Perturbed dimension

Similarly the RMS radius of gyration for perturbed dimensions in real chains, is greater than that for unperturbed dimensions. 

The perturbed dimensions will differ from unperturbed dimensions by the expansion a of the molecule arising from the long-range effects. Thus, we may write... 

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. 

Problem 3.14 Show that the perturbed dimensions of highly expanded polymer coils are proportional to where n is the number of backbone bonds in the polymer chain. 

There are two basic ways in which measurements of the unperturbed dimensions are obtained (1) determination of unperturbed dimensions directly, by measurements in theta solvents and (2) determination of the perturbed dimensions in a good solvent and extrapolation of the values to the unperturbed state using one of the existing theories. Both methods have been widely used as will be shown. 

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 variables T ax and represent the hmits of the time interval the second or indirect dimension is recorded in. The index inc refers to the indirect spectral dimension. As wiU later be seen, the direct dimension can originate from two different data sets, then A B, or from the same data set, hence A = B. In the latter case, one data set is merely the transpose of the other one. Within a representation such as Eq. (5.5), the two data sets— which contain mixed time—frequency data before and frequency—frequency data after transformation—are correlated by a shared indirect dimension. The common feature can be understood as a perturbation, the dimension hence called perturbation dimension. 

The generalization of Eq. (5.3) is given as Eq. (5.14). The variables / and (O denote spectral variables such as frequencies, that may be obtained from any kind of spectroscopy. They are only related by the common time domain t, which can also be substituted by another perturbation dimension, such as a series of samples. 

In an NMR experiment, the perturbation dimension usually comprises the nuclear spin frame instead of a time frame or sample frame of macroscopic phenomena such as a chemical reaction. Different phases thus depend on the individual nuclear spins rather than on the molecular assembly of spins. Therefore, the exploitation of phase information within the covariance processing described above might seem little attractive. 

The exponent a of the [riJ-M-relationship increases with the solvent quality. In a good solvent, a coil has its so-called perturbed dimensions. In a good solvent the exponent a can reach a theoretical value of 0.76 according to the mean-field theory [47]. Slopes between 0.5 and 0.76 are also experimentally observed, for example for poly(acrylamide) in H2O at 25 °C in Fig. 6.7. Exponents a greater than 0.76 are experimentally not observed for flexible coils in good solvents [47]. Therefore, exponents a greater than 0.76 are not solely caused by the solvation of the polymer chain. In these cases, the coil expansion is also determined by the chemical structure of the polymer chain as shown in the following. 

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). 

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