Theory of interacting dislocations on cylinders

We study the mechanics and statistical physics of dislocations interacting on cylinders, motivated by the elongation of rod-shaped bacterial cell walls and cylindrical assemblies of colloidal particles subject to external stresses. The interaction energy and forces between dislocations are solved analytically, and analyzed asymptotically. The results of continuum elastic theory agree well with numerical simulations on finite lattices even for relatively small systems. Isolated dislocations on a cylinder act like grain boundaries. With colloidal crystals in mind, we show that saddle points are created by a Peach-Koehler force on the dislocations in the circumferential direction, causing dislocation pairs to unbind. The thermal nucleation rate of dislocation unbinding is calculated, for an arbitrary mobility tensor and external stress, including the case of a twist-induced Peach-Koehler force along the cylinder axis. Surprisingly rich phenomena arise for dislocations on cylinders, despite their vanishing Gaussian curvature.

Figure 1

Lattice configuration corresponding to a single dislocation with Burgers vector in the positive $x$ direction for a square (left) and triangular (right) lattice of masses and harmonic springs relaxed to their equilibrium configuration. The Burgers vector is defined as the deficit (red arrow) of a clockwise loop that would be closed in the absence of a dislocation; hence $\vec{b}=b\widehat{x}$ in both cases, where $b$ is the lattice constant. The “T” symbol encodes both the position and orientation of the dislocation; the leg of the inverted T points in the direction of added material.

Figure 2

Schematic illustration of insertions of new glycan strands into the peptidoglycan mesh of a bacterial cell wall. The new material is inserted in the vicinity of the blue arrow, which points azimuthally around the cylinder and sits near the core of the dislocation. The axis of the cylinder runs horizontally, which is also the direction of the Burgers vector of this dislocation. Although the the region around the new insertions is distorted, the connectivity of the structure is preserved locally due to the simultaneous insertions of two extra glycan strands into the structure, respecting the lattice geometry. The rectangular (dashed-line) box shows the biologically relevant unit cell, which is however not the minimal unit cell of the underlying lattice [8]. The red, dashed arrow shows the Burgers vector.

Figure 3

Schematic of dislocations (i.e., glycan strand ends) on the cylindrical portion of a bacterial cell wall of radius $R_0$. A subset of these defects rotates circumferentially when propelled by the addition of material from inside the bacterium, mediated by strand elongation machinery on the dislocation cores (not shown). Constant velocity climb motion of such dislocations leads to exponential elongation of the cylinder length $L\left(t\right)$; see Refs. [8, 9].

Figure 4

Model of a dislocation pair on a cylinder. This structure can systematically elongate if new particles (green) are added by two counter-rotating dislocations moving via climb. For one of the dislocations, the climb direction is indicated by a blue arrow. For the other, the Burgers circuit around the dislocation is shown (yellow) as well as the resulting Burgers vector $\vec{b}$. Alternatively, the dislocation pair could separate by glide (motion parallel to $\vec{b}$), with motion predominately along the cylinder, with negligible elongation. In both cases, however, a triangular lattice with slightly tilted Bragg rows is created in the center, relative to the purely azimuthal Bragg rows made of silver particles on the two sides. Although the model was constructed with magnetic beads interacting via dipole-dipole interactions, we expect similar configurations for micron-sized colloids assembled on a cylindrical substrate.

Figure 5

Equipotential contours for a dislocation on a cylinder at position $(x,y)$ with $\vec{b}=-b\widehat{x}$, interacting with another dislocation at the origin with $\vec{b}=b\widehat{x}$. Energy is measured in units of $A{b}^2.$ Note the much lower energies than in Fig. 8 where the Burgers vectors are rotated by ${90}^{\circ }$.

Figure 6

Slice along the energy contours of Fig. 5 showing interaction energy as a function of $x$, for several ratios of $y/W$. Note the double minima, which become a single maximum for $y\ge W/4$.

Figure 7

Equipotential contours for a dislocation at $(x,y)$ on a cylinder with $\vec{b}=b\widehat{y}$, interacting with another dislocation at the origin with $\vec{b}=b\widehat{x}$. Energy is measured in units of $A{b}^2.$ Note that the large distance core energy contribution, analogous to that appearing in Eq. (32), is undefined without additional dislocations on the cylinder, since the sum of the Burgers vectors should vanish. Nevertheless, this expression is useful for charge-neutral dislocation configurations on a cylinder, whose interactions can be decomposed into pairwise interactions involving $E_{\widehat{x},-\widehat{x}}$, $E_{\widehat{y},-\widehat{y}}$, and $E_{\widehat{x},\widehat{y}}$; see Eqs. (35), (40), and (38).

Figure 8

Equipotential contours for a dislocation at $(x,y)$ on a cylinder with $\vec{b}=-b\widehat{y}$, interacting with another dislocation at the origin with $\vec{b}=b\widehat{y}$. Energy is measured in units of $A{b}^2.$ Note that for $x\gg R=W/2\pi$ the energy contours become parallel to the $y$ axis, so that the forces are primarily in the $x$ direction. This trend, and the associated linear potential, can be seen from Eqs. (12), (13), (30), and (31).

