## 3 Results and Discussion  
### Experimental Results  
Figure 4a presents a characteristic TEM image of the GNRs used in this study. As can be seen from this image, most of the particles are single nanorods of dimensions near the manufacturer specifications, although several appear to have non-horizontal orientations relative to the detector. In addition, some GNR assemblies were also present. Figure 4b shows the dispersion averaged size (as a percentage) of GNR or GNR obtained from three DLS measurements. Significant deviation in sizes was obtained from repeated DLS analysis, which also indicates that some dynamic clustering may occur for small nanoparticles [58; 59]. As can be seen from the size distribution shown in Figure 4b, the majority (\(\sim\)22%) of the detected structures had an effective hydraulic diameter around 55 nm. According to the other experimental studies [58; 59], for a gold nanoparticle with a similar size, the discrepancy between the actual and DLS measured size was around 12 nm. Therefore, we concluded that the measured GNRs size appears to be as specified from Sigma-Aldrich(St. Louis, MO, USA). In Figure 5, from the left to the right, the samples were labelled A, B, C, D, E, and F. For gold nanorods the plasmonic absorption is characterized by two bands, corresponding to the oscillation of the free electrons along and perpendicular to their long axis [60; 61]. The transverse mode shows a resonance at about 520 nm, which is co-incident with the plasmon resonance of spherical particles of the same diameter, while the resonance of the longitudinal mode is red-shifted and strongly depends on the AR of the nanorods. The presence of these two modes can be clearly observed in the results for the absorbance at 520 nm and 780 nm measured in the optical experiment and shown in Figure 6. The yellow solid line in Figure 6 indicates the maximum of absorbance depending on wavelength, which was slightly red-shifted. This red shift of the SPR occurred has been reported to be due to aggregation of nanoparticles [29; 62]. Therefore, a slight red shift in the longitudinal SPR (see Figure 6) can be attributed to GNR aggregations, which showed an increasing (albeit diminishing return) trend with the GNR volume fraction. The difference in absorbance between Sample E and Sample D was 0.19, while the increase for Sample F was 0.16. These results suggested that for medical treatment an appropriate volume fraction of GNRs should be identified and applied to generate the necessary amount of heat with a minimum total injected mass of gold. Therefore, it is important to quantify the relationship between the capability of heat generation and the volume fraction of gold nanorods solutions. For comparison, the numerically predicted absorbance averaged for different orientations of nanorods for the solution with the same volume fraction as in the sample F was also shown in Figure 6. The discrepancy between experimental and numerical results can be explained from the fact that in the numerical calculation, a single GNR in the size of exact 10 \(\times\) 38 nm was analyzed. However, the width and length of the GNRs in the solution as specified by manufacturer, could vary by up to 10% of this size, thus the aspect ratio may not be exactly 3.8 and could vary from 3.36 to 4.33. As will be shown later, such a variation can cause a change in both absorption efficiency and the wavelength of the SPR. More specifically, for GNRs with AR of 3.36 and 4.33, the wavelength of SPR will be around 767 nm and 864 nm, respectively. The difference could also be caused by the aggregations of GNRs in the samples, which are clearly present in the dispersion as shown in Figure 4b. Figure 4: (**a**) Transmission electron microscopy (TEM) and (**b**) dynamic light scattering (DLS) measurements of the GNRs, with an original concentration of 5.20 \(\times\) 10\({}^{11}\) GNRs/mL.