Stress Relaxation in a Perfect Nanocrystal by Coherent Ejection of Lattice Layers

We show that a small crystal trapped within a potential well and in contact with its own fluid responds to large compressive stresses by a novel mechanism—the transfer of complete lattice layers across the solid-fluid interface. Further, when the solid is impacted by a momentum impulse set up in the fluid, a coherently ejected lattice layer carries away a definite quantity of energy and momentum, resulting in a sharp peak in the calculated phonon absorption spectrum. Apart from its relevance to studies of stability and failure of small sized solids, such coherent nanospallation may be used to make atomic wires or monolayer films.

Figure 1

(a) A 2D crystal of $N_s<N$ atoms confined to a central region $\mathcal{S}$ of area $A_s$ by means of an optical trap with a potential $\varphi (\mathbf{r})=-\mu$ for $\mathbf{r}\in \mathcal{S}$, which increases sharply to zero elsewhere over a width $\delta =\sigma /4$. The atoms interact with a hard disk potential [4, 23, 24] $V_{ij}=0$ for $|{\mathbf{r}}_{ij}|>\sigma$ and $V_{ij}=\infty$ for $|{\mathbf{r}}_{ij}|\le \sigma$, where ${\mathbf{r}}_{ij}={\mathbf{r}}_j-{\mathbf{r}}_i$ the relative position vector of the particles. (b) Equilibrium behavior for different $\mu$ at fixed $\eta =0.699$ (density at freezing ${\eta }_f=0.706$ [23]) obtained by Monte Carlo simulations with periodic boundary conditions in both directions ($N=1200$ particles occupy an area $A=22.78\times 59.18$ with the solid occupying the central third of the cell of size $L_s=19.73$). The trap depth $\mu =6$, supports an equilibrium solid of density ${\eta }_s=0.753$ in contact with a fluid of density ${\eta }_{\ell }=0.672$. The closest-packed lines of the solid in $\mathcal{S}$ are parallel to the solid-fluid interfaces that lie, at all times, along the lines where $\varphi (y)\to 0$. The density profile $\eta (y)$ coarse grained over strips of width $\sigma$ (averages taken over ${10}^3$ MC configurations each separated by ${10}^3$ MCS), varies from ${\eta }_{\ell }$ to ${\eta }_s$ as we move into $\mathcal{S}$. The horizontal lines are predictions of a simple free-volume based theory (see Fig. 2) for ${\eta }_s$ and ${\eta }_{\ell }$. (c) Superposition of particle configurations from the MC run in (b) showing a solid-like order (red: high $\eta$) gradually vanishing into the fluid (yellow: low $\eta$) across a well defined solid-fluid interface.

Figure 2

Plot of the equilibrium fractional density change $\Delta \eta /\eta$ as a function of $\mu$ [points (MC data), thick solid line (approximate theory)], showing a discontinuous jump at $\mu \approx 8$. The MC data are obtained by averaging over ${10}^3$ configurations each separated by ${10}^3$ MCS, while the system is equilibrated for $>{10}^7$ MCS starting from a uniform fluid with density $\eta =0.699$. The approximate theory is based on the assumption that a change in $\mu$ produces a uniform geometric strain ${\epsilon }_d$ from a reference triangular lattice with the same number of atomic layers. The geometric strain ${\epsilon }_d$ is an oscillatory function [4] of $L_s$, with an amplitude that decays as $1/L_s$. The Helmholtz free energy of the harmonic solid is then given by $f_s=f_{\Delta }+\frac{1}{2}{K}_{\Delta }{\epsilon }_d^2$, where $f_{\Delta }$ and $K_{\Delta }$ are the free energy and Young’s modulus, respectively, of an undistorted triangular lattice, which may be obtained from simple free-volume theory [4, 25]. Minimizing the total free energy density of the $\mathrm{\text{fluid}}+\mathrm{\text{solid}}$ regions, $f=x[f_s({\eta }_s,L_s)-4{\eta }_s\mu /\pi ]+(1-x)f_l({\eta }_{\ell })$ with the constraint $\eta =x{\eta }_s+(1-x){\eta }_{\ell }$, where $x$ is the area fraction occupied by $\mathcal{S}$ and $f_l({\eta }_{\ell })$ is the free energy of the hard disk fluid [26] that produces the jump in $\Delta \eta (\mu )/\eta$. Inset (top): A cycle-averaged hysteresis loop as $\mu$ is cycled at the rate of 0.2 per ${10}^6$ MCS. Inset (bottom): A plot of the tensile stress ${\gamma }_d$ against strain ${\epsilon }_d$. The arrows show the behavior of these quantities as $\mu$ is increased from the points marked A–D. The corresponding points in the $\Delta \eta /\eta$ vs $\mu$ plot is also marked for comparison. The state of stress in the solid jumps discontinuously from tensile to compressive from $\mathrm{B}\to \mathrm{C}$ due to an increase in the number of solid layers by one accomplished by incorporating particles from the fluid. This transition is reversible, and the system relaxes from a state of compression to tension by ejecting this layer as $\mu$ is decreased.

