Theory of mobility of inhomogeneous-polymer-grafted particles

We develop a theory for the motion of a particle grafted with inhomogeneous bead-spring Rouse chains via the generalized Langevin equation (GLE), where individual grafted polymers are allowed to take different bead friction coefficients, spring constants, and chain lengths. An exact solution of the memory kernel K(t) is obtained for the particle in the time (t) domain in the GLE, which depends only on the relaxation of the grafted chains. The t-dependent mean square displacement g(t) of the polymer-grafted particle is then derived as a function of the friction coefficient γ0 of the bare particle and K(t). Our theory offers a direct way to quantify the contributions of the grafted chain relaxation to the mobility of the particle in terms of K(t). This powerful feature enables us to clarify the effect on g(t) of dynamical coupling between the particle and grafted chains, leading to the identification of a relaxation time of fundamental importance in polymer-grafted particles, namely, the particle relaxation time. This timescale quantifies the competition between the contributions of the solvent and grafted chains to the friction of the grafted particle and separates g(t) into the particle- and chain-dominated regimes. The monomer relaxation time and the grafted chain relaxation time further divide the chain-dominated regime of g(t) into subdiffusive and diffusive regimes. Analysis of the asymptotic behaviors of K(t) and g(t) provides a clear physical picture of the mobility of the particle in different dynamical regimes, shedding light on the complex dynamics of polymer-grafted particles.