Burst-by-Burst Measurement of Rotational Diffusion at Nanosecond Resolution Reveals Hot-Brownian Motion and Single-Chain Binding

We record dark-field scattering bursts of individual gold nanorods, 52 × 15 nm2 in average size, freely diffusing in water suspension. We deduce their Brownian rotational diffusion constant from autocorrelation functions on a single-event basis. Due to spectral selection by the plasmonic resonance with the excitation laser, the distribution of rotational diffusion constants is much narrower than expected from the size distribution measured by TEM. As rotational diffusion depends on particle hydrodynamic volume, viscosity, and temperature, it can sense those parameters at the single-particle level. We demonstrate measurements of hot Brownian rotational diffusion of nanorods in temperature and viscosity gradients caused by plasmonic heating. Further, we monitor hydrodynamic volumes of gold nanorods upon addition of very low concentrations of the water-soluble polymer PVA, which binds to the particles, leading to measurable changes in their diffusion constant corresponding to binding of one to a few polymer coils. We propose this analysis technique for very low concentrations of biomolecules in solution.


S1 TEM
To determine the physical dimensions of GNR sample we have performed TEM microscopy.
Transmission electron microscopy (TEM) images were taken at a magnification of 73, 000.
GNR samples we prepared on a EM grid (CF200-Cu from Electron Microscopy Sciences).

S2 Estimation of events overlap
To estimate the probability for one event overlapping with another we performed Monte-Carlo Simulations. Specifically we simulate traces with 10 6 points length and randomly seed events with a given duration into these traces until 10 percent of all points (as found experimentally) are occupied by at least one event. We then determine the ratio of the numbers of events that possess occupancy ≥ 2 and the total number of events. We run each simulation 30 times to minimize statistical noise. We find that in this situation 10% of detected events will result from the overlap of at least two events.

S3 Simulations of rotational diffusion
In latter sections of this SI we want to determine how several experimental conditions would influence the analysis, in particular the determination of rotational diffusion times. Such experimental parameters are different polarizer configurations but also the length of events as determined by the analyte's translational diffusion properties and the measurement geometry (detection volume). In order to gain a basic understanding of the influence of these parameters we have performed simulations of rotational diffusion, which we describe in the following: In order to understand how the scattering signal changes according to the random orientation of a GNR, we simulated the orientation of the rod's long axis with a random walk on the surface of a unit sphere. To perform the random walk, we generate a small rotation vectorû with a random direction determined by two random numbers η 1 and η 2 uniformly distributed between 0 and 1, giving isotropically distributed directions with polar (θ = [−π/2, π/2]) and azimuthal (ϕ = [0, 2π]) angles according to: and determine a unit vector Rotation of the initial vector r i by a small angle (ω) around vectorû gives a shift vector and the next position ⃗ r i+1 = ⃗ r i + ⃗ ∆r i . We normalize this vector to avoid numerical errors.
As starting value we use ⃗ r 0 = [0, 0, 1]. We have found that ω = 0.1 provides a good enough resolution with a reasonable number of steps to simulate long rotational diffusion events. To allow for direct correlation with our measurements one may convert step size to time units by using the conversion factor 0.12 µs/step, which was determined by comparing the autocorrelation decay time of random walk simulations with the average experimentally determined decay times.

S3.1 Rotational correlation function with polarization analysis of the detection
We want to investigate the influence of the incident and analyzed polarization configuration on the autocorrelation decay. For this we consider random orientations of the GNR axis (θ, ϕ) as described above and compute the respective intensity traces (neglecting the numerical aperture of the objective). We consider the GNR with polarizability along its main axis and write the components of polarizability ← → α relative to the rotated coordinates as where ← → R is the rotation matrix in 3D. Then, by consideringŷ as the optical axis and E 0ẑ as the incident electric field ( fig.S4), we find the scattered field as: Consequently, the detected intensity in parallel (I || ) and cross (I ⊥ ) polarization configurations can be written as: Figure S4: Polar θ and azimuthal ϕ angles are shown in cartesian coordinates of the optical measurement.ŷ is the optical axis (xz is focal plane), the incident electric field is alongẑ with propagation direction ⃗ k alongŷ.
From these simulated intensity traces, we then compute the autocorrelation curves and fit them with a double exponential decay as: Then we can compare the obtained fitting parameters with Pecora's decays (eq. 2 main manuscript). Examples of simulated traces and autocorrelations are shown in fig. S5a Figure S5: a) Excerpts of simulated intensity traces (blue:parallel, orange:crossed configuration) versus step number of the random walk. b) The autocorrelation functions (dots) of the intensity traces in (a), calculated for 2 × 10 6 steps. Solid lines: Fits to double-exponential decays.   fig. S5b).

