First cumulant

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

The crossover from 0- to good solvent conditions leads at constant x = (x — 0)/0 to increasing Q(Q,x)/Q3 with decreasing Q. Qualitatively, this effect is well described in the framework of the blob model using the method of the first cumulant, proposed by Akcasu and coworkers... 

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

The incorporation of non-Gaussian effects in the Rouse theory can only be accomplished in an approximate way. For instance, the optimized Rouse-Zimm local dynamics approach has been applied by Guenza et al. [55] for linear and star chains. They were able to obtain correlation times and results related to dynamic light scattering experiments as the dynamic structure factor and its first cumulant [88]. A similar approach has also been applied by Ganazzoli et al. [87] for viscosity calculations. They obtained the generalized ZK results for ratio g already discussed. 

At large Q quantitative agreement between experiment and RPA prediction is found. Here the first cumulant is proportional to - we are in the Rouse... 

Figure 6.25 shows the dispersion of the effective diffusion Fq/Q with Fq the first cumulant of the relaxation function. Three branches are visible. A collec-... 

Mobility matrix Tube survival probability Neutron wavelength First cumulant matrix Rouse variable Osmometic pressure Conductivity... 

In this section some details of the static and dynamic structure factors and on the first cumulant of the time correlation function are given. Hie quoted equations are needed before the cascade theory can be applied. This section may be skipped on a first reading if the reader is concerned only with the application of the branching theory. 

As was shown by Bixon75, Ackerson79 and Akcasu and Gurol80, the first cumulant can be calculated exactly without knowing the space-time distribution function with the following result... 

Note, that non only the ensemble average is needed. In the following we confine ourselves to the first cumulant, i.e. to the initial part of the TCF, and drop the subscript 1 in Eq. (B.37). 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

In Chap. B II.4 we have shown that the angular dependence of the first cumulant of the electric field correlation function can be obtained by integration over the particle-scattering factor. This rule remains valid also for copolymers but is restricted to Gaussian behavior of the subchains. Although the whole q-region can be covered by this integration, which in most cases has to be carried out numerically, it is useful to discuss the... 

Fig. 35. Dependence of the reduced and normalized first cumulant I7D2 on u2 = (S2)q2 for regular stars. Insert behavior at small u2102)...

Fig. 38. Theoretically predicted u2-dependence of the normalized first cumulant Dapp/D for the soft sphere model93 ...

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