Effects of confinement on the dynamics and correlation scales in kinesin-microtubule active fluids

We study the influence of solid boundaries on dynamics and structure of kinesin-driven microtubule active fluids as the height of the container, $H$, increases from hundreds of micrometers to several millimeters. By three-dimensional tracking of passive tracers dispersed in the active fluid, we observe that the activity level, characterized by velocity fluctuations, increases as system size increases and retains a small-scale isotropy. Concomitantly, as the confinement level decreases, the velocity-velocity temporal correlation develops a strong positive correlation at longer times, suggesting the establishment of a “memory”. We estimate the characteristic size of the flow structures from the spatial correlation function and find that, as the confinement becomes weaker, the correlation length, $l_c$, saturates at approximately 400 microns. This saturation suggests an intrinsic length scale which, along with the small-scale isotropy, demonstrates the multiscale nature of this kinesin-driven bundled microtubule active system.

Figure 1

Example of particle tracking in three dimensions. [(a) and (b)] Raw images taken at two successive times illustrating the sparse seeding of tracer particles. Particle positions are indicated by the Airy rings. (c) The center and size of each ring is detected using Sobel edge detection and the Hough circle transform. (d) Particles are tracked according to a nearest-neighbor algorithm. (e) The $z$ location of each particle is localized using the library of ring sizes vs. distance from the focal plane obtained from a calibration. [(f) and (g)] Two samples of particle tracks, taken from a shallow and a deep test geometry ($H=100$, $2000\mu \mathrm{m}$) illustrating the qualitative change in particle mobility due to confinement. The scale bar represents $40\mu \mathrm{m}$.

Figure 2

Normalized mean-square displacements (MSDs) of 1- and 2.9-$\mu \mathrm{m}$ particles diffusing in water and tracked using the Airy-disk three-dimensional tracking technique. The MSDs are normalized by the size of particle (1-$\mu \mathrm{m}$ or 2.9-$\mu \mathrm{m}$ diameter), and the error bars indicate standard errors of the averaging over all tracked particles.

Figure 3

(a) Lifetime of active systems in varying geometries, indicated by the parallel component of the specific kinetic energy $e_{\parallel }$. The highlighted area defines a time period, $T=[t_s,t_e]$, over which temporal averaging was calculated. (b) Mean value of parallel component of the specific kinetic energy ${\overline{e}}_{\parallel }$. The Roman numerals (i-vii, vi-a, and vi-b) represent different channel geometries (Table 1).

Figure 4

(a) Mean velocity ${\overline{u}}_i$ of all tracked particles over the period of $[t_s,t_e]$. Blue: ${\overline{u}}_x$; red: ${\overline{u}}_y$; yellow: ${\overline{u}}_z$. (b) Velocity fluctuation strength ${\sigma }_i$ of all tracked particles over the period of $[t_s,t_e]$. Blue: ${\sigma }_x$; red: ${\sigma }_y$; yellow: ${\sigma }_z$. Duplicate symbols represent two repeated tests in identical geometries.

Figure 5

[(a) and (b)] Scaled mean-square displacements of tracers in the $x\text{−}y$ plane (${\mathrm{MSD}}_{\parallel }$) and in the $z$ direction (${\mathrm{MSD}}_{\perp }$) in different geometries. Two repeated experiments from the same geometry are averaged together. Dashed lines indicate the crossover time as slope equals 1. [(c) and (d)] Velocity-velocity temporal correlation in $x\text{−}y$ plane and in $z$ direction from different geometries. The Roman numerals (i-vii, vi-a, and vi-b) represent different channel geometries (Table 1).

Figure 6

(a) Normalized equal-time spatial velocity-velocity correlation as a function of separation distance for varying confinements. Circles denote average spatial correlation function measured from two duplicate experiments in each geometry and solid lines present exponential fitting. Brown circles specify the correlation in an ATP-depleted (“dead”) system measured in a 2-mm cubic channel. Solid lines indicate exponential fit to the data. (b) Estimated correlation length as a function of channel height computed from the exponential fits. Duplicate symbols represent two repeated tests in identical geometries. The gray dotted line is solely to guide the eye for the measurements made at the center plane of the channel, $\mathrm{\Delta }z/H\sim 0.5$. The Roman numerals (i–vii, vi-a, and vi-b) represent different channel geometries (Table 1).