Thermally Driven Order-Disorder Transition in Two-Dimensional Soft Cellular Systems

Marc Durand and Julien Heu

Phys. Rev. Lett. 123, 188001 – Published 29 October 2019

Abstract

Many systems, including biological tissues and foams, are made of highly packed units having high deformability but low compressibility. At two dimensions, these systems offer natural tesselations of a plane with fixed density, in which transitions from ordered to disordered patterns are often observed, in both directions. Using a modified cellular Potts model algorithm that allows rapid thermalization of extensive systems, we numerically explore the order-disorder transition of monodisperse, two-dimensional cellular systems driven by thermal agitation. We show that the transition follows most of the predictions of Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory developed for melting of 2D solids, extending the validity of this theory to systems with many-body interactions. In particular, we show the existence of an intermediate hexatic phase, which preserves the orientational order of the regular hexagonal tiling but loses its positional order. In addition to shedding light on the structural changes observed in experimental systems, our study shows that soft cellular systems offer macroscopic systems in which the KTHNY melting scenario can be explored, in the continuation of Bragg’s experiments on bubble rafts.

Received 23 July 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.188001

Physics Subject Headings (PhySH)

BKT transition Crystallization Emergence of patterns

Emulsions Epithelial cells Foams

Metropolis algorithm Monte Carlo methods

Interdisciplinary PhysicsPhysics of Living SystemsStatistical Physics & ThermodynamicsPolymers & Soft MatterCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Marc Durand ^* and Julien Heu

Matière et Systèmes Complexes (MSC), UMR 7057 CNRS & Université Paris Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France

^*marc.durand@univ-paris-diderot.fr

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Vol. 123, Iss. 18 — 1 November 2019

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Images

Figure 1
Evolution of defect populations and orientational order with temperatures. Only a small portion of the system is shown. Columns from left to right: solid, hexatic, and liquid phases. Top row (a)–(c): Population of topological defects. Inset: Sketch of a $T 1$ event: In a hexagonal lattice, this topological change corresponds to the creation of a bounded pair of dislocations. Middle row (d)–(f): Phase of the local orientational parameter order $\arg ψ_{6} (r_{i})$ . Insets: Normalized histograms of $\arg ψ_{6} (r_{i})$ , calculated over the whole simulation time. Bottom row (g)–(i): Norm of the local orientational parameter order $| ψ_{6} (r_{i}) |$ . Insets: Normalized histograms $| ψ_{6} (r_{i}) |$ , calculated over the whole simulation time.
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Figure 2
(a) Pair correlation function $g (r)$ at different reduced temperatures. Plots are evenly shifted vertically for clarity. Peaks are smeared out as the temperature increases, indicating the loss of translational order. (b) Variation of $| Ψ_{t} |$ and $| Ψ_{6} |$ with the temperature. The drop of $| Ψ_{t} |$ slightly precedes the $| Ψ_{6} |$ drop, indicating two distinct transitions. The shaded area indicates the existence domain of the hexatic phase as determined from the peaks of the two susceptibilities [see Fig. 4]. Standard errors of the mean are smaller than the size of the markers.
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Figure 3
(a) Translational correlation function $C_{t} (r)$ in a log-log plot. The curve evolves from a power-law decay (pink area) to an exponential decay as the temperature increases. The $r^{- 1 / 3}$ decaying behavior at the solid-hexatic transition is predicted by the KTHNY theory. (b) Orientational correlation function $C_{6} (r)$ in a log-log plot. At low temperatures, the curves decay to a plateau. At intermediate temperatures, curves follow a power-law decay (gray area). At higher temperatures, the curves exhibit an exponential decay. The $r^{- 1 / 4}$ decaying behavior at the hexatic-liquid transition is predicted by the KTHNY theory.
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Figure 4
The susceptibilities $χ_{t}$ and $χ_{6}$ as a function of the reduced temperature. The peaks of the translational susceptibility $χ_{t}$ and the orientational susceptibility $χ_{6}$ clearly indicate two transition points, at $T_{m}^{⋆} = 2.549 \times 10^{- 2}$ and $T_{i}^{⋆} = 2.609 \times 10^{- 2}$ , respectively.
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Figure 5
Number of bound dislocation pairs, dislocations, disclinations, and charged aggregates as a function of the reduced temperature. Right panel: Close-up view showing that the two phase transitions coincide with the rapid increase of dislocations and $disclinations + charged$ aggregates, respectively.
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