Elasticity, shape fluctuations, and phase transitions in the new tubule phase of anisotropic tethered membranes
Abstract
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 ν and an anisotropy exponent z. We derive scaling laws for the radius of gyration RG(L⊥,Ly) (i.e., the average thickness) of the tubule about a spontaneously selected straight axis and for the tubule undulations hrms(L⊥,Ly) transverse to its average extension. We show that for square membranes (with intrinsic size L⊥=Ly=L), RG~Lν, and hrms~L1-ηκz/2, with ηκ a bending rigidity anomalous elasticity exponent related to ν and z. For phantom (i.e., non-self-avoiding) membranes, we predict ν=14, z=12, and ηκ=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 ν, z, and ηκ. For a D-dimensional membrane embedded in d>d* [d*(D=2)>72], ηκ=0 and z=(D-1+2ν)/3, while for d<d*, ηκ>0 and z=(D-1+2ν)/(3-ηκ). ``Flory'' theory yields, in the physical case of D=2 and d=3, ν=3/4, while the recent 11-ɛ expansion results yield ν=0.52. The actual value of ν 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 ν 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.
- Publication:
-
Physical Review E
- Pub Date:
- February 1998
- DOI:
- 10.1103/PhysRevE.57.1832
- arXiv:
- arXiv:cond-mat/9708046
- Bibcode:
- 1998PhRvE..57.1832R
- Keywords:
-
- 82.65.Dp;
- 64.60.Fr;
- 05.40.+j;
- Equilibrium properties near critical points critical exponents;
- Condensed Matter - Soft Condensed Matter;
- Condensed Matter - Disordered Systems and Neural Networks;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 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)