Statistical Mechanics and Shape Transitions in Microscopic Plates

Unlike macroscopic multistable mechanical systems such as snap bracelets or elastic shells that must be physically manipulated into various conformations, microscopic systems can undergo spontaneous conformation switching between multistable states due to thermal fluctuations. Here we investigate the statistical mechanics of shape transitions in small elastic elliptical plates and shells driven by noise. By assuming that the effects of edges are small, which we justify exactly for plates and shells with a lenticular section, we decompose the shapes into a few geometric modes whose dynamics are easy to follow. We use Monte Carlo simulations to characterize the shape transitions between conformational minimal as a function of noise strength, and corroborate our results using a Fokker-Planck formalism to study the stationary distribution and the mean first passage time problem. Our results are applicable to objects such as graphene flakes or protein $\beta$ sheets, where fluctuations, geometry, and finite size effects are important.

Figure 1

(a) “Closed” and “open” state of a slap bracelet. (b) Model of the shape of a microscopic sheet such as a $\beta$ sheet by a differentiable coarse-grained map $w(x,y)$ [see Eq. (5))] with two bending modes ${\kappa }_x$, ${\kappa }_y$ and a twisting mode ${\kappa }_{xy}$. (c) Saddle with $\mathrm{\kappa ⃗}={({\kappa }_x,{\kappa }_y,{\kappa }_{xy})}^T={(-1,1,0)}^T$.

Figure 2

(a) Contour plot of the elastic energy as a function of ${\kappa }_x$ and ${\kappa }_y$. Point 1 is the local minimum, point 2 is the saddle point, and point 3 is the global minimum. (b) Two separate stochastic realizations with plate initially at point 1 and $T=0.1/1$ (blue and red histogram). (c) Histogram of equilibrium statistics consisting of ${10}^3$ stochastic realizations under cold conditions ($T=0.1$) with the plate initially at point 1. (d) Same as (c) but under hot conditions ($T=1$).

Figure 3

The steady-state PDF $p({\kappa }_x)$, $p({\kappa }_y)$, and $p({\kappa }_{xy})$, at (a) cold conditions ($T=0.1$), (b) mild conditions ($T=0.5$), and (c) hot conditions ($T=5$) for a circular plate with ${\mathrm{\kappa ⃗}}^0={(3,3,-1)}^T$ and ${\mathrm{\kappa ⃗}}^m={(-2.2,-2.2,-0.2)}^T$.

Figure 4

Plot of $\log (\mathrm{\Pi })$ as a function of $1/T$. $\mathrm{\Pi }$ is the mean first passage time (MFPT) of the microscopic plate to escape over a barrier from initial curvature ${\mathrm{\kappa ⃗}}^m$ to the global minimum ${\mathrm{\kappa ⃗}}^0$. The curves are lines of best fit to data or theory. (see text) Inset: Histogram of the FPT for $T=0.2$. The FPT has an exponential distribution with parameter $\lambda =1/\mathrm{\Pi }\approx 0.1$, as shown in the blue curve.