E2 Chain Configuration: Kuhn Segment and Turning Radius

Topic

The molecular chain of this polymer—whose actual contour length would be unwieldy to describe segment by chemical segment—is reduced, in the ideal chain model, to a random walk of Kuhn segments; each Kuhn segment encompasses a section of the actual molecular chain long enough for its orientation to be statistically independent of that of the preceding Kuhn segment. It is the length of this Kuhn segment, combined with the number of such segments making up the molecular chain, that determines—based on this random walk—the chain's typical end-to-end distance. A longer Kuhn segment, or a chain composed of a greater number of such segments, results in a correspondingly larger typical end-to-end distance within this random walk framework.

For an ideal chain treated as a Gaussian chain, this end-to-end distance is related to the molecular chain's radius of gyration—defined as the root-mean-square distance of the Kuhn segments from the chain's center of mass. The greater the end-to-end distance resulting from the random walk of Kuhn segments, the greater the radius of gyration becomes, in the proportion dictated by the Gaussian chain relationship; conversely, a smaller end-to-end distance results in a correspondingly smaller radius of gyration. For this reason, the model—initially defined by Kuhn segments undergoing a random walk and characterized by a specific radius of gyration—predicts that when the molecular chain is dissolved in a "good" solvent rather than existing in an ideal state, it deviates from that ideal random walk in accordance with Flory scaling: in a good solvent, different segments of the chain repel one another instead of crossing freely (as the ideal random walk allows), and this repulsion causes the chain to swell to a size larger than that predicted by the ideal random walk for the same number of Kuhn segments. If the chain is placed in an even better solvent—resulting in even stronger repulsion between its segments—the Flory scaling causes it to deviate even further from the ideal random walk upon which the model was originally based.

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