#### 2.1 Metallic nanoparticles and localized surface plasmons

Metal nano-objects support electronic resonances known as LSPs that can be excited upon illumination. The frequency of LSP resonances strongly depends on the morphology of the metal nano-object and its dielectric environment. For instance, elongating a sphere into a rod-like shape tends to red-shift the LSP resonance. For noble metals, such as gold, silver or copper, this property allows accurate tuning of LSP resonances from the visible to the near-infrared (NIR) frequency range.

Recent advances in both *bottom-up* and *top-down* fabrication techniques offer a tremendous variety of metal NP sizes and shapes. On the one hand, chemists have developed synthesis procedures to produce colloidal noble metal NPs with numerous geometries including rods, cubes, triangles, shells, stars, etc. [3]. On the other hand, techniques such as e-beam lithography and focused ion beam milling are convenient means to design planar metal nanostructures on a flat substrate with a resolution down to a few tens of nanometers. Examples of colloidal gold NPs and lithographically prepared gold nanostructures are presented in Fig. 1.

The origin of LSP resonances in metal NPs can be simply derived for a metal sphere that is much smaller than the illumination wavelength and can be considered as an electromagnetic dipole. In this case, the sphere polarizability reads

- (1)

where *R* is th radius of the sphere. In this expression, the polarizability α is defined such that the complex amplitude of the polarization vector of the NP reads . Equation (1) shows that a resonance occurs at the frequency ω at which . For a gold sphere smaller than ∼30 nm in water, this occurs for nm. However, for larger spheres, this dipolar approximation is no longer valid and more complex models, such as Mie theory [4, 5], [6], accounting for retardation effects, are required. For more sophisticated geometries, numerical simulations are needed (see Section 'Numerical methods').

Note that this conclusion is valid when considering the *respective maxima* of both cross-section spectra, but not the cross-sections at an arbitrary wavelength. Indeed, as shown in Fig. 2 b, for an 88 nm NP, even though the respective maxima of the absorption and scattering spectra are equal, absorption can be either dominant or negligible depending on the wavelength. This is the consequence of the spectral shift that usually occurs between absorption and scattering spectra for large or non-spherical NPs.

Consequently, even though spherical gold NPs are usually better absorbers than scatterers, the illumination wavelength must be specified to determine what is the actual dominant energy conversion pathway. It is worth noticing that, for this reason, considering experimental *extinction* spectra to estimate the *absorption* efficiency of a plasmonic structure, as sometimes seen in the literature [9], is not always reliable.

Tuning the plasmonic resonance frequency of a NP can be easily achieved by changing its morphology. Any deviation from the spherical shape tends to red-shift the resonance. Experimental results presented in Fig. 3 illustrate the red-shift of the plasmon resonance of a gold nanorod while increasing its aspect ratio.

In the following, we focus on the *absorption* processes and the subsequent heat generation.

#### 2.2 Delivered heat power

The power absorbed (and delivered) by a NP can be simply expressed using the absorption cross-section σ_{abs} introduced in the previous section:

- (5)

were *I* is the irradiance of the incoming light (power per unit surface). In the case of a plane wave, .

#### 2.3 Temperature profile under continuous-wave illumination

While the computation of the delivered heat power *Q* turns out to be a full-optical problem as explained in the previous section, the determination of the steady-state temperature distribution inside and outside the NP is based on the resolution of the heat diffusion equation:

- (8)

- (9)

where is the thermal conductivity. For a spherical NP of radius *R*, simple calculations lead to a temperature increase [13]:

- (10)

- (11)

where is the temperature increase of the NP. Interestingly, while the heat power density can be highly non-uniform within the NP as clearly observed in Fig. 4 [12], the temperature at equilibrium is, on the contrary, generally perfectly uniform inside the NP [13]. This is due to the much larger thermal conductivity of metals as compared with that of the surroundings (liquid, glass, etc.). The actual temperature increase experienced by a NP is dependent on numerous parameters, namely its absorption cross-section, its shape, the thermal conductivity of the surrounding medium and the wavelength and irradiance of the incoming light. For a spherical NP, the NP temperature increase is related to the absorbed power according to [13]

- (12)

where κ_{s} is the thermal conductivity of the surrounding medium.

The establishment of this steady-state temperature profile is usually very fast when working with NPs. The typical duration τ_{tr} of the transient regime is not dependent on the temperature increase but on the characteristic size *L* of the system (for instance the radius *R* for a sphere) [6]:

- (13)

where ρ is the mass density of the NP and *c*_{p} its specific heat capacity at constant pressure. For example, for spherical NPs of diameters 10 nm, 100 nm and 1 μm, one gets τ_{tr} of the order of 0.1 ns, 10 ns and 1 μs, respectively.

