Path-space ratio as a molecular shape descriptor of polymer conformation

Polymers at interfaces exhibit properties that cannot be completely captured by descriptors of mean molecular size. Recent work in the literature shows that a combined analysis of mean size and chain entanglement provides a more discriminating approach to understanding the onset of configurational transitions in these systems. Usually, chain entanglement is characterized by properties such as the mean overcrossing number or the chain's writhe; these are powerful properties but their evaluation can be computationally demanding. In this work, we introduce a geometrical descriptor of polymer shape, termed the path-space ratio zeta, aimed at quantifying essential features of chain complexity, but at a lower computational cost. The descriptor includes information on chain geometry and topology. The path-space ratio zeta is built by taking into account two key ideas: (a) a dimensionless measure of length along the backbone of the polymer, and (b) the behavior of topological "knot energies". Here, we compare zeta with other approaches to quantify polymer geometry and connectivity. Particular attention is devoted to the ability of these descriptors to discriminate and quantify conformational changes in grafted polymers under compression. We show that, for these types of applications, the path-space ratio presents a fast alternative to the mean overcrossing number.