Computation of Transport Coefficients

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

The EMA method is similar to the volume-averaging technique in the sense that an effective transport coefficient is determined. However, it is less empirical and more general, an assessment that will become clear in a moment. Taking mass diffusion as an example, the fundamental equation to solve is 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t — oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

As a second method to determine effective transport coefficients in porous media, the position-space renormalization group method will be briefly discussed. 

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Abstract. In this chapter we discuss approaches to solving quantum dynamics in the condensed phase based on the quantum-classical Liouville method. Several representations of the quantum-classical Liouville equation (QCLE) of motion have been investigated and subsequently simulated. We discuss the benefits and limitations of these approaches. By making further approximations to the QCLE, we show that standard approaches to this problem, i.e., mean-field and surface-hopping methods, can be derived. The computation of transport coefficients, such as chemical rate constants, represent an important class of problems where the QCL method is applicable. We present a general quantum-classical expression for a time-dependent transport coefficient which incorporates the full system s initial quantum equilibrium structure. As an example of the formalism, the computation of a reaction rate coefficient for a simple reactive model is presented. These results are compared to illuminate the similarities and differences between various approaches discussed in this chapter. 

Luo, H. Hoheisel, C. (1992). Computation of transport coefficients of liquid benzene and cyclohexane using rigid pair interaction models. J. Chem. Phys., 96, 3173-3176. 

A modification of the Newtonian MD scheme with the purpose of generating a thermodynamical ensemble at constant temperature is called a thermostat algorithm. The use of a thermostat ean be motivated by one (or a number) of the following reasons (i) to match experimental conditions (most condensed-phase experiments are performed on thermostatized rather than isolated systems) (ii) to study temperature-dependent processes (e.g., determination of thermal coefficients, investigation of temperature-dependent conformational or phase transitions) (iii) to evacuate the heat in dissipative non-equilibrium MD simulations (e.g., computation of transport coefficients by viscous-flow or heat-flow simulations) (iv) to enhance the efficiency of a conformational search (e.g., high-temperature dynamics, simulated annealing) (v)... 

Selected entries from Methods in Enzymology [vol, page(s)j Boundary analysis [baseline correction, 240, 479, 485-486, 492, 501 second moment, 240, 482-483 time derivative, 240, 479, 485-486, 492, 501 transport method, 240, 483-486] computation of sedimentation coefficient distribution functions, 240, 492-497 diffusion effects, correction [differential distribution functions, 240, 500-501 integral distribution functions, 240, 501] weight average sedimentation coefficient estimation, 240, 497, 499-500. 

Our main focus in computing thermal transport coefficients is calculation of the frequency-dependent energy diffusion coefficient, D go), which appears in Eq. (12). Computation of Dim) is relatively straightforward if we express the vibrations of the object in terms of its normal modes. We shall compute Dim) with wave packets expressed as superpositions of normal modes, which we then filter to a range of frequencies near go to determine D co). 

In the previous section we computed thermal transport coefficients for a water cluster whose size is reasonably similar to that of a typical globular protein. The calculation of thermal transport properties of proteins turns out not to be so simple. For one thing, there is considerable computational and experimental evidence to suggest that energy transport in proteins is non-Brownian. 

We have explored in this chapter how quantum mechanical energy flow in moderate-sized to large molecules influences kinetics of unimolecular reactions and thermal conduction. In the first part of this chapter we addressed vibrational energy flow in moderate-sized molecules, and we also discussed its influence on kinetics of conformational isomerization. In the second part we examined the dynamics of vibrational energy flow through clusters of water molecules and through proteins, and we computed thermal transport coefficients for these objects. 

In the second part of the chapter, we have examined the spread of vibrational energy through coordinate space in systems that are large on the molecular scale—in particular, clusters of hundreds of water molecules and proteins—and computed thermal transport coefficients for these systems. The coefficient of thermal conductivity is given by the product of the heat capacity per unit volume and the energy diffusion coefficient summed over all vibrational modes. For the water clusters, the frequency-dependent energy diffusion coefficient was... 

The topic of this chapter is the description of a quantum-classical approach to compute transport coefficients. Transport coefficients are most often expressed in terms of time correlation functions whose evaluation involves two aspects sampling initial conditions from suitable equilibrium distributions and evolution of dynamical variables or operators representing observables of the system. The schemes we describe for the computation of transport properties pertain to quantum many-body systems that can usefully be partitioned into two subsystems, a quantum subsystem S and its environment . We shall be interested in the limiting situation where the dynamics of the environmental degrees of freedom, in isolation from the quantum subsystem [Pg.521]

We show how the quantum-classical evolution equations of motion can be obtained as an approximation to the full quantum evolution and point out some of the difficulties that arise because of the lack of a Lie algebraic structure. The computation of transport properties is discussed from two different perspectives. Transport coefficient formulas may be derived by starting from an approximate quantum-classical description of the system. Alternatively, the exact quantum transport coefficients may be taken as the starting point of the computation with quantum-classical approximations made only to the dynamics while retaining the full quantum equilibrium structure. The utility of quantum-classical Liouville methods is illustrated by considering the computation of the rate constants of quantum chemical reactions in the condensed phase. 

