Pinning Force in Active Media

Pinning of vortices by defects plays an important role in various physical (superconductivity, superfluidity, etc.) or biological (propagation in cardiac muscle) situations. Which defects act as pinning centers? We propose a way to study this general problem by using an advection field to quantify the attraction between an obstacle and a vortex. A full solution is obtained for the real Ginzburg-Landau equation (RGLE). Two pinning mechanisms are found in excitable media. Our results suggest strong analogies with the RGLE when the heterogeneity is excitable. Unpinning from an unexcitable obstacle is qualitatively harder, resulting in a stronger pinning force. We discuss the implications of our results to control vortices and propose experiments in a chemical active medium and in cardiac tissue.

Figure 1

Pinning and depinning in the GL model. (a) A small drift term (arrow) shifts the vortex (dotted line); see Eq. (1). (b) Normalized unpinning velocity ${\tilde{c}}_u=c_u/c_u^{\infty }$.

Figure 2

An obstacle with $b_{\mathrm{i}\mathrm{n}}<b_{\mathrm{o}\mathrm{u}\mathrm{t}}$ does not pin a spiral wave (a),(b), whereas with $b_{\mathrm{i}\mathrm{n}}>b_{\mathrm{o}\mathrm{u}\mathrm{t}}$ pins it (c),(d). (a) A rotating wave is expelled from an obstacle (thick dashed line) with $b_{\mathrm{i}\mathrm{n}}=0.099$ and $b_{\mathrm{o}\mathrm{u}\mathrm{t}}=0.1$(corresponding to $r_{c,\mathrm{i}\mathrm{n}}=2.18$ and $r_{c,\mathrm{o}\mathrm{u}\mathrm{t}}=2.32$). The trajectory of the tip for four rotations at different times is shown. (b) Maximal distance per period to the center of the obstacle as a function of time for three values of $b_{\mathrm{i}\mathrm{n}}$. (c) Same as (a) with $b_{\mathrm{i}\mathrm{n}}=0.101$ (corresponding to $r_{c,\mathrm{i}\mathrm{n}}=2.50$); the spiral wave is captured by the obstacle. (d) Same as (b). The position of the tip is defined by the intersection of isolines: $u=0.5$ and $v=0.5a-b_{\mathrm{o}\mathrm{u}\mathrm{t}}$. Period of rotation of the spiral $\sim 8\mathrm{\text{time units}}$. Obstacle radius $R=1.2$ ($=10\mathrm{\text{grid units}}$). Parameters in Eq. (9): $ϵ=0.02$ and $a=0.7$.

Figure 3

Unpinning by an advective field. (a) Unpinning velocity as a function of the obstacle radius ($b_{\mathrm{o}\mathrm{u}\mathrm{t}}=0.1$). The dashed line is a straight line with slope $2r_c/T$. (b),(c) The routes from a pinned rotating wave to a free one. ($R/r_{c,\mathrm{o}\mathrm{u}\mathrm{t}}=0.67$). $r_c(b=0.1)=2.32$ and $r_c(b=0.11)=6.13$.

Figure 4

A pinned vortex becomes free when the gradient of parameters is increased. (a) Drift of a rotating wave due to the excitability gradient. (b) Pinning to an obstacle. (c) The excitability gradient is increased by 20%. The vortex becomes free and drifts as in (a). $b(x=100)=0.1$; $b$ increases from left to right: $\Delta b/\Delta x={10}^{-4}$ in (a),(b); inside the obstacle (b),(c): $b_{\mathrm{i}\mathrm{n}}=0.11$. Arrows indicate gradient. Axes scale is grid units.