# Nanoscale temperature determination of individual gold nano-heaters in colloids by confocal dynamic light scattering

María Luz Martinez Ricci, Andrea Montero-Oleas, Stéphanie Kodjikian, Xavier Cattoën, Aude Barbara

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María Luz Martinez Ricci, Andrea Montero-Oleas, Stéphanie Kodjikian, Xavier Cattoën, Aude Bar-
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Distributed under a Creative Commons Attribution 4.0 International License
Nanoscale temperature determination of
individual gold nano-heaters in colloids by
confocal dynamic light scattering

María Luz Martínez Ricci,*,† Andrea Montero-Oleas,†,‡ Stéphanie Kodjikian,‡
Xavier Cattoën,‡ and Aude Barbara*,‡

†Instituto de Química Física de los Materiales, Medio Ambiente y Energía (INQUIMAE),
CONICET, Universidad de Buenos Aires, Ciudad Universitaria, Pab. II, Ciudad
Autónoma de Buenos Aires, Argentina

‡Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France

E-mail: mricci@qi.fcen.uba.ar; aude.barbara@neel.cnrs.fr

Abstract

Accurate temperature determination at the nanoscale is crucial for understanding
and optimizing the photothermal properties of plasmonic nanoparticles (NPs). Here,
we introduce confocal dynamic light scattering (confocal DLS) as a novel method for
*in-situ* measurement of the temperature of individual colloidal NPs. By coupling a
tightly focused pump-probe configuration with single-particle DLS, we quantify NPs
heating from their Brownian motion, exploiting the temperature dependence of the
solvent viscosity. Experiments on Au NPs (20–50 nm) reveal excellent agreement with
the hot Brownian motion theory, enabling reliable temperature increases determina-
tion up to ~150 K, without any adjustable parameters. Importantly, the method
extends to Au@SiO₂ core-shell nano-heaters, where a transition from the high-density
to the low-density hot Brownian regime is experimentally observed for the first time,
demonstrating that the density ratio between the NPs and the solvent governs the
motion of the heated NPs. Confocal DLS thus provides unprecedented access to the
photothermal response of nanoscale heaters at the single-particle level, including meso-
porous core–shell architectures, paving the way for applications in nanoscale catalysis,
thermally responsive nanomaterials and controlled drug release.

Introduction

Plasmonic nanoparticles (NPs) have attracted significant interest due to their unique optical
properties, mainly arising from the localized surface plasmon resonance, a collective oscilla-
tion of conduction electrons of metal nanoparticles, when illuminated with light at specific
wavelengths within the visible spectrum. This resonance leads to highly efficient light ab-
sorption within confined volumes, with the absorption characteristics depending on the size,
shape, and configuration of the plasmonic NPs. The absorbed light energy is subsequently
converted into heat, raising the temperature of the NP and its surrounding environment.
Owing to these photothermal properties, plasmonic NPs are promising candidates for a wide
range of temperature-responsive nanoscale applications, such as catalysis,¹ tuning of material
properties,²,³ or biomedical applications,⁴ in particular for photothermal cancer treatments.⁵
In this context, core–shell nanoparticles offer a versatile platform for molecular transport in
biological environments. These structures typically feature a gold core, acting as a localized
heat source, and a mesoporous silica shell, that provides a high loading capacity for guest
molecules. The ability to precisely determine and regulate temperature at the nanoscale is
essential for optimizing the release and functionality of such nanosystems.

