Time dependence self diffusion coefficient

FIGURE 4.12 Volume fraction dependence of the short- (squares) and long-time (circles) self-diffusion coefficients. Both coefficients are normalized by the free diffusion coefficient... 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

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...

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

The practical advantage of these relations is that, in MD simulations, single molecule properties like the self-diffusion coefficient and rotational relaxation times converge much faster than system properties due to additional averaging over the number of molecules in the ensemble. We applied eqs. 10 and 11 to our MD results using data at 800 K as a reference point in order to predict the viscosity over the entire temperature interval. In Fig. 7 we compare the predicted values with those obtained from simulation. It appears that in the temperature interval 600 K to 800 K predictions of Eq. (10) are more consistent with MD results than are the predictions of Eq. (11). This leads us to conclude that the viscosity temperature dependence in liquid HMX is more correlated... 

The ApBq and A Bn layers are seen to grow parabolically, whereas the thickness of the ArBs layer will gradually decrease with passing time. Eventually, this layer will disappear. It is easy to notice that in this case the values of the diffusional constants k[A2 and kim can readily be determined from the experimental dependences x2- t and z2- t, respectively, using an artificially prepared specimen A-ApBq-ArBs-A iB,-B or A-A,B-B. It is essential to mention that both the ApBq and AtBn layers must be the first to occur at the A-B interface. The diffusional constant k[A2 thus obtained is the reaction-diffusion coefficient of the A atoms in the ApBq lattice, while the diffusional constant klB3 is the reaction-diffusion coefficient of the B atoms in the Afin lattice, to be compared with respective self-diffusion coefficients determined using radioactive tracers. 

Powerful methods for the determination of diffusion coefficients relate to the use of tracers, typically radioactive isotopes. A diffusion profile and/or time dependence of the isotope concentration near a gas/solid, liq-uid/solid, or solid/solid interface, can be analyzed using an appropriate solution of - Fick s laws for given boundary conditions [i-iii]. These methods require, however, complex analytic equipment. Also, the calculation of self-diffusion coefficients from the tracer diffusion coefficients makes it necessary to postulate the so-called correlation factors, accounting for nonrandom migration of isotope particles. The correlation factors are known for a limited number of lattices, whilst their calculation requires exact knowledge on the microscopic diffusion mechanisms. 

In both cases, an analysis of the diffusion front as a function of time shows a t2 dependence of the front line which indicates Fickian diffusion and allows for the determination of a diffusion coefficient according to x2 = 2Dt from the slope of the curves in Fig. 24. The inter-diffusion coefficient was measured at [1.15 + 0.05] x 10 9m2/s, while the self-diffusion coefficient was measured at [8.4 + 0.5] x 10 10m2/s, which is the same order of magnitude as that recorded for non-swelling technical-grade kaolinite at similar water content.97 This indicates... 

The dynamic characteristics of adsorbed molecules can be determined in terms of temperature dependences of relaxation times [14-16] and by measurements of self-diffusion coefficients applying the pulsed-gradient spin-echo method [ 17-20]. Both methods enable one to estimate the mobility of molecules in adsorbent pores and the rotational mobility of separate molecular groups. The methods are based on the fact that the nuclear spin relaxation time of a molecule depends on the feasibility for adsorbed molecules to move in adsorbent pores. The lower the molecule s mobility, the more effective is the interaction between nuclear magnetic dipoles of adsorbed molecules and the shorter is the nuclear spin relaxation time. The results of measuring relaxation times at various temperatures may form the basis for calculations of activation characteristics of molecular motions of adsorbed molecules in an adsorption layer. These characteristics are of utmost importance for application of adsorbents as catalyst carriers. They determine the diffusion of reagent molecules towards the active sites of a catalyst and the rate of removal of reaction products. Sometimes the data on the temperature dependence of a diffusion coefficient allow one to ascertain subtle mechanisms of filling of micropores in activated carbons [17]. 

Sturz and DoUe measured the temperature dependent dipolar spin-lattice relaxation rates and cross-correlation rates between the dipolar and the chemical-shift anisotropy relaxation mechanisms for different nuclei in toluene. They found that the reorientation about the axis in the molecular plane is approximately 2 to 3 times slower than the one perpendicular to the C-2 axis. Suchanski et al measured spin-lattice relaxation times Ti and NOE factors of chemically non-equivalent carbons in meta-fluoroanihne. The analysis showed that the correlation function describing molecular dynamics could be well described in terms of an asymmetric distribution of correlation times predicted by the Cole-Davidson model. In a comprehensive simulation study of neat formic acid Minary et al found good agreement with NMR relaxation time experiments in the liquid phase. Iwahashi et al measured self-diffusion coefficients and spin-lattice relaxation times to study the dynamical conformation of n-saturated and unsaturated fatty acids. 

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