Ju l 1 99 8 Glass transition in self organizing cellular patterns

We have considered the dynamical evolution of cellular patterns controlled by a stochastic Glauber process determined energetically by the squares of the deviations of local cell topol-ogy from hexagons. Above a critical temperature evolution is towards a common equilibrium state from any initial configuration, but beneath this temperature there is a dynamical phase transition, with a start from a quasi-random state leading to non-equilibrium glassy freezing, whereas an ordered start leads to the thermodynamically optimal state. A temporal persistence function decays exponentially in the high temperature equilibrating state but has a characteristic slow aging-like behaviour in the low temperature glassy phase. Comparisons are made between simulations and analytical approximations for both thermodynamical and dynamical observables. The hexagonal tiling is the best partition of the plane in equal cells. It solves both the packing and the covering problems. It is the space-filling assembly of equal sized cells with the minimal interfacial extension. It is regular, perfect and beautiful. Surprisingly, it is never realized in natural biological tissues where a relevant amount of disorder is always present [1, 2, 3, 4]. This is reminiscent of the situation found in many covalently bonded solids, where, despite the lowest energy state being crystalline, in practice amorphous glassy structures are the common quasi-stable states resulting from normal preparation [5, 6, 7, 8]. We show that a simple cellular pattern, subject to a local dynamics which globally favours such a periodic hexagonal configuration, fails to reach it in a finite time but rather freezes into a glassy state. More particularly, we show that when the local dynamics is stochastic, jamming is observed beneath a characteristic temperature to an amorphous state with an amount of disorder similar to that observed in biological tissue (like human epidermis, vegetable leaves, human amnion or cucumber skin). This low temperature glassy state also exhibits two-time correlation features reminiscent of the aging phenomena found in other glasses and spin glasses. In the hexagonal tiling each cell (hexagon) has six neighbours and three hexagons meet at a common vertex. It is a regular two-dimensional, minimally-connected pattern (3 edges 1 incident on a vertex, 2 faces incident on a edge). These patterns (froths) are topologically stable: a vertex with higher connectivity can be split into several minimally-connected ver-tices by infinitesimal local rearrangements. The average number of sides per cell is imposed to be equal to 6 by the Euler theorem [9, 10]. Locally, the …