Memory-aware acceleration of orientational dynamics in nanoparticle suspensions

Abstract

The relaxation of stochastic systems after sudden perturbations is constrained by speed limits and often reveals memory effects that hinder attempts to accelerate their dynamics. Here we demonstrate Kovacs-type nonmonotonic relaxation in the electro-orientation of non-spherical nanoparticles and show how this memory effect limits simple acceleration protocols. Experimentally, the orientational dynamics is monitored optically through field-induced birefringence, which is proportional to the nematic order parameter. When an AC electric field is first set to an extreme value until the birefringence reaches its target and is then switched to the target field (matched two-step protocol), the relaxation exhibits a characteristic Kovacs shoulder. We interpret this behavior within a theoretical framework based on the Smoluchowski equation for the orientational probability density. In the high-frequency AC regime, orientational relaxation is governed by induced dipoles, and the observed memory effect originates from polydispersity, which generates a spectrum of rotational diffusion coefficients and hence multiscale relaxation. Building on this insight, we design protocols that mitigate the detrimental effect of memory by sequentially suppressing the slowest active relaxation mode. Experiments on nanoparticle suspensions with different properties confirm these mechanisms, and we demonstrate substantial reductions in relaxation time compared with single quenches and matched two-step protocols with NaMt suspensions. More broadly, these results illustrate how memory effects emerge when many degrees of freedom are steered with a single control parameter and provide an experimentally accessible strategy for controlling multiscale stochastic dynamics.

Figures (6)

• Figure 1: Optimizing the alignment dynamics of Brownian particles in electric fields. Schematic representation of the experimental system in a poorly aligned (left) and a strongly aligned (right) configuration. The upper panels show the time evolution of the electric field and the normalized order parameter $\bar{S}(t)$ during alignment processes (AP) using a direct protocol (purple) and a near time-optimal, three-step bang–bang protocol (red). The lower panels depict the corresponding misalignment processes (MP). In both cases, the three-step bang–bang protocol substantially reduces the response time compared with the direct one.
• Figure 2: Alignment (AP) and misalignment (MP) processes under direct and matched two-step protocols.(a) and (b): Time evolution of the normalized orientational order parameter $\bar{S}(t)$ of NaMt nanoparticles immersed in an aqueous solution under direct (purple) and matched two-step (green) protocols for AP (a) and MP (b) at different final electric fields ($E_{\rm f}=1.96,\ 2.94,\ 3.91,\ 4.66,\ 5.29~\mathrm{V\,mm^{-1}}$). Initial conditions are $E_{\rm i}=0$ for AP and $E_{\rm i}=5.88~\mathrm{V\,mm^{-1}}$ for MP. Gray curves represent direct protocols to the limit fields, $E_{\rm i}\!\to\!E_{\max}=5.88~\mathrm{V\,mm^{-1}}$ (AP) and $E_{\rm i}\!\to\!E=0$ (MP). As an example, the matched two-step protocol for the target value $\bar{S}_{\rm f}=0.055$ ($E_{\rm f}=3.91~\mathrm{V\,mm^{-1}}$; horizontal dashed line) is shown in the shaded area above each panel. In this protocol, the system is first subjected to the limit field $E_{\max}$ (AP) or $E=0$ (MP) until the time instant $t_{\rm c}^{(\mathrm{K})}$ (vertical dashed line), where $\bar{S}(t_{\rm c}^{(\mathrm{K})})=\bar{S}_{\rm f}$, after which the field is switched to $E_{\rm f}$. Error bars are smaller than the symbol size. (c): Amplitude of the Kovacs anomaly, $|\Delta\bar{S}|$, defined as the maximum deviation of $\bar{S}(t)$ from $\bar{S}_{\rm f}$ after $t_{\rm c}^{(\mathrm{K})}$, plotted as a function of the distance in $\bar{S}$ between the initial and final states, $|\bar{S}_{\rm f}-\bar{S}_{\rm i}|$. Filled symbols correspond to AP and open symbols to MP.
• Figure 3: Transient orientational dynamics for different particle systems under a matched two-step AP. Time evolution of the normalized order parameter $\bar{S}(t)/\bar{S}_{\rm f}$ for four representative materials, indicated in each panel: gold nanorods, gibbsite, goethite, and silver nanowires. All measurements were performed under 100 kHz AC fields following alignment processes. Insets show electron microscopy images of the corresponding samples, illustrating their morphology and characteristic sizes (scale bars as indicated). Nearly monodisperse systems such as gold nanorods exhibit an almost single-exponential relaxation, while more polydisperse materials (e.g., silver nanowires) display a pronounced Kovacs shoulder.
• Figure 4: Alignment (AP, left column) and misalignment (MP, right column) processes under different two-step protocols.(a) and (b) RMS electric field, $E_{\mathrm{RMS}}(t)$, as a function of time for AP and MP. Initial conditions are $E_{\rm i}=0$ for AP and $E_{\rm i}=5.88~\mathrm{V\,mm^{-1}}$ for MP, with a common final state corresponding to $E_{\rm f}=3.91~\mathrm{V\,mm^{-1}}$. Three protocols are shown: a matched case ($t_{\rm c}=t_{\rm c}^{(\mathrm{K})}$), an improved case ($t_{\rm c}$ chosen so that $b_{\rm slow}\!\to\!0$), and an intermediate two-step protocol. (c) and (d) Time evolution of the normalized orientational order parameter $\bar{S}(t)$ under the protocols shown above. The horizontal dashed line marks the target value $\bar{S}_{\rm f}=0.055$, and vertical dashed lines indicate the field-switching times. Gray curves correspond to the direct protocols $E_{\rm i}\!\to\!E_{\max}=5.88~\mathrm{V\,mm^{-1}}$ (AP) and $E_{\rm i}\!\to\!E=0$ (MP), shown for comparison. Error bars are smaller than the symbol size.
• Figure 5: Evolution of the fast and slow mode amplitudes with protocol duration. Amplitudes of the fast (squares) and slow (circles) decay modes as a function of the difference between the duration of the first time window, $t_{\rm c}$, and its matched value, $t_{\rm c}^{(\mathrm{K})}$, for AP (panel (a)) and MP (panel (b)). Colors encode the distance in the order parameter between the initial and final steady states, $|\bar{S}_{\rm f}-\bar{S}_{\rm i}|$, as indicated in the color bar below. Error bars are smaller than the symbol size.