Time scales mode coupling theory

At the heart of the mode coupling theory of liquids is the assumption that a separation of time scale exists between different dynamical events. While the time scale separation between the fast collisional events and the slower collective relaxation is explicitly exploited in the formulation of the theory, there is also an underlying assumption of the separation of length scales between different relaxation modes. Much of the success of MCT depends on the validity of this separation of length and time scales. 

It should be noted that although in Eq. (90) only the connected motion of the solute and the solvent is retained, in the argument presented on the time scale it is the disconnected parts which have been considered. This is because in the latter part, for the derivation of the expression of Ci. the solute and the solvent motions are assumed to be disconnected. This assumption is the same as those made in the density functional theory and also in mode coupling theories where a four-point correlation function is approximated as the product of two two-point correlation functions. This approximation when incorporated in Ci. means that after the binary collision takes place, the disturbances in the medium will propagate independently. A more exact calculation would be to consider the whole four-point correlation function, thus considering the dynamics of the solute and the solvent to be correlated even after the binary collision is over. Such a calculation is quite cumbersome and has not been performed yet. 

Figure 19. Time-dependent diffusion D i) of a two-dimensional system plotted against reduced time. The solid line represents the D t) obtained from the mode coupling theory (MCT) calculation, and the short-dashed line and the long-dashed line represent the D(t) obtained from simulated VACF and MSD, respectively. In the inset, fits to long-time D(t) to Eq. (351) are also shown. The plots are at p = 0.7932 and T = 0.7. The time is scaled by TJC = Jma2/c. D(t) is scaled by o2/. This figure has been taken from Ref. 175.

Crossover Temperature for Various Glass Formers as Reported by the Different Methods From the Temperature Dependence of the Stretching Parameter y(T), Scaling the Time Constant xa — xa(r) [cf. Eq. (42)], Non-ergodicity Parameter 1 —f(T) Obtained from Spectra Analysis, Electron Paramagnetic Resonance (EPR), and from Tests of the Asymptotic Laws of Mode Coupling Theory ... 

Also similar to percolation is the existence of a critical exponent in the mode-coupling theory the longest relaxation time r scales as... 

Thus, in the fluid state, there are two relaxation processes, the a and the with relaxation times that scale with proximity to the critical point with differing exponents, -y and — l/2fl, respectively. For spherical particles, y — 2.58 and l/2a = 1.66 thus the a process is predicted to slow more dramatically as the transition is approached than the process. Figure 4-22 shows the relaxation times t and extracted from the relaxation data of Fig. 4-20 for the colloidal fluids. The power laws given by Eqns. (4-33) and (4-34) fit these experimental concentration dependencies well, supporting the mode-coupling theory of this transition. 

Thus, for molecular glass formers, mode-coupling theory might be helpful in understanding how the very fast molecular vibrations on the picosecond time scale begin to slow... 

Noteworthy is the strong evidence for the coincidence of Jg with the mode coupling theory (MCT) ergodic - non-ergodic crossover, critical , temperature MCT Moreover, the system-independent time-scale for the dynamic... 

Gottke et al. [5] offered a theoretical treatment of collective motions of mesogens in the isotropic phase at short to intermediate time scales within the framework of the Mode coupling theory (MCT). The wavenumber-dependent collective orientational time correlation function C/m(, t) is defined as... 

The current picture of relaxation behavior of supercooled liquids is complex. Mode coupling theory introduced slow a- and fast -processes, whose existence has been confirmed experimentally in almost all known glass forming liquids. The fast -process takes place on the picosecond time scale at all... 

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). " ... 

When an atomic system is cooled below its glass temperature, it vitrifies, that is, it forms an amorphous solid [1]. Upon decreasing the temperature, the viscosity of the fluid increases dramatically, as well as the time scale for structural relaxation, until the solid forms concomitantly, the diffusion coefficient vanishes. This process is observed in atomic or molecular systems and is widely used in material processing. Several theories have been developed to rationalize this behavior, in particular, the mode coupling theory (MCT) that describes the fluid-to-glass transition kinetically, as the arrest of the local dynamics of particles. This becomes manifest in (metastable) nondecaying amplitudes in the correlation functions of density fluctuations, which are due to a feedback mechanism that has been called cage effect [2],... 

Hinze, G., Brace, D.D., Gottke, S.D. and Payer, M.D. (2000). A detailed test of mode-coupling theory on all time scales Time domain studies of structural relaxation in a supercooled liquid. J. Chem. Phys. 113 3723-3733. 

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