Photoacoustic generation by a gold nanosphere: From linear to nonlinear thermoelastics in the long-pulse illumination regime

TL;DR

This paper models photoacoustic wave generation by a gold nanosphere in water under long-pulse illumination, revealing nonlinear effects at high fluences and providing quantitative estimates of the critical energy where nonlinearity becomes significant.

Contribution

It introduces a detailed theoretical model accounting for finite size and temperature-dependent properties, highlighting nonlinear regimes and correcting previous overestimations by simpler models.

Findings

Over 90% of photoacoustic energy is generated in water for nanosecond pulses.

Nonlinear effects become significant at high fluences, altering the amplitude-fluence relationship.

The critical energy for nonlinearity scales with the water diffusion volume during illumination.

Abstract

We investigate theoretically the photoacoustic generation by a gold nanosphere in water in the thermoelastic regime. Specifically, we consider the long-pulse illumination regime, in which the time for electron-phonon thermalisation can be neglected and photoacoustic wave generation arises solely from the thermo-elastic stress caused by the temperature increase of the nanosphere or its liquid environment. Photoacoustic signals are predicted computed based on the successive resolution of a thermal diffusion problem and a thermoelastic problem, taking into account the finite size of the gold nanosphere and the temperature-dependence of the thermal expansion coefficient of water. For sufficiently high illumination fluences, this temperature dependence yields a nonlinear relationship between the photoacoustic amplitude and the fluence. For nanosecond pulses in the linear regime, we show that more than 90 % of the emitted photoacoustic energy is generated in water, and the thickness of the generating layer around the particle scales close to the square root of the pulse duration. Our results demonstrate that the point-absorber model introduced by Calasso et al.[17] significantly overestimates the amplitude of photoacoustic waves in the nonlinear regime. We therefore provide quantitative estimates of a critical energy, defined as the absorbed energy required such that the nonlinear contribution is equal to that of the linear contribution. Our results suggest that the critical energy scales as the volume of water over which heat diffuses during the illumination pulse. Moreover, thermal nonlinearity is shown to be expected only for sufficiently high ultrasound frequency. Finally, we show that the relationship between the photoacoustic amplitude and the equilibrium temperature at sufficiently high fluence reflects the thermal diffusion at the nanoscale around the gold nanosphere.