The characterization of the photothermal properties of plasmonic NPs, in particular the
determination of their surface temperature and that of their surrounding media, has been
the focus of several studies (Table S1). On the one hand, the collective heating effect result-
ing in a macroscopic temperature increase of the system and induced by a myriad of heated
plasmonic NPs has been investigated through diverse observations of heat transfer to the surrounding medium. These strategies range from measuring the macroscopic temperature of the liquid,⁶,⁷ to more sophisticated in-situ methods such as monitoring thermally induced chemical reactions,⁸ phase transitions,⁹ Raman shifts of molecular vibrations¹⁰ or changes of the properties of the solvent itself (thermal conductivity, refractive index, or viscosity).¹¹,¹² On the other hand, studies have been dedicated to measuring the temperature on individual plasmonic NP heaters, which requires more complex configurations. For this purpose, indirect and direct observations have been achieved. Indirect approaches in colloidal suspensions have been studied using some labeling molecules, such as up-conversion NPs¹³ or SERS-active species,¹⁰ to estimate the temperature, unavoidably altering the plasmonic NP environment. Direct observations have been allowed to circumvent this limitation by studying label-free systems, often achieved by immobilizing plasmonic nanoparticles on a substrate and observing them under a microscope, where their heating is quantified through transient absorption or radiation measurements.¹⁴⁻¹⁹

Despite these various experimental approaches, accurate measurements of the temperature of individual and small plasmonic NPs in their colloidal state are scarce, although they are of particular interest for potential applications. Theoretical studies, referred to as Hot Brownian Motion (HBM), analyzed the Brownian motion of heated plasmonic NPs in colloidal suspension and derived expressions for their translational²⁰ and rotational²¹ diffusion coefficient from which the temperature of the NPs can be accessed. Proof of principle validation of this theoretical approach was given for both the translational motion, by photothermal correlation spectroscopy²²,²³ and the rotational motion, by dark-field scattering, on gold NP and nano-rods of 40 to 60 nm.²⁴ Furthermore, the temperature of optically trapped large single heated NPs (2.15 µm) was also reported.²⁵

In this study, we report on the in-situ temperature determination of small plasmonic NPs by confocal Dynamic Light Scattering (DLS).²⁶ Our approach merges the properties of DLS, which is a widely used method to measure the diffusion coefficient of colloids, to the
confocal microscopy, that probes a very small volume of the solution (~1fL) and thereby offers the possibility to measure the NPs individually, as they cross the observation volume. To this end, we have developed a confocal pump-probe DLS setup using a pump wavelength tuned close to the plasmon resonance of NPs, to heat them, and a probe wavelength far from the resonance, to measure their translational diffusion coefficient. The temperature determination of the NPs relies on the solvent viscosity changes with temperature, which are typically very significant, thus offering a simple and sensitive mean to study local heating effects at the nanoscale. Photothermal studies of plasmonic NPs were previously reported using standard DLS, where the collective response of the particles was investigated in the context of superdiffusion regimes.¹²

Our work firstly presents the proof-of-principle for this novel approach by successfully retrieving the temperature of model systems, namely spherical gold NPs of known sizes, through the analysis of their Brownian motion under controlled excitation. The results show excellent agreement with previously reported theoretical predictions,²⁰ validating the underlying principle of the method. In particular, our study enables the photothermal study of individual plasmonic NPs as small as 20 nm, which broadens the range of accessible sizes in colloids to much smaller values than those typically addressed in previous studies (Table S1). Finally, we extend the applicability of the technique to diverse Au@SiO₂ core–shell nano-heaters, with core of only 15 nm, which revealed a different regime of HBM as compared to bare Au NPs. The excellent agreement with theoretical results for these NPs also allows accurate prediction of the temperature in the vicinity of each nano-heater, which is fundamental for diverse biological applications.
# Methods

## Theoretical Background

In thermodynamic equilibrium conditions, i.e, for NPs at the same temperature $T_0$ than the solvent, the translational diffusion coefficient of spherical NPs is given by the well-known Stokes-Einstein relation:²⁷

$$D = \frac{k_B T_0}{6\pi\eta(T_0)R_H}, \qquad (1)$$

where $\eta(T_0)$ is the viscosity of the solvent, $k_B$ is the Boltzmann constant and $R_H$ is the hydrodynamic radius. $R_H$ corresponds to a slightly increased value of the physical radius $R$, due to the molecules surrounding the particles in solution. In most standard DLS studies, experiments are conducted at a fixed temperature $T_0$ and the determination of $D$ is meant to access the value of $R_H$. Alternatively, if $R_H$ is known, it is possible to determine $T_0$, by calculating the ratio $T_0/\eta(T_0)$ and using the temperature dependence of $\eta(T_0)$ of the solvent, which is usually well referenced in the literature.

