Emergent dynamic stress regulators via coordinated thermal fluctuations and stress in harmonic crystalline lattices

Abstract

Understanding thermal fluctuations yields insights into a wide range of behaviors in many-body systems. In this work, we analyze the dynamical adaptation of two-dimensional crystalline lattice system under harmonic interaction in response to the intricate interplay of thermal agitation and mechanical stress by developing the characteristic stress-absorbing quadrupole structures and stress-releasing fold structures. These thermally driven stress regulator structures serve as a tangible embodiment of thermal fluctuations, offering a unique perspective on the characterization and manipulation of the elusive fluctuations. Specifically, we reveal the stretch-driven alignment and linear accumulation of quadrupoles, characterize the formation and proliferation of folds, and present the phase diagram of the dynamical states defined by these characteristic structures. This work demonstrates the promising avenue of re-examining classical mechanical systems subject to thermal agitation, which is of fundamental physical interest and has potential practical significance in the design of mechanical devices in thermal environments.

Figures (3)

• Figure 1: Thermalized crystalline lattice confined on the cylindrical surface. (a) Schematic plot of the lattice-on-cylinder model. The lattice is generated by the periodicity vector $\vec{V}=p \vec{a} + q \vec{b}$. $(p,q)=(8, 4)$ (left) and $(16, 8)$ (right). The $x$ axis is along the $\vec{V}$ vector. (b)-(d) show the thermalization of the disturbed lattice system. (b) The convergence of the speed distribution towards the Maxwell distribution (dashed red curve) at $v_0=0.1$. The inset figure shows the rapid convergence process as characterized by $\delta f$ (the deviation from the Maxwell distribution) at $v_0=0.1$ (blue), $0.25$ (red) and $0.5$ (black). (c) Plot of temperature (as derived from the Maxwell distribution) versus the initial disturbance strength $v_0$ at $\epsilon_0=-20\%$ (red), $0$ (blue) and $20\%$ (black). (d) Instantaneous distribution of the length of bonds parallel to the $x$ axis. The magnitude of the dispersion derived in theory is indicated by the red arrow. $v_0=0.3$. $t=25,000\tau_0$ and $\epsilon_0=0$ [(b) and (d)].
• Figure 2: Characterization of thermally driven quadrupoles in a typical stretched crystalline lattice. (a) Emergence of quadrupoles by the combined effects of stress and thermal fluctuation. A quadrupole consists of four disclinations of opposite signs organized in a square configuration; the five- and seven-fold disclinations as indicated by red and blue dots. The zoomed-in plots of the highlighted isolated quadrupole and quadrupole pile are presented and analyzed in (a) and (b). $\epsilon_0=10\%$, $v_0=0.3$, $t=7,500\tau_0$, and $(p,q)=(39,0)$. The lower panels show the histogram of $\theta$ (the angle between the $\vec{q}$ vector and the $x$ axis) at $\epsilon_0=10\%$ (as indicated by the red arrow in the right panel), and the plot of the relative number of horizontal quadrupoles ($\gamma$) versus the initial strain $\epsilon_0$. The statistical analysis is based on the sampling during $t\in [500\tau_0, 10,000 \tau_0]$ at the resolution of $\delta t=200\tau_0$. (b) Demonstration of the formation of a quadrupole via the flip of the geometric bond from horizontal ($AB$ in configuration $H$) to vertical ($C'D'$ in configuration $V$). (c) Illustration of the connection of the quadrupole pile and the shear band. (d) Analysis of the lifespan $\tau$ of quadrupoles at $\epsilon_0=0$ (green), $\epsilon_0=5\%$ (blue), and $\epsilon_0=10\%$ (black). The histograms of lifespans are presented in the lower panels. $v_0=0.3$. The sampling is during $t\in [40\tau_0, 100 \tau_0]$ with a temporal resolution of $\delta t = 0.02\tau_0$, which is much shorter than the quadrupole lifespan. Note that the initial sampling time is chosen to be one or two orders of magnitude longer than the relaxation time to ensure that the velocity distribution is fully equilibrated.
• Figure 3: Thermally driven fold structure and the fold-driven collapse of the crystalline lattice. (a) Identification of the fold structure in thermalized, pre-compressed lattice. $\epsilon_0=-20\%$, $v_0=0.2$, $t=1,000\tau_0$, and $(p,q)=(20,20)$. The red arrows indicate the growth direction of the folds initially appearing on the edges at the indicated times. The upper part of the lattice is removed for visual convenience. (b) The paper model to illustrate the formation of the fold via folding with respect to the reference line (in red). (c) The energy barrier during the associated quasi-static mechanical deformation process characterized by the displacement $x$, as demonstrated in the inset figure. $\Delta E$ is the variation of the elastic energy of the highlighted rhombs in the inset figure. (d) Collapse of the lattice as characterized by its sudden shrinking under strong thermal agitation. $\epsilon_0=0$. (e) Plot of the relative time-averaged width $\langle W \rangle /W_0$ versus $L_0$ at typical aspect ratio $L_0/W_0$. $\epsilon_0=0$, $v_0=0.7$, and $p=q$. (f) The phase diagram for the dynamical states of the system. The symbol "0" refers to the fold-free and defect-free state, "D" for the state with defects, "1" for the folded state, "C" for the collapsed state [$\langle W\rangle /W_0<0.5$], and "PC" for the partially collapsed state [$\langle W\rangle /W_0 \in (0.5, 0.9)$]. $(p, q) = (20, 20)$. $W_0=30\ell$. In (e) and (f), the sampling is during $t\in [500\tau_0, 10,000 \tau_0]$ at the resolution of $\delta t = 100\tau_0$.