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Statistical mechanics and dynamics of semiflexible filaments, and the role of curvature in elastic low dimensional soft matter

  • Author(s): Kernes, Jonathan Matthew
  • Advisor(s): Levine, Alex J
  • et al.
Abstract

In Part I, we examine both the statistical mechanics and dynamics of semiflexible filaments that are coupled to an external system. Chapters 2 and 3 look at semiflexible filaments in network, where the external system is the elastic response of the filamentous network itself. The linear elastic compliance of the network is modeled by attaching a Hookean spring at the boundary of the filament, which in turn, introduces a nonlinearity into to the Hamiltonian. Chapter 2 uses this model to propose a method for noninvasive microrheology measurements of semiflexible filament networks based on thermal fluctuations of transverse undulations. The external force is seen to counteract bending strain, broadening fluctuations at the boundaries.

Chapter 3 considers the dynamics of this model, using the Martin-Siggia-Rose-Janssen-De-Dominicis formalism to compute the time-dependent correlation functions of transverse undulations and of the filament's end-to-end distance. The spring serves to renormalize the filament's tension, altering the cross-over frequency between tension- and bending-dominated modes of the system.

In Chapter 4, we look at the dynamics of bundles of semiflexible filaments, bound together by transient crosslinkers. We explore bundle dynamics across a broad range of unbinding times $\tau_\text{off}$, finding the behavior is determined by whether crosslinkers prefer to relax via diffusion or unbinding. In both cases, the linker stiffness is effectively reduced. Linker unbinding introduces a frequency scale $\tau_\text{off}^{-1}$, set by the unbinding rate, at which the bundle dynamics are affected by cross-linkers

In Part II, we explore the properties of lower dimensional elastic structures whose stress-free state is curved. The mechanics of these structures depends strongly on geometry. Chapter~\ref{ch: FU} studies the coupling of in-plane to out-of-plane elastic modes, focusing on the simplest nontrivial structure -- a curved elastic rod. We find undulatory waves becomes gapped in the presence of finite curvature. Bending modes are absent below a frequency proportional to the curvature of the rod. Undulatory waves with frequencies in the gap associated with the curved region, may tunnel through that curved region via conversion into compression waves.

Finally, motivated by the results of Chapter 5, Chapter 6 explores the fate of undulatory waves across a nearly flat, thin membrane, roughened by a Gaussian quenched disordered height field with power-law correlations. We adopt the Donnell-Mushtari-Vlasov theory of membrane elasticity in which the membrane responds instantaneously to relieve in-plane stress. The amplitude of undulatory waves is related to its energy density, a conserved quantity, and shown to obey a diffusion equation at long length/time scales. Time-reversed interference corrections, the so-called coherent backscattering effect, are shown to reduce the diffusion coefficient logarithmically with respect to system size, in agreement with general results for 2D localization.

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