In the case of heated NPs diffusing in a cooler solvent at a temperature $T_0$, the previous approach is not valid any more as the system is in a non-equilibrium regime. The hot NPs dissipate their heat into the solvent, creating a radially symmetric gradient of temperature around them, as depicted in Figure 1. As the thermal conductivity of Au is very high, the steady state temperature is reached in a few ps,²⁸ which corresponds to much faster time scale than the motion of the NPs. Consequently, the NPs diffuse along with their hot halo and the temperature in the solvent which writes as:

$$T(r) = \frac{\Delta T \cdot R}{r} + T_0, \quad \text{with} \quad r \ge R \qquad (2)$$

where $r$ is the radial distance from the center of the NP, $R$ correspond to the total radius of the NP and $\Delta T = T_s - T_0$ with $T_s$ the temperature of the NPs at their surface. The temperature profile is schematized in Figure 1 for bare and core-shell nano-heaters.
<figure><img src="image_2.png" /><figcaption>Figure 1: Typical temperature profile for (a) bare Au NPs and (b) Au@SiO₂ core-shell NPs. *T*<sub>s</sub> is the temperature at the surface of the NP and *T*<sub>0</sub> is the solvent equilibrium temperature. In (b) *T*<sub>c</sub> is the temperature of the Au core.</figcaption></figure>

This rise in temperature results in a decrease of the viscosity in the vicinity of the particles which becomes dependent on r (η(T(r)) = η(r)). Theoretical models have been developed to generalize and solve the Einstein,²⁹,³⁰ or the Langevin³¹,³² equations in order to describe the motion of the particles in such a case. In particular, it was shown³¹ that for temperature increases of the NP ΔT < 150K, the so-called Hot Brownian Motion (HBM) of heated NPs could be described by an effective diffusion coefficient *D*<sub>eff</sub> which still obeys Eq. 1, but considering effective temperature and viscosity, that both depend on the temperature elevation ΔT of the NPs. Complimentary, Falasco et al.³³ predicted a continuous transition between two limiting cases of effective temperatures, depending on the volumetric mass ratio ρ<sub>NP</sub>/ρ<sub>s</sub> of the NP and the solvent: (i) the HBM high-density (HBM-HD) regime where ρ<sub>NP</sub>/ρ<sub>s</sub> ≫ 1 and T<sub>eff</sub> ~ T<sub>0</sub> + 5/12ΔT and (ii) the HBM low-density (HBM-LD) regime where ρ<sub>NP</sub>/ρ<sub>s</sub> ≪ 1 and T<sub>eff</sub> ~ T<sub>0</sub> + 0.86ΔT. The few experimental demonstrations of HBM
described in the literature<sup>21,22,24</sup> use Au nano-heaters in water for which $\rho_{NP}/\rho_s = 19.3$ and whose motion is thus characterized by the high-density regime, which has led to the generalization, in the literature, of the term HBM to this regime. Strictly speaking, however, the HBM term applies to all the different effective temperatures, from the high to the low-density regime, where the effective temperature is almost that of the surface of the NPs. To our knowledge, the latter has never been observed.

