Dynamic structure factor long-time

The long-time behavior (Q(Q)t) > 1 of the coherent dynamic structure factors for both relaxations shows the same time dependence as the corresponding incoherent ones... 

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

It is noteworthy that the neutron work in the merging region, which demonstrated the statistical independence of a- and j8-relaxations, also opened a new approach for a better understanding of results from dielectric spectroscopy on polymers. For the dielectric response such an approach was in fact proposed by G. Wilhams a long time ago [200] and only recently has been quantitatively tested [133,201-203]. As for the density fluctuations that are seen by the neutrons, it is assumed that the polarization is partially relaxed via local motions, which conform to the jS-relaxation. While the dipoles are participating in these motions, they are surrounded by temporary local environments. The decaying from these local environments is what we call the a-process. This causes the subsequent total relaxation of the polarization. Note that as the atoms in the density fluctuations, all dipoles participate at the same time in both relaxation processes. An important success of this attempt was its application to PB dielectric results [133] allowing the isolation of the a-relaxation contribution from that of the j0-processes in the dielectric response. Only in this way could the universality of the a-process be proven for dielectric results - the deduced temperature dependence of the timescale for the a-relaxation follows that observed for the structural relaxation (dynamic structure factor at Q ax) and also for the timescale associated with the viscosity (see Fig. 4.8). This feature remains masked if one identifies the main peak of the dielectric susceptibility with the a-relaxation. 

Thus we note that the memory kernel has a short-time and a long-time part. It is the long-time part which is not present in the viscoelastic model, becomes important in the supercooled-liquid-near-glass transition, and gives rise to the long-time tail of the dynamic structure factor. 

It has been discussed in the previous section that the long-time part in the memory function gives rise to the slow long-time tail in the dynamic structure factor. In the case of a hard-sphere system the short-time part is considered to be delta-correlated in time. In a Lennard-Jones system a Gaussian approximation is assumed for the short-time part. Near the glass transition the short-time part in a Lennard-Jones system can also be approximated by a delta correlation, since the time scale of decay of Tn(q, t) is very large compared to the Gaussian time scale. Thus the binary term can be written as... 

As discussed by Kirkpatrick [30], T( ) can be replaced by T( m) since there is a marked softening near this wavenumber. The maximum contribution from Eq. (225) in the long time comes when both the dynamic structure factors are evaluated near qm. The Gaussian part of the dynamic structure factor can also be neglected in the asymptotic limit. Thus the long-time part of the memory function now contains the vertex function and a bilinear product of the dynamic structure factors, all evaluated at or near qm. To make the analysis simpler, the wavenumber dependence of all the quantities are not written explicitly. 

In determining the dynamic structure factor, the value of Do (7o) is needed to be specified. Here the decoupling is studied as a function of the change of density. A noticeable long-time tail in density relaxation appears only very near to the glass transition line. This makes the choice of 7o or p0 rather easy. It is found that, for the reduced temperature, T = 0.8, the reduced density is 0.91. 

The long-time tail in the solvent dynamic structure factor is responsible for the large value of the viscosity in supercooled liquid. The solute dynamics for smaller solutes are faster than that of the solvent. Thus, the solute dynamics is decoupled from this long-time tail of the solvent dynamic structure factor. 

The behavior of VACF and of D in one-dimensional systems are, therefore, of special interest. The transverse current mode of course does not exist here, and the decay of the longitudinal current mode (related to the dynamic structure factor by a trivial time differentiation) is sufficiently fast to preclude the existence of any "dangerous" long-time tail. Actually, Jepsen [181] was the first to derive die closed-form expression for the VACF and the diffusion coeffident for hard rods. His study showed that in the long time VACF decays as 1/f3, in contrast to the t d 2 dependence reported for the two and three dimensions. Lebowitz and Percus [182] studied the short-time behavior of VACF and made an exponential approximation for VACF [i.e, Cv(f) = e 2 ], for the short times. Haus and Raveche [183] carried out the extensive molecular dynamic simulations to study relaxation of an initially ordered array in one dimension. This study also investigated the 1/f3 behavior of VACF. However, none of the above studies provides a physical explanation of the 1/f3 dependence of VACF at long times, of the type that exists for two and three dimensions. 

