From stiff to rigid spring polymer part I: dumbbell dynamics

Abstract

Elastic and rigid dumbbells are two of the simplest models in polymer kinetic theory. It is natural to expect that an elastic dumbbell with a finite natural length becomes a rigid dumbbell when the spring stiffness tends to infinity. The limit, however, is singular. For a spring of stiffness H , natural length L , and connector length Q , rigidity requires Q → L , while the spring force H ( Q − L ) need not vanish. Instead, the extension tends to zero in such a way that H ( Q − L ) approaches the finite constraint tension of the rigid dumbbell. Thus, setting Q = L before taking the stiff-spring limit removes precisely the force required to maintain rigidity. In this Tutorial, we develop the stiff-spring limit of a finite-rest-length elastic dumbbell step by step. The deterministic problem first shows that the radial motion becomes a fast variable, with relaxation time τ _H = ζ /(2 H ), while the flow-induced extension is O ( H ⁻¹ ). The product H ( Q − L ) remains finite and becomes the rigid-dumbbell constraint tension, while projection perpendicular to the connector recovers the rigid orientation equation. Brownian motion is then included consistently with the fluctuation-dissipation theorem. Introducing