Relaxation times and diffusion coefficients

Let us now consider an example of how to calculate dynamic properties of pure liquids by computational methodologies. As already said in Section 8.7.2, molecular dynamics simulations are able to take into account flie time-dependence in the calculation of liquid properties. 

Paying attention to pure acetone, Brodka and Zerda have calculated rotational relaxation times and translational diffusion coefficients by molecular dynamics simulations. In particular, the calculated rotational times of flie dipole moment can be compared with a molecular relaxation time Xm obtained from the experimentally determined Xb by using the following expression, which considers a local field factor  

Diffusion eoeffieients can be calculated directly from the velocity correlation functions or from mean square displacements, as  

The values obtained by using the [8.140] or the [8.141] are almost the same, and properly deseribe the experimental temperature and density dependeneies of the diffusion coefficients, even if about 30% smaller than obtained using NMR spin-eeho techniques. 

Among the pure solvents we have treated so far, other available data regard the calculation of the self diffusion eoefficient (D) for liquid ethanol at different temperatures. The D parameter was obtained from the long-time slope of the mean-square displacements of the eenter of mass experimental ehanges of D over the 285-320K range of temperatures were acceptably reproduced by moleeular dynamies simulations. 

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We have applied the discrete-exchange model to these data. An exchange between Na ion under a particular slow-motion condition and in the extreme narrowing limit is assumed. Transverse relaxation time and diffusion coefficient are written as follows ... 

Consider a molecule made out of two /-arm stars with Kuhn segments per arm with junction points connected by a central linear strand of Abb Kuhn monomers. This molecule is called a pom-pom polymer. If/= 1, this molecule is linear, while the H-polymer corresponds to /=2. Estimate the /-dependence of relaxation time and diffusion coefficient of a melt of monodisperse pom-pom polymers for /> 1. Consider only single-chain modes and assume that the coordination number of an entanglement network is z. 

The investigation of relaxation times and diffusion coefficients requires the determination of the eigenvalues of the matrix corresponding to the system of algebraic equations obtained from Eqs. (18) after Fourier-Laplace transformation (s, Laplace transform of time q, Fourier transform of the space coordinate). The roots of the secular equation are... 

In many polymer systems, polydispersity is a feature that must be accounted for in analysing the results of PGSE NMR measurements. Allowing for both relaxation time and diffusion coefficient dependence on molar mass, the echo attenuation expression in equation (9.6) must be modified to read... 

Average Relaxation Time (s) Diffusion Coefficients (s ) and Energy Barriers (kj/mol) for Compounds Having Isotropic Motion ... 

For a given sequence, Bloch equations give the relationship between the explanatory variables, x, and the true response, i]. The / -dimensional vector, 0, corresponds to the unknown parameters that have to be estimated x stands for the m-dimensional vector of experimental factors, i.e., the sequence parameters, that have an effect on the response. These factors may be scalar (m — 1), as previously described in the TVmapping protocol, or vector (m > 1) e.g., the direction of diffusion gradients in a diffusion tensor experiment.2 The model >](x 0) is generally non-linear and depends on the considered sequence. Non-linearity is due to the dependence of at least one first derivative 5 (x 0)/50, on the value of at least one parameter, 6t. The model integrates intrinsic parameters of the tissue (e.g., relaxation times, apparent diffusion coefficient), and also experimental nuclear magnetic resonance (NMR) factors which are not sufficiently controlled and so are unknown. 

There are other physical measurements which reflect molecular mobility and can be related to relaxation times and friction coefficients similar to those which characterize the rates of viscoelastic relaxations. Although such phenomena are outside the scope of this book, they are mentioned here because in some cases their dependence on temperature and other variables can be described by reduced variables and, by means of equation 49 or modifications of it, free volume parameters can be deduced which are closely related to those obtained from viscoelastic data. These include measurements of dispersion of the dielectric constant, nuclear magnetic resonance relaxation, diffusion of small molecules through polymers, and diffusion-controlled aspects of crystallization and polymerization. 

In contrast, the dynamic properties such as viscosity, viscoelastic relaxation time, gel diffusion coefficient, which depend on the entanglements and thus on the length f, do not scale with C/C. ... 

The entanglement criterion for necklaces is similar to that for a normal PE. Also, the number of entanglement sites is the same, and the entanglement concentration can be determined by Ce = n c [94]. The only difference is the tube diameter and the correlation length defining it. The relaxation time, self-diffusion coefficient and viscosity of entangled PE necklaces can be determined by [94] ... 

The skin layers from the palm of the hand were scanned in vivo. A CPMG sequence was applied to sample the echo train decays as a function of depth. The decay was determined by both the relaxation time and the diffusion coefficient. To improve the contrast between the layers, a set of profiles was measured as a function of the echo... 

In the forthcoming sections we will consider several methods that have been used to derive different integral relaxation times for cases where both drift and diffusion coefficients do not depend on time, ranging from the considered mean transition time and to correlation times and time scales of evolution of different averages. 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

These experiments suggest that as the long time self-diffusion coefficient approaches zero the relaxation time becomes infinite, suggesting an elastic structure. In an important study of the diffusion coefficients for a wide range of concentrations, Ottewill and Williams14 showed that it does indeed reduce toward zero as the hard sphere transition is approached. This is shown in Figure 5.6, where the ratio of the long time diffusion coefficient to the diffusion coefficient in the dilute limit is plotted as a function of concentration. 

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