Table 1.

Analytical solution of single gold nanosphere heated by a rectangular pulse

Heating process	$${\left.T_{\mathrm{Au}}(r,t)\right|}_{t<2\mu }=T_{\mathrm{ref}}+\frac{{R_{\mathrm{NP}}}^2}{k_{\mathrm{Au}}}\frac{C_{\mathrm{abs}}F}{V_{\mathrm{NP}}2\mu }\left\{\frac{1}{3}\frac{k_{Au}}{k_m}+\frac{1}{6}\left(1-\frac{r^2}{{R_{\mathrm{NP}}}^2}\right)-\frac{2R_{\mathrm{NP}}b}{r\pi }{\int }_0^{\infty }\frac{\exp (-\frac{y^{2}t}{{\gamma }_1})}{y^2}\frac{\left(\text{sin}y-y\text{cos}y\right)\text{sin}(\frac{ry}{R_{NP}})}{[{\left(c\text{sin}y-y\text{cos}y\right)}^2+b^2{y}^2{\text{sin}}^{2}y]}dy\right\}$$ (19)
		$${\left.T_{\mathrm{Au}}(r=0,t)\right|}_{t<2\mu }=T_{\mathrm{ref}}+\frac{{R_{\mathrm{NP}}}^2}{k_{\mathrm{Au}}}\frac{C_{\mathrm{abs}}F}{V_{\mathrm{NP}}2\mu }\left\{\frac{1}{3}\frac{k_{\mathrm{Au}}}{k_m}+\frac{1}{6}-\frac{2b}{\pi }{\int }_0^{\infty }\frac{\exp (-\frac{y^{2}t}{{\gamma }_1})}{y}\frac{\left(\text{sin}y-y\text{cos}y\right)}{[{\left(c\text{sin}y-y\text{cos}y\right)}^2+b^2{y}^2{\text{sin}}^{2}y]}dy\right\}$$ (20)
	$${\left.T_{\mathrm{water}}(r,t)\right|}_{t<2\mu }=T_{\mathrm{ref}}+\frac{{R_{\mathrm{NP}}}^3}{r{k}_{\mathrm{Au}}}\frac{C_{\mathrm{abs}}F}{V_{\mathrm{NP}}2\mu }\left\{\begin{array}{c}\frac{1}{3}\frac{k_{Au}}{k_m}-\frac{2}{\pi }{\int }_0^{\infty }\frac{\exp \left(-\frac{y^{2}t}{{\gamma }_1}\right)}{y^3}\\ \frac{\left(\text{sin}y-y\text{cos}y\right)\left[by\text{sin}y\text{cos}\sigma y-\left(c\text{sin}y-y\text{cos}y\right)\text{sin}\sigma y\right]}{[{\left(c\text{sin}y-y\text{cos}y\right)}^2+b^2{y}^2{\text{sin}}^{2}y]}dy\end{array}\right\}$$ (21)
Cooling process	$${\left.T_{\mathrm{Au}}\left(r,t\right)\right|}_{t\ge 2\mu }=T_{\mathrm{ref}}+\frac{2{R_{\mathrm{NP}}}^{3}b}{r\pi k_{\mathrm{Au}}}\frac{C_{\mathrm{abs}}F}{V_{\mathrm{NP}}2\mu }{\int }_0^{\infty }\frac{\exp \left[-\frac{y^2\left(t-2\mu \right)}{{\gamma }_1}\right]-\exp \left(\frac{-y^{2}t}{{\gamma }_1}\right)}{y^2}\frac{\left(\text{sin}y-y\text{cos}y\right)\text{sin}(\frac{ry}{R_{\mathrm{NP}}})}{[{\left(c\text{sin}y-y\text{cos}y\right)}^2+b^2{y}^2{\text{sin}}^{2}y]}dy\hspace{0.17em}$$ (22)
	$${\left.T_{\mathrm{Au}}\left(r=0,t\right)\right|}_{t\ge 2\mu }=T_{\mathrm{ref}}+\frac{2{R_{\mathrm{NP}}}^{2}b}{\pi k_{\mathrm{Au}}}\frac{C_{\mathrm{abs}}F}{V_{\mathrm{NP}}2\mu }{\int }_0^{\infty }\frac{\exp \left[-\frac{y^2\left(t-2\mu \right)}{{\gamma }_1}\right]-\exp \left(\frac{{-y}^{2}t}{{\gamma }_1}\right)}{y}\frac{\left(\text{sin}y-y\text{cos}y\right)}{[{\left(c\text{sin}y-y\text{cos}y\right)}^2+b^2{y}^2\hspace{0.17em}{\text{sin}}^{2}y]}dy$$ (23)
	$${\left.T_m(r,t)\right|}_{t\ge 2\mu }=T_{\mathrm{ref}}+\frac{{R_{\mathrm{NP}}}^3}{r{k}_{\mathrm{Au}}}\frac{C_{\mathrm{abs}}F}{V_{\mathrm{NP}}2\mu }\left\{\begin{array}{c}\frac{1}{3}\frac{k_{\mathrm{Au}}}{k_m}-\frac{2}{\pi }{\int }_0^{\infty }\frac{\exp \left[-\frac{y^2\left(t-2\mu \right)}{{\gamma }_1}\right]-\exp \left(\frac{y^{2}t}{{\gamma }_1}\right)}{y^3}\\ \frac{\left(\text{sin}y-y\text{cos}y\right)\left[by\text{sin}y\text{cos}\sigma y-\left(c\text{sin}y-y\text{cos}y\right)\text{sin}\sigma y\right]}{[{\left(c\text{sin}y-y\text{cos}y\right)}^2+b^2{y}^2\hspace{0.17em}{\text{sin}}^{2}y]}dy\end{array}\right\}$$ (24)
	$$b=\frac{k_m}{k_{Au}}\sqrt{\frac{{\alpha }_{\mathrm{Au}}}{{\alpha }_m}},\hspace{0.17em}c=1-\frac{k_m}{k_{\mathrm{Au}}},\hspace{0.17em}{\gamma }_1=\frac{{R_{\mathrm{NP}}}^2}{{\alpha }_{\mathrm{Au}}},\hspace{0.17em}\sigma =\left(\frac{r}{R_{\mathrm{NP}}}-1\right)\sqrt{\frac{{\alpha }_{\mathrm{Au}}}{{\alpha }_m}}$$ (25)