Figure 9

Equipotential contours for a dislocation on a cylinder with a Burgers vector $\vec{b}=b(\frac{\sqrt{3}}{2}\widehat{x}+\frac{1}{2}\widehat{y})$, forming an angle of $\pi /6$ with respect to the $\widehat{x}$ axis, interacting with another dislocation with an opposite Burgers vector. The interaction energy is given by Eq. (46). Note the two skewed saddle points. Energy is measured in units of $A{b}^2.$

Figure 10

Stress components ${\sigma }_{xx}$ (left) and ${\sigma }_{yy}$(right) as a function of $x$ at a constant offset around the cylinder $y=W/4$, for a dislocation with $\vec{b}=b(\frac{\sqrt{3}}{2}\widehat{x}-\frac{1}{2}\widehat{y})$ at the origin. To account for a charge-neutral dislocation configuration, another dislocation with an antiparallel Burgers vector is placed along the negative $\widehat{x}$ end of the cylinder, far enough from the origin to create a constant ${\sigma }_{yy}$ stress [see Eq. (30)] and a negligible contribution to the other components of the stress tensor. The symbols are for simulations of cylinders with $W$ increasing from $8b$ to $40b$ where $b$ is the equilibrium lattice constant of the simulation lattice. In all cases, the length of the cylinder is $8.66W$ and the dislocation is situated at the center of the cylinder along the long axis to minimize edge effects. The solid line is the theoretical prediction from Eq. (43) with $\theta =-\pi /6$, where for the right figure one has to add the constant ${\sigma }_{yy}$ stress mentioned above. Except for one or two points closest to the dislocation, where discrete effects become important, the simulations agree with the results of continuum elasticity even for relatively small system sizes ($W\gtrsim 16b).$ For negative values of $x$, the convergence to the continuum limit results is slower, and there are corrections to the stresses which scale as $\sim$$b/W.$ These will be discussed in more detail in future work.

Figure 11

Dislocations on a cylinder are analogous to grain boundaries. The images show configurations obtained numerically for a hexagonal lattice of masses and harmonic springs whose energy is minimized as the dislocation pair is separated by a glide. On this triangular lattice, a dislocation corresponds to a point with five neighbors adjacent to a point with seven neighbors. Starting with two nearby dislocations (a), we glide one of the dislocations to the right in steps of the lattice spacing [Supplemental Material [40]; panels (b)–(d) show intermediate snapshots during the glide]. The finite section of the cylinder between the two dislocations displays a helical structure, similar to the model with magnetic beads shown in Fig. 4. The analogy with a flat polycrystalline material is illustrated by unrolling the cylindrical crystal in (d) and duplicating it several times in the $y$ (circumferential) direction to replicate the periodic boundary conditions (e). The resulting infinite columns of dislocations (dotted lines) define two grain boundaries separating the middle region, rotated by an angle $\theta =b/W$, from sections with the original crystal orientation on either side.

Figure 12

Equipotential contours for a dislocation pair with $\frac{GW}{A{b}^2}=1,$ with $W=2\pi R$. Note that the force $G$ caused by a macroscopic stress ${\sigma }_{xx}$ has pulled the maximum away from $y=W/2$. Note also the two saddle points (denoted by the letter $S$) near $y\approx W/4.$

Figure 13

Equipotential contours for a dislocation pair with $\frac{GW}{A{b}^2}=10$. Note the maximum and two saddle points near the line $y=W/10.$

Figure 14

The rescaled $y$ coordinate of the saddle point $\frac{yG}{A{b}^2}$ is shown as a function of the dimensionless parameter $\frac{GW}{A{b}^2}$. The horizontal line is the result in flat space, where $y=\frac{A{b}^2}{G}.$

Figure 15

The saddle point excitation energy $U$, given by Eq. (35), is rescaled and plotted as a function of $\frac{GW}{A{b}^2}$. In the flat space limit the scaling function is given by $\eta \left(D\right)=-\ln \left(D\right),$ corresponding to $U=A{b}^2\ln (Ab/G)$. For the opposite case, when $G$ is small and the cylindrical geometry is important, the energy $U$ saturates at a constant value even as $G\to 0$, $\frac{U}{A{b}^2}\approx \text{const}+\ln \left(\frac{W}{b}\right)$.

Figure 16

A mobile dislocation, initially at point $A$, interacting with a pinned dislocation with antiparallel Burgers vector at the origin $O$. The first dislocation is placed just beyond the saddle point, at $(x,y)=(\frac{W}{10},\frac{W}{10})$, for $\frac{GW}{A{b}^2}=10$ (for a flat space, this would be the exact position of the saddle, however on a cylinder this point lies beyond it). The temperature is close to zero, although we assume that motion over the Peierls potential [4] is possible. The isotropic mobility tensor components are equal. After completing one loop, the dislocation is “recaptured” and annihilates with the dislocation at the origin.