Radius of Gyration
In solution, polymeric chains can form different conformations, depending upon the solvent. When the solvent is such that the chains are fully solvated, they are relatively extended and the molecules are randomly coiled. The polymer–solvent interaction forces determine the amount of space that the molecular coil of the polymer occupies in solution. While quite extended in a good solvent, if the solvent is a “poor” one, the chains are curled up. A measure of the size of the polymer molecule in solution, or the amount of space that a polymer molecule occupies in solution is determined as radius of gyration., or root mean square radius of gyration, S. Qualitatively, it is the average distance of the mass of the molecule from the center of its mass (from its center of gravity). The following equation
Fig. 2.16 Illustration of a molecular coil
defines this relationship [7]. To put it in other words, it is the square of the distances between various masses and the center of the mass:
where m is the mass associated with each of the N chain bonds, and S is the vector distance from the center of the mass to the terminal chain bond. The size of randomly coiled polymer molecules is commonly designated by the root-mean square distance between the ends, R2. A molecular coil is illustrated in Fig. 2.16. The distancebetween the chainends is of ten expresse dinterms of unperturbed dimensions (S0orR0) and an expansion factor (a) that is the result of interaction between the solvent and the polymer
The unperturbed dimensions refers to molecular size exclusive of solvent effects. It arises from intramolecular polar and steric interactions and free rotation. The expansion factor is the result of solvent and polymer molecule interaction. For linear polymers, the square of the radius of gyration is related to the mean-square end-to-end distance by the following relationship:
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 volume that these molecules occupy in solution is important in determinations of molecular weights It is referred to as the hydrodynamic volume. This volume depends upon a variety of factors.
These are interactions between the polymer molecule and the solvent, chain branching , conformational factors arising from polarity, restricted rotation due to resonance, and the bulk of substituents. The above, of course, assumes that the polymer molecules are fully separated from each other.