S3.2 Effect of event length on accuracy of the decay times
We investigate the clipping of the rotational diffusion trace by the duration of each event, which is limited by translational diffusion of the rod through the confocal volume. Short enough events will lead to a broadening of the rotational diffusion histogram due to finite sampling of the trace. As a simple model of clipping by translational diffusion, we multiply the rotational diffusion trace by a Gaussian envelope, which simulates the dwell time of the rod in the confocal volume. The Gaussian envelope here is just meant as a convenient way of varying the sampling time of rotational diffusion. In real measurements, however, the random diffusion path would lead to more statistical fluctuations, which we ignore here. Fig. S7 is an example of how we simulate the events. To see the effect of statistical fluctuations clearly, here, we have considered the events much shorter than the translational diffusion through the confocal volume. By increasing the width of the events, we expect to have higher statistics of sub-bursts in each event which yields a narrower histogram of rotational diffusion times (or τ d ). Here, we investigate the influence of the event duration on the width of the histogram. We introduce as a parameter the ratio N of the event duration to the tumbling time as equation 7, which is roughly the number of sub-bursts in an event: where τ T is the translational diffusion time through the confocal volume (corresponding to the duration of events) and τ d is the decay time of the rotational autocorrelation. Fig. S8a shows the simulated histograms for different numbers of sub-bursts (N ). We have fitted a log-normal function: to each histogram, where µ and σ respectively are expected value (or mean) and standard deviation of the variable's natural logarithm, and σ has been plotted as a function of N in fig. S8b. Increasing N reduces the width of the histogram, according to the central limit theorem: where A is a fit parameter and σ is the standard deviation of the log-normal distribution in equation 8. In our confocal measurement, a typical event duration is around 2 ms. By considering our experimental value of 11 µs as the mean rotational diffusion time, we find N ≥ 180, even higher than the highest N = 108 in fig

S3.3 Influence of polarization configuration on decay time accuracy
In our experiments, we found that τ d and Θ-distributions obtained in cross-polarized configuration are narrower than their parallel-polarized counterparts. Therefore, here we compare simulated sub-bursts in parallel-and cross-polarization as obtained from the same random walks and repeat the analysis as performed above for different Gaussian windows ( fig. S9).
We find that the cross-polarized configuration provides narrower distributions due to its overall faster fluctuations and therefore higher number of sub-bursts. This difference, however, quickly becomes very small for longer event durations and is therefore negligible in our measurements. The main narrowing effect in the cross-polarized configuration is due to stronger selection by the LSPR, as discussed below.

