Layering in crumpled sheets

We introduce a toy model of crumpled sheets, and use Monte Carlo simulation to show there is a first-order phase transition in the model, from a disordered dilute phase to a mixture with a layered phase. We demonstrate the transition through two order parameters, corr and lay, the first of which measures orientational order while the second measures bulk layering. An important feature of the argument is the behavior of the system as its size is increased. Copyright c © EPLA, 2010 Introduction. – When a sheet of stiff paper is crumpled into a compact ball, creases and folds appear. This storage of energy, especially in the irreversibly distorted creases, has been widely studied, for instance in [1–4]. Our interest here is in geometric changes associated with the (reversible) folds, which are less well understood. Consider the densest possible state of the material, in which the stiff sheet is carefully layered into a compact stack of parallel leaves. We will be concerned with the transition between states of varying density. Imagine the process of compacting the sheet within a contracting sphere, from a typical initial state of low volume fraction near 0 to a typical state of high volume fraction near 1. These two regimes are noted for instance in [3,4]. Our paper focuses on whether as compaction proceeds, the connection between these extreme regimes is smooth. We in fact suggest that the connection is not smooth, but instead singular in a manner which is commonly called a phase transition, that is, a change in behavior at a precise degree of compaction in the infinite volume limit of the system. Justification of this would need two components, theoretical and experimental. We discuss here mainly the theoretical aspect, through a model, but note some experimental issues in the last section. We are interested in matter progressively confined as when a sheet is crumpled in one’s hands. A sheet is thin in one of its dimensions. There is a natural aspect of this subject in which the material is thin in two of its dimensions, for instance compacted wires or linear polymers. In one sense these materials are simpler; sheets of paper (a)E-mail: [email protected] cannot easily deform into a spherical cap, but instead form irreversible creases, while wires are not subject to this complication. For wires however there is a question of the manner of confinement; without any special constraint a wire could for instance produce a dense phase by coiling like a spring [5], which is irrelevant for sheets. To restrict to the essentials of both types of material we will discuss a model of a wire which is confined in a thin box as it is compacted, as has been done for instance in [6] where confining plates eliminate the possibility of coiling. Crumpled materials are a form of soft matter, in the sense that they can be macroscopically deformed with much less energy than required to similarly deform a typical equilibrium solid. De Gennes [7], Flory [8], Edwards [9], and others have long championed the use of statistical mechanics methods to model various forms of soft matter, especially polymers, colloids and granular media. Recently, this approach has also been used to model sheets (see [4]), wires (see [10]) and polymers (see [5,11]) under variable confinement, even to model a possible phase transition, in the strict sense we are using, between the highand low-density regimes. We note in particular two different types of modelling, a meanfield approach by Boué and Katzav [10] for a model of wires, and one using self-avoiding walks on a lattice by Jacobsen and Kondev [11] to model folded polymers. Both predict a second-order phase transition. We use a closely related approach but find a first-order phase transition, in contrast with the results of those papers. The difference is significant. Our results are evidence within this form of soft matter of an order/disorder transition with coexisting phases, known already in both the fluid/solid and liquid crystal transitions of equilibrium fluids, as well as in