## Confocal DLS

The motion of the NPs upon heating was studied by confocal DLS which gives access to the translational diffusion coefficient $D$ of the NPs, from which the temperature at their surface will be determined. Similarly to standard DLS, the latter coefficient is derived from the normalized auto-correlation function (ACF) of their Rayleigh scattering $I(t)$, which writes as:<sup>34,35</sup>

$$G(\tau) = \frac{\langle I(t)I(t + \tau) \rangle}{\langle I(t) \rangle^2} = 1 + \beta e^{-t/\tau_c} + \frac{c^2}{\langle N \rangle} \frac{1}{(1 + t/\tau_D)(1 + t/w^2\tau_D)^{1/2}},$$

considering freely diffusing NPs and a 3D Gaussian laser intensity distribution $I(r, z) = I_0 e^{-\frac{2r^2}{\omega_0^2}} e^{-\frac{2z^2}{z_0^2}}$ in the detection volume, where $\omega_0$ and $z_0$ correspond to the in-plane and out-of-plane waists of the confocal (or detection) volume. This equation takes its origin in the dependence of the scattered intensity $I(t)$, on the interferences of the electric fields scattered by the different NPs and also on the number $N(t)$ of NPs in the observation volume. Due to the motion of the NPs, both the local concentration and the total phase vary, giving rise to intensities fluctuating over time with two characteristic times: (i) $\tau_c = 1/2q^2D$, which corresponds to the classical term in standard DLS and is related to the time needed to diffuse within half a wavelength distance, with $q = 4\pi n_m \sin(\theta/2)/\lambda$ the scattering vector, $\theta$ the scattering angle, $n_m$ the refractive index of the solvent and $\lambda$ the laser wavelength, and (ii) $\tau_D = \omega_0^2/4D$, related to the time needed to diffuse across the observation volume.
The term $0 \le \beta \le 1$, is an experimental factor proportional to the number of measured speckles,³⁵ $\langle N \rangle$ is the average number of NPs in the detection volume, $c$ is the signal to noise ratio³⁶ and $w = z_0/\omega_0$ is the shape factor of the detection volume. $\beta$, $c^2/\langle N \rangle$ and the characteristic times are adjustable parameters, while $w = z_0/\omega_0$ is deduced from calibrations measurements (Figures 4-SI and 5-SI). In the present study, $z_0 \gg \omega_0$, such that $w \gg 1$ and the correlation function depends essentially on the in-plane diffusion of the particles as shown in Equation 3:

$$G(\tau) = 1 + \beta e^{-t/\tau_c} + \frac{c^2}{\langle N \rangle} \frac{1}{1 + t/\tau_D} \quad (3)$$

Contrarily to standard DLS where the detection volume is a few mm³ and $1/\langle N \rangle \to 0$, the key element of confocal DLS is a very small detection volume ($\sim 1fL$). $\langle N \rangle$ is typically close to unity or below and the NPs are measured sequentially rather than in an ensemble average. A typical ACF curve is exhibited in Figure 1-SI where $\tau_c$ and $\tau_D$ are denoted. The diffusion coefficient $D$ can be derived from the experimental values of both diffusion times $\tau_c$ and $\tau_D$, obtained from the adjustment of the experimental ACFs with Eq.3. However, it is more reliable to only use the value of $D$ determined from $\tau_D$ as the interference term does not probe a unique NP. In the present study, this provides an additional advantage by allowing us to neglect the temperature dependence of the refractive index, which only affects $\tau_c$.

# Experimental procedures

## Optical setup

The experimental setup is displayed in Figure 2. For the DLS experiments, the colloidal suspensions are illuminated with a tightly focused CW probe laser beam at $\lambda = 671$ nm (CNI, MLR-III), using a $\times 60$ water immersion objective (Olympus, NA = 1.2) and a power
laser of 0.5 mW. The image of the focus point is formed by a $f = 150$ mm lens tube and is spatially filtered by a pinhole whose aperture defines the detection volume.

<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic representation of the optical setup. The ×60 objective lens represents the microscope, used both for the illumination and the detection of the light. The pump and probe laser are represented by green and red lines, respectively. M: Mirror, BS: Beam Splitter, APD: Avalanche Photodiode.</figcaption></figure>

The scattering of the NPs is collected in a back-scattering configuration, using the same objective as for the illumination, and is detected by an avalanche photo-diode (APD) (Perkin Elmer, SPCM-AQR-15). Each arrival of a photon on the APD is output as a TTL pulse which is measured by a National Instrument card (NI-6602) with different integrating times (ranging from 1 µs to 1 ms) to access the temporal fluctuations of the scattered intensity. Each measurement lasts 10 s and is realized 20 to 30 times in a row to obtain reliable statistics. The auto-correlation functions are then calculated by inverse Fourier transform of the spectral density $S(f) = \lim_{T→∞} \frac{1}{T} |I(f)|^2$, where $I(f)$ is the Fourier transform of the signal $I(t)$.