S4 LSPR-based detection
In our measurements, detection of the GNRs is greatly facilitated by resonance of their Localized Surface Plasmon (LSP) with our probe laser at 785 nm. We select those GNRs which have a resonance wavelength close to that value. Here, we investigate the effect of selection by LSPR on the width of rotational diffusion histograms. We calculate the scattering cross sections σ scat at 785 nm for many individual GNRs drawn at random from the population determined from TEM images (blue dots in fig. S2a), and supposing diameter and length of the rods to be independent random variables. The results are displayed in fig.S10, where each dot is color-coded by its scattering cross section in a length-diameter scatter plot. Because of a comparatively narrow distribution of diameters, we see that the distribution of GNRs in resonance with the laser is considerably narrower than the general distribution. The scattering cross sections of the best rods are more than 3 times larger than the average cross section. Only these best rods will be detectable optically, leading to a very narrow histogram. Such narrow histograms are the cornerstone of our method for detecting small changes in rotational diffusion constant. Such changes would be much more difficult to detect on the un-selected histogram (blue histogram in fig.S10c).
By comparing the simulated data with the experimental histogram of parallel polarization in fig.2h of the main text, we choose a minimum scattering cross section of 1100 nm 2 as detection threshold for the measured GNR bursts (fig. S10).  fig. S2). The scattering cross sections have been calculated by MNPBEM, 1 for linearly polarized light along the main axis of each GNR and wavelength of 785 nm (same wavelength as our laser) in water as medium. To remove the nanorods which are out of resonance, we set a threshold on the scattering cross section σ sca at (1100 nm 2 ). 440 GNRs have scattering cross section higher than the threshold. b) Histogram of calculated rotational diffusion coefficients of GNRs. In the blue histogram, all the GNRs have been considered and in the dark red histogram, only those with cross section higher than threshold have been considered. The red histogram is narrower than the blue one which indicates the selectivity of our method (based on localized surface plasmon resonance (LSPR)). The small difference of the mean values in these histograms is due to the small difference of the average LSPR peak in comparison to our laser wavelength (785 nm), as shown in fig. 1h in the main text. The blue and red histograms are equivalent to the gray and yellow histograms of fig. 2j in the main text. In that figure, a 2.2 nm CTAB (Cetyltrimethylammonium Bromide) layer has been considered. c, d) Histograms of lengths and diameters of GNRs after thresholding shown in (a) with mean values of L mean = 55.4 nm and D mean = 14.8 nm.  fig. S10a) without any selection on the GNRs. b) Histogram of the selected population of GNRs with scattering cross section higher than 300 nm 2 and a log-normal fit. c) Histogram of Θ for selected GNRs with 1100 nm 2 as threshold, same population as fig. S10b. d) Histogram of Θ by setting 1600 nm 2 as the threshold. The width of the histograms decreases markedly as the threshold is raised. All the histograms have been plotted considering a 2.2 nm CTAB layer.  Also, we did a calculation of the selected rotational diffusion coefficients for two differ-ent incident wavelengths 785 nm and 795 nm and compare these thresholded populations in fig. S13. The histograms of rotational diffusion coefficients Θ overlap perfectly, which proves that biasing the plasmonic selection will not change the rotational diffusion coefficients.

S6.1 Theory and simulation
The temperature increase δT of a gold nanorod with length L and diameter D under illumination with light intensity (or irradiance) I is given by: where δT is the temperature change of the nanoparticle T NP − T 0 , the correction factor β is β ≈ 1 + 0.096587 ln 2 (L/D) and σ abs is the absorption cross section of the GNR. I is the power per unit area, κ is the thermal conductivity of the surrounding medium and a 0 is the radius of a sphere of equal volume.
A hot particle performing a translational Brownian motion carries a higher-temperature halo with itself, characterized by an effective temperature T x HBM . A hot spherical particle with radius R has an effective translational diffusion coefficient D HBM similar to the wellknown Stokes-Einstein coefficient, but modified by a different temperature and the associated different viscosity, as: 3 where k B is Boltzmann's constant and η x HBM is an effective viscosity of the surrounding medium in this temperature gradient. The effective temperature and viscosity are determined so as to keep the above form of the Stokes-Einstein equation, by taking the temperature dependence of the viscosity into account.
For the case of rotational diffusion, the flow field is more localized around the particle which yields a higher effective temperature in comparison to the translational one: where T HBM is the effective temperature for the rotation of the nanoparticle. We have where T 0 is the laboratory temperature. The effective viscosity around the heated particle follows from the well-known temperature dependence of the viscosity of water, modelled by a Vogel-Fulcher equation: 3,5 In water, η ∞ = 0.0298376 × 10 −3 P a.s, A = 496.889 • K, and T V F = 152.0 • K. The effective viscosity and temperature are chosen such that η HBM = η(T HBM ).
Then, by using Tirado's model in eq. 1 (main text) we can write the effective rotational diffusion constant as: By considering the size distribution of our GNRs as obtained from TEM images and different incident powers, we can calculate Θ HBM and compare it with the rotational diffusion coefficient in the absence of any heating (Θ). Without any heating, the histogram of rotational diffusion coefficients Θ shows a Gaussian distribution. In the presence of heating, different GNRs will experience different effective temperatures T HBM and local viscosities because of their different dimensions, aspect ratios and absorption cross sections. Therefore their rotational constants Θ HBM will also vary. The histogram of rotational diffusion coefficient Θ HBM displays two populations, which split apart upon increasing the incident power. Those GNRs whose resonance is close to the laser 785 nm and which have larger volumes (same population as fig. S10b) will have higher effective temperature and lower local viscosity compared to the rest of the GNRs. Therefore, they will tend to tumble faster than the rest of the population (higher Θ HBM ).
In order to visualize this process, we have calculated Θ HBM for the whole population of our GNRs (blue circles in fig. S2a) for different powers. We apply the same powers we used in   fig.3h in the main text, the two events in fig. S15c and fig. S16c belong to two populations of GNRs with different rotational diffusion times τ d . We assign this deviation to a higher absorption cross section of these GNRs.  fig.3j in the main text, with 369 µW as incident power. g) Zoom-in on one of the fast sub-bursts that shows the high time resolution of our measurements. The rise time in this sub-burst is less than 200 ns, and it is not broadened by insufficient sampling. h, i) Autocorrelation of the two highlighted events in (a) and a single exponential decay (red) as a fitting function which shows 5.8 µs and 6.6 µs decay times for the first and the second events, respectively.

