Equilibrium and dynamic pleating of a crystalline bonded network

Abstract

We describe a phase transition that gives rise to structurally non-trivial states in a two-dimensional ordered network of particles connected by harmonic bonds. Monte Carlo simulations reveal that the network supports, apart from the homogeneous phase, a number of heterogeneous "pleated" phases, which can be stabilised by an external field. This field is conjugate to a global collective variable quantifying "non-affineness", i.e. the deviation of local particle displacements from local affine deformation. In the pleated phase, stress is localised in ordered rows of pleats and eliminated from the rest of the lattice. The kinetics of the phase transition is unobservably slow in molecular dynamics simulation near coexistence, due to very large free energy barriers. When the external field is increased further to lower these barriers, the network exhibits rich dynamic behaviour: it transforms into a metastable phase with the stress now localised in a disordered arrangement of pleats. The pattern of pleats shows ageing dynamics and slow relaxation to equilibrium. Our predictions may be checked by experiments on tethered colloidal solids in dynamic laser traps.

On the existence of thermodynamically stable rigid solids

“…For solids where individual particles can be distinguished and tracked, one should be able to realise h X in the laboratory and check our predictions in the full h X − ε plane. Indeed, it has already been discussed in detail [15,17,20] how this may be accomplished in the future for colloidal particles in 2d using dynamic laser traps. Briefly, the set of reference coordinates is read in and a laser tweezer is used to exert additional forces, F χ (r i ) = −∂H X /∂r i to each particle i which bias displacement fluctuations.…”

“…For our results of the equilibrium structures and transitions at T > 0 we employed the sequential umbrella sampling (SUS) technique (see Appendix C) coupled to Monte Carlo [22,23] in the constant number N , area A = L x × L y , ε and temperature T ensemble. Advanced sampling techniques such as SUS are necessary to overcome the large barriers between the N and M phases, enabling the equilibrium transition to be observed [17]. We show results for T = 0.8 and density ρ = 1.1547, corresponding to the choice a = 1.0 for the lattice parameter.…”

“…The deformation is measured from a set of fixed reference coordinates. In a series of papers [14][15][16][17][18][19] (see also Appendix A for details) some of us have worked out in detail the statistical thermodynamics of χ i , which quantifies the "non-affine" component of the particle displacements, analytically and numerically for a number of two dimensional (2d) crystals at finite temperature. At any T > 0, X behaves as a regular thermodynamic variable with a well-defined mean and variance ∼ N −1 .…”

Wrinkles, folds and ripplocations: unusual deformation structures of confined elastic sheets at non-zero temperatures

“…Our calculations explicitly take into account the effect of finite temperature T > 0 and therefore represent a definite advance on earlier work. We show that ripplocation like structures, which are the generalization of the purely two-dimensional (2D) pleats studied earlier [12,13], readily form following a phase transition from an essentially flat sheet containing at most thermally generated ripples. These phases co-exist at a first-order boundary.…”

“…Recently, it has been observed that spatially confined flexible membranes deform by reorganizing their morphology to form hierarchical structures of great complexity [6]. Wrinkles, ripples, folds [7][8][9][10][11], as well as higher order structures like pleats [12,13] are ubiquitous both in nature and in emergent technology. They have been observed in biological tissues [14,15], polymer sheets [16], and many other lowdimensional materials involving flexible sheets and membranes [17][18][19][20][21][22][23][24][25][26][27].…”