Concerning the dynamic structure factor, we shall confine our attention to the incoherent case, where the self-correlation function B(0, ) only is required since we have q -4 1, it may be shown that the results are essentially valid for the coherent case as well (long-time limit) [86]. From,Eqn. (3.1.18) we get... 

It is noteworthy that in the Rouse limit the diffusion constant is independent of the coil s size.) From the preceding, in the long-time limit, the characteristic time ti/2 of the incoherent dynamic structure factor exp[- Q B(0, t)] is linked to Q by the power law... 

When the bead-and-spring chain is not in the ideal state, the intramolecular force is given in Eqn. (3.1.3). As it may be seen, in general, the force is not simply transmitted by first-neighboring atoms, but it has a long-range character. The relaxation times are given by Eqn. (3.1.11) after they are known, the dynamic viscosity i (cu) and the atomic correlation function B(k, t) are obtained from Eqs. (3.1.15) and (3.1.18) (for the periodic chain), and the complex modulus and dynamic structure factors are easily constructed. 

Dynamic light scattering (DLS) is an effective technique to measure the collective diffusion coefficients by measuring the time correlation of the concentration fluctuations. These experiments allow us to can determine the collective modes of the system that couple to concentration fluctuations. In binary systems this connection is quite useful since there is only one independent concentration variable. Then one can obtain the collective diffusion coefficient theoretically from the dynamic structure factor in the long wavelength limit. 

In the polymer problem, the validity of this equation is not obvious since the des ption of the polymeric system by c disregards the chain connectivity and, therefore, neglects the entanglement effect. However, as far as the dynamics in the short time-scale is concerned, this will not be a serious problemt since, as we shall discuss later, the topological constraints are not important in the short time-scale dynamics. Indeed it will be shown that the initial slope in the dynamical structure factor is correctly described in this approach. In the long time-scale, on the other hand, the validity of eqn (5.88) is not clear, and it may well be that the theory has to be modified in future. Fortunately, many experiments related to concentration fluctuations are concerned with the short time-scale motion, so that it is worthwhile to pursue the idea in detail. 

SuiQ, E) = SiiQ, E) 0 Sr Q, E), that is, a convolution of the translational dynamic structure factor, Si(Q,E), and the rotational one, 5r(<2, ) In addition, for small Q spectra, Q < 1 A the 5r(<2, E) can be made negligibly small, hence 5 h((2, E) Si(Q, E) and its Fourier transform will give the self-intermediate scattering function F Q, t) that have a stretched exponential FniQj) = exp [ - r (g) r] long-time decay. When the T is above the room temperature, P 1. A situation for which the exponential form Eh(Q, t) exp(—r(g)/) can be approximately used, or equivalently, in frequency domain theSnCg, E) of water is approximated as a Lorentzian shape function [67],... 

We consider the dynamic structure factor of a rodlike molecule. The long-time behavior is rather trivial. The orientational distribution will be averaged, and the center-of-mass diffusion alone will survive. Then,... 

If the AIMD simulation is performed over a sufficiently long time, it is possible to compute the dynamical structure factor S q, w) for frequencies ranging from l/T,T being the total simulation time, and 1/t, t being the simulation time step, and for q... 

The discussion of the large- tail in S(q) in Section 7.4.1.1, which is characteristic for a short-ranged attraction, enables one to formulate a simplified theory of bond formation within MCT with the result that the long-time limit of the dynamic structure factor is controlled by a single interaction parameter, F = fP-(p/b. Bond formation occurs at T, = 3.02... [34]. For small values of F, the dynamic structure factor decays to zero for all wavevectors. Physically, this means that concentration fluctuations decay into equilibrium at long times, just as expected for a colloidal fluid. However, for F > F, the solutions yield a nonzero glass form factor, namely, the system arrests in a metastable state. This simple result requires the approximate expression for S(q) given above and needs to be replaced by a full numerical solution whenever this approximation fails. 

Detailed high-frequency (terahertz) dynamical studies of glasses have been probed by inelastic X-ray scattering (IXS) [139], The advantage of this technique is that with reliable measurements it allows determination of the so-called nonergodicity parameter f(q, T) as a function of wavevector q this quantity is defined by the long time limit of the density-density correlation function F(q, t) divided by the static structure factor [15],... 

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