S6.3 Translational hot Brownian motion
Whereas many rotational diffusion sub-events are sampled during the passage of a nanorod in the confocal volume, only one translational diffusion event is recorded for each nanorod.
Therefore, much better statistics can be accumulated in rotation, whereas translational times are subject to a large statistical distribution depending on the specific trajectory of each rod in the confocal volume. Moreover, fluctuations due to rotational diffusion are in principle independent of the optical probing geometry defined by the confocal volume. To illustrate this drawback of translational diffusion, we have acquired histograms of translational times from our events (one event per GNR). We have extracted the translational diffusion times from the slow component of our single-event autocorrelation (only events not overlapping with either the begin or the end of our time traces were considered). Expectedly, we find that the distribution of translational diffusion times is relatively much broader than that of rotational times, particularly at high intensities (see fig. S17). By comparing the translational diffusion histograms of fig. S17 with the rotational diffusion histograms of fig. 3 of the main text, we see that rotational diffusion is more sensitive than translational diffusion to heating effects.

S7 Optical torque
To check the effect of forces applied by the laser onto our GNRs and their possible effect on their Brownian rotational diffusion, we performed calculations considering the average dimension of our GNR (D=14.9 nm and L=51.7 nm).
The optical forces and torques acting on the nanoparticles in an electric field can be calculated from the polarizability of the nanoparticle. We approximate our GNR as an ellipsoid. The polarizability of an ellipsoid is given by: where α ∥ and α ⊥ are the polarizability of the ellipsoid when the applied field is parallel and perpendicular to the longitudinal axis of the ellipsoid, respectively. V p is the volume The anisotropic GNR polarizability leads to an optical potential energy in an electric field with amplitude E 0 , which depends on the orientation of the GNR longitudinal axis with respect to the electric field: where Re[∆α] is real part of the difference between longitudinal and transverse polarizabil-  fig. S18. We find that the maximum optical energy is around 8 × 10 −21 N m at the highest power 369 µW (red) and for the case that the GNR is aligned along the electric field. At the effective temperature (around 420 K) found for this case this potential is 1.38 times higher than k B T . That means the applied torque can have a significant effect on the Brownian rotational motion of a large and resonant rod.
(b) (a) Figure S18: a) Effective temperature T HBM of a GNR with 14.9 nm as diameter and 51.7 nm length as a function of incident wavelength, which shows a peak at 750 nm (average resonance of our GNRs, see fig. 1h). Different colors correspond to 4 different powers: 49 (blue), 147 (green), 262 (orange) and 369 µW (red). By increasing the illumination intensity, the effective temperature increases, especially at around the resonance wavelength. b) Optical potential of the GNR in an applied electric field as a function of the angle θ (the angle between the GNR main axis and the incident electric field). The colors correspond to the different powers mentioned in (a). The minimum of the potential energy is obtained in the case of a positive real polarizability for the highest power (red) and at θ=0, π and 2π. It is around 8 × 10 −21 N m for an average resonant GNR. If we compare it with the value of k B T at this temperature (that is corresponding to the Brownian motion) which is k B T = (1.38×10 −23 )420 = 5.79×10 −21 N m, we find that maximum absolute potential for on the resonance GNR is 1.38 times more than k B T . It means the applied torque has a significant effect on the rotational Brownian motion at our highest excitation power (369 µW).

