Crossover length

The dynamic crossover occurs at fcross 40 ns. From r (tcross))= 50 a crossover length A is deduced. This length scale may be identified with the... 

Ferri, F., Greco, M., Arcovito, G., De Spirito, M., and Rocco, M. (2002). Structure of fibrin gels studied by elastic light scattering techniques Dependence of fractal dimension, gel crossover length, fiber diameter, and fiber density on monomer concentration. Phys. Rev. E. 66(1 Pt 1), 011913. 

For the material and field strength described in Section III.C.l, is on the order of 500 nm and A is approximately 1.9 nm. Therefore, the crossover length is on order of 100 /am. This value is far greater than the typical distance between defects. 

This relation illustrates the general trend that long-ranged fluctuations are suppressed in the limit of large invariant degree of polymerization, N —> The crossover length scale. A,... 

Fig. 5 Viscosity deteraiined using diffusion measurements of differently sized probes in PEG 20000 solution. Small probes experience nanoviscosity, whereas large probes follow the macroviscosity which is shown as the dashed black line as a guide to the eye. The crossover length scale between nano- and macroviscosity in PEG20000 is larger than 3.1 nm and smaller than 13 nm (reproduced and adapted from Holyst et al. [106] with permission of the PCCP Owner Societies)...

Raman and Brillouin studies were extended to silica aerogels produced with different procedures, having different microstructures, coimectivities and thus different vibrational dynamics. In particular, a series of samples were prepared with different densities p and thus different crossover lengths co, but with nearly identical structures on fractal scales, i.e. for distances smaller than fee SANS measurements showed a fractal behavior, allowing one... 

As is evident from the fomi of the square gradient temi in the free energy fiinctional, equation (A3.3.52). k is like the square of the effective range of interaction. Thus, the dimensionless crossover time depends only weakly on the range of interaction as In (k). For polymer chains of length A, k A. Thus for practical purposes, the dimensionless crossover time is not very different for polymeric systems as compared to the small molecule case. On the other hand, the scaling of to is tln-ough a characteristic time which itself increases linearly with k, and one has... 

In Fig. 20 we show the MSQ of a system of GM [66] with different mean chain lengths (depending on 7, cf. Eq. (12)) for three values of LO=l, 0.1, 0. 01. Since the individual chains have only transient identity, it is meaningless to discuss their center of mass diffusion. It is evident from Fig. 20 that the MSQ of the segments, g t) = ([x( ) - x(O)j ), follows an intermediate sub-diffusive regime, g(t) oc which is later replaced by conventional diffusion at some characteristic crossover time which grows... 

N = 32. (b) Crossover scaling description of i g and (c) vs the crossover scaling variable [AjA, where A = e/e — I for different chain lengths. The flat branch of each curve refers to [e] < [ed while the other branch refers to [e] > [ed. Ideally, the slopes of the latter branches should yield the theoretical exponents expected for large A N, 2 u2 — u)/ip 0.55, and -lujip a -2.03, respectively [13]. 

A crossover scaling for and R of a polymer chain in a quenched random medium may be developed if one introduces a correlation length... 

Figure 1. Crossover scaling plot for tlie order parameter ( m > = ( ( ia - Bl / (<1>a + B)> of a symmetrical polymer mixture simulated by tlie bond fluctiiatioii model on tlie simple cubic lattice, with a concentration (jiv = 0.5 of vacant sites. Here N " ( m > is plotted vs. N t, and chain lengths from N = 32 to N = 512 are...

A lower bound on the overall effect of crossover, which can both create and destroy instances of a given schema, can be estimated by calculating the probability, Pc S), that crossover leaves a schema S unaltered. Let be the probability that the crossover operation will be applied to a string. Since a schema S will be destroyed by crossover if the operation is applied anywhere within its defining length, the probability that S will be destroyed is equal to Pc x 6 S)/ K — 1), where 6 S) is the defining length of S. Hence, the probability of survival ps = 1 — PcS S)/ K — 1), and equation 11.9 takes the updated form ... 

Figure 7. Experimental data (symbols) for TNB s viscosity [78] superimposed on the results of the fitting procedure (line) from Lubchenko and Wolynes [47] are shown. Ta is diown by a tickmark. (TNB = trinaphthyl benzene). The temperature Ter signifies a crossover from activated to collisional viscosity, dominant at the lower and higher temperatures, respectively (see text). The temperature is varied between the boiling point and the glass transition. The right-hand side panel depicts the temperature dependence of the length scales of cooperative motions in the liquid. The thick solid and dashed lines are the critical radius and the cooperativity length respectively. Taken from Ref. [47] with permission.

Fig. 39. Crossover from 0- to good solvent conditions in dilute solutions. Calculated characteristic frequencies, normalized to 0-conditions, as dependent on reduced temperature for two different chain lengths N at various values of (QS). To the right of each curve the increase in Qred (q,x) between t = 0 and t = 0.9 is given...

For a given chain length, the sharpness of the crossover as well as the increase in Qred(Q, t) are diminished with increasing (Q/)-values ... 

If the crossover points Q (x) are determined from Fig. 45, taking the x-values at half-step height, Q (x) = 1/1 (x) = (0.7 + 0.2)x is obtained in the case of the PS system. This has to be compared with static value Qs (x) = 1.6x, derived from the same polymer solvent system by elastic neutron scattering [103], As long as no corresponding data from other polymer solvent systems are available, the final decision as to whether static and dynamic scaling lengths coincide or not, is still open. 

On macroscopic length scales, as probed for example by dynamic mechanical relaxation experiments, the crossover from 0- to good solvent conditions in dilute solutions is accompanied by a gradual variation from Zimm to Rouse behavior [1,126]. As has been pointed out earlier, this effect is completely due to the coil expansion, resulting from the presence of excluded volume interactions. 

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