Polymers perturbed/unperturbed dimension

The second virial coefiicient A2, which is related to the Flory dilute solution parameters by Eq. (3.121), is a measure of solvent-polymer compatibility. Thus, a large positive value of A% indicates a good solvent for the polymer favoring expansion of its size, while a low value (sometimes even negative) shows that the solvent is relatively poor. The value of A2 will thus tell us whether or not the size of the polymer coil, which is dissolved in a particular solvent, will be perturbed or expanded over that of the unperturbed state, but the extent of this expansion is best estimated by calculating the expansion factor a. As defined by Eqs. (3.123) and (3.124), a represents the ratio of perturbed dimension of the polymer coil to its unperturbed dimension. 

Measurements on polymers of given constitution and configuration dissolved in randomly chosen solvents generally do not lead to unperturbed dimensions of the polymers, since the dimensions are perturbed by both long-range and short-range forces ... 

This follows from the expansion factor, a is greater than unity in a good solvent where the actual perturbed dimensions exceed the unperturbed ones. The greater the value of the unperturbed dimensions the better is the solvent. The above relationship is an average derived at experimentally from numerous computations. Because branched chains have multiple ends it is simpler to describe them in terms of the radius of gyration. The volume that a branched polymer molecule occupies in solution is smaller than a linear one, which equals it in molecular weight and in number of segments. 

The unperturbed dimension refers to the size of the molecule, exclusive of solvent effects. It arises from a combination of free rotations and intermolecular steric and polar interactions. The expansion factor arises from interactions between the polymer and the solvent. In a good solvent a will be greater than 1 and the actual (perturbed) dimensions of the polymer will exceed its unperturbed dimensions. The greater the value of a, the better the solvent. For the special case where a = 1, the polymer adopts its unperturbed dimensions and behaves as an ideal statistical coil . 

In the models for polymer chain conformation that we have considered so far, the polymer chain is allowed to intersect itself, because each link is a vector that takes up no volume. This is clearly unrealistic for real polymer molecules, where the segments occupy a certain volume and the chain cannot cross itself. This leads to excluded volume, which cannot be occupied by other segments. Polymer coils which have excluded volume are said to be perturbed, whereas (r )J gives the unperturbed dimensions of the coil assuming volumeless links. The perturbed dimensions (r ) / are related to the unperturbed dimensions by the expansion factor, a ... 

The statistics of non-intersecting chains are described by self-avoiding walks instead of random walks, but we do not go into the details here. Suffice it to say that, in general, polymer chains have perturbed dimensions. However, polymers can adopt unperturbed dimensions in solutions in a so-called theta solvent (Section 2.5.1). 

The ratio between perturbed and unperturbed dimensions defines the expansion factor, a (Eq. 2.6). In a theta solvent, a = 1, in a good solvent a > 1, whereas in a poor solvent a < 1. In a poor solvent, polymers will often precipitate to avoid contact with the solvent, rather than adopt a very compact conformation. 

C can take values ranging between 4 and 10, depending on the type of polymer expression (5.39) thus corresponds to the unperturbed dimensions of a macro-molecular chain. The term unperturbed conveys the idea that the chain dimension is not perturbed by long-range interactions, as in the presence of a good solvent which would cause the chain to expand further. Unperturbed dimensions are characteristic of the so-called 6 conditions and are observed at temperature 0. 

Next we consider the situation of a coil which is unperturbed in the hydro-dynamic sense of being effectively nondraining, yet having dimensions which are perturbed away from those under 0 conditions. As far as the hydrodynamics are concerned, a polymer coil can be expanded above its random flight dimensions and still be nondraining. In this case, what is needed is to correct the coil dimension parameters by multiplying with the coil expansion factor a, defined by Eq. (1.63). Under non-0 conditions (no subscript), = a(rg)Q therefore under these conditions we write... 

Perturbed and unperturbed polymer dimensions deduced from intrinsic viscosity measurements, according to procedures which will be discussed later, are given in Table XXXIX of Chapter XIV. The... 

Although these semiempirical treatments can be useful in predicting interfacial tensions, they are not successful from a fundamental standpoint and cannot be used to predict the interfacial composition profile. Furthermore, these theories neglect the entropy effects associated with the configurational constraints on polymer chains in the interfacial region. These effects are unique in polymers and arise because the typical thickness of the interfacial region between polymer phases is less than the unperturbed molecular coil dimensions of a high polymer. Major perturbations of... 

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