The heating of the NPs is ensured by second CW laser (CNI, MSL-FN) aligned on the first one and at $λ = 532$ nm, which acts as the pump laser since its wavelength has been chosen close to the electrostatic plasmon resonance of the particles to have an efficient absorption
(Figure 6-SI). The pump laser power at the sample was varied from 0 to 45 mW using optical density filters with a step of 0.1. The power was measured at the sample, with a calibrated photodiode power sensor (Thorlabs- S120C). The samples consist in 150 µL of the colloidal suspension deposited in an aluminum cuvette that evacuates the heat easily in order to avoid any global heating of the solution.

# Synthesis of NPs

## Material and reagents

O-[2-(3-Mercaptopropionylamino)ethyl]-O'-methylpolyethylene glycol (Mw 5000, PEG-SH), sodium citrate dihydrate, sodium chloride, tetrachloroauric acid, TEOS and ammonia were purchased from Sigma Aldrich.

**Au NPs:** Gold nanoparticles of nominal values 20, 30, 40 and 50 nm in diameter were purchased from Sigma-Aldrich (Ref. #741965, #741973, #741981, #742007, respectively). TEM analyses were carried out for each sample size and after a statistical analysis, their physical size was determined and resumed in Table 1:

<table><caption>Table 1: Morphological parameters of Au NPs for each nominal value indicated in the reference name. More analysis can be found in Figure 2-SI</caption><thead><tr><th colspan="5">Au NPs diameter</th></tr><tr><th>Reference name</th><th>Au20</th><th>Au30</th><th>Au40</th><th>Au50</th></tr></thead><tbody><tr><td>Measured (nm)</td><td>(20 ± 6)</td><td>(28 ± 5)</td><td>(36 ± 5)</td><td>(47 ± 15)</td></tr></tbody></table>

**Control polystyrene NPs:** Polystyrene nanoparticles of 100 nm in diameter nominal value were purchased from Polysciences (Ref. #00876).
# Synthesis

## Synthesis of Au@SiO₂

**CS15@50:** The synthesis was adapted from Penelas et al.⁷ Au NPs were prepared following a Turkevitch-Frens synthesis. In a 500 mL round bottom flask, water (250 mL) was boiled under vigorous stirring with an olive stir bar (25×45 mm). A tetrachloroauric acid solution (0.10 M, 1.18 mL, 0.12 mmol) was added. Then a solution of citric acid (32.6 mM, 12 mL, 0.39 mmol) was added. The mixture was left under reflux for 30 min. During this time, the solution discolored before a violet solution was obtained. After cooling, the suspension was stored at 4°C overnight. To 40 mL of the above suspension, PEG-SH (10 mg) was added at 20°C. The mixture was sonicated for 30 s, then left for 30 min at room temperature. After centrifugation for 15 min at 41000 g at 8°C, the supernatant was removed. 20 mL of ethanol was added, then the mixture was briefly sonicated. The mixture was centrifuged under the same conditions, then the pellet was taken in 15 mL of ethanol. Water (3.8 mL), aqueous ammonia (30%, 0.30 mL) then TEOS (130 µL of a 10% v/v solution in absolute ethanol) were added. After overnight stirring, the NPs were obtained by centrifugation, then resuspended in water.

