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This section outlines the major experimental steps involved in using GNRs as OCT based diffusion probes.

GNRs concentration estimation:

The first step in using GNRs for OCT imaging involves the estimation of the number density of GNRs in the original solution. For this purpose, absorbance from a dilute suspension of GNRs is measured using a spectrophotometer (solution is diluted enough to allow majority of the light to pass through so that multiple scattering events of the optical beam are avoided, which would otherwise change the optical path length of the beam). The Beer-Lambert law relates the absorbance in a dilute suspension to the extinction cross-section of GNRs (σt), number density (N), and optical path length in

the solution (l):

Using the size distribution of GNRs measured by TEM, a prediction for absorbance is also computed using Mie Gans theory. The computed absorbance is matched with the

experimental absorbance at the LSPR wavelength by adjusting N (l is known) to get

an estimate for N in the dilute solution used in the spectrophotometer. Thus, using

this estimate forN and the dilution factor of GNRs used during spectrophotometry, an estimate for the number density of GNRs in the original solution can be made.

Addition of GNRs to sample:

After the number density of GNRs is estimated, a small quantity of GNRs is topically added to the sample to have an ensemble of GNRs in each coherence volume (typically a few hundred; coherence volume estimated to be ∼375 µm3 _{in air). This quantity is} chosen to be of low enough concentration that the GNRs don’t physically interact with

each other in the sample (chosen such that the average separation between the GNRs

maximum translational distance GNRs diffuse during the duration of the measurement). Depending on the nature of the sample, the added GNRs can be gently mixed using a pipette, left on a rotator for slow mixing, or allowed to diffuse over time without any disturbance to the sample.

PS-OCT data acquisition:

The custom-built PS-OCT system described earlier in subsection 3.2 is used to collect temporal data (M-mode) from the same region in the sample containing GNRs. To avoid unintentionally heating the GNRs in the sample, power in the sample beam is limited to 3 mW. Due to buffer size limitations in the current control PC, the maximum number of A-lines comprised of 4096 CCD array is presently limited to 12000 for each time series. Thus, using the maximum CCD linerate of 25 kHz (i.e., each A-line collected in 40 µs), M-mode data can be collected for 480 ms in a single temporal frame. When

a longer observation time is necessary, sampling rates of 10 kHz, 5 kHz, 2 kHz, and 1 kHz result in an observation time of 1.2 s, 2.4 s, 6 s, and 12 s respectively, with the tradeoff being that the temporal spacing gets sparser with decreasing sampling rates. To improve the accuracy of the evaluated autocorrelations, the choice of sampling rate for each experiment is based on the following experimental criteria:

1. The fastest autocorrelation decay time to be measured must be at least twice the sampling time (Nyquist sampling criterion).

2. The total observation time (Tobs) must be such that Tobs τ1/e, where τ1/e is

the 1/e decay time g(1)(τ). This allows g(1)(τ) sufficient time to decay off to a state of no correlation. (Note that, in this thesis, Tobs > 100× τ1/e,HV and

Tobs >25×τ1/e,ISO are chosen for DR and DT estimations respectively).

Representative B-mode and the corresponding M-mode HH, HV images of a 0.2%

agarose gel with GNRs (premixed before gelation) is shown in figure 4.3. Each M-mode scan is collected at the center of the contextual B-scan, which shows temporal intensity streaks in both the co-polarized (HH) and cross-polarized (HV) channels. The intensity streaks in the co-polarized channel are observed to be longer than those in the cross- polarized channel. Qualitatively, this suggests that the temporal co-polarized signal decays over a longer timescale than the cross-polarized signal.

Diffusion coefficients estimation:

To evaluate the ensemble averaged rotational and translational diffusion coefficients from M-mode scans, the following steps are used (associated MATLAB code is included in appendix A.1):

1. Both the real and imaginary parts of the complex OCT signals ˜SHH(z, t) =

Figure 4.3: RepresentativeHH andHV B-mode images and the corresponding M-mode images acquired with the beam temporally probing the same position in the 0.2% agarose sample premixed with GNRs. All images were acquired at a sampling rate of 10 kHz. The M-mode images consist of 12,000 A-lines and thus the temporal range extends to 1.2 seconds.

