Dynamic exponent

The exponents a and a+ depend not only on the relaxation exponent n, but also on the dynamic exponents s and z for the steady shear viscosity of the sol and the equilibrium modulus of the gel. 

The classical theory predicts values for the dynamic exponents of s = 0 and z = 3. Since s = 0, the viscosity diverges at most logarithmically at the gel point. Using Eq. 1-14, a relaxation exponent of n = 1 can be attributed to classical theory [34], Dynamic scaling based on percolation theory [34,40] does not yield unique results for the dynamic exponents as it does for the static exponents. Several models can be found that result in different values for n, s and z. These models use either Rouse and Zimm limits of hydrodynamic interactions or Electrical Network analogies. The following values were reported [34,39] (Rouse, no hydrodynamic interactions) n = 0.66, s = 1.35, and z = 2.7, (Zimm, hydrodynamic interactions accounted for) n = 1, s = 0, and z = 2.7, and (Electrical Network) n = 0.71, s = 0.75 and z = 1.94. 

The dynamic properties depend strongly on the material composition and structure. This is not included in current theories, which seem much too ideal in view of the complexity of the experimentally found relaxation patterns. Experimental studies involving concurrent determination of the static exponents, df and t, and the dynamic exponent, n, are required to find limiting situations to which one of the theories might apply. 

Immediately when the dynamic interpretation of Monte Carlo sampling in terms of the master equation, Eq. (31), was realized an application to study the critical divergence of the relaxation time in the two-dimensional Ising nearest-neighbor ferromagnet was attempted . For kinetic Ising and Potts models without any conservation laws, the consideration of dynamic universality classespredicts where z is the dynamic exponent , but the... 

The dynamic exponent depends on the transport mechanism (Mullins 1963),... 

Thus, depending on the mode of transport which is operative on the length and time scales of interest, any value for the dynamic exponent z between 5 and 8 can be expected for the surface diffusion case. Smaller values of z are also conceivable if the rare-event dominated top terrace dissociation or a miscut enters the game. A detailed analysis, however, is beyond the scope of present article. 

Thus, there is a continuous variation in the dynamical exponent for 1 < a < 3, while the attachment-detachment universality class holds for a < 1 and the step-edge universality class holds for a > 3. 

These critical exponents are often referred to as the static exponents because they are associated with equilibrium thermodynamic quantities. A second set, referred to as the dynamic exponents are associated with relaxation phenomena, e.g. correlation phenomena. 

By relating yA to the dynamic exponent z with yA = zv, we can have z 0.91. It should be meaningful to compare the obtained results here with those from other kinds of MIT in 2D systems. 

As indicated at the outset of the RG analysis, our primary interest centers upon the dynamical exponent z. Tlie relevance or irrelevance of the fluctuations of the potential <)6(x, t) is circumscribed by the explicit value of the dynamical exponent z. If the fluctuations of the potential have no effect on the transport and reaction kinetics of the solute, then the parameter /z in the above equations will flow successively toward smaller values on rescaling, eventually vanishing at the longest length scales (Z -> oo). In such a case, we can set z = 2, thereby leaving D and u invariant upon rescaling. Tliis would then preserve the kinetics as obtained in the absence of the potential (x, r) = 0. In the following (refer to Ref. 25 for details) we discuss the flow... 

Thus oj(q = 0) vanishes as co q = 0) oc Xt % Yl" — % <2 n>> and eq. (201) hence implies the classical value Zd = 2 — ij. Although eq. (206) thus suggests a relationship between the dynamic exponent and static ones, this is not true if effects due to non-mean-field critical fluctuations are taken into account. In fact, for the kinetic Ising model (Kawasaki, 1972) extensive numerical calculations imply that z. 2.18 in d = 2 dimensions (Dammann and Reger, 1993 Stauffer, 1992 Landau et al., 1988) rather than Zc = 2 - r) = 1.75. Note also [this is already evident from eq. (206)] that not all fluctuations slow down as Tc is approached but only those associated with long wavelength order parameter variations. One can express this fact in terms of a dynamic scaling principle... 

Critical Dynamical Exponents for the Gelation Threshold of Gelatin... 

This problem of comparing of the dynamic exponents ji and n seems to be not settled at all and is still a controversial issue. It must be emphasized that in several studies (Takeda et al. 2000 Norisuye et al. 1999, 2000) the power law exponent in DLS was discussed in the inconsistent context to the viscoelastic exponent n, while no rheological experiments were performed. To demonstrate this we carried out oscillatory shear rheology experiments. In Figure 21 the frequency-dependent storage and loss moduli for three selected temperatures are shown. The power law behaviour regarding Eq. (12) (G (co) oc ftj° G"(co) oc ftj° ) can be observed at 25°C with an averaged exponent of 0.7. 

Figure 9 shows a log-log plot of S(t) for the three oil concentrations studied. The black lines are linear fits to the data and suggest that the chain growth has a power-law behavior just like in ER fluids. The dynamic exponents range from... 

In Figure 4 we have plotted on logarithmic scales the correlation time of the Wolff algorithm for the 2D Ising model at the critical temperature, over a range of different system sizes. The slope of the line gives us an estimate of the dynamic exponent. Our best fit, given the errors on the data points is... 

Figure 4. The correlation time for the 2D Ising model simulated using the Wolff algorithm. The measurements deviate from a straight line for small system sizes L, but a fit to the larger sizes, indicated by the dashed line, gives a reasonable figure of z = 0.25 0.02 for the dynamic exponent of the algorithm.

In fact, in studies of the Wolff algorithm for the 2D Ising model, one does not usually bother to make use of Eq. (2.5) to calculate r. If we measure time in Monte Carlo steps (i.e., simple cluster flips), we can define the corresponding dynamic exponent zsteps in terms of the correlation time rsteps of Eq. (2.5) thus ... 

The exponent zsteps is related to the real dynamic exponent z for the algorithm by... 

Figure 6. The correlation time T,tepl of the 20 Ising model simulated using the Wolff algorithm, measured in units of Monte Carlo steps (i.e., cluster flips). The fit gives us a value of zstep, = 0.50 0.01 for the corresponding dynamic exponent.

Comparison of Dynamic Exponent z for the Metropolis, Wolff and Swendsen-Wang Algorithms in Various Numbers of Dimensions11... 

Figures are taken from Coddington and Baillie [5], Matz et al. [10], and Nightingale and Blote [2]. To our knowledge, the dynamic exponent of the Metropolis algorithm has not been measured in four dimensions. 

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