**CS20@65:** AuNPs were prepared following a Turkevitch-Frens synthesis, with the addition of sodium chloride to reach larger AuNP sizes.³⁷ In a 250 mL round bottom flask, a tetrachloroauric solution (0.25 mM, 100 mL) containing sodium chloride (29.2 mg, 0.50 mmol) was brought to reflux. An aqueous trisodium citrate solution (5% w/w, 0.70 mL, 0.12 mmol) was added. After 30 minutes at reflux, the purple mixture was cooled to room temperature, then stored at 4°C. To this suspension (40 mL) was added PEG-SH (10 mg). The mixture was briefly sonicated then stirred for 30 min at 20°C. After mild centrifugation at 2150 g for 3 h, the supernatant was removed. Ethanol (40 mL) was added, then the suspension was centrifuged again (2150 g, 3 h). The supernatant was discarded, then the pellet was suspended in absolute ethanol (22.7 mL). Water (5.4 mL) and ammonia (30%, 433 µL) were
added successively under stirring. A solution of TEOS in absolute ethanol (10% v/v, 187 µL) was then added. After overnight stirring, part of the NPs suspension was dialyzed in water (7 mL in 2.5 L of water, membrane with 12k MWCO).

**CSm15@85:** This synthesis was reproduced from Montero-Oleas et al.³⁸ using a 1:5 BrijC10:CTAB ratio.

<table><thead><tr><th colspan="4">Au@SiO<sub>2</sub> NPs diameter</th></tr><tr><th>Reference name</th><th>CS15@50</th><th>CS20@65</th><th>CSm15@85</th></tr></thead><tbody><tr><td>d<sub>c</sub> (nm)</td><td>(15 ± 3)</td><td>(21 ± 2)</td><td>(15 ± 4)</td></tr><tr><td>d<sub>s</sub> (nm)</td><td>(49 ± 6)</td><td>(66 ± 5)</td><td>(85 ± 8)</td></tr><tr><td>Porosity %</td><td>0</td><td>0</td><td>70</td></tr></tbody></table>

Table 2: Morphological parameters of Au@SiO₂ NPs and nominal values indicated in the reference name. Diameters were obtained by SEM and porosity by N₂-Sorption analysis, as detailed in Figure 3-SI.

## Synthesis of control NPs

**Control dense silica NPs:** The synthesis followed the reference Montero-Oleas et al.³⁹

**Control mesoporous silica NPs:** Particles were obtained following reference Guerrero-Florez et al.,⁴⁰ sample k-MSN0.2-SH.

## Results and discussion

### Confocal DLS for probing the photothermal properties of individual NPs

To determine the photothermal properties of colloidal suspensions by confocal DLS, the change of the diffusion coefficient *D* of the particles must be measured and analyzed upon heating of the NPs by the pump excitation laser. The diffusion coefficient *D* is obtained from the diffusion time $\tau_D = \omega_0^2/4D$, determined from the fitting of experimental ACFs according to Eq. 3. As shows Eq. 1, *D* (resp. $\tau_D$) is expected to increase (resp. decrease) with rising
the temperature of the NPs, due to both an increased value of *T* and a decreased value of the solvent viscosity surrounding the particles (see *η*(*T*) behavior in Figure 11-SI).

Figure 3(a) shows various confocal DLS ACFs recorded at 4 different pump laser powers (which governs the NP temperatures) for Au NPs of diameter of *d* = (36 ± 5) nm (named Au40). As expected, the ACFs shift to shorter times (orange arrow in Fig.3(a)) as the pump laser power increases. The decay of the diffusion time τ<sub>D</sub>, as a function of the heating laser power, is evidenced with a fourfold decrease (Figure 3(b)). Control experiments were carried out on diverse non-resonant particles under the same conditions, showing no decay on τ<sub>D</sub> (Figure 7-SI). The same absence of decay was also observed for Au NPs excited at a non-resonant wavelength λ = 671 nm (Figure 8-SI). These results demonstrate that the observed effect is not an artifact due to the presence of the pump laser and confirms the role of the absorbance of the NPs in the observed photothermal effect.

