Shape-specific fluctuations of an active colloidal interface

TL;DR

This work analyzes a moving, phoretically coupled 1+1D active interface formed by a rotoptranslationally coupled colloidal chain. The authors identify a C-shape dynamical steady state and demonstrate a novel Family–Vicsek scaling for height fluctuations, with $z_h \approx 0.5$, $\alpha_h \approx 0.9$, and $\beta_h \approx 1.7$, alongside a negative orientational roughness exponent $\alpha_\theta \approx -0.5$ indicating smoothening with system size. They also reveal a ballistic-to-diffusive crossover in orientational dynamics, a locally flat regime with distinct exponents, and discuss a potential continuum hydrodynamic description of coupled height and polarization fields. Overall, the results point to a new non-equilibrium universality class for active interfaces with non-standard topology and show how roto-translational coupling reshapes interfacial fluctuations and scaling.

Abstract

Motivated by a recently synthesizable class of active interfaces formed by linked self--propelled colloids, we investigate the dynamics and fluctuations of a phoretically (chemically) interacting active interface with roto--translational coupling. We enumerate all steady--state shapes of the interface across parameter space and identify a regime where the interface acquires a finite curvature, leading to a characteristic ''C--shaped'' topology, along with persistent self--propulsion. In this phase, the interface height fluctuations obey Family--Vicsek scaling but with novel exponents: a dynamic exponent $z_h \approx 0.5$, a roughness exponent $α_h \approx 0.9$ and a super--ballistic growth exponent $β_h \approx 1.7$. In contrast, the orientational fluctuations of the colloidal monomers exhibit a negative roughness exponent, reflecting a surprising smoothness law, where steady--state fluctuations diminish with increasing system size. Together, these findings point towards a unique non--equilibrium universality class associated with self--propelled interfaces of non--standard shape.

Figures (9)

• Figure 1: A) Phase diagram in $\Lambda$-$\tilde{\Lambda}$ plane, see Eq.\ref{['eq:peclet_nums']}. The log absolute value of curvature has been used to delineate the phases. The phase diagram has been drawn with for a chain with $N=256$ number of monomers. B) Representative images for each phase is shown along with the marker key for a smaller chain for clarity. Values of $(\Lambda, \tilde{\Lambda})$ are I: $(10^{4},10^{2})$, II: $(10^{1},10^{0})$, III: $(10^{-3},10^{2})$, III: $(10^{1},10^{2})$. C) Displays the effect of selection statistics of the C-shape under two separate protocols, where the absolute value of the curvature $| \kappa |$ has been plotted against the control parameter. Top corresponds to that of variation of $\Lambda$ starting in Phase I, whilst bottom is starting in Phase IV. Note that $| \kappa| \approx 0.0797$ indicates the C-shape absolute curvature. Light points denote the beginning of the protocol whilst dark the end for the forward direction, with the opposite color shading for the reverse. Color shade and parameter values correspond to arrows shown in panel A for transitions between states I and IV of the phase diagram. D) An example snapshot of the evolution in Phase IV with $N=12$ and $\tau_f = 0.2$. The rightmost snapshot shows the steady-state shape with an example mean height (vertical dotted line) and definition of $\Delta h$ in Eq.(\ref{['eq:d2h']}).
• Figure 2: Scaling of height fluctuations. A$W_h$ for different chain lengths $N$, with the three exponents $\beta_h$ indicated via dashed black line, fit within $(t/\tau N^{z_h}) \in [1, 20 ]$. The same scaling is plotted in the inset, with $t_{h}^{*}(N)$ labelled in dashed vertical line. B Roughness scaling for $W_h^{\text{SS}}(N)$, with effective exponent in inset. Note that for the effective exponents $N \in [ 64, 312]$ (excluding the smallest $N=32$, due to forward-difference) Here, $\tau_f = 0.2$, $\Lambda=0.2$, $\tilde{\Lambda}=0.01$.
• Figure 3: Scaling of the orientation fluctuations. A$W_{\theta}$ for different chain lengths $N$, with the dynamic $\beta_{\theta}$ labelled (fit between $(t/\tau N^{z_h}) \in [10, 200 ]$). The same scaling is plotted in the inset, with $t^{*}$ labelled in dashed vertical line. B (Anti-) Roughness scaling for $W_{\theta}^{\text{SS}}(N)$, with effective exponent in the inset. C Orientation profile of the C-shape, taken at different time points. D Dynamical evolution of $\theta_i$ for selected monomers $i$. Here, $\tau_f = 0.2$, $\Lambda=0.2$, $\tilde{\Lambda}=0.01$, and $N=311$ (for panels C and D).
• Figure 4: Height and orientational fluctuations for finite $\tau_r$. A$W_{h}$ for different chain lengths $N$, with the dynamic $\beta_{h}$ labelled (fitting window is $t/(\tau N^{z_h}) \in [0.5,5 ]$). Scaling for $W_h^{\text{SS}}(N)$, with effective exponent in the inset. C$W_{\theta}$ for different chain lengths $N$, with the dynamic $\beta_{\theta}$ labelled (fitting window is $t/(\tau N^{z_{\theta}}) \in [4,80 ]$). D (Anti-) Roughness scaling for $W_{\theta}^{\text{SS}}(N)$, with effective exponent in the inset. Here, $\tau_f = 0.2$, $\Lambda=0.2$, $\tilde{\Lambda}=0.01$ and $\tau_r = 70$.
• Figure 5: Scaling of height fluctuations in locally flat region. A$W_{h}$ for different locally flat lengths $N'$, with the dynamic $\beta_{h}'\approx 0.25$ indicated via dashed black line, fit between $(t/\tau N'^{z_{h}'}) \in [5\times 10^{-6}, 10^{-5}]$. B Roughness scaling for $W_{h}^{\text{SS}}(N')$. Here, $N=312$ and $\tau_f = 0.2$, $\Lambda=0.2$, $\tilde{\Lambda}=0.01$.