Transport coefficients, mode coupling theory

There have been several treatments to calculate correlation functions and the transport coefficients near the critical point (Fixman [35], Kawasaki [36] and Kadanoff and Swift [37]). All these treatments embody essentially the same physical ideas and contains the genesis of the modem mode coupling theory. Here we discuss the treatment of Kadanoff and Swift [37] because this is physically the most transparent one and seems to have influenced the latter development of the mode coupling theory in a more significant manner. 

In a similar way the contribution for all the different modes to the three transport coefficients can be calculated. Equations (58) and (61) are the classic mode coupling theory expressions that provide general expressions for the shear viscosity and thermal conductivity, respectively. Using these general expressions and the ideas of static scaling laws, Kadanoff and Swift have calculated the transport coefficients near the critical point. 

For many typical cases, the dimensionless ratio (AL A k)/(ALA k), which appears in Eq. (44) due to Eq. (46), simply serves to cut off the Xk k K. Thus, changing the Xk- to an integral, we find that the bilinear contribution to A varies as (Ajj + Ayyj V. Near a critical point, A,j Ayy, so the bilinear part of A goes as (Ay,) /c, k This part of A will dominate A near a critical point, and the classical behavior of A° breaks down. We emphasize that the demonstrations given of both aspects of the breakdown of classical theory hold for fc- 0 the results for kc>k K require a somewhat more sophisticated analysis. The third difficulty mentioned earlier, that of the appearance of transport coefficients in transport coefficients, is also explained by mode-mode coupling theory. Since contains [Pg.270]

Beautiful and useful though they are, Eqs. (130) and (136) do not represent the complete solution of a mode-mode coupling problem. Recall that the aim of mode-mode coupling is to predict all dynamical critical phenomena and long-time tails from equilibrium properties and bare transport coefficients. Thus, we have not really completed our calculation of D until 17 is eliminated. To do this, we calculate 17 via mode-mode coupling theory. 

To what extent the schematic model systems A and B for a polymer melt show this typical relaxation behavior will be addressed in this subsection, by calculating various structural correlation functions that probe the dynamical changes of the melt on different length scales (Section 6.3.2.1). From these correlation functions it is possible to extract relaxation times the temperature dependence of which can be studied and compared to that of transport coefficients, such as the diffusion coefficient. This will be done in Section 6.3.2.2. The final paragraph of this subsection then deals with the calculation of the incoherent intermediate scattering function and its quantitative interpretation in the framework of the idealized mode coupling theory (MCT). " ... 

We now wish to illustrate another of the three main consequences of mode-mode coupling mentioned in the introduction, that of the existence of expressions for transport coefiidents in terms of other transport coefficients such expressions could not exist if classical theory held. Of course, the results derived in the previous section are examples of such expressions. However, we wish to discuss cases that have nothing to do with critical phenomena. The best known example of the expressions of interest is the Stokes-Einstein law for the self-diffusion coefficient of a large spherical particle. 

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