To measure the temperature increases at the vicinity of each particle, it is important
to verify that no collective heating effect occurs. Sample dilution was performed (Fig. 9-
SI), leaving the effect observed in Fig. 3b unchanged, which confirms that the scattering
measurements correspond to independent NPs. This result is corroborated by estimating
the average inter-particle distance: at the investigated concentrations of (3 – 60) × 10¹⁰
p/mL, the distance between the particles is of the order of 100 R, where the temperature
increase of the solvent is only ~ 1% of that of the NP (Eq.2). Moreover, Baffou et al.⁴¹,⁴²
introduced a dimensionless factor to evaluate the significance of photothermal collective
effects which considers the mean nearest-neighbor interparticle distance *p*, the NP radius *R*,
and the characteristic size the illumination, here *ω*<sub>H</sub>. Considering the in-plane configuration
of our setup, we calculated the 2D dimensionless factor ξ = p² / (3Rω<sub>H</sub>), obtaining for all particles
considered in this study, ξ ≫ 1, which confirms that the photothermal effects of individual
nanoparticles are dominant.
<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Confocal DLS ACFs for Au40 NPs for 4 different pump laser powers, showing a decrease in the diffusion time $\tau_D$, denoted by the orange arrow. The shorter time decay of the ACFs corresponds to the interference term in Eq. 3. (b) $\tau_D$ evolution as a function of the pump laser power, exhibiting a decrease up to a factor 4. Dash line is a line guide, not a fit. Inset: SEM of Au40 NPs</figcaption></figure>

Finally, it should be noted that the aperture of the pinhole selects the size of the detection volume.⁴³ The latter can thus be adjusted to adapt the confocal volume size to the sample concentration, so as to keep a mean number of particles under observation around 1. In the case of the present setup, the pinhole selection also modifies the photothermal measurement since the intensity of the pump-laser is not spatially uniform. As it is shown in Figure 10-SI, the smaller the size of the detection volume, the more the measured NPs will be those that scatter near the center of the focus point, where the pump-laser is the most intense. Thus,
the average irradiance to which the detected NPs are exposed was defined in Eq. SI-1 as a function of the pinhole size and will be hereafter regarded as the irradiance incident on each NP.

## Temperature determination of Au NPs

Quantitative analyses, to determine the temperature at the surface $T_s$ of the plasmonic NPs from the measured ACFs, were first performed on a colloidal suspension of small spherical gold NPs (Au20) in water. The experimental diffusion times, measured at different irradiances, were normalized as shown in Figure 10-SI(b). According to the definition of $\tau_D = \omega_0^2/4D$, its normalization by its value $\tau_{D0}$ in the absence of pumping ($I = 0$) writes as:

$$ \frac{\tau_D}{\tau_{D0}} = \frac{D(T_0)}{D(T)} = \frac{T_0 \eta(T)}{\eta_0 T}, \qquad (4) $$

where $\eta_0 = \eta(T_0)$ and $T_0$ are the viscosity and the temperature of the solvent respectively, in the absence of pumping ($I = 0$).

Considering the HBM theory, $T$ and $\eta(T)$ refer to effective temperature and viscosity respectively, under heating of the NPs (see Theoretical Background section). In particular, in the case of the high density HBM expected for the Au NPs, the full analytical expression of $T$ and $\eta(T)$ as a function of the NPs temperature elevation $\Delta T$ is:²⁰

