Time-dependent diffusion, mode coupling

In order to understand the above questions/paradoxes, a mode coupling theoretical (MCT) analysis of time-dependent diffusion for two-dimensional systems has been performed. The study is motivated by the success of the MCT in describing the diffusion in 3-D systems. The main concern in this study is to extend the MCT for 2-D systems and study the diffusion in a Lennard-Jones fluid. An attempt has also been made to answer the anomaly in the computer simulation studies. 

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.

Collective Modes and Time-Correlation Functions. Our linear equations (6.15) and (6.16) describe two characteristic kinds of collective modes in gels the longitudinal part of u obeys Tanaka s equation (4.16) and g = V u is governed by the diffusion equation (4.18), while the transverse parts of u and are coupled to form a slow transverse sound at small wave numbers. By assuming the space-time dependence as exp(i[Pg.99]

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

The most important prediction of the mode coupling theory is the temperature or the density dependence of the relaxation time, tmc(< )- MCT predicts that this relaxation time grows as a power law as the glass transition is approached (from the supercooled liquid side). This is because the diffusion coefficient Do of the liquid goes to zero in the following fashion ... 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

Finally, the measurements of Chu and collaborators [140] reject the reptation model for their system. In the reptation model, polymer chains only perform free diffusion after escaping from their tubes at long times t > Td- At times t < Td, polymer chains are confined to their tubes and perform confined diffusion. For t < Td, the reptation model (and Schweizer s mode coupling calculation) [143] predict that a chain s mean-square displacement should satisfy (5x ) t where s is 0.5 or so. Ref. [140] reports that their chains remain within the hypothesized tubes for times Td 1.2 or > 2 minutes, depending on which estimator of is applied. However, ref. [140] measured directly (a (t)) from Fig, 2 of ref. [140], for times as short as Td/l. That is, these... 

The generation/collection (G/C) modes constitute a different SECM procedure that expands the applicability of the technique to a wide range of situations, hi these modes, the collector (either tip or substrate) works as an amperometric sensor that collects the products produced at the generator surface (either substrate or tip, respectively). Thus, the collector potential is controlled to electrochemically reaet with the generator-produced species. Typical collector responses used in G/C experiments are (a) voltammetric curves, where the collector potential is swept, and (b) diffusion-controlled limiting current vs. time curves. In contrast to the feedback mode where steady-sate responses are monitored, in G/C experiments, the current-time dependence is an important set of data to evaluate. The timescale of most of G/C transient experiments is much wider, possibly up to 100 sec. Moreover, as the tip-substrate distances increase, typical coupling and distortion of transient responses are not significant. 

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). " ... 

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