Thermal transport coefficients

Eq. (437) may be transformed into a true transport equation for f this transport equation is the generalized linearized Boltzmann equation for f, as it also appears in the theory of thermal transport coefficients. More precisely, we get ... 

Section II provides a summary of Local Random Matrix Theory (LRMT) and its use in locating the quantum ergodicity transition, how this transition is approached, rates of energy transfer above the transition, and how we use this information to estimate rates of unimolecular reactions. As an illustration, we use LRMT to correct RRKM results for the rate of cyclohexane ring inversion in gas and liquid phases. Section III addresses thermal transport in clusters of water molecules and proteins. We present calculations of the coefficient of thermal conductivity and thermal diffusivity as a function of temperature for a cluster of glassy water and for the protein myoglobin. For the calculation of thermal transport coefficients in proteins, we build on and develop further the theory for thermal conduction in fractal objects of Alexander, Orbach, and coworkers [36,37] mentioned above. Part IV presents a summary. 

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. 

While we have only carried out calculations for myoglobin, we note that for the two other proteins studied in this chapter, cytochrome c and GFP, the latter structurally very different than myoglobin, the diffusion of energy and sound velocities scale with frequency in similar (though not identical) ways. Since rates of anharmonic decay have a similar frequency dependence, and since these proteins are not all that different in size, we expect the thermal transport coefficients for these proteins to be similar. 

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... 

In myoglobin, we find that the anharmonic contribution significantly enhances thermal conduction over that in the harmonic limit, by more than a factor of 2 at 300 K. Moreover, the thermal conductivity rises with temperature for temperatures beyond 300 K as a result of anharmonicity, whereas it appears to saturate around 100 K if we neglect the contribution of anharmonic coupling of vibrational modes. The value for the thermal conductivity of myoglobin at 300 K is about half the value for water. The value for the thermal diffusivity that we calculate for myoglobin is the same as the value for water. Thermal transport coefficients for other proteins will be presented elsewhere. 

Since we submitted this chapter we have slightly refined our calculation of thermal transport coefficients for myoglobin and green fluorescent protein [163], as well as for water [164]. 

This observation constitutes the basic idea of the local equilibrium model of Prigogine, Nicolis, and Misguich (hereafter referred to as PNM). One considers the case of a spatially nonuniform system and deduces from (3) an integral equation for the pair correlation function that is linear in the gradients. This equation is then approximated in a simple way that enables one to derive explicit expressions for all thermal transport coefficients (viscosities, thermal conductivity), both in simple liquids and in binary mixtures, excluding of course the diffusion coefficient. The latter is a purely kinetic quantity, which cannot be obtained from a local equilibrium hypothesis. 

We turn first to computation of thermal transport coefficients, which provides a description of heat flow in the linear response regime. We compute the coefficient of thermal conductivity, from which we obtain the thermal diffusivity that appears in Fourier s heat law. Starting with the kinetic theory of gases, the main focus of the computation of the thermal conductivity is the frequency-dependent energy diffusion coefficient, or mode diffusivity. In previous woik, we computed this quantity by propagating wave packets filtered to contain only vibrational modes around a particular mode frequency [26]. This approach has the advantage that one can place the wave packets in a particular region of interest, for instance the core of the protein to avoid surface effects. Another approach, which we apply in this chapter, is via the heat current operator [27], and this method is detailed in Section 11.2. 

Thermal transport coefficients can be computed in the harmonic approximation with the mode diffusivities plotted in Figure 11.1 using Equations 11.1 through 11.3. 

FIGURE 11.2 Thermal conductivity (below) and thermal diffusivity (above) computed for myoglobin as a function of temperature. Dashed curves indicate values obtained in the harmonic approximation, whereas solid curves correspond to calculations where anharmonicity is accounted for in the calculation of the thermal transport coefficient. 

These are plotted for myoglobin in Figure 11.2, where we have used for the volumetric heat capacity the volume 2.1 x 10 used in previous calculations [26,30]. The results are similar to but about 20%-30% smaller than the thermal transport coefficients we computed in the harmonic approximation previously, where the... 

We compute effective thermal transport coefficients for proteins using linear response theory and beginning in the harmonic approximation, with anharmonic contributions included as a correction. The correction can in fact be rather large, as we compute anharmonicity to nearly double the magnitude of the thermal conductivity and thermal diffusivity of myoglobin. We expect that anharmonicity will generally enhance thermal transport in proteins, in contrast, for example, to crystals, where anharmonicity leads to thermal resistance, since most of the harmonic modes of the protein are spatially localized and transport heat only inefficiently. 

Yu X, Leitner DM. 2005. Heat flow in proteins Computation of thermal transport coefficients. J. Chem. Phys. 122 054902-1-11. 

Reaction rate equation and aU physical properties as heats of reactions, thermal transport coefficient, density of packed bed and fluid mixture, fluid mixture viscosity, reactor and catalyst particles diameters, void fraction, etc., have to be known for the specific system. 

It turns out that Eq. (35) can be used to derive time correlation function expressions for thermal transport coefficients, but here we will... 

Within the framework of linear response theory a phenomenological thermal transport coefficient Lij can be shown to have the form of a Green-Kubo relation... 

The time correlation functions for the various thermal transport coefficients can be efficiently computed by MD with low statistical uncertainty. The results for transport coefficients are usually accurate to within 5-10%. One-particle time correlation functions are in general more accurate than collective functions due to the possible averaging over each single particle trajectory (Hansen McDonald 1986 Hoheisel Vogelsang 1988). So self-diffusion coefficients are, for instance, more accurately computable than mutual-diffusion coefficients. 

In the following, the Green-Kubo formulas for self-diffusion, shear viscosity and thermal conductivity coefficients are compiled. A complete list of all the thermal transport coefficients of one- and two-component systems was given by Hoheisel Vogelsang (1988). 

Only the transport properties of pure substances are considered here. However, also for binary mixture systems, there are a lot of interesting results for transport coefficients of atomic liquids and a few studies for molecular liquids. Obviously, for atomic liquids all the thermal transport coefficients, including the thermal diffusion coefficient, can be obtained by MD calculations with reasonable accuracy. For molecular mixtures, the presently available theoretical investigations are too rare to give a sufficient picture of the transport phenomena. To be more specific, even the sign of the thermal diffusion coefficient of a molecular liquid mixture of about equal masses of the component molecules is difficult to obtain by MD calculations. 

Borgelt, R, Hoheisel, C. Stell, G. (1990). Exact molecular dynamics and kinetic theory results for thermal transport coefficients of the Lennard-Jones argon fluid in a wide range of states. Phys. Rev., A42, 789-794. 

Hoheisel, C. Vogelsang, R. (1988). Thermal transport coefficients for one- and two-component liquids from time correlation functions computed by molecular dynamics. Comput. Phys. Rep., 8,1-69. 

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. 

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