• Open Access

Statistical Mechanics of Thin Spherical Shells

Andrej Košmrlj and David R. Nelson
Phys. Rev. X 7, 011002 – Published 11 January 2017

Abstract

We explore how thermal fluctuations affect the mechanics of thin amorphous spherical shells. In flat membranes with a shear modulus, thermal fluctuations increase the bending rigidity and reduce the in-plane elastic moduli in a scale-dependent fashion. This is still true for spherical shells. However, the additional coupling between the shell curvature, the local in-plane stretching modes, and the local out-of-plane undulations leads to novel phenomena. In spherical shells, thermal fluctuations produce a radius-dependent negative effective surface tension, equivalent to applying an inward external pressure. By adapting renormalization group calculations to allow for a spherical background curvature, we show that while small spherical shells are stable, sufficiently large shells are crushed by this thermally generated “pressure.” Such shells can be stabilized by an outward osmotic pressure, but the effective shell size grows nonlinearly with increasing outward pressure, with the same universal power-law exponent that characterizes the response of fluctuating flat membranes to a uniform tension.

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  • Received 20 June 2016

DOI:https://doi.org/10.1103/PhysRevX.7.011002

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & ThermodynamicsPolymers & Soft MatterPhysics of Living SystemsInterdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Andrej Košmrlj1,* and David R. Nelson2,†

  • 1Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
  • 2Department of Physics, Department of Molecular and Cellular Biology and School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

  • *andrej@princeton.edu
  • nelson@physics.harvard.edu

Popular Summary

Tiny objects floating in a liquid are constantly buffeted by the random motion of molecules. This thermal agitation can transform threadlike proteins, polymers, and microscopic filaments into random coils. Thin two-dimensional sheets of molecules, on the other hand, remain flat and behave qualitatively like a wrinkled sheet of paper. These thermally wrinkled sheets are much harder to bend than their smooth counterparts but much easier to stretch. Not much is known, however, about how this molecular motion affects the mechanical properties of curved thin shells such as cell membranes, protein coats around viruses, and bacterial cell walls, as well as many artificial structures. We find an entirely new phenomenon: This thermal motion can crush large shells even in the absence of external pressure.

We use statistical mechanics to calculate how the elastic properties of spherical shells respond to thermal motion from surrounding molecules. This motion can effectively create a negative surface tension, generating a kind of spontaneous inward pressure. Tiny shells remain stable, but shells larger than a certain temperature-dependent size will be squashed. Given enough outward pressure, these large shells can be stabilized, but their elastic properties are highly nonlinear, representing a breakdown of a 17th century law of mechanics discovered by Robert Hooke.

We expect that our results will stimulate numerical and experimental tests of our predictions in both biological and artificial shells by tuning shell size, temperature, and osmotic pressure.

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Vol. 7, Iss. 1 — January - March 2017

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