External Potential-Induced Enhancement of Monomer Mobility in Trapped Polymers: A Generalized Langevin Equation Study

Monomer friction in trapped polymers is a key determinant of the fundamental time scales governing a wide range of dynamic and kinetic processes in physics, chemistry, and biology. In this paper, we present a theoretical investigation of the monomer friction of individual Rouse chains trapped in a harmonic potential. Our analysis is based on the friction memory function within the framework of the generalized Langevin equation (GLE). The friction memory function comprises contributions from both solvent friction (assumed to be constant and independent of the external potential) and intra-chain friction. We demonstrate that the external potential induces an additional harmonic potential by modifying the correlations between monomers. This modification, in turn, alters the friction memory function. Our analyses of the induced potentials and relaxation time spectra reveal a characteristic chain length Nc, which describes the unperturbed size of the harmonically confined chain. In the long chain limit (chain length N ≫ Nc ≫ 1), the time-dependent monomer friction function exhibits universal power-law behaviors in the mediate time scale. Specifically, the exponent decreases from 1/2 to 0 as the ratio k′/k (where k′ is the external harmonic potential strength and k is the bond spring strength) increases. This behavior aligns with dynamic scaling arguments. In the long time limit, the friction function reaches a plateau, with the value varying from Ncγ/2 for k′/k ≪ 1 to the solvent friction γ for k′/k ≫ 1. These results indicate that the external potential enhances monomer diffusion by reducing intra-chain friction. This finding provides a fundamental understanding of the dynamics of trapped polymers and highlights the crucial role of external potentials in modifying intra-chain interactions and monomer mobility.