Chain statistics theta temperature

Flory and Huggins developed an interaction parameter that may be used as a measure of the solvent power of solvents for amorphous polymers. Flory and Krigbaum introduced the idea of a theta temperature, which is the temperature at which an infinitely long polymer chain exists as a statistical coil in a solvent. 

Theta solvents ru Solvents for a particular polymer in dilute solution, and at a temperature called the theta temperature that slows the polymer chains to assume their unperturbed, random-coil configurations with theoretical root-mean-square distances between chain ends. Kamide K, Dobashi T (2000) Physical chemistry of polymer solutions. Elsevier, New York. Flory PJ (1969) Statistical mechanics of chain molecules. Interscience Publishers... 

So far our description of the random-coil chain basically assumes a dilute solution and we have not yet defined the term dilute solution. It has been discovered that when the concentration increases to a certain point, interesting phenomena occur chain crossover and chain entanglement. Chain crossover refers to the transition in configuration from randomness to some kind of order, and chain entanglement refers to the new statistical discovery of the self-similar property of the random coil (e.g., supercritical conductance and percolation theory in physics). Such phenomena also occur to the chain near the theta temperature. In this section, we describe the concentration effect on chain configurations on the basis of the theories advanced by Edwards (1965) and de Gennes (1979). In the next section, we describe the temperature effect, which is parallel to the concentration effect. 

It is important to remember that the effective segment-segment interaction potential in a polymer solution is by no means the same as for the hypothetical case of the chain in a vacuum. In a real system the net effect includes contributions from polymer-solvent and solvent-solvent interactions. In a given polymer -h solvent system there may be a theta temperature (see above) at or near which there is a delicate balance among these effects such that the net excluded volume effect vanishes and random flight statistics prevail, i.e. R and Rq both become proportional asymptotically to the square root of the polymer molecular weight. 

Substituting T = 0, in the above equation yields a = 1 and(r ) = (to). At the theta tanperature, polymer chain statistics are therefore well described by the random flight model. As the temperature is progressively increased above 0f, o increases toward o and polymer chains adopt more expanded structures in solution. These conditions are termed good solvent conditions and correspond to large z values. Under these circumstances, it can be shown that the expansion factor increases weakly with jc, a x , and... 

It is however possible to find conditions, called unperturbed or theta conditions (because for each polymer-solvent pair they correspond to a well-defined temperature called d temperature) in which a tends to 1 and the mean-square distance reduces to Q. In 6 conditions well-separated chain segments experience neither attraction nor repulsion. In other words, there are no long-range interactions and the conformational statistics of the macromolecule may be derived from the energy of interaction between neighboring monomer units. For a high molecular weight chain in unperturbed conditions there is a simple relationship between the mean-square end-to-end distance < > and the mean-... 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

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