Stratifications of cellular patterns

TL;DR

This paper studies the geometric layering of cellular patterns like foams, revealing hysteresis and convergence phenomena influenced by disorder, with implications for understanding the structural dynamics of such materials.

Contribution

It introduces the concepts of hysteresis and convergence in layer sequences of cellular patterns, highlighting the effects of disorder and initial conditions.

Findings

Layers exhibit hysteresis under reversed construction directions.

Layer sequences converge from different initial conditions in cylindrical geometry.

Dislocations in layers move erratically and annihilate upon collision.

Abstract

Geometrically, foams or covalent graphs can be decomposed into successive layers or strata. Disorder of the underlying structure imposes a characteristic roughening of the layers. Our main results are hysteresis and convergence in the layer sequences. 1) If the direction of construction is reversed, the layers are different in the up and down sequences (irreversibility); nevertheless, under suitable but non-restrictive conditions, the layers come back, exactly, to the initial profile, a hysteresis phenomenon. 2) Layer sequences based on different initial conditions (e.g. different starting cells) converge, at least in the cylindrical geometry. Jogs in layers may be represented as pairs of opposite dislocations, moving erratically due to the disorder of the underlying structure and ending up annihilating when colliding.