For high Da the column is dose to chemical equilibrium and behaves very similar to a non-RD column with n -n -l components. This is due to the fact that the chemical equilibrium conditions reduce the dynamic degrees of freedom by tip the number of reversible reactions in chemical equilibrium. In fact, a rigorous analysis [52] for a column model assuming an ideal mixture, chemical equilibrium and kinetically controlled mass transfer with a diagonal matrix of transport coefficients shows that there are n -rip- 1 constant pattern fronts connecting two pinches in the space of transformed coordinates [108]. The propagation velocity is computed as in the case of non-reactive systems if the physical concentrations are replaced by the transformed concentrations. In contrast to non-RD, the wave type will depend on the properties of the vapor-liquid and the reaction equilibrium as well as of the mass transfer law. 

When we construct normal solutions of the generalized Boltzmann equation using the resummed collision operator and computes the transport coefficients for a moderately dense gas (in three dimensions), we find that the viscosity, say, has the expansion ... 

Then in Section 10.3, we address the problem of computing mass transport coefficients in porous materials called zeolites. Zeolites are materials with a wide range of applications, such as petrochemical separation, water purification, and catalysis. Understanding and predictably computing mass transport coefficients for a variety of molecules in these molecular sieves instruct optimal use of the appropriate zeolite for an application. We describe the methodology used to compute... 

To calculate a trajectory requires knowledge of the intermolecular potential, which is not readily measured. To circumvent this difficulty a guess is made as to the form of the potential which is then used to compute the transport coefficients, the second virial coefficient, and a few other properties dependent upon two-particle interaction only. Good agreement between theory and experiment provides a posteriori justification for the assumed potential. A simple model potential was suggested by Lennard-Jones. It is... 

Trajectory analysis is used to compute the transport coefficients. Since momentum transfer in collisions accounts for transport, it is apparent that calculations based upon the Lennard-Jones potential (or any other that incorporates intermolecular attraction) lead to rather different predictions than did the hard-sphere model. The differences are naturally most significant at low temperatures, where the hard-sphere model is particularly inappropriate, a phenomenon which is marked by the temperature dependence of the apparent value of d (Figs. 2.7 and 2.8). The greater variation of d at low temperature reflects the fact that the mean collision energy... 

Dynamic processes in atomic liquids are nowadays well understood in terms of kinetic theory and computer calculations. Transport coefficients and dynamic scattering functions of liquid argon, for example, are well reflected by kinetic theories and equilibrium (MD) as well as nonequilibrium molecular dynamics (NEMD) calculations using Lennard-Jones (U) pair potentials. 

Table 20.10 provides a compilation of ac impedance studies on conducting polymer films. The aspects under investigation include modeling of the ac impedance response of these materials [348-350,354,368.372], the separation of ionic and electronic contributions to the total conductivity [290,368], overdoping [352], the relative contribution of Faradaic and capacitive components to the total measured charge [221,351], the computation of diffusion coefficients associated with the oxidation of these polymers and the transport of dopant ions... 

The theory coimecting transport coefficients with the intemiolecular potential is much more complicated for polyatomic molecules because the internal states of the molecules must be accounted for. Both quantum mechanical and semi-classical theories have been developed. McCourt and his coworkers [113. 114] have brought these theories to computational fruition and transport properties now constitute a valuable test of proposed potential energy surfaces that... 

The attractive feature of LADM Is that once the fluid structure Is known (e.g., by solution of the YBG equations given In the previous section or by a computer simulation) then theoretical or empirical formulas for the transport coefficients of homogeneous fluids can be used to predict flow and transport In Inhomogeneous fluid. For diffusion and Couette flow In planar pores LADM turns out to be a surprisingly good approximation, as will be shown In a later section. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Absorption, in general, is treated as a physicochemical transport process based on computations of logP (the octanol/water partition coefficient) and solubility governed by factors such as polar surface area on the molecule. It is conceivable that SNPs in drug transporter genes will affect the pharmacokinetic properties of compounds and, therefore, these may have to be taken into consideration in the design process. 

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

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