Role of Friction in Multidefect Ordering

We use continuum simulations to study the impact of friction on the ordering of defects in an active nematic. Even in a frictionless system, $+1/2$ defects tend to align side by side and orient antiparallel reflecting their propensity to form, and circulate with, flow vortices. Increasing friction enhances the effectiveness of the defect-defect interactions, and defects form dynamically evolving, large-scale, positionally, and orientationally ordered structures, which can be explained as a competition between hexagonal packing, preferred by the $-1/2$ defects, and rectangular packing, preferred by the $+1/2$ defects.

Figure 1

Defect ordering in wet active turbulence: (a) Snapshot of active turbulence for very low friction, $F=0.023$. The white (magenta) symbols are $+1/2$ $(-1/2)$ defects. Background color denotes the vorticity. (b) Schematic representation of $+1/2$ and $-1/2$ defects. $+1/2$ defects have a single polar axis (blue line) and $-1/2$ defects have three axes. (c),(d) For a reference $+$ (c) or $-$ (d) defect $i$ we define an associated polar coordinate system. (e) Spacing between $+1/2$ defects [defined as the position of the maximum in $g_{++}$ (in lattice units) as a function of the active length scale $\sqrt{K/\zeta }$]. Activity $\zeta$ and elasticity $K$ were varied. (f)–(i) Pair distribution function $g_{ab}(r,\theta )$ where $a$ and $b$ represent $+$ and/or $-$ defects showing the positional distribution of $b$-type defects around an $a$-type defect. The white arrows represent the orientational distribution vector $\mathit{S}$ with arrow size normalized by the magnitude of $\mathit{S}$ and axes are in lattice units.

Figure 2

(a) Snapshot of the defect structures at intermediate friction $F=0.023$. $+1/2$ ($-1/2$) defects are shown in white (magenta). There is transient local defect ordering into a rectangular (green outline) or a hexagonal (magenta outline) pattern. The background color represents the vorticity field. (b) Schematic of the rectangular ordering. (c) Schematic of the hexagonal ordering. This is chiral: $-1/2$ defects (in gray) have either a left or right neighboring $-1/2$ defect (in green). The other position is filled by rotating $+1/2$ defects resulting in local zero charge.

Figure 3

Ordering of the $+1/2$ defects at high friction (a),(b) Pair distribution function $g_{++}(r,\theta )$ and $g_{+-}(r,\theta )$ (color map) and the orientation distribution vector (white arrows) for high friction $F=0.103$. Axes are in lattice units. (c),(d) $g_{++}(r,0)$ and $g_{++}(r,\pi /2)$ showing the buildup of order along the $x$ and $y$ axes (in the direction of the red dotted line) with increasing friction. (e) Nematic defect ordering $S_d$, as a function of dimensionless friction $F=\sqrt{(K/\zeta )(f/\eta )}$, for varying elastic constant $K$.

Figure 4

Ordering around a $-1/2$ defect at high friction. (a) Pair distribution function $g_{--}(r,\theta )$ (color map) and the orientation distribution vector (white arrows) for intermediate friction $F=0.083$. Axes are in lattice units. (b),(c) Pair distribution functions $g_{--}(r,\theta )$ and $g_{-+}(r,\theta )$ for high friction $F=0.103$. (d)–(f) $g_{--}(r,-\pi /2)$, $g_{--}(r,-\pi /6)$, and $g_{-+}(r,-\pi /6)$ (along the red dotted line) with increasing friction.