Branching structure factor

In order to determine the branching structure factor e, Foster ( ) studied a large qroup of high pressure low density polyethylene resins (HP-LDPE). Using the MWBD method, he calculated the whole polymer number average number of branch points per JOOO carbon atoms from SEC data as a function of e. Then the Xfj values were compared with those obtained by nMR. 

Figure 3. Effect of the branching structure factor (e) on the LCB frequency (X j) calculated for three LDPE resins using the MWBD method ((l ) values obtained...

The MWBD method also requires an independent measure of the branching structure factor e. For our analysfs of polyvinyl acetate, it was obtained by comparing M and Bf values calculated from SEC data, analyz d using the MWBD method and various epsilons, and the Mfj and Bj values predicted by Graessley s (21) kinetic model. An epsilon value of 1.0 was found to fit best. 

Keywords. Solution properties. Regularly branched structures. Randomly and hyperbranched polymers. Shrinking factors. Fractal dimensions. Osmotic modulus of semi-di-lute solutions. Molar mass distributions, SEC/MALLS/VISC chromatography... 

The simplest case of comb polymer is the H-shaped structure in which two side arms of equal length are grafted onto each end of a linear cross-bar [6]. In this case the backbones may reptate, but the reptation time is proportional to the square of Mj, rather than the cube, because the drag is dominated by the dumb-bell-like frictional branch points at the chain ends [45,46]. In this case the dependence on is not a signature of Rouse motion - the relaxation spectrum itself exhibits a characteristic reptation form. The dynamic structure factor would also point to entangled rather than free motion. 

In Fig. 3.16 dynamic structure factor data from a A =36 kg/mol PE melt are displayed as a function of the Rouse variable VWt (Eq. 3.25) [4]. In Fig. 3.6 the scaled data followed a common master curve but here they spht into different branches which come close together only at small values of the scahng variable. This splitting is a consequence of the existing dynamic length scale, which invalidates the Rouse scaling properties. We note that this length is of purely dynamic character and cannot be observed in static equilibrium experiments. 

As has been emphasized previously (IJ), the level of crystallinity is not the major determinant of the linewidth in the semicrystalline state. Rather the supermolecular structure or morphology is a major factor in governing the magnitude of the linewidth. Structural factors and crystallization conditions under which low density (branched) polyethylene forms... 

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. 

M. M. A. Olsthoorn, I. M. Lopez-Lara, B. O. Petersen, K. Bock, J. Haverkamp, H. P. Spaink, and J. E. Thomas-Oates, Novel branched nod factor structure results from a-(l->3) fucosyl transferase activity The major lipo-chitin oligosaccharides from Mesorhizobium loti strain NZP2213 bear an a-(l->3) fucosyl substituent on a nonterminal backbone residue, Biochemistry, 37 (1998) 9024-9032. 

Olvera de la Cruz and Sanchez [76] were first to report theoretical calculations concerning the phase stability of graft and miktoarm AnBn star copolymers with equal numbers of A and B branches. The static structure factor S(q) was calculated for the disordered phase (melt) by expanding the free energy, in terms of the Fourier transform of the order parameter. They applied path integral methods which are equivalent to the random phase approximation method used by Leibler. For the copolymers considered S(q) had the functional form S(q) 1 = (Q(q)/N)-2% where N is the total number of units of the copolymer chain, % the Flory interaction parameter and Q a function that depends specifically on the copolymer type. S(q) has a maximum at q which is determined by the equation dQ/dQ=0. 

Structural factors characterizing the graft copolymer are basically the number of branches and the length (molar mass) of the backbone and of the branch. The reported values are mostly average values, since the precise determination of their distribution is much more complicated or sometimes impossible. 

The calculations of the various structure factors for a dendrimer are rather straightforward [25], but somewhat tedious. There are four main contributions to these correlations (1) one intrabranch self-correlations part, S b, (2) one intra-branch cross-correlations part between blocks that originate from the same stem, S[b, (3) one intra-branch cross-correlations part between blocks that originate from different stems, S b, and (4) one interbranch correlations part Sjb. These various correlations are sketched in Fig. 4. 

The second virial coefficient decreases with increasing molecular weight of the solute and with increased branching. Both factors tend to result in more compact structures which are less swollen by solvent, and it is generally true that better solvents result in more highly swollen maeromoleeules and higher virial coefficients. 

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