Monodisperse molecular weight distribution

The results above are only valid for tetrafunctional crosslinking of monodisperse polymer. However, in many thermoreversible systems the crosslinks have functionalities that are much larger than four. Moreover, the polymers used are not monodisperse in general. In order to be able to calculate network parameters the present author [39—44] extended the Flory-Stockmayer model for polydisperse polymer which is crosslinked with f-functional crosslinks. It was possible to calculate network parameters for polymers of various molecular weight distributions (monodisperse polymer with D s M, /r3 = 1, a Schulz-Flory distribution with D = 1.5, a Flory distribution with D = 2, a cumulative... 

It may be shown that M > M. The two are equal only for a monodisperse material, in which all molecules are the same sise. The ratio MI /MI is known as the polydispersity index and is a measure of the breadth of the molecular weight distribution. Values range from about 1.02 for carefully fractionated samples or certain polymers produced by anionic polymerization, to 20 or more for some commercial polyethylenes. 

This dimensionless number measures the breadth of the molecular weight distribution. It is 1 for a monodisperse population (e.g., for monomers before reaction) and is 2 for several common polymerization mechanisms. 

An important by-product of the development of this approach is that Orthogonal Chromatography provides a direct method of estimating the shape of the chromatogram for extremely narrow molecular weight distributions. This shape function is fundamental information for axial dispersion evaluation and is not otherwise easily obtained. Even commercially available monodisperse standards synthesized by anionic polymerization are too polydisperse. 

Propylene oxide is a surface active monomer structurally similar to ethylene oxide and therefore of interest as a SHM W-SP, but with more than ten repeating units this polymer is not water soluble. A compositional isomer methyl vinyl ether is water soluble the adsorption behavior of this polymer (PMVE) is illustrated in Figure 4. At 1 ppm the rate of 7T increase is linear over three hours. The diffusion rate could be calculated if the W-SP s molecular weight were monodispersed. The polymer studied had a Gaussian molecular weight distribution, which is true of essentially all W-SPs even after attempts have been made to... 

For example, we have described that nearly monodisperse PEs can be formed by 2/ MAO (1 min polymerization, atmospheric pressure 25 °C Mn 52,000, MJMn 1.12 50 °C Mn 65,000, MJMn 1.17) and 38 (Fig. 25)/MAO (1 min polymerization, atmospheric pressure W25n°C M 8000, M /M 1.05 50 °C M 9000, M IM 1.08) [28, 68, 69]. Additionally, Coates and coworkers subsequently reported that Ti-FI catalysts 34 (Fig. 22) and 39 (Fig. 25) can form nearly monodisperse PEs under controlled conditions [70]. With these Ti-FI catalysts, however, synthesizing high molecular weight and narrow molecular weight distribution PEs is generally difficult (e.g., 5 min polymerization, atmospheric pressure, 50 °C 2Mn 132,000, MJMn 1.83 38Mn 24,000, MJMn 1.46) [28, 68]. Moreover, normally, these catalysts cannot be applied to block copolymer formation. 

There are many facets of this study which we feel merit further investigation. In particular it is necessary to consider am extension of the proposed model, which in its present form is confined to the performance of a simple column, to cover the behaviour of any set of columns since it is column sets which are normally used. In addition, it is important to consider the input to the model which should be truly representative of polymers with a molecular weight distribution and not merely a concentration pulse of perfectly monodisperse polymer. In relation to this latter suggestion it would be significant if it were possible to link this model to the very real problem of deconvolution, i.e. the removal of instrumental and column broadening from the observed chromatogram to produce the true molecular weight distri-... 

Now we compare the above osmotic pressure data with the scaled particle theory. The relevant equation is Eq. (27) for polydisperse polymers. In the isotropic state, it can be shown that Eq. (27) takes the same form as Eq. (20) for the monodisperse system though the parameters (B, C, v, and c ) have to be calculated from the number-average molecular weight M and the total polymer mass concentration c of a polydisperse system pSI in the parameters B and C is unity in the isotropic state. No information is needed for the molecular weight distribution of the sample. On the other hand, in the liquid crystal state2, Eq. (27) does not necessarily take the same form as Eq. (20), because p5I depends on the molecular weight distribution. 

Figure 4A. Molecular weight distribution of essentially monodisperse polymer.

The mechanism of the polymerization of this monomer has been studied in far greater detail than any other. It is clear from the outset that a much more complex mechanism is involved than is the case for olefins. A large proportion of the initiator is used to form polymer whose molecular weight is only a few hundreds and the overall molecular weight distribution is so broad as to be rivalled only by those found in polyethylene produced by the high pressure process (19, 39). The initiator disappears almost instantaneously on mixing the reactants (19, 38). Under these conditions, an almost monodisperse polymer would be expected if chain transfer or termination processes are absent. 

The complete balance of the upturn by the polydispersity is only obtained for random branching processes. Often the reaction is impeded by serious constraints, or the primary chains before cross-linking are monodisperse. Then the resultant final molecular-weight distribution is narrower than in the random case, and the characteristic upturn as a result of branching, develops again. A strange coincidence in behavior is observed with star-molecules, where the rays are polydisperse, and with the ABC-type polycondensates. In both cases the particle-scattering factors can be expressed as ... 

No restriction is made to the same molecular weight distribution. Instead of this, the natural distributions for f > 2 and f = 2 are taken. For star-molecules, f = 2 corresponds to the monodisperse linear chain or to a linear chain that obeys the most probable distribution, and in the case of random polycondensates, f > 2 corresponds to the branched non-fractionated sample, and f = 2 to the linear polycondensate. The g and h-factors so defined no longer have the appearance of shrinking factors in all cases, as may be recognized from Figs. 43 and 44. For star-molecules, both factors decrease as... 

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