can specify a large number of nonequilibrium states by a small number of reproducible properties such as the mass, density, velocity, and energy density of a fluid conserved during the collision of molecules. Therefore, the hydrodynamic equations can describe a wide range of relaxation processes of nonequilibrium states to equilibrium state. We call such processes decay processes represented by phenomenological equations, such as Fourier s law of heat conduction. The decay rates are determined by the transport coefficients. Reliable transport coefficients provide microscopic and macroscopic information, and validate the results of molecular dynamics. 

Another well-known example is the coupling between mass flow and heat flow. As a result, an induced effect known as thermal diffusion (Soret effect) may occur because of the temperature gradient. This indicates that a mass flow of component A may occur without the concentration gradient of component A. Dufour effect is an induced heat flow caused by the concentration gradient. These effects represent examples of couplings between two vectorial flows. The cross-phenomenological coefficients relate the Dufour and Soret effects. In order to describe the coupling effects, the thermal diffusion ratio is introduced besides the transport coefficients of thermal conductivity and dififusivity. 

The phenomenological equations (6.9.1) have thus been reexpressed in (6.9.9) solely in terms of the measurable transport coefficients a, k, and o. The Seebeck coefficient may be interpreted as the entropy carried per electronic charge. Equation (6.9.9a) represents a further generalization of Ohm s Law, showing how the current density behaves in the presence of a temperature gradient see also Exercise 6.9.3. Equation (6.9.9b) specifies the entropy flux under the joint action of a gradient in electrochemical potential and in temperature this represents a generalization of Fourier s Law. 

The transport of mass, momentum and energy through a fluid are the consequences of molecular motion and molecular interaction. At the macroscopic level, associated with the transport of each dynamic variable is a transport coefficient or property, denoted by X, such that the flux, J, of each variable is proportional to the gradient of a thermodynamic state variable such as concentration or temperature. This notion leads to the simple phenomenological laws such as those of Pick, Newton and Fourier for mass, momentum or energy transport, respectively. 

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