Slow mode hydrodynamic modes

A third kind of slowness, that due to hydrodynamic modes, has been discussed already. It is difficult to do anything about these slow collective modes, but fortunately they cannot cost very many orders of magnitude in a system of a few thousand atoms or less. 

II. Hydrodynamic Modes The Basic Dynamical Variables in Liquids HI. Slow Dynamics at Large Wavenumbers de Gennes Narrowing... 

The slow (diffusive) mode, whose diffusion coefficient is 1 -2 orders of magnitude lower than that of the fast mode, may be interpreted as arising from the presence of large-scale heterogeneities [87], The diffusion coefficient of the slow mode, Ds, is 5 x 10 s cm2/s for the ionomer solution (as seen in Figure 11). This corresponds to a hydrodynamic radius, Rh, of 475 A, 3.6 times that of the unmodified PS, 130 A (with a diffusion coefficient of 1.8 X 10 7 cm2/s). Here, the Stokes-Einstein relation,... 

This surprising result prompted Mazenko, Ramaswamy and Toner to examine the anharmonic fluctuation effects in the hydrodynamics of smectics. We have already shown that the undulation modes are purely dissipative with a relaxation rate given by (5.3.39). To calculate the effect of these slow, thermally excited modes on the viscosities, we recall that a distortion u results in a force normal to the layers given by (5.3.32). This is the divergence of a stress, which, from (5.3.53), contains the non-linear term 0,(Vj uf. Thus, there is a non-linear contribution (Vj uf to the stress. Now the viscosity at frequency co is the Fourier transform of a stress autocorrelation function, so that At (co), the contribution of the undulations to the viscosity, can be evaluated. It was shown by Mazenko et that Atj(co) 1 /co. In other words, the damping of first and second sounds in smectics, which should go as >/(oo)oo , will now vary linearly as co at low frequencies. 

It should be noticed that the time evolution of these (quasi-conserved) hydrodynamic mode variables becomes very slow as A — 0. 

It is well established that the principal results of the generalized kinetic theory, especially the functional form of the slow portion of the memory function, can be obtained also by a direct mode-coupling approach [18, 19, 20]. The basic idea behind the mode-coupling theory is that the fluctuation of a given dynamical variable decays, at intermediate and long times, predominantly into pairs of hydrodynamic modes associated with quasi-conserved dynamical variables. The possible decay channels of a fluctuation are determined by selection rules based, for example, on time-reversal symmetry or on physical considerations. 

For finite wavelengths, the collective dynamics of bulk nematics can be described within the hydrodynamic equations of motion introduced by Ericksen [4-8] and Leslie [9-11]. A number of alternate formulations of hydrodynamics [12-18] leads essentially to the equivalent results [19]. The spectrum of the eigenmodes is composed of one branch of propagating acoustic waves and of two pairs of overdamped, nonpropagating modes. These can be further separated into a low- and high-frequency branches. The branch of slow modes corresponds to slow collective orientational relaxations of elastically deformed nematic structure, whereas the fast modes correspond to overdamped shear waves, which are similar to the shear wave modes in ordinary liquids. 

The dynamics of concentration fluctuations expressed by eqn [161] (or eqn [34]) of the Omstdn-Zemike type exhibits the diflrrsion coeflrdent D =feT/(6nj 0), eqn [72], which can also be derived from fluctuations of gel-like networks with screening of hydrodynamic interactions. Thus, it is often rderred to as the gd mode. A decrease in mesh size increases the rdaxation rate of the gel mode. Experimentally, DLS for semidilute solutions rrsually exhibits another mode of motions with a very slow decay rate. This slow mode may be assodated with some inhomogeneity depending on solvent quality and sample preparation, or may be rdated to the translational... 

It only operates in continuous mode and uses catalyst particles of a slightly larger size than in BCR an upward flow of L maintains S in suspension, but the L velocity should be slower than the S settling velocity. Stability also requires a very narrow particle size distribution. Hydrodynamics and mass transfer depend on G/L flow ratio. G velocity is usually rather slow, with bubbles rising through a continuous L phase. Heat removal is restricted to use of wall exchangers. 

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

Turning our attention to the fast portion of the memory function, [Fl(A , t)]fast it is clear that this part is badly defined since it consists of a strictly binary part Mb,h and the term mu representing couplings to all the fast non-hydrodynamic (i.e., H > A) modes. However, it turns out in practice that a detailed knowledge of the functional forms of Mb,11 and mu is hardly necessary, and one can proceed as follows. As mentioned above, the effects of non-binary slow contributions start as As a result, up to O(t ) in the short-time region, the full and the fast portion of the memory function coincide, and the functional form of the fast portion can be deduced by analyzing the short-time behavior of the full memory function. This idea can be employed to write... 

---