Statistical mechanics of dislocation pileups in two dimensions

Dislocation pileups directly impact the material properties of crystalline solids through the arrangement and collective motion of interacting dislocations. We study the statistical mechanics of these ordered defect structures embedded in two-dimensional crystals, where the dislocations themselves form one-dimensional lattices. In particular, pileups exemplify a new class of inhomogeneous crystals characterized by spatially varying lattice spacings. By analytically formulating key statistical quantities and comparing our theory to numerical experiments using an intriguing mapping of dislocation positions onto the eigenvalues of recently studied random matrix ensembles, we uncover two types of one-dimensional phase transitions in dislocation pileups: A thermal depinning transition out of long-range translational order from the pinned-defect phase, due to a periodic Peierls potential, to a floating-defect state, and finally the melting out of a quasi-long-range ordered floating-defect solid phase to a defect liquid. We also find the set of transition temperatures at which these transitions can be directly observed through the one-dimensional structure factor, where the delta function Bragg peaks, at the pinned-defect to floating-defect transition, broaden into algebraically diverging Bragg peaks, which then sequentially disappear as one approaches the two-dimensional melting transition of the host crystal. We calculate a set of temperature-dependent critical exponents for the structure factor and radial distribution function, and obtain their exact forms for both uniform and inhomogeneous pileups using random matrix theory.

Figure 1

Schematic of conditions under which one-dimensional dislocation pileups, consisting of (orange) short edge dislocation lines (these become pointlike when the $y$ dimension is extremely narrow) with Burgers vectors aligned with a glide direction (turquoise strips), embedded in a two-dimensional surface (gray). (a) When a thin two-dimensional crystalline slab experiences applied shear stress $\sigma \left(x\right)$, short dislocation lines pile up along the direction of the shear stress. The spatial profile of the applied stress $\sigma \left(x\right)$ directly determines the density distribution of the dislocations via Eq. (8). (b) When a two-dimensional crystal is curved, residual stresses due to the Gaussian curvature leads to dislocation pileups that wrap around the spherical cap. Schematic in (b) adapted from Refs. [21, 22].

Figure 2

(a) Phase diagram for dislocation pileups as a function of temperature $T$. A pinned defect crystal with Delta function Bragg peaks appears at low temperatures, with a floating-defect solid phase that gradually melts at intermediate temperatures. (b) Melting of a semicircular dislocation pileup in the floating-defect solid phase as revealed by random matrix simulations of a semicircular density distribution of defects. Black downward arrow on the left side indicates the direction of increasing temperature $T$. At each temperature, the structure factor $S\left(q\right)$ extracted from one random matrix simulation of $N=5000$ total dislocations is shown on the left, and snapshots of the dislocations (with positions given by the eigenvalues of a random matrix) near the lattice center and the dislocations closer to the lattice edge (as indicated in the top schematic) are shown on the right. Here $n\left(x\right)$ denotes the 1D dislocation density profile and $G_1$ is the location of the first Bragg peak.

Figure 3

(a) Illustration of the average dislocation density of the semicircular pileup (red) and approximately uniform pileup (semicircle pileup truncated to include only those dislocations within $25\%$ of the center) (green). (b)–(d) Structure factor $S\left(q\right)$ averaged over 500 realizations of rank $N=5000$ random matrices. The first, second, and third Bragg peaks form at $\beta =4$, 16, 36, where $\beta$ is the dimensionless random matrix inverse temperature parameter $\beta =\frac{Y{b}^2}{4\pi k_{B}T}$ in Eq. (18), as predicted by Eq. (57). The wave vectors $q$ on the $x$ axes are scaled by the first reciprocal lattice vector $G_1=2\pi /D$ where $D$ is the eigenvalue spacing at the center of the semicircular lattice. Straight lines connecting the dots are there to guide the eye.

Figure 4

Structure factor $S\left(q\right)$ of the truncated semicircle lattice [i.e., approximately uniform lattice, see schematic on top right of Fig. 3] near the first reciprocal lattice vector at different temperatures in range $T_c^{\left(2\right)}>T\ge T_c^{\left(1\right)}$ ($16>\beta \ge 4$). Green dots show the results from random matrix simulations while blue lines are from Eq. (73); simulations and theory show good agreement.

Figure 5

Structure factor $S\left(q\right)$ of the full semicircular pileup at different temperatures in range $T_c^{\left(2\right)}>T\ge T_c^{\left(1\right)}$ ($16>\beta \ge 4$). Red dots show the results from random matrix simulations while blue lines are Eq. (76); simulations and theory show good agreement. Deviations above the first Bragg peak $q>G_1$ likely come from the smeared out contribution of the $S_2\left(q\right)$ term, neglected in Eq. (63), which contributes a noise floor of $S(q\to \infty )=1$ in the absence of higher order Bragg peaks. Here $G_1$ is defined as $G_1\equiv 2\pi /D(x=0)$, where $D(x=0)$ is the average dislocation spacing at the center of the semicircular pileup.

Figure 6

Radial distribution function $g\left(r\right)$ at $\beta =1,2,4,6$ for uniform pileups. This quantity always approaches unity for large $r$. Green dots are data from random matrix simulations (RMS) of the truncated semicircle pileup [i.e., an approximately uniform pileup, see see inset of (a)] averaged over 500 realizations of rank $N=5000$ random matrices. Large $\frac{r}{D}$ behaviors from classical random matrix theory at $\beta =1,2,4$ (Table 3) are plotted in blue in (a)–(c) [$D=\frac{\pi }{2N}$ is known from random matrix theory to be the mean lattice spacing at the middle of the semicircle lattice spanning $(-1,1)$.] Large $\frac{r}{D}$ behavior according to Eq. (84) at $\beta =6$ is plotted in blue in (d).

Figure 7

Local radial distribution functions $g_R\left(r\right)$ at $T=\frac{1}{2}{T}_c^{\left(1\right)}$ ($\beta =8$) for inhomogeneous pileups at three different mean locations $R=\{0,0.47,0.85\}$ in the semicircular pileup of length $L=2$ spanning $(-1,1)$, as indicated in the top graph. Dots are data averaged over 500 random matrix simulations using rank $N=5000$ matrices, colored according to the mean location $R$ indicated by the colored slices in the top schematic. Blue lines are the predictions of Eq. (87) with the mean lattice spacing $\overline{D}\left(R\right)$ extracted from simulations by averaging the positions of dislocations (eigenvalues) within each region $R\pm 0.01$. Theory and simulations show good agreement.

Figure 8

Illustrations of a pileup commensurate with the underlying Peierls potential (gray) with $D/a=3$ (a) and a pileup incommensurate with the underlying Peierls potential with $D/a=3.5$ (b), where $a$ is the host lattice constant and $D$ is the average dislocation spacing.