Analytical temperature profiles for laser-irradiated gold nanorod immersed in an optically non-absorbing medium

Abstract

Laser heating of gold nanorods immersed in an optically non-absorbing medium is important in both therapeutic and diagnostic medical procedures. For both continuous wave and pulsed lasers, we derive analytical temperature solutions by modeling a nanorod as a prolate spheroid. For the continuous wave laser irradiation, an exact analytical solution for the steady state heat transfer case using integer-degree Legendre function expansions in prolate spheroidal coordinates is presented. The steady state solution reveals that for the case of gold nanorod immersed in water, there is insignificant temperature variations on any single spheroidal surface that is coincident or confocal to the prolate spheroid surface, and the temperature only changes across spheroidal surfaces within the immediate surrounding water. This is found to be due to the very high thermal conductivity of gold which is more than a hundred-fold higher than that of water. The insight that the temperature rise is nearly invariant on any single spheroidal surface, along with a dimensional analysis, helps reduce the complexity of the heat equation in the unsteady regime of a pulsed laser illumination, and leads to a simple, lower dimensional model of unsteady heat equation, which takes into account the temporal dependency and spatial dependency in the direction across spheroidal surfaces. For the reduced unsteady heat equation, a simple, efficient, and insightful solution algorithm based on the Laplace transform and complex-degree Legendre functions is proposed. The maximum temperature and temperature profiles as functions of laser power, and nanorod geometries are obtained, revealing that for a fixed laser fluence and fixed nanorod mass, the more stretched nanorods yield higher temperatures in the spheroid and its immediate surroundings. In other words, more stretched nanorods are more efficient in converting the laser energy into localized, elevated temperatures. The obtained temperature solutions are useful in designing a parameter space for optimum therapeutic procedures for ablating cancer cells using high temperatures generated in the immediate vicinity of nanoparticles. The unsteady temperature solutions can also be used to obtain solutions to thermo-elastic models which are used to compute the photoacoustic signals that can, in turn, be used for cellular imaging and diagnostics.