Dynamics of a polymer attached to a surface: Bead and spring model

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Abstract
The Smoluchowski formalism is used to solve the problem of a bead of frictional resistance β attached to a surface with a spring of force constant k over which a linear shear field of strenght α flows. The power dissipation is given by βα2kT/k. k and T have their usual meanings. The result is generalized to an n‐bead polymer. It is found that the power dissipation of a Rouse model polymer attached to a surface at one end is twice that of an identical polymer flowing freely in solution. If the force constant k arises from an entropy force, then, because of the effect of the surface on the number of polymer configurations, there is an additional factor of two. The same relationship is expected to also hold for the frequency‐dependent power dissipation. It is argued that a net circulation exists in the beads above the surface and that the magnitude of the circulation is roughly comparable to that which exists in a polymer freely rotating in solution under a shear field of the same magnitude.