## 2 Methods
### Numerical Methods  
The discrete dipole approximation (DDA), introduced by [PERSON] [42], has been considered as one of the most flexible and reliable methods for calculating the optical properties of particles with arbitrary geometries. In using this approximation, the target is replaced by an array of point dipoles (polarizable points). The electromagnetic scattering problem for an incident periodic wave interacting with this array of point dipoles is then solved essentially exactly. Figure 3 demonstrates the schematic of the scattering process of a single dipole with \(\overrightarrow{E}\) indicating the electric field of the incident light, propagating along the \(\overrightarrow{k}\) direction. In the current study, a DDA code named DDSCAT 7.3 [43] was applied to calculate the absorption and scattering efficiency GNRs. [PERSON] and [PERSON] [44] developed  
Figure 2: (**a**) The GNR solution sample was wrapped by insulation and reflective material and (**b**) Equipment set-up of the thermal experiment. and validated the DDSCAT code based on Mie theory. This code has been adopted and validated in several previous studies of gold nanoparticles and assemblies [24; 45; 46; 47]. The amount of absorption and scattering is usually identified by the absorption and scattering coefficients \(C_{abs}\) and \(C_{sca}\). The total amount of absorption and scattering is the extinction, which is expressed in the term of the extinction cross section \(C_{ext}\). The dimensionless efficiency factors of these optical properties are commonly used as specified below:  
\[ \begin{split} Q_{abs}&=\frac{C_{abs}}{\pi r^{2}} \\ Q_{sca}&=\frac{C_{sca}}{\pi r^{2}}\\ Q_{ext}&=\frac{C_{ext}}{\pi r^{2}}=Q_{abs}+Q_{sca }\\ \end{split}  \tag{3}\]  
The analytical expressions of the Mie theory for these efficiency factors and for other characteristics of absorption and anisotropic scattering can be found in several texts; e.g., [48; 49; 50; 51; 52]. It is important to note that the scattering problem is characterized by the so-called diffraction or size parameter:  
\[x=\frac{2\pi r}{\lambda} \tag{4}\]  
where \(r\) is the radius of a particle and \(\lambda\) is the wavelength of the incident light. In DDSCAT, the arbitrary target consists of \(N\) dipoles with lattice spacing \(d\). \[V=Nd^{3} \tag{5}\]  
The size of the target is characterized by the \"effective radius\" to calculate the arbitrary geometry. \[r_{eff}=\sqrt[3]{\frac{3V}{4\pi}} \tag{6}\]  
Since the solutions used in this study are aqueous, the refractive index of water, \(1.33+0i\), was applied as the refractive index of the ambient at all wavelengths. The real and imaginary components of the bulk dielectric constant \(\varepsilon_{bulk}\) can be calculated using the refractive index of the bulk material:  
\[ \begin{split}\varepsilon_{r}&=n^{2}-k^{2}\\ \varepsilon_{i}&=2 nk\end{split}  \tag{7}\]  
where \(n\) is the index of refraction and \(k\) is the index of absorption. Figure 3: Schematic of the scattering process of a spherical particle. [PERSON], et al. [53] measured the values of the complex dielectric function of gold at various wavelengths using a thin film of gold. However, when the target dimension was smaller than the mean free path length of electrons in the bulk material, the limitations to electron collisions within the particle would change the dampening constant of conduction electrons. This would result in size-dependent dielectric constants which are different from the dielectric constants of the bulk material. To account for this, [PERSON] and [PERSON] [54] developed a model to modify the real and imaginary components of the dielectric constant based on size via an effective relaxation frequency, \(\gamma_{eff}\). This size effect needs to be considered to match the complex dielectric function measured by [PERSON], et al. [53]. This is critical for the particles used in this paper, since the effective radius \(r_{eff}\) of the GNR was only 8.9 nm.