Phase Separation by Entanglement of Active Polymerlike Worms

We investigate the aggregation and phase separation of thin, living T. tubifex worms that behave as active polymers. Randomly dispersed active worms spontaneously aggregate to form compact, highly entangled blobs, a process similar to polymer phase separation, and for which we observe power-law growth kinetics. We find that the phase separation of active polymerlike worms does not occur through Ostwald ripening, but through active motion and coalescence of the phase domains. Interestingly, the growth mechanism differs from conventional growth by droplet coalescence: the diffusion constant characterizing the random motion of a worm blob is independent of its size, a phenomenon that can be explained from the fact that the active random motion arises from the worms at the surface of the blob. This leads to a fundamentally different phase-separation mechanism that may be unique to active polymers.

Figure 1

Aggregation and phase separation of T. tubifex. (a)–(c) Snapshots of active-worm aggregation in a $25\times 25\times 2.5\mathrm{cm}$ volume at (a) $t=0$, (b) 9.5, and (c) 60 min. (d) Snapshots from a 1D experiment in a square tube of $51\times 1\times 1\times 1\mathrm{cm}$. (e)–(h) Close-ups from another experiment at (e) $t=25$, (f) 28.5, (g) 29, and (h) 31.5 min, showing the coalescence of two blobs. See Supplemental Material [21] for the videos.

Figure 2

Phase-separation dynamics. (a) (i) Modulus squared of the Fourier transform and (ii) Fourier power spectrum of the image of the spatial worm distribution at different times. The $q$ value at which the intensity is maximal shifts to lower values with increasing time (2D experiment, $T=30°\mathrm{C}$). (b) Average size $⟨R_{nD}⟩\sim q_{\max }^{-1}$ (left axis for 2D experiments and right axis for 1D experiments) as a function of time, showing approximate power-law behavior with a power of $1/3$ for the 2D experiment at all temperatures (circles) and $1/4$ for the 1D experiment at ambient temperature (squares). The solid lines are guides to the eye. (c) Trajectories of a single worm time at a controlled temperature $T$. Each solid trace of a different color represents the center-of-mass trajectory of one-hour duration for the same worm at a different temperature [$T=5$, 20, $30°\mathrm{C}$; color coding as in (b)]. The origin of all trajectories is set to $(x_0,y_0)=(0,0)$. (d) Mean square displacement (MSD) as a function of time of ten worms in water at $T=5$, 20, $30°\mathrm{C}$ [same color coding as in (b)]. The dashed line of slope 1 in the shaded area shows the expected scaling for Brownian motion. The dashed line in the nonshaded area has slope 2. (e) Diffusion coefficient $D_{\mathrm{worm}}$ of a worm extracted from the MSD($t$) as a function of temperature. Each measurement is an average over ten trajectories of the same worms. (f) Average size of all the 2D experiments at the different temperatures collapsed onto a single master curve by rescaling the time $\tau =D_{\mathrm{worm}}t$.

Figure 3

Simulation of phase-separation dynamics. (a) Snapshots of simulated blob growth by coalescence of randomly moving spherical blobs, with blob-diffusion constant $D_{\mathrm{blob}}$ inversely proportional to the blob radius (top) and independent of blob radius (bottom). (b) Average blob radius $⟨R⟩$ as a function of time obtained from the simulations. In all cases, the growth follows a power law (indicated by lines), with exponents of 0.2 and 0.3, respectively.

Figure 4

Blob diffusion. Effective diffusion constant as a function of average blob size $⟨R_{\mathrm{blob}}⟩$ at $T=20°\mathrm{C}$ as determined from the slopes of experimental MSDs (see Supplemental Material [21]). The error bars are mostly due to sample-to-sample variability. The purple line shows the expected scaling for particles undergoing Brownian random motion ($D_{\mathrm{blob}}\sim {⟨R_{\mathrm{blob}}⟩}^{-1}$). The experimental data (blue symbols) indicate a diffusion constant independent of blob size (dotted line). (Lower insets) Photographs of blobs of different sizes from which we measured the diffusion coefficient.