Figure 3

(a)–(c) Plots of the absolute value of the momentum $|v_y(y)|$ for molecular dynamics times $t=0.0007$ (a), 0.2828 (b), and 2.8284 (c). The green lines show the positions of the solid-fluid interfaces. The parameters $L_s=19.73$, ${\eta }_s=0.789$, and $\mu =4.8$. The fit to a Gaussian (blue line) is also shown in (c). The initial momentum pulse with strength $V_0=6$ is given within a narrow strip of size $\sim \sigma$, just to the left of the solid region and the Gaussian fitted (and the width ${\Delta }^2$ extracted) when the maximum of the pulse reaches a fixed distance of 44.1 from the source. A reflected pulse can also be seen. To reduce interference from the reflected pulse through periodic boundary conditions, we increase the fluid regions on either side, so that for the MD calculations we have a cell of size $22.78\times 186.98$ comprising 3600 particles. (d) A plot of the time development of the Fourier component of the local density correlation ${\rho }_{\mathbf{G}}(y,t)$ obtained by averaging, at each time slice $t$, the sum $\sum _{j=1,N}\exp \mathbf{(}-i\mathbf{G}·({\mathbf{r}}_{\mathbf{j}}-{\mathbf{r}}_{\mathbf{i}})\mathbf{)}$ over all particle positions ${\mathbf{r}}_{\mathbf{i}}$ within a strip of width $\sim \sigma$ centered about $y$ and spanning the system in $x$. The wave number $\mathbf{G}=(2\pi /d)\widehat{\mathbf{n}}$ where $d=0.92$ is the distance between crystal lines in the direction $\widehat{\mathbf{n}}$ normal to the fluid-solid interface. The solid [central region with ${\rho }_{\mathbf{G}}(y,t)\ne 0$] ejects a layer (shown by an arrow) that subsequently dissolves in the fluid. The curves from bottom to top correspond to time slices at intervals of $\Delta t=0.07$ starting from $t=1.06$ (bottom). We have shifted each curve upward by $0.03t/\Delta t$ for clarity. Curves such as in (a)–(d) are obtained by averaging over 100–300 separate runs using different realizations of the initial momentum distribution.

Figure 4

(a) Configuration snapshot from a portion of our MD cell showing hard disk atoms (green circles) at the solid (bottom)-liquid interface (yellow line) as a weak momentum pulse ($V_0=2$) emerges into the liquid. The pulse initially ejects a few atoms of the interfacial crystalline layer (red circles) of a metastable 22 layered solid at $\mu =4.8$. The resulting large nonuniform elastic strain evidenced by the bending of lattice layers, however, causes these atoms to be subsequently pulled back into the solid. Only a stronger pulse capable of ejecting a complete lattice layer succeeds in reducing the number of solid layers by one leading to overall lower elastic energy. (b) Plot of the squared width ${\Delta }^2$ of the momentum pulse after it emerges from the solid as a function of $V_0$ for $\mu =4.8$ (◇) and 9.6 (□). The solid line is a guide to the eye. The absorption of momentum is largest when the available kinetic energy of the pulse exactly matches the potential energy required to eject a layer. The peak in ${\Delta }^2(V_0)$ so produced is more prominent for the metastable 22 layered solid $\mu =4.8$ than for the stable ($\mu =9.6$) system showing a more coherent momentum transfer in the former case.