Polydisperse Chains

If the growth front is frozen, the crystallization at the growth front depends crucially on the commensurability between the thickness of the growth front and the length of crystallizing chains. For details, see Ref. 49. A collection of polydisperse chains gets effectively fractionated at the growth front. 

We now consider the properties of the moment free energy with the (chain number) density p0 retained as an extra moment. This provides additional geometrical insight into the properties of polydisperse chains (while for the numerical determination of the CPC and shadow curve, the above one-moment free energy is preferable). To construct the two-moment free energy, we proceed as before. The family (39) is now... 

LCs were the earliest studied structures, in which polypeptide homopolymer rods pack in an ordered manner to form smectic, nematic, and cholesteric phases. The smectic LCs are mainly formed by polypeptide homopolymers with identical polymer length. The cholesteric phase can be prepared by synthetic polypeptides with polydisperse chain length. The nematic phase can be regarded as a special example of the cholesteric phase with an infinite cholesteric pitch. The cholesteric pitch and chirahty in the polypeptide LCs are dependent on many factors, such as temperature, polymer concentration, solvent nature, and polypeptide cOTiformation. Deep understanding of such phenomena is necessary for preparation of ordered polypeptide assembles with delicate stmctures. The addition of denaturing solvent to polypeptide solution can lead to an anisotropic-isotropic reentrant transition at low temperatures where the intramolecular helix-coil transformation occurs. However, the helical structure is more stable in LC phase than in dilute solution due to the conformational ordering effect. 

After reviewing various earlier explanations for an adsorption maximum, Trogus, Schechter, and Wade [244] proposed perhaps the most satisfactory one so far (see also Ref. 243). Qualitatively, an adsorption maximum can occur if the surfactant consists of at least two species (which can be closely related) what is necessary is that species 2 (say) preferentially forms micelles (has a lower CMC) relative to species 1 and also adsorbs more strongly. The adsorbed state may also consist of aggregates or hemi-micelles, and even for a pure component the situation can be complex (see Section XI-6 for recent AFM evidence of surface micelle formation and [246] for polymeric surface micelles). Similar adsorption maxima found in adsorption of nonionic surfactants can be attributed to polydispersity in the surfactant chain lengths [247], Surface-active impuri-... 

For preparative purposes batch fractionation is often employed. Although fractional crystallization may be included in a list of batch fractionation methods, we shall consider only those methods based on the phase separation of polymer solutions fractional precipitation and coacervate extraction. The general principles for these methods were presented in the last section. In this section we shall develop these ideas more fully with the objective of obtaining a more narrow distribution of molecular weights from a polydisperse system. Note that the final product of fractionation still contains a distribution of chain lengths however, the ratio M /M is smaller than for the unfractionated sample. 

The primary polymerization product ia these processes has a relatively wide molecular weight distribution, and a separate step is often used to narrow the polydispersity. Such a narrowkig step may consist of high vacuum stripping to remove volatile polymer chains, often followed by a solvent fractionation step (35,36), sometimes a solvent fractionation step alone (37,38), or a fractional precipitation from organic solvent (32). The molecular weight distribution can also be narrowed by depolymerization at elevated temperatures ia the presence of a depolymerization catalyst (217—220). 

The minimum polydispersity index from a free-radical polymerization is 1.5 if termination is by combination, or 2.0 if chains ate terminated by disproportionation and/or transfer. Changes in concentrations and temperature during the reaction can lead to much greater polydispersities, however. These concepts of polymerization reaction engineering have been introduced in more detail elsewhere (6). 

Issues to be considered in selecting the best stabilizing system are polymeric chain branching which increases with high temperature and the presence of some stabilizers, polydispersity of the particles produced, and grafting copolymerization, which may occur because of the reaction of vinyl acetate with emulsifiers such as poly(vinyl alcohol) (43,44). 

A further feature of anionic polymerisation is that, under very carefully controlled eonditions, it may be possible to produee a polymer sample which is virtually monodisperse, i.e. the molecules are all of the same size. This is in contrast to free-radical polymerisations which, because of the randomness of both chain initiation and termination, yield polymers with a wide molecular size distribution, i.e. they are said to be polydisperse. In order to produce monodisperse polymers it is necessary that the following requirements be met ... 

This result should be vahd for sufficiently high density 0 where correlations, brought about by the mutual avoidance of the chains, are negligible. Due to the recombination-scission process a polydisperse solution of living polymers should absorb or release energy as the temperature is varied. This is reflected by the specific heat Cy, which can be readily obtained from Eq. (9) as a derivative of the internal energy U... 

The changes in the average chain length of a solution of semi-flexible selfassembling chains confined between two hard repulsive walls as the width of the sht T> is varied, have been studied [61] using two different Monte Carlo models for fast equihbration of the system, that of a shthering snake and of the independent monomer states. A polydisperse system of chain molecules in conditions of equilibrium polymerization, confined in a gap which is either closed (with fixed total density) or open and in contact with an external reservoir, has been considered. 

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