2. Fluctuations in the above signals are isolated by subtracting their average value at each depth, as in equation (2.8).

3. G(1)_HH(τ) _Re and G (1) HV(τ)

_Re are computed at each pixel in z (Note: ‘|Re’ is added to emphasize the use of the real part of the OCT signal). The zero-lag value (τ = 0) in the autocorrelations contains non-deterministic noise, and thus is replaced by a value extrapolated using the next two data points in the autocorrelation (i.e.,lag of τ and lag of 2τ). 4. Normalized autocorrelations g_HH(1) (τ) Re and g (1) HV(τ)

Re at each pixel inz are ob-

tained by normalization of the above G(1)_HH(τ) Re and G (1) HV(τ) Re by their cor-

responding maximum values (i.e., τ = 0 values) at each pixel in z. Normalized isotropic autocorrelation g_ISO(1) (τ)

_Re at each pixel in z is evaluated by combining

g_HH(1) (τ) Re and g (1) HV(τ) Re using equation (4.2).

5. Steps (3) and (4) are repeated using the imaginary part of the OCT signals to get g_HH(1) (τ) Im, g (1) HV(τ) Im, and g (1) ISO(τ) Im.

6. For all HH, HV, and ISO autocorrelations, g(1)(τ) at each pixel in z is obtained by adding g(1)(τ)

Re and g

(1)_(τ)

Im.

7. g_HH(1) (τ), g(1)_HV(τ), and g_ISO(1) (τ) at each pixel in z are averaged over 10-25 pixels in depth (corresponds to a depth-section of∼15µm to 38 µm), and this averaging is performed sequentially at multiple depth-sections in the M-mode image. The ob- tained autocorrelations represent the depth-resolved autocorrelations: g(1)_HH(z, τ), g_HV(1) (z, τ), and g_ISO(1) (z, τ). In heterogeneous samples, the averaging is performed over depth-sections of 3 pixels only (so, each depth-section is 4.65 µm, which is comparable to the axial resolution of the OCT system).

8. To estimateτ1/e, the 1/e decay ofg(1)(τ), (unweighted) linear least-squared fittings

of ln[g(1)_{(τ)] over a region of initial time lags (i.e.,fromτ = 0 to τ}

1/e) to−t/τ+c

are carried out in each depth-section. Representative inverse-exponential fittings of the form e−t/τ+c _tog(1)_{(τ) are shown in figure 5.6.}

9. For the estimation of DR and DT in each depth-section, τ1/e values of g (1) HV(τ)

and g_ISO(1) (τ) are used based on equation (4.3) and equation (4.2) respectively. In a homogeneous sample, the average and standard deviation of DR(z) and DT(z)

computed at several depth-sections are reported.

Having outlined the experimental method in this section, the estimation of rotational and translational diffusion coefficients of GNRs in Newtonian fluids and non-Newtonian

fluids is outlined in the next chapter. More importantly, the depth-resolved autocor- relations, g_HH(1) (z, τ) and g_HV(1) (z, τ), offer a unique opportunity to understand the het- erogeneity present in various samples, which is also shown in the following chapters. In non-Newtonian fluids, it should be noted that the autocorrelations deviate slightly from pure exponentials at longer time lags due to the elastic memory in the samples. Performing inverse-exponential fittings (as outlined above) to such autocorrelations over a region of initial time lags (i.e.,fromτ = 0 toτ1/e) can yet describe the short timescale

dynamics of the GNRs and the outlined method lends itself as a semi-quantitive tool to understand diffusion in complex fluids. It should be noted however that the reported DT for the ensemble of GNRs in such instances represent an “on average” estimate be-

tween the timescale of τ = 0 andτ1/e, which assumes the viscous drag as the dominant

force behind the diffusion of GNRs and thus ignores any non-viscous contributions to the autocorrelations during that duration.

Chapter 5

Diffusion of GNRs using OCT