Relaxation of the order parameter

Suppose that for t 0, the polymers are oriented by an external field. If the eld is switched off at t = 0, the polymers will return to the equilibrium state. We shall first study this relaxation process using the kinetic equation (10.48). 

Let M be the dhection towards whidi the polymers have been oriented by the external field. This direction will coindde with the director if the system is in the nematic phase. Since the system will retain uniaxial symmetry around m during the relaxation process, the order parameter tensor is vnritten as 

In this case the system can become either an isotropic state or a nematic state depending on the initial value of S. 

Though this analysis does not tell where phase separation takes place (since the system is assumed to be homogeneous in this treatment), the general behaviour of the phase transition is described by the approximate kinetic equation (10.48). 

Equation (10.58) gives the equilibrium order parameter 5 q as a function of the concentration. Since 

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The results (6.33) and (6.35) are of principal importance. We already know that at the second order transition the structural susceptibility diverges (Curie law). Now we see that relaxation times diverge as well, i.e., on approaching the transition from any side, the relaxation of the order parameter becomes slower and slower and, at... 

The dynamics of the electroclinic effect is, in fact, the dynamics of the elastic soft mode. From Eqs. (13.18) and (13.19) follows that the switching time of the effect is defined only by viscosity and the term a(T — T ) and is independent of any characteristic size of the cell or material. It means that the relaxation of the order parameter amplitude is not of the hydrodynamic type controlled by term Kq (K is elastic coefficient). For the same reason Xg is independent of the electric field in agreement with the experimental data, shown in Fig. 13.9b. At present, the electroclinic effect is the fastest one among the other electro-optical effects in liquid crystals. 

For low-Reynolds-number fluids the second term in the right-hand side of the Navier-Stokes equation can be neglected. Additionally, assuming that the viscous relaxation occurs more rapidly than the change of the order parameter, the acceleration term in Eq. (65) can be also omitted. Such approximations are validated in the case of polymer blends, for which they become exact in the limit of infinite polymer length, N —> oo. After these approximations, the NS equation can be easily solved in the Fourier space [160]. 

In this nonlinear relaxation time, < is the order parameter for the initial condition, in which the Monte Carlo run is started, and t = is simply the equilibrium value of the order parameter. 

Deuterium nmr describes in equally clear-cut and elegant a manner the static (order parameter) and the microdynamic (correlation time t ) properties of hydrocarbon chains in aggregated surfactants 13). The example of octanoate, in various phases, resembles the membrane behavior of phospholipids a decrease of the order parameter is found for deuterium relaxation rates,... 

Since the total concentration a + r + s follows the time evolution d(a + r + s)/ dt = F - k(a + r + s), it approaches the steady state value F/k with a relaxation time 1 /k. This is a consequence of unbiased outflow (Eq. 47) of all reactants with the same rate k. Consequently, even though we are dealing with an open system under a flow, the analysis is similar to the closed system by replacing the total concentration c with the steady state value F/k. Instead of recycling, therefore, constant supply of the substrate allows the system to reach a certain fixed point with a definite value of the order parameter 0i, independent of the initial condition. 

Halle et al. (1981) measured NMR relaxation for solutions of several proteins as a function of frequency and protein concentration. They estimated hydration by use of a two-state fast-exchange model with local anisotropy and with assumed values of the order parameter and several other variables. The hydration values ranged from 0.43 to 0.98 h for five proteins, corresponding approximately to a double layer of water about a protein. The correlation time for water reorientation was, averaged over the set of proteins, 20 psec, about eight times slower than that for bulk water. A slow correlation time of about 10 nsec was attributed to an ordering of water by protein at very high concentration. 

An important advantage of the depolarization technique is that it allows one to measure the molecular ordering, as well as the motional parameters. For this purpose, it is necessary to detect the time dependence of the anisotropy. In the presence of ordering constraints, the r value does not decay to zero, but to some limiting value foo r = (ro — roo)e / c - - poo. The rate of decay defines a rotational correlation time, and Poo is a direct measure of the order parameter through the following relation s = Poo/ o (29). The fluorescence depolarization method works well as long as fluorescence lifetimes, which are typically 10 s, are not too different from the rotation relaxation times to be measured. When the rotational correlation time... 

The relation between W and Ws, the domain wall widths in the bulk and at the surface, can be seen in Figure 8. The effect of the surface relaxation is clearly visible as the order parameter at the surface Qs never reaches the bulk value Qo- The distribution of the square of the order parameter at the surface shows the structure that some of the related experimental works have been reported (Tsunekawa et al. 1995, Tung Hsu and Cowley 1994), namely a groove centred at the twin domain wall with two ridges, one on each side. 

The grand-canonical ensemble is particularly well suited for studies of liquid-vapor phase coexistence (i) Fluctuations of the order parameter, i.e., the density, are efficiently relaxed. Since the density is not conserved, spatial fluctuations do not decay via slow diffusion of polymers but relax much faster through insertion/deletion moves. In the grand-canonical ensemble one controls the temperature, T, the volume, V, and the chemical potential, p,... 

Suitable definitions of regions A and B may require considerable trial and error. Fortunately, it is straightforward to diagnose an unsuccessful order parameter. For example, most short trajectories initiated from the state Xq will quickly visit states with values of q characteristic of state B. In other words, the probability to relax into B is close to one. (This relaxation probability plays an important role in the analysis of transition pathways, as will be discussed in detail in Section V.) In contrast, the probability to relax into B from is negligible. When relaxation probabilities indicate that definitions of A and B do not exclude nonreactive trajectories, the nature and/or ranges of the order parameter must be refined. 

The scattered light intensity correlation function C (/) has been measured over a very wide range of temperature and wave vectors for various systems. Typical intensity correlation functions C-(/) are depicted in Fig. 4 for the sake of illustration. These graphs show systematic deviations from the usual exponential decay, which are also observed for most of the systems we report in this part. As a remark, nonexponential decays that are small at low concentration, close to the critical point, become large for dense systems. From the initial slope of the time-dependent intensity correlation function, one can deduce the first cumulant F, which is the relaxation rate of the order parameter fluctuations. 

Because the equilibrium order in heterophase systems is characterized by only one nonzero degree of freedom of the order parameter tensor, the fluctuation modes of all five degrees of freedom are uncoupled. Due to the uniaxial symmetry of the phase the two biaxial modes are degenerate and so are the two director modes. If a nematic layer is bounded by walls characterized by a strong surface interaction and a bulk-like value of the preferred degree of order, the fluctuation modes /3j s are sine waves, and their relaxation rates may be cast into... 

We neglect fluctuations of the order parameter, and assume that there is a simple linear relationship between a torque dg dr and a relaxation rate dr ldt. Physically it means that the steeper potential well g(ri) (larger dg dr ), the faster is relaxatimi (larger dr ldf) of the induced order parameter. Hence the Landau-Khalatnikov equation reads... 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x—>oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L—>oo), the distortion wavevector 0 and the... 

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