Wavevector dependence

The Ti q) behave as wavevector-dependent relaxation times and the form of the wavevector dependence can provide a useful check on the consistency of models. Table 5 shows a comparison of the experimental coefficients for fresh apple tissue with those calculated with the numerical cell model. The agreement is quite reasonable and supports the general theoretical framework. It would be interesting to apply this approach to mealy apple and to other types of fruit and vegetable. 

Figure 1.5. (a) Wavevector dependence of scattering intensity [10] at different times (b) time dependence of integrated intensity [10] (c) wavevector... 

These conclusions were later supported by time-resolved SAXS experiments by Stiihn et al. (1994) who studied the ordering of a PS-PI diblock with /PS = 0.44 following quenches from the disordered phase into the lamellar phase. They found that the relaxation times of the structure factor were wavevector dependent, and consistent with the Cahn-Hilliard from (Cahn and Hilliard 1958)... 

H. Mode Coupling Theory Calculation of Wavevector-Dependent Transport Properties... 

The projection operator Q in X eliminates all the local equilibrium states, which leads to the elimination of the eigenstates of L with small eigenvalues (the transport states). Q leaves the remaining states almost untouched. Hence, the wavevector dependent X can be written as... 

The mode coupling theory of molecular liquids could be a rich area of research because there are a large number of experimental results that are still unexplained. For example, there is still no fully self-consistent theory of orientational relaxation in dense dipolar liquids. Preliminary work in this area indicated that the long-time dynamics of the orientational time correlation functions can show highly non-exponential dynamics as a result of strong in-termolecular correlations [189, 190]. The formulation of this problem, however, poses formidable difficulties. First, we need to derive an expression for the wavevector-dependent orientational correlation functions C >m(k, t), which are defined as... 

Recently, an interesting study of the molecular dynamics calculation of the wavevector dependence of the viscosity and the thermal conductivity of a Lennard-Jones fluid was reported [202]. The transport properties were found to decrease rapidly as the value of the wavevector k was increased from zero, and they were nearly zero when kxs is larger than 5. However, we are not aware of any mode coupling theory calculation of this interesting behavior. In fact, most of the theoretical expressions exist, but the numerical calculation is formidable. 

The remainder of this contribution is organized as follows In the next section, the connection between the experimentally observed dynamic Stokes shift in the fluorescence spectrum and its representation in terms of intermolecular interactions will be given. The use of MD simulation to obtain the SD response will be described and a few results presented. In Section 3.4.3 continuum dielectric theories for the SD response, focusing on the recent developments and comparison with experiments, will be discussed. Section 3.4.4 will be devoted to MD simulation results for e(k, w) of polar liquids. In Section 3.4.5 the relevance of wavevector-dependent dielectric relaxation to SD will be further explored and the factors influencing the range of validity of continuum approaches to SD discussed. 

Figure 3.21 MD simulation results for (a) wavevector-dependent dielectric relaxation and (b) solvation dynamics in acetonitrile at room temperature. The charge density TCF qq(k, t) is separated into single-molecule <3>qq(k, f) and pair q(/c, t) contributions. The results for

B. M. Ladanyi and B.-C. Perng, Computer simulation of wavevector-dependent dielectric properties of polar and nondipolar liquids, in L. R. Pratt and G. Hummer (eds) Simulation and Theory of Electrostatic Interactions in Solution, AIP Conf. Proc., Melville, NY, 1999, Vol. 492, pp 250-264. 

The electromagnetic fields of the right- and left-propagating polaritons, respectively, follow the wave equations with the speeds and damping rates of the different frequency components dispersed according to the frequency- and wavevector-dependent complex refractive index n = v/e(k, oj). A typical example of the dispersion of these modes is shown in Fig. 1 for the case of a real permittivity e. The term Ao(r,t) represents the envelope of the wavepacket on the phonon-polariton coordinate A. Note that this phonon-polariton coordinate is a linear combination of ionic and electromagnetic displacements, which both contribute to the polarization... 

Equation (2.162) is a special case for a form factor of a fractal (with fractal dimension T> = 2). For any fractal, the wavevector dependence of the form factor gives a direct measure of the fractal dimension V ... 

Fig. 6 Computer simulations results of the scattering wavevector dependence of the form factors P( ) of a star with / = 24 arms (solid line) and a (compact lattice) hard sphere with nearly the same number of beads (dotted line). Taken from [41]... 

In the previous section, we have seen that it cannot suffice to consider the order parameter alone. A crucial role is played by order parameter fluctuations that are intimately connected to the various singularities sketched in fig. 11. We first consider critical fluctuations in the framework of Landau s theory itself, and return to the simplest case of a scalar order parameter (j ) with no third-order term, and u > 0 [eq. (14)], but add a weak wavevector dependent field <5 H(x) = SHqexp(iq x) to the homogeneous field H. Then the problem of minimizing the free energy functional is equivalent to the task of solving the Ginzburg-Landau differential equation... 

It is easy to see that the above treatment of the critical scattering and correlations in lerms of the wavevector dependent susceptibility goes through as previously, i.e. we still have y, = 1, i>t = 1/2, = 0 as in the standard Landau theory. But the behavior of the specific heat changes, since [cf. eq. (46)]... 

Here Xeff(q) is a wavevector-dependent generalization of the Flory-Huggins X parameter [78]. For q-> 0, Eq. (34) yields... 

These expressions can be used to derive the nearly total reflectance of metals below their plasma frequency. A similar characteristic frequency dependence of a(o)) and e((o) may be seen in semiconductors where oip depends the electron density in the fliled valence band. The conduction electrons can oscillate as a collective mode (plasma oscillation). A plasmon is a quantized plasma oscillation. The frequency and wavevector dependence of plasmons in one-dimensional metals have been predicted (458, 576) to be qualitatively different from those of three-dimensional metals. Recent direct measurements (552) of plasmons in the one-dimensional organic metal tetrathiofulvalinium-tetracyano-quinodimethanide (TTF)(TCNQ) are qualitatively consistent with some of the predictions assuming a tight-binding band (576). 

It is reasonable to expect that for dense fluids the decay of the memory function at intermediate and long times is dominated by those mode correlations which have the longest relaxation times. The sluggishness of the structural relaxation processes typical of dense liquids suggests that the slow decay of the memory function at long times is basically due to couplings to wavevector-dependent density modes of the form... 

Here p refers to a wavevector, not a momentum variable.) The slow portion of the memory function can then be expressed as a sum of products of the wavevector-dependent density modes provided a decoupling (or factorization) approximation is made (see below). 

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