When Soft Crystals Defy Newton’s Third Law: Nonreciprocal Mechanics and Dislocation Motility

The effective interactions between the constituents of driven soft matter generically defy Newton’s third law. Combining theory and numerical simulations, we establish that six classes of mechanics with no counterparts in equilibrium systems emerge in elastic crystals challenged by nonreciprocal interactions. Going beyond linear deformations, we reveal that interactions violating Newton’s third law generically turn otherwise quiescent dislocations into motile singularities which steadily glide though periodic lattices.

Figure 1

Hydrodynamic interactions do not obey Newton’s third law. We give four examples of nonreciprocal forces between pairs of identical particles driven in a viscous fluid. Parity-symmetric forces (green arrows) correspond to equal drag forces acting on the particles. (a) Two identical particles sedimenting in a viscous fluid. The interaction force field decomposes on the $n=0$, 2 modes of Eq. (3). (b) Two particles advected in a shallow microfluidic channel interact hydrodynamically via the sole $n=2$ mode. Parity-antisymmetric forces (red arrows) correspond to opposite drag forces acting on the particles. (c) Two particles spinning in a viscous fluid interact via transverse forces ($n=1$ mode only). (d) When swimming in the same direction two so-called squirmers interact hydrodynamically via the superposition of forces with $n=-1$, 1, and 3 angular modes [30].

Figure 2

Mechanics of elastic lattices challenged by nonreciprocal interactions. (a) When a hexagonal lattice undergoes a deformation, parity-antisymmetric forces induce a stress $\sigma$ (red) that can be computed using the Irving-Kirkwood formula [Eq. (5)]. The macroscopic response to a deformation $D$ reads ${\sigma }_{\alpha }={\mathrm{\Pi }}_{\alpha }+K_{\alpha \beta }{D}_{\beta }$ where $\alpha$, $\beta$ denote the elementary stress and deformation modes defined in (b) and $\mathrm{\Pi }$ is a so-called prestress supported by the crystal even in absence of any deformations. In the case of parity-symmetric interactions, a deformation $D$ induces a net force $\mathcal{F}$ defined as the net force experienced by any test particle (green arrow). $\mathcal{F}$ and $D$ are related by the linear yet noninvertible relation ${\mathcal{F}}_i={\mathcal{F}}_i^0+A_{i\beta }{D}_{\beta }$ where $i=x$, $y$. The six possible structures for the matrices $K$ and $A$ are given in (c). (b) Elementary modes of 2D deformations (and stresses). ${\tau }^{\delta }$ corresponds to a dilation (isotropic pressure), ${\tau }^{\omega }$ to a rotation (torque), while ${\tau }^{s_1}$ and ${\tau }^{s_2}$ are the two pure-shear deformations (shear stresses). (c) On a hexagonal lattice, parity-antisymmetric interactions define three classes of stress-strain relations. These constitutive relations are determined by the angular symmetry of the interactions ($n\equiv 1,3,5[6]$ where “[6]” stands for “modulo 6”), see Eq. (3). The prestresses $\mathrm{\Pi }$ are given: $\mathrm{\Pi }=(P_{\delta },T_{\omega },0,0)$ when $n\equiv 1[6]$, $\mathrm{\Pi }=0$ when $n\equiv 3[6]$, and $\mathrm{\Pi }=(0,0,S_1,S_2)$ when $n\equiv 5[6]$. Parity-symmetric interactions define three classes of force-strain relations determined by the angular symmetry of the interactions ($n\equiv 2,4,0[6]$). The three possible constitutive relations correspond to the three possible structures of the matrix $A$. A net force ${\mathcal{F}}_i^0$ can exist even in the absence of deformation under the action of nonreciprocal interactions when $n\equiv 0[6]$, it is given by ${\mathcal{F}}_i^0=(c_1,c_2)\ne 0$. The analytic expression of all material parameters are expressed in terms of the microscopic interactions both for parity-antisymmetric and symmetric forces in the SM [30].

Figure 3

Nonreciprocal interactions power dislocation glide. In our numerical simulations [30], we choose the nonreciprocal interactions to correspond to pure multipoles. When $n\ge 2$, $f_n(r)=1/r^n$ [see Eq. (3)]. The case $n=2$ then corresponds to hydrodynamic interactions in shallow channels. For $n=1$, we arbitrarily choose $f_1(r)=e^{-r^2/2}$ to reflect the short range coupling between colloidal spinners. Furthermore, the crystal is stabilized by reciprocal interactions $\mathbf{F}(\mathbf{r})=-A_{\mathrm{rep}}\mathbf{r}/\parallel \mathbf{r}{\parallel }^5$ (dipole-dipole repulsion). (a) Sketch of dipolar field ($n=2$) induced by a given particle. ${\alpha }_2$ is the phase of mode $n=2$ and $\mathbf{p}=(\cos {\alpha }_2,\sin {\alpha }_2)$. (b) Glide of a dislocation in a crystal. The nonreciprocal interactions correspond to $n=2$, ${\alpha }_2=20°$ ($A_{\mathrm{rep}}=10$). The three snapshots correspond to $t=0$, 0.25, and 0.5. (c) Left: numerical computation of the forces on the particles induced by dipolar pairwise interactions (${\alpha }_2=20°$). Right: theoretical prediction for the force field according to Eq (8), for the same microscopic parameters. (d) The variations of the gliding speed of a dislocation with the phase ${\alpha }_n$ measured from our numerical simulations (red dots) are in excellent agreement with our theoretical predictions (black solid line). From left to right $n=1$, 3, and 5 ($A_{\mathrm{rep}}=1$, 10, 50). (e) Same plots as in (d) in the case of parity-even forces. From left to right $n=2$, 4, and 6 ($A_{\mathrm{rep}}=10$, 20, 50).