Zimm approximation

Zimm approximation [4], introduced in Chapter 11, valid for > 1, was used to compute absolute weight-... 

Zero corresponds to a finite limit of the viscosity, zero with a prime to its logarithmic divergence. The first least reliable line refers to the Rouse approximation, Eq. (17 f), the second line to the excluded volume (Zimm) approximation, Eq. (17 b), the last and most reliable line to the superconductor analogy of Eq. (18)... 

Figure 13 (a) Zimm representation within the Zimm approximation the plot of 1/P versus CF gives a straight line. The slope of the straight line is proportional to Rg. (b) Kratky plot of Debye s form factor at large (3 the representation of (fP[Q) versus 0 gives a constant value. 

For small QRg, the stmcture factor can be written in Zimm approximation as... 

To generate the necessary distribution functions, the ratio of is used to approximate the true molecular weight distribution by a Schulz-Zimm distribution. It is also assumed that the reactive functional groups are distributed randomly on the polymer chain. The Schulz-Zimm parameters used to calculate distribution functions and probability generating functions (see below) are defined as follows ... 

The weight-average Molecular weightin the inverse of the intercept at c = 0 and q = 0 in the Zimm plot. The second virial coefficient can be calculated from the slope of the lines at constant angle by eq. 8.34, and to a first approximation is independent of angle. The radius of gyration is... 

Millar et al,(20) originally assumed that 0 = 90° and that internal motion of the dye does not affect the amplitude of the n = 0 term. Subsequent work partially remedied these deficiencies, but the resulting torsion constants are invalid due to use of the incorrect anisotropy formula of Barkley and Zimm for nonvanishing < Jx(/)2>.(19) Ashikawa and co-workers also assumed that eo = 90° and used a simple anisotropy expression that is an approximation to the incorrect formula of Barkley and Zimm.(21 52 a lt8) Consequently, absolute values of the torsion constants reported by both groups need to be corrected before any detailed comparison with results from other laboratories is possible. In principle, the relative changes in torsion constant are more reliable. 

Ashikawa and co-workers attempted to determine both the bending and torsional rigidities simultaneously from their FPA data by fitting a simple approximate form that was then compared with the incorrect anisotropy formula of Barkley and Zimm.(21 58,61,108) Neither their claim to distinguish the bending contribution nor their reported bending rigidities can be taken seriously. 

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. 

Thus a simple power law behavior with an exponent of 1.5 would result if 0"= 1 [125]. Zimm and Kilb [128] made a first attempt to calculate g for star branched macromolecules on the basis of the Kirkwood-Riseman approximation for the hydrodynamic interaction. They came to the conclusion that... 

The PDMS and the PIB chains consist of around 78 monomers rendering the usually applied long wavelength approximation of the Zimm model... 

This formulation is used in the literature however, see also Eq. 6.48 which yields a better approximation of the Zimm dynamics. 

Fig. 6.23 a Comparison of the universal Zimm relaxation function F(x) to stretched exponentials. The dotted line is the residual error xlO to the best fit (a=1.354, =0.852). The p value of 2/3 only applies at large values of x with F(x)< 10 , which are irrelevant for NSE data. The dashed blue line in the right part of the figure corresponds to the asymptotic form given in [6] exp(-1.35x ). b Comparison of the integration result solid line), the approximation Eq. 6.48 dashed line) and the asymptotic form dashed-dotted line). Only for very large values of x>30 does the asymptotic value of p emerge... 

In 1959, Zimm and Kilb (34) made some calculations of the intrinsic viscosities of certain branched polymer molecules, taking into account the hydrodynamic interaction between portions of the polymer chain, using a modification of the Rouse procedure. They carried out these difficult calculations for a quite restricted range of models, obtaining numerical results for equalarmed stars with 3, 4, and 8 branches, and for one modified star with 2 long branches and 8 shorter branches. They found that their numerical results for this set of structures could be approximately represented by ... 

Since the Zimm-Bragg parameters o and s of the naturally occurring amino acids (In water) cannot be obtained from studies of the helix-coil transition in homopolymers, because of experimental difficulties, a technique Is developed to circumvent these problems. It involves the study of the thermally induced transition curves for random copolymers of "guest amino acid residues in a water-soluble host" po y(amino acid). The data may be interpreted with the aid of suitable theories for the helix-coil transition in random copolymers to obtain a and s for the "guest" residues. It is shown in this paper that, for the usual ranges of parameters found for polylamino acids), one of the two lowest order approximations (corresponding to earlier treatments by Lifson and Allegra) is completely adequate. In essence, the low-order approximations hoid if o and s for the two constituents of the copolymer do not differ appreciably from each other. 

The synthesis and characterization of water-soluble, random copolymers containing L-isoleucine with AAO-hydroxypropyD-L-glutamine, and the thermally induced helix-coil transitions of these copolymers in water, are described. The incorporation of L-isoleucine is found to increase the helix content of the copolymers in water at all temperatures considered. The Zimm-Bragg parameters helix-coil transition in poly(L-isoleucine) in water are deduced from an analysis of the melting curves. Values of the Zimm-Bragg parameters s for poly(L-lsoleucine) in water are computed with the Lifson (o = 5.1 x 10 3) / Aiiegra (o = 5.5 x 10 3) approximation s = 1.20/1.25 (273 K), 1.11/1.14 (293 K), 1.07/1.08 (313 K), and 1.01/1.01 (333 K). 

Zimm 100) has extended the Rouse model to allow for intramolecular hydro-dynamic interaction, i.e., changes in medium velocity near each bead caused by the flow disturbance from other beads on the same chain. The Oseen approximation, evaluated with the beads located at their mean equilibrium positions, was used to estimate the velocity disturbances. The intensity of the disturbance depends on the parameter h ... 

The first seven eigen values XP of eq. (3.56), as calculated by Zimm, Roe and Epstein (89), are given in Table 3.1 together with values of an equivalent quantity, I1(p) + Is(ft), as given in the mentioned treatment of Pyun and Fixman (92). At large values of subscript ft the following approximate formula is given by Zimm et al. (89) for Xp ... 

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