Plasmonic nanostructures exhibit unique interactions with light due to localized surface plasmon resonance (LSPR), leading to enhanced electromagnetic fields and significant heat generation. Understanding the theoretical models governing heat generation and dissipation in these systems is critical for optimizing their performance in photothermal applications. This article explores the fundamental mechanisms of Joule heating, heat diffusion, and photothermal efficiency calculations in plasmonic nanostructures under illumination.

When illuminated at their resonant frequency, plasmonic nanostructures absorb photons, converting them into energetic electron oscillations. These oscillations decay through electron-electron and electron-phonon scattering, producing heat. The primary mechanism of heat generation is Joule heating, where the absorbed optical energy is converted into thermal energy. The power density of heat generation can be described by the following expression:

\[ Q = \frac{1}{2} \omega \epsilon_0 \text{Im}(\epsilon) |E|^2 \]

Here, \( Q \) is the heat generation rate per unit volume, \( \omega \) is the angular frequency of incident light, \( \epsilon_0 \) is the vacuum permittivity, \( \text{Im}(\epsilon) \) is the imaginary part of the nanostructure’s dielectric function, and \( |E|^2 \) is the electric field intensity enhancement factor. This equation highlights that heat generation depends on the material’s optical properties and the local field enhancement.

The temperature distribution in and around the plasmonic nanostructure is governed by the heat diffusion equation. For a steady-state system, the equation simplifies to:

\[ \nabla \cdot (k \nabla T) + Q = 0 \]

where \( k \) is the thermal conductivity of the material and \( T \) is the temperature. In transient cases, the time-dependent heat diffusion equation includes a term for heat capacity:

\[ \rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q \]

Here, \( \rho \) is the mass density and \( C_p \) is the specific heat capacity. Solving these equations requires boundary conditions, such as the temperature at the nanostructure’s surface and the surrounding medium’s thermal properties. Analytical solutions are often limited to simple geometries, while numerical methods like finite element analysis (FEA) are employed for complex structures.

For spherical nanoparticles, an analytical solution to the steady-state heat equation can be derived. The temperature rise at the nanoparticle’s surface is given by:

\[ \Delta T = \frac{P_{\text{abs}}}{4 \pi k R} \]

where \( P_{\text{abs}} \) is the absorbed power, \( k \) is the thermal conductivity of the surrounding medium, and \( R \) is the nanoparticle radius. This approximation assumes uniform heating and neglects thermal boundary resistance, which may become significant for very small nanoparticles or high heating rates.

Photothermal efficiency quantifies the fraction of incident light converted into heat. It is defined as:

\[ \eta = \frac{P_{\text{abs}}}{P_{\text{inc}}} \]

where \( P_{\text{inc}} \) is the incident power. The absorbed power \( P_{\text{abs}} \) can be calculated using the absorption cross-section \( C_{\text{abs}} \):

\[ P_{\text{abs}} = I_0 C_{\text{abs}} \]

Here, \( I_0 \) is the incident light intensity. The absorption cross-section depends on the nanostructure’s size, shape, material, and surrounding medium. For gold nanoparticles, typical \( C_{\text{abs}} \) values range from \( 10^{-14} \) to \( 10^{-12} \) m² under resonant conditions.

The efficiency of heat dissipation is influenced by the nanostructure’s thermal conductivity and the surrounding medium. Metals like gold and silver have high thermal conductivities (~300 W/m·K), facilitating rapid heat dissipation. However, at the nanoscale, reduced phonon mean free paths can lower effective thermal conductivity. The surrounding medium also plays a crucial role; water (k ≈ 0.6 W/m·K) dissipates heat slower than air (k ≈ 0.026 W/m·K), leading to higher localized temperatures.

Theoretical models also account for collective heating effects in nanoparticle assemblies. When multiple nanoparticles are closely spaced, near-field coupling alters both the electromagnetic and thermal responses. The temperature distribution becomes more complex due to interparticle heat exchange. Numerical simulations incorporating coupled electromagnetic and thermal solvers are necessary to accurately predict temperature profiles in such systems.

Nonlinear effects may arise under high-intensity illumination, where the nanostructure’s optical properties become temperature-dependent. For example, the dielectric function of gold changes with temperature, modifying the LSPR frequency and absorption efficiency. This feedback mechanism must be included in models for high-power applications to avoid underestimating or overestimating heating effects.

Theoretical frameworks also explore pulsed illumination scenarios, where short laser pulses induce transient heating. The timescales of electron-phonon relaxation (~1 ps) and phonon-phonon coupling (~100 ps) dictate the temperature evolution. Two-temperature models (TTM) are often employed, separating electron and lattice temperatures:

\[ C_e \frac{\partial T_e}{\partial t} = -G (T_e - T_l) + Q \]
\[ C_l \frac{\partial T_l}{\partial t} = G (T_e - T_l) \]

Here, \( T_e \) and \( T_l \) are electron and lattice temperatures, \( C_e \) and \( C_l \) are their respective heat capacities, and \( G \) is the electron-phonon coupling constant. These equations capture the ultrafast dynamics of heat generation and redistribution.

In summary, theoretical models of heat generation and dissipation in plasmonic nanostructures involve coupled electromagnetic and thermal analyses. Joule heating, governed by the material’s dielectric properties and field enhancement, initiates thermal processes. Heat diffusion equations, solved analytically or numerically, predict temperature distributions, while photothermal efficiency calculations quantify energy conversion. Advanced models incorporate collective effects, nonlinearities, and ultrafast dynamics for comprehensive understanding. These theoretical insights guide the design of plasmonic systems for applications requiring precise thermal control.