$$ \left\{ \begin{array}{l} T = T_{eff} \sim T_0 + \frac{\Delta T}{2} - \left[1 - \ln\left(\frac{\eta_0}{0.029}\right)\right] \frac{\Delta T^2}{24T_0} \\ \sim T_0 + \frac{5}{12}\Delta T, \\ \\ \eta(T) = \eta(T_{eff}) \sim \eta_0 \left[1 + \frac{193}{486} \ln\left(\frac{\eta_0}{0.029}\right) \left(\frac{\Delta T}{T_0 - 152}\right) \right. \\ \left. - \left[\frac{56}{243} \ln\left(\frac{\eta_0}{0.029}\right) - \frac{12563}{118098} \ln^2\left(\frac{\eta_0}{0.029}\right)\right] \left(\frac{\Delta T}{T_0 - 152}\right)^2 \right]^{-1} \end{array} \right. \qquad (5) $$
The ratio $\frac{\tau_D}{\tau_{D0}}$ was calculated according to Eqs. 4 and 5 and is plotted in Fig.4(a), as a function of $\Delta T$. As a comparison, the ratio was also calculated and plotted, considering a classical isothermal case (iso-T), i.e using the standard Stokes-Einstein relationship and the phenomenological expression of the viscosity of water as a function of temperature³¹ (Table SI-2). It is possible to clearly observe that the expected temperatures in each regime differ considerably. For instance, an experimental ratio of $\frac{\tau_D}{\tau_{D0}} = 0.4$, represented by a gray line in Fig.4(a), leads to $\Delta T$ differences of ~40% from one model to another.

The experimental temperature increases $\Delta T$ were obtained from the experimental value of $\frac{\tau_D}{\tau_{D0}}$ considering both models, and are plotted in Fig.4(b) as a function of the irradiance $I$.

In the present case of a model spherical system, the NPs theoretical temperature changes $\Delta T$ at the surface of the plasmonic NP can also be calculated according to the electromagnetically (EM) classical heat dissipation equations which lead to:

$$ \Delta T = T_s - T_0 = \frac{\sigma_{abs} I}{4\pi\kappa R}, \qquad (6) $$

where $\sigma_{abs} I$ corresponds to absorbed power by the plasmonic NP, being $\sigma_{abs}$ the absorption cross section⁴⁴ defined in Eq.SI-4 and $\kappa$ the thermal conductivity of the solvent. These theoretical values are also reported in Figure 4(b) (blue line), using $R$ from Table 1, except for Au20 where $R=9$ nm was used, and $\kappa = \kappa_w = 0.598$ nW/(nm K). Comparison of the calculated values of $\Delta T$ to the experimentally determined ones shows that considering the HBM-HD theory leads to a correct determination of the particle's temperature, unlike the isothermal model which underestimates the temperatures. Moreover, the experimentally determined temperatures fit with the theoretical ones over a large range of irradiances and thus for significant temperature increases ($\Delta T \sim 150K$). We also note that the agreement between calculated and experimental $\Delta T$ was obtained without any adjustable parameters. These results validate both the experimental approach of confocal DLS and the HBM theory for the study of heated Au NPs.
<figure><img src="image_5.png" /><figcaption>Figure 4: (a) $\frac{\tau_D}{\tau_{D0}}$ ratio calculated considering the HBM model in the high density regime (red line) and considering an isothermal solvent (black line). For each experimental value of $\frac{\tau_D}{\tau_{D0}}$, here for instance 0.4, the temperature increase of the NP $\Delta T$ is determined. (b) Experimental $\Delta T$ values as a function of the pump laser irradiance, obtained for Au20, for the two regimes considered in (a). The EM theoretical surface temperatures of the particles (blue line) are calculated from Eq.6 and clearly validate the HBM-HD model.</figcaption></figure>

Analogous experiments and data analyses were conducted on larger AuNPs specified in Table 1. The obtained results are shown in Figs. 5(a)-(c). As for the smaller NPs, analyzing the data with the HBM-HD model allows access to the correct temperatures of the particles. However, above a critical irradiance value $I_c$, the temperatures determined from the experiments start to significantly deviate from the theoretical values. Moreover,
the value of $I_c$ diminishes as the diameter of the AuNP increases.

<figure><img src="image_6.png" /><figcaption>Figure 5: Comparison of theoretical temperature increase (blue line) and the experimentally determined ones (red dots) using the HBM-HD model, for (a) Au30, (b) Au40 and (c) Au50 NPs, as a function of the pump-laser irradiance. Vertical dotted lines indicate the limit irradiance $I_c$ above which the experimental temperature deviates from the theoretical temperatures. