Nanoscale Rheology and Anisotropic Diffusion Using Single Gold Nanorod Probes

The complex rotational and translational Brownian motion of anisotropic particles depends on their shape and the viscoelasticity of their surroundings. Because of their strong optical scattering and chemical versatility, gold nanorods would seem to provide the ultimate probes of rheology at the nanoscale, but the suitably accurate orientational tracking required to compute rheology has not been demonstrated. Here we image single gold nanorods with a laser-illuminated dark-field microscope and use optical polarization to determine their three-dimensional orientation to better than one degree. We convert the rotational diffusion of single nanorods in viscoelastic polyethylene glycol solutions to rheology and obtain excellent agreement with bulk measurements. Extensions of earlier models of anisotropic translational diffusion to three dimensions and viscoelastic fluids give excellent agreement with the observed motion of single nanorods. We find that nanorod tracking provides a uniquely capable approach to microrheology and provides a powerful tool for probing nanoscale dynamics and structure in a range of soft materials.

Figure 1

(a) GNR scattering and imaged by a dark-field microscope. (b) GNR embedded in viscoelastic polymer solution. (c) Coordinate system of the GNR defined by polar $\beta$ and azimuthal angles $\phi$.

Figure 2

Integrated intensity of a GNR in glycerol in orthogonal $x$ and $y$ polarization images (a), with the corresponding sum of intensities (b) and the inferred orientation angles (c). Time-dependent 3D GNR orientation mapped into a single octant (d). Orientations of selected time points (1–7) marked in (e), corresponding image pairs ($1.6\times 1.6\mu {\mathrm{m}}^2$) in (f). Trajectory (g) of the GNR showing translational motion. Frequency distributions of orientation angles (h) from ${10}^5$ image pairs (symbols) match a random orientation model (curves).

Figure 3

MSADs of simulated (a) and measured (b) GNR in glycerol (circles), stretched exponential fit (dashed line), and unbounded MSAD (curve). Measured MSADs (c) of a GNR in 200 kDa PEO, $6.7\%\mathrm{w}/\mathrm{w}$ (squares) and $12.2\%\mathrm{w}/\mathrm{w}$ (circles). (d) Scaled unbound MSAD (open symbols), translational MSD (closed circles) of the GNR, and rescaled microsphere MSD [1] (lines) at two PEO concentrations. Storage and loss moduli for the PEO solution obtained from GNR rotational nanorheology (symbols) and literature microrheology [1] (lines) at $6.7\%\mathrm{w}/\mathrm{w}$ (e) and $12.2\%\mathrm{w}/\mathrm{w}$ (f).

Figure 4

Anisotropic translational displacement distributions (a), at $\tau =2\mathrm{ms}$, of a GNR in 90% glycerol (symbols) with a Gaussian fit (lines) and corresponding anisotropic MSDs (b). Diffusion coefficients of the GNR (c), corresponding to the data in (a) and (b), in the fixed lab frame with the $x$ axis aligned with the rod’s initial position with the model fit [Eq. (3)]. Mixed translational and orientational correlations (d) in a fixed frame (symbols) and MC simulation (curves). Rescaled anisotropic MSDs (e) of GNR in PEO 6.7% [as in Fig. 3] in the fixed lab frame (symbols) with a viscoelastic model (curves); unscaled MSDs shown in Fig. S14 [45]. Diffusion anisotropy (f), ${\mathcal{A}}_{ii}=(D_{ii}-D_t)/\mathrm{\Delta }D$, for a GNR in glycerol and 6.7% PEO solution (symbols), with the model [Eq. (4)] (curves).