Realizing microrheological response of configurable viscoelastic media with a dynamic optical trap

Abstract

The local viscoelastic (VE) environment governs the motion of an embedded microsphere and consequently, pertinent dynamical phenomena. However, studying such phenomena with varying VE properties remains challenging for various reasons, including the strong coupling among the VE parameters and their dependence on experimental conditions, such as temperature. Here, we demonstrate the experimental realization of configurable VE media with broad variations, wherein the VE properties can be systematically and independently tuned, employing a dynamic optical trap. Specifically, the dynamics of a particle in a slowly diffusing optical trap provides the linear microrheological response of single-relaxation VE fluids, namely, Jeffreys or Maxwell-Voigt (MV) fluids, where the trap strength and its diffusion coefficient regulate the elastic response and the low-frequency viscosity, respectively. We validate this approach by comparing the experimentally observed dynamics of the trapped bead with those of a probe particle in real single-relaxation complex fluids, analytical predictions, and simulation results following harmonically bound Brownian particle with long-time diffusion model describing MV fluids. We extend the applicability of this scheme for realizing the microrheological response of double-relaxation VE media by incorporating appropriately correlated noise in the trap trajectory, signifying its validity for any linear VE media with multiple relaxations. Our scheme can be further extended to realize probe particle dynamics in an active VE environment, e.g., an entangled network of active polymers, by translating the trap along an active Brownian trajectory. Therefore, our scheme enables systematic microrheological studies in VE regimes that are otherwise challenging to realize or not readily accessible with real materials.

Figures (5)

