Long-Time Diffusion Coefficient

Figure 2. Short time diffusion coefficient. Computed by sequential averages of T steps from single trajectory iterated for 1 x 1010 time steps. K = 0.8 and b = 0.001, 0.002, 0.004, 0.010, 0.020, 0.040, 0.100. We have no anomalous diffusion in long time steps.

This experiment thus affords two independent ways to test the reptation, or any other, model of polymer diffusion. At long times we can extract the center-of-mass diffusion coefficient (from a plot of I vs. t ) and determine, for... 

This equation (4.30) shows that a straight line is obtained when plotting the ratio M/M versus the square root of time. But, as already stated in Chapter 1, this relationship is of value only when the coefficient of convection is infinite, with a liquid strongly stirred, thus, this fact reduces the interest of this equation in calculating the diffusivity. In fact. Equation (4.30) can be used only as a first approach to obtain an approximate value of the diffusivity. With the same approximation. Equation (4.28) can be useful for determining an approximate value of the diffusivity for long times, as it reduces to the simple Equation (4.31) ... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

Aqueous diffusion coefficients are usually on the order of 5 x 10 cm /s. A second is typically a long time to an electrochemist, so 6 = 30 fim. The definition of far is then 30 ]lni. Short is less than a second, perhaps a few milliseconds. Microseconds are not uncommon. Small, referring to the diameter of the electrode, is about a millimeter for microelectrodes, or perhaps only a few micrometers for ultramicroelectrodes (13). 

Figure 13. Voltage relaxation method for the determination of the diffusion coefficients (mobilities) of electrons and holes in solid electrolytes. The various possibilities for calculating the diffusion coefficients and from the behavior over short (t L2 /De ) and long (/ L2 /Dc ll ) times are indicated cc h is the concentration of the electrons and holes respectively, q is the elementary charge, k is the Boltzmann constant and T is the absolute temperature.

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

The long-time behaviour of K (t) is obviously exponential, and the rate is proportional to the diffusion coefficient defined in Eq. (2.42). Therefore the expression... 

A number of bulk simulations have attempted to study the dynamic properties of liquid crystal phases. The simplest property to calculate is the translational diffusion coefficient D, that can be found through the Einstein relation, which applies at long times t ... 

Ionic, polar and amphiphilic solubilizates are forced to reside for relatively long times in very small compartments within the micelle (intramicellar confinement, compart-mentalization) involving low translational diffusion coefficients and enhancement of correlation times. 

MC simulations and semianalytical theories for diffusion of flexible polymers in random porous media, which have been summarized [35], indicate that the diffusion coefficient in random three-dimensional media follows the Rouse behavior (D N dependence) at short times, and approaches the reptation limit (D dependence) for long times. By contrast, the diffusion coefficient follows the reptation limit for a highly ordered media made from infinitely long rectangular rods connected at right angles in three-dimensional space (Uke a 3D grid). 

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. 

Under conditions of nonlimiting interfacial kinetics the normalized steady-state current is governed primarily by the value of K y, which is the relative permeability of the solute in phase 2 compared to phase 1, rather than the actual value of or y. In contrast, the current time characteristics are found to be highly dependent on the individual K. and y values. Figure 16 illustrates the chronoamperometric behavior for K = 10, log(L) = —0.8 and for a fixed value of Kf.y = 2. It can be seen clearly from this plot that whereas the current-time behavior is sensitive to the value of Kg and y, in all cases the curves tend to be the same steady-state current in the long-time limit. This difference between the steady-state and chronoamperometric characteristics could, in principle, be exploited in determining the concentration and diffusion coefficient of a solute in a phase that is not in direct contact with the UME probe. 

Since it was proposed in the early 1980s [6, 7], spin-relaxation has been extensively used to determine the surface-to-volume ratio of porous materials [8-10]. Pore structure has been probed by the effect on the diffusion coefficient [11, 12] and the diffusion propagator [13,14], Self-diffusion coefficient measurements as a function of diffusion time provide surface-to-volume ratio information for the early times, and tortuosity for the long times. Recent techniques of two-dimensional NMR of relaxation and diffusion [15-21] have proven particularly interesting for several applications. The development of portable NMR sensors (e.g., NMR logging devices [22] and NMR-MOUSE [23]) and novel concepts for ex situ NMR [24, 25] demonstrate the potential to extend the NMR technology to a broad application of field material testing. 

Pore shape is a characteristic of pore geometry, which is important for fluid flow and especially multi-phase flow. It can be studied by analyzing three-dimensional images of the pore space [2, 3]. Also, long time diffusion coefficient measurements on rocks have been used to argue that the shapes of pores in many rocks are sheetlike and tube-like [16]. It has been shown in a recent study [57] that a combination of DDIF, mercury intrusion porosimetry and a simple analysis of two-dimensional thin-section images provides a characterization of pore shape (described below) from just the geometric properties. 

The last issue we address concerns the existence of long-time tails in the discrete-time velocity correlation function. The diffusion coefficient can be written in terms of the velocity correlation function as... 

We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

Consider a long circular cylinder in which a solute diffuses radially. The concentration is a function of radial position (r) and time (t). In the case of constant diffusion coefficient, the diffusion equation is... 

In this fitting procedure the time-dependent diffusion coefficient as discussed above has been taken explicitly into account. As can be seen, with only two parameters both sets of spectra are well described. This holds also for the other two samples. For the model parameters, the fit reveals Nc = 150 and A (0)/ = 0.18 ns. Nc is very close to Nc = 138 as obtained from the NSE experiments on long-chain PE (see next section). 

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