S8 Rotational diffusion in PVA solutions
Binding affinity of PVA to the GNRs at very low concentrations The adsorption process can be described by Langmuir's adsorption (isotherm) model 7 as a reversible chemical process: where P is the adsorbate molecule, S is an empty adsorption site and SP the same site with the adsorbed species. By considering a and d as adsorption and desorption rates, respectively, we write the fraction of occupied surface sites as: where closed brackets are the concentrations or populations of each species.
is the total population of adsorption sites. We assume that the hydrodynamic volume of the diffuser, V H , increases linearly upon binding polymer chains. By assuming that the rotational correlation decay time (τ d ) scales linearly with the volume of the object, which itself varies linearly with the amount of polymer bound, we can write: where τ d 0 is the decay time of the rotational component of the autocorrelation function of the GNR without any polymer bound to it. Consequently, the y axis of the plot in fig. 4d in the main text is proportional to the ratio of occupied sites to the total number of sites [SP ]/[S 0 ]. It means that, by increasing the concentration of PVA, the number of occupied sites increases and at very high concentration, all the sites will be occupied by PVA.
For the lowest concentrations of PVA in our measurement (62 and 125 ppb, fig.4 a and b in the main text) by considering the population of the fast decay times as the free rods and the second population as the occupied ones and ignoring the heterogeneity of binding sites, we can approximately write: where p 1 is the probability of finding a GNR occupied by at least one polymer coil bound and p 0 is the probability of detecting a GNR without any polymer. We assign the two Gaussian distributions in the histograms of fig.4 a and b to the populations of free rods and occupied rods. p 0 and p 1 are proportional to the areas of these histogram components.   Figure S20: Histograms of decay times τ d for different concentrations of PVA: 0.5 ppm (blue), 62 ppb (red) and 125 ppb (gray). a) Shows an overlap between the second population of the red histogram and the blue one, which is corresponding to the GNRs with one polymer coil bound to them. b) There is a big overlap between the blue histogram and the gray one which shows a big ratio of the GNRs in the gray histogram are corresponding to the rods with at least one polymer coil on them.
By considering a persistence length of 3Å for PVA, and a Flory exponent of 0.6, we estimate the radius of each polymer coil to 37 nm on average.
Upon adsorption of a PVA chain (molecular weight: 125 000 g/mol) we expect a redshift on the order of few nm similar to the values obtained by P. Zijlstra et al. for Streptavidin-RPE (300 kDa, max. 2 nm shift). 8 According to sect. S5, the LSPR changes in the range of a few nanometers do not affect the rotational diffusion coefficient.
We also investigate the distribution of translational diffusion times by extracting the decay time of slow component of autocorrelations τ D . We compare the results for three measurements: GNR without PVA, GNR with 0.5 ppm PVA and 100 ppm PVA, and present them in fig. S21. As discussed in sect. 6.3, we find broader histograms than for rotational diffusion. Moreover, these histograms overlap considerably, which confirms that rotational diffusion is a better tool to study binding of biomolecules to GNRs. indicating an increase of hydrodynamic volume by 16%, i.e., by ≈ 1100 nm 3 . From this value we can approximate the average number of BSA molecules adsorbed to GNR (assuming BSA molecule as spheres with diameters of 7 nm and volumes of ≈ 180 nm 3 ) as N BSA ≈ 6.