• Figure 1: Typical microrheological response of single-relaxation viscoelastic (VE) fluids. (a) The schematic shows a probe microsphere embedded in a complex fluid represented by entangled wormlike micelles. (b) Microrheological responses of cetyltrimethylammonium bromide--sodium salicylate (CTAB-NaSal) and cetylpyridinium bromide--sodium salicylate (CPyBr-NaSal) are shown at various concentrations, characterized by the MSD ($\left\langle \Delta r^2 \right\rangle$, left axis) of an embedded 1.98µm polystyrene (PS) probe particle and derived creep compliance ($J$, right axis) using Eq. \ref{['eq:gser']}. Open symbols represent the experimentally captured data, and the solid lines of the same color are fits to Eq. \ref{['eq:msd-pbp']}. The MSDs show three distinct regimes: free diffusion at short time-lags ($\tau$), elastic confinement manifested as plateaus at intermediate $\tau$, and diffusive dynamics at $\tau > \lambda$, where $\lambda$ is the relaxation time (marked by vertical dotted lines).
• Figure 2: Harmonically bound Brownian particle (HBBP) with long-time diffusion model describing the microrheological response of an MV fluid. (a) A schematic shows the spring-dashpot model of a Maxwell-Voigt (MV) fluid, constituted by two springs with a common spring constant ($k=6\pi a G_\mathrm{p}$) and dashpots with different damping coefficients ($\gamma_\mathrm{s}$ and $\gamma$) corresponding to the high- and low-frequency viscosities ($\eta_{\mathrm{s}}$ and $\eta$), respectively. (b) Pictorial illustration of the HBBP with long-time diffusion model exhibits the typical confined dynamics of a HBBP (magenta filled circle) with instantaneous position $x_\mathrm{HBBP} (t)$ (green), and the long-time diffusion of the harmonic well (HW) by a representative trajectory $x_\mathrm{HW} (t)$ (red). The friction coefficient $\gamma_\mathrm{s}$ governs the short-time diffusion of the particle, whereas $\gamma_\mathrm{HW}$, a tuneable parameter that regulates the long-time diffusion of the HW center, corresponds to $\gamma$ of the MV model in (a) and the post-relaxation low-frequency viscosity of the MV fluid $\eta = \gamma_\mathrm{HW}/6\pi a$, which is same as the viscosity associated with the diffusion of the HW, $\eta_{\mathrm{HW}} = \gamma_\mathrm{HW}/6\pi a$. (c) Time evolution of the position distribution of the HBBP in reference to the HW, $p (x_{\mathrm{HBBP}}, t)$ (green), and that of the HW in the laboratory frame, $p (x_{\mathrm{HW}}, t)$ (red), are shown at four progressing times (lighter shades indicate later times). (d) Filled symbols show the simulated MSD ($\left\langle \Delta r^2 \right\rangle$, left axis) of a $1.98~\unit{\micro\meter}$ probe sphere in an MV fluid from Langevin dynamics (Eq. \ref{['eq:le-pbp-mv']}) and the corresponding creep compliance ($J$, right axis), with black line showing fits to the theoretical prediction of the model (Eq. \ref{['eq:msd-pbp']}). The dashed and dotted vertical lines mark the timescales $\tau_k$ and $\lambda$ at which the plateau is reached and the elastic confinement is relaxed, respectively. The blue trajectories represent the probe dynamics at three different times ($t = \{1, 10, 100\}\;\unit{\second}$), with the slowly diffusing HW overlaid in red.
• Figure 3: Experimental realization of the microrheological response of a configurable VE fluid with a dynamic optical trap. (a) The orange gradients show the axial and radial intensity profiles of a tightly focused laser beam propagating along $\hat{z}$, creating harmonic confinement at the focal point in the $x$-$y$ plane. (b) The MSD ($\langle \Delta r^2 \rangle$) of a 1.98µm PS probe sphere in a static optical trap exhibits the microrheological response of a Voigt solid, with the corresponding bound trajectory and overlaid long-time position distribution $p(x_{\mathrm{HBBP}})$ shown in the inset. (c) The ray diagram shows a steerable optical trap. L and M denote the lenses and mirrors in the laser beam path (orange lines). The trap is steered in the focal ($x$-$y$) plane by regulating the tilt ($\theta_x$, $\theta_y$) of the piezo-controlled mirror, which is placed at the conjugate image plane of the back focal plane (BFP) of the objective, following Eq. \ref{['eq:back-focal']}. CON, OBJ, DM, and CCD in the illumination beam path (cyan lines) denote condenser, objective, dichroic mirror, which reflects the trapping laser and transmits the illumination, and a CCD camera, respectively. The bottom left inset illustrates the tilt angles $\theta_x$ and $\theta_y$ around the $x$- and $y$-axis of the mirror mount. The top inset shows the resultant displacement of the trap (orange gradient) at the focal plane.
• Figure 4: Systematic experimental tuning of the VE parameters in the microrheological response of a configurable single-relaxation VE environment. The response is exhibited by the MSD of a 1.98µm diameter PS probe particle in a diffusive optical trap. The experimental data are shown with open symbols, whereas the solid lines of the same color represent the corresponding theoretical predictions of our model (Eq. \ref{['eq:msd-pbp']}). (a) Systematic variation of the high-frequency viscosity $\eta_{\mathrm{s}}$ of the emergent VE environment is achieved by using viscous media of different viscosities. Here, $\eta_{\mathrm{s}}$ is varied while keeping $G_{\mathrm{p}}$ and $\lambda$ unchanged, as the trap stiffness $k = 6\pi a G_\mathrm{p}$ ($\propto$ laser power $P$) and the viscosity of the diffusing trap $\eta_{\mathrm{HW}}$ remain constant. (b) The plateau modulus $G_\mathrm{p}$ is regulated by varying the laser power $P$ while keeping $\eta_{\mathrm{s}}$ and $\eta_{\mathrm{HW}}$ unchanged. An increase in $G_\mathrm{p}$ diminishes the MSD plateau value and shifts both $\tau_k = \eta_{\mathrm{s}}/G_{\mathrm{p}}$ and $\lambda = \eta_{\mathrm{HW}}/G_{\mathrm{p}}$ to shorter time-lags. (c) Independent variation of $\lambda$ is achieved by varying $\gamma_{\mathrm{HW}}$, i.e., $\eta_{\mathrm{HW}}$, while keeping $G_\mathrm{p}$ and $\eta_\mathrm{s}$ unchanged. The MSDs show identical short-time diffusion but systematically different relaxation times.
• Figure 5: Realization of double-relaxation VE media and active VE environments employing a dynamic trap scheme. The MSDs of a 1.98µm diameter PS probe particle in a dynamic trap with correlated and persistent translational noise, exhibiting microrheological response of configurable double-relaxation and active VE environments. The experimental data and simulation results are shown with open and filled symbols, respectively, whereas the solid lines of the same color represent the corresponding theoretical predictions. All relevant parameters are given in the insets. (a) Microrheological response of double-relaxation VE environments is generated by translating the trap with exponentially correlated noise, providing equilibration timescales $\tau_{k,1}$ and $\tau_{k,2}$. The MSDs exhibit two distinct plateaus separated by an intermediate relaxation, demonstrating hierarchical VE response consistent with Eq. \ref{['eq:msd-pbp-double']} (solid line). (b) Probe particle dynamics in active VE baths is realized by moving the trap along an ABP trajectory with persistent translational noise. The MSDs exhibit features similar to those of probe particles in a single-relaxation VE fluid, with distinct post-relaxation ($\tau > \lambda$) behavior showing super-diffusive and diffusive dynamics before and after the persistence time $\tau_{\mathrm{R}}$, respectively, as predicted by Eq. \ref{['eq:msd-pbp-active']}.