1997/08/31 by Leo Radzihovsky, John Toner · 4 citations
Biochemistry, Genetics and Molecular Biology · Chemistry · Physics and Astronomy · #Force Microscopy Techniques and Applications #Lipid Membrane Structure and Behavior #Surfactants and Colloidal Systems #cond-mat.dis-nn #cond-mat.soft #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.57.1832
published as Phys.Rev.E57:1832-1863,1998 · 34 PRE pages, RevTex and 11 postscript figures, also available at http://lulu.colorado.edu/~radzihov/ version to appear in Phys. Rev. E, 57, 1 (1998); minor changes
arxiv created 1997/12/11 · openalex publication_date 1998/02/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We study the shape, elasticity, and fluctuations of the recently predicted [L. Radzihovsky and J. Toner, Phys. Rev. Lett. 75, 4752 (1995)] and subsequently observed (in numerical simulations) [M. Bowick, M. Falcioni, and G. Thorleifsson, Phys. Rev. Lett. 79, 885 (1997); tubule phase of anisotropic membranes, as well as the phase transitions into and out of it. This novel phase lies between the previously predicted flat and crumpled phases, both in temperature and in its physical properties: it is crumpled in one direction, and extended in the other. Its shape and elastic properties are characterized by a radius of gyration exponent \ensuremathν and an anisotropy exponent z. We derive scaling laws for the radius of gyration RG(L_\ensuremath⊥,Ly) (i.e., the average thickness) of the tubule about a spontaneously selected straight axis and for the tubule undulations hrms(L_\ensuremath⊥,Ly) transverse to its average extension. We show that for square membranes (with intrinsic size L_\ensuremath⊥=Ly=L), RG\ensuremath∝L^\ensuremathν, and hrms\ensuremath∝L^1\ensuremath-\ensuremathη_\ensuremathκz/2, with \ensuremathη_\ensuremathκ a bending rigidity anomalous elasticity exponent related to \ensuremathν and z. For phantom (i.e., non-self-avoiding) membranes, we predict \ensuremathν=(1)/(4), z=(1)/(2), and \ensuremathη_\ensuremathκ=0, exactly, in excellent agreement with simulations. For D=2 dimensional membranes embedded in the space of dimension d<11, self-avoidance greatly swells the tubule and suppresses its wild transverse undulations, changing its shape exponents \ensuremathν, z, and \ensuremathη_\ensuremathκ. For a D-dimensional membrane embedded in d>d* [d*(D=2)>(7)/(2)], \ensuremathη_\ensuremathκ=0 and z=(D\ensuremath-1+2\ensuremathν)/3, while for d<d*, \ensuremathη_\ensuremathκ>0 and z=(D\ensuremath-1+2\ensuremathν)/(3\ensuremath-\ensuremathη_\ensuremathκ). ``Flory'' theory yields, in the physical case of D=2 and d=3, \ensuremathν=3/4, while the recent 11\ensuremath-\ensuremathε expansion results yield \ensuremathν=0.52. The actual value of \ensuremathν probably lies closer to the Flory estimate, between these two limits. We give detailed scaling results for the shape of the tubule of an arbitrary aspect ratio, i.e., for the tubule thickness, its transverse undulations, and a variety of other correlation functions, as well as for the anomalous elasticity of the tubules, in terms of \ensuremathν and z. Finally we present a scaling theory for the shape and specific heat near the continuous transitions into and out of the tubule phase, and perform detailed renormalization group calculations for the crumpled-to-tubule transition for phantom membranes.