Nanoparticles heat through light localization
Nanoparticles heat through light localization
Nathaniel J. Hogan1,2†, Alexander S. Urban2,3†, Ciceron Ayala-Orozco2,5, Alberto Pimpinelli4, Peter Nordlander1,2,3,4, and Naomi J. Halas1,2,3,4,5,* 1
Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA. 2
3
Laboratory for Nanophotonics, Rice University, Houston, TX 77005, USA.
Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA. 4 5
Rice Quantum Institute, Rice University, Houston, TX 77005, USA.
Department of Chemistry, Rice University, Houston, TX 77005, USA.
† These authors contributed equally to this work *Naomi J. Halas: [email protected]
Supporting Information We show in Fig. S1 a comparison between the analytic solutions of the diffusion approximation and Beer-Lambert law and the numerical solutions of our Monte-Carlo simulations. This comparison validates the use of the Monte-Carlo simulations for the case of particles which have comparable absorption and scattering efficiencies. A common geometry that lends itself to analytic solutions of the diffusion approximation of the radiative transfer equation is a point source located at the origin radiating into a homogeneous medium of particles. The solution of the diffusion approximation is given by 17:
1 + ̂ 1 4 4 where Fd is the radiant flux at a given radius, P0 is the radiant power of the point source, and κd is the transfer coefficient. The transfer coefficient is defined in terms of the particle concentration as well as the absorption and scattering cross sections σa and σs by the following equations: =
= 3 2
= ! 1 − #̅ + 3
where ρ is the nanoparticle concentration. The factor #̅ is zero for the case of dipole scatterers, whose scattering phase function is a cosine-squared function of the scattering angle. We can obtain the power passing through a spherical surface of radius r centered about the point source by integrating Fd over the surface, which yields:
= %1 + & 4 A corresponding solution in the case of purely absorbing particles is also easily calculated and merely decays exponentially according to the value of the absorption coefficient. Both of these analytic solutions are plotted in Fig. S1A. For comparison, we performed Monte-Carlo simulations for three cases in which the transfer coefficient remains constant but the ratio of absorption to scattering is varied. The results provide two important conclusions. First, they provide a control for validating the Monte-Carlo simulation itself. Second, they illustrate the limitations of the diffusion approximation and Beer-Lambert law in accurately modeling the transfer of light in systems with comparable absorption and scattering cross sections, such as the plasmonic particles used in this study. For this reason, we chose to analyze the light transport with the Monte-Carlo approach.
Supplemental Figures:
2
3
4
5
Nathaniel J. Hogan1,2†, Alexander S. Urban2,3†, Ciceron Ayala-Orozco2,5, Alberto Pimpinelli4, Peter Nordlander1,2,3,4, and Naomi J. Halas1,2,3,4,5,* 1
Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA. 2
3
Laboratory for Nanophotonics, Rice University, Houston, TX 77005, USA.
Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA. 4 5
Rice Quantum Institute, Rice University, Houston, TX 77005, USA.
Department of Chemistry, Rice University, Houston, TX 77005, USA.
† These authors contributed equally to this work *Naomi J. Halas: [email protected]
Supporting Information We show in Fig. S1 a comparison between the analytic solutions of the diffusion approximation and Beer-Lambert law and the numerical solutions of our Monte-Carlo simulations. This comparison validates the use of the Monte-Carlo simulations for the case of particles which have comparable absorption and scattering efficiencies. A common geometry that lends itself to analytic solutions of the diffusion approximation of the radiative transfer equation is a point source located at the origin radiating into a homogeneous medium of particles. The solution of the diffusion approximation is given by 17:
1 + ̂ 1 4 4 where Fd is the radiant flux at a given radius, P0 is the radiant power of the point source, and κd is the transfer coefficient. The transfer coefficient is defined in terms of the particle concentration as well as the absorption and scattering cross sections σa and σs by the following equations: =
= 3 2
= ! 1 − #̅ + 3
where ρ is the nanoparticle concentration. The factor #̅ is zero for the case of dipole scatterers, whose scattering phase function is a cosine-squared function of the scattering angle. We can obtain the power passing through a spherical surface of radius r centered about the point source by integrating Fd over the surface, which yields:
= %1 + & 4 A corresponding solution in the case of purely absorbing particles is also easily calculated and merely decays exponentially according to the value of the absorption coefficient. Both of these analytic solutions are plotted in Fig. S1A. For comparison, we performed Monte-Carlo simulations for three cases in which the transfer coefficient remains constant but the ratio of absorption to scattering is varied. The results provide two important conclusions. First, they provide a control for validating the Monte-Carlo simulation itself. Second, they illustrate the limitations of the diffusion approximation and Beer-Lambert law in accurately modeling the transfer of light in systems with comparable absorption and scattering cross sections, such as the plasmonic particles used in this study. For this reason, we chose to analyze the light transport with the Monte-Carlo approach.
Supplemental Figures:
2
3
4
5