For non-spherical NPs, there is no simple analytical expression giving the NP temperature increase as a function of the absorbed heat power *Q* and numerical simulations are required. However, Baffou et al. [13] have recently proposed to use a dimensionless geometrical correction factor β defined such that the NP temperature increase reads

- (14)

where *R*_{eq} is the equivalent NP radius. The value of beta for a large set of geometries with axial symmetry (namely rods, ellipsoids, discs and tori) are given in Ref. [13].

A last feature that is worth discussing is the possible influence of the surface thermal resistivity at the surface of a NP immersed in a liquid [17, 18], [19, 20]. A thermal resistivity may occur because of the material discontinuity, which acts as a thermal impedance. This resistivity can play a significant role in the heat release since it can reach appreciable values when the liquid does not wet the solid. The wetting depends on the nature of the interface and, in particular, on a possible molecular coating. Namely, hydrophobic coatings are associated with poor thermal conductivities. The direct consequence of a finite interface conductivity *G* (or resistivity ) is a temperature discontinuity at the NP interface such that

- (19)

However, in the steady-state regime, this surface resistivity has no effect on the temperature *outside* the NP, in the surrounding medium. As evidenced in Eq. (10), the temperature outside the medium is only dependent on the heat power *Q* released by the NP. A finite surface conductivity *G* has only an effect on the temperature within the NP. It can also contribute in making any transient regime longer. The incidence of such a resistivity can be quantified using the Kapitza number λ_{K} defined such that

- (20)

The surface resistivity can be neglected in any thermal process if . For gold NPs in water, usual values of the interface conductivity *G* range from 50 MW/(m^{2}K) to ∞ [20, 17]. For NP radii ranging from 5 to 50 nm, typical Kapitza numbers λ_{K} thus range from 2 to ∼0. As an example, gold nanorods coated with cetyltrimethylammonium bromide (CTAB) molecules are endowed with a typical surface conductivity of MW/(m^{2}K) [17]. If the nanorods are, say, 50 × 12 nm in size, the effective radius is nm and Eq. (20) yields . Hence, in this case, the surface resistivity does not have any significant effect.

#### 2.4 Nanoparticle heating under pulsed illumination

The use of pulsed illumination (from the femtosecond to the nanosecond range) to heat metal NPs markedly increases the range of applications of noble metal NPs. Compared to continuous-wave (CW) illumination, a new class of effects can be triggered, such as shorter temperature and pressure variations [21-23], further temperature confinement [6], acoustic wave generation [22], vibration modes [24-26], bubble formation [27-31], NP shape modification [32-35] and melting [32, 36], [37] and extreme thermodynamics conditions [38-42]. In this section, we briefly discuss the thermodynamics of metal NPs under pulsed illumination and in particular the influence of parameters such as pulse duration, pulsation rate and size of the NP.

The absorption of laser pulse energy by a gold NP can be described as a three-step process [43, 44], each of these steps involving different time scales as follows.

*Step 1. Electronic absorption*: During the interaction with the laser pulse, part of the incident pulse energy is absorbed by the gas of free electrons of the NP, much lighter and reactive than the ion lattice. The electronic gas thermalizes very fast to a Fermi–Dirac distribution over a time scale 100 fs [44]. This leads to a state of non-equilibrium within the NP: the electronic temperature *T*_{e} of the electronic gas increases while the temperature of the lattice (phonons) *T*_{p} remains unchanged.

*Step 3. External heat diffusion*: The energy diffusion from the NP to the surroundings usually occurs at longer characteristic time scale τ_{tr} (see Eq. (13)) and leads to a cooling of the NP and a heating of the surrounding medium. The time scale of this process depends on the size of the NP and ranges from 100 ps to a few nanoseconds. For small NPs (<20 nm), this third step can overlap in time with the electron–phonon thermalization [23] (as discussed hereafter).

During this process, the total absorbed energy reads

- (21)

where *f* is the pulsation rate and *F* the fluence of the laser pulse (energy per unit area).

Different regimes can be observed depending on the pulse duration compared with τ_{tr} (see Eq. (13)). When the pulse duration is short enough (typically <0.1 ns) and/or the NP is small enough (typically <100 nm in diameter), the three steps can be considered to happen *successively*. In this regime, the initial temperature increase reaches its maximal value [6]:

- (22)

where *V* is the volume of the NP. (The expression is only valid for moderate temperature increase so that *c*_{Au} remains constant.) In this case, the duration of the heating of the medium is of the order of τ_{tr}.

If we consider now the regime were the pulse duration exceeds τ_{tr}, the three steps will overlap in time. In other words, heat gets absorbed and delivered in the surroundings simultaneously. In this case, the maximum temperature increase inside the NP will not reach and the heating duration will simply equal the pulse duration. This situation is usually obtained when using *nanosecond* pulses on gold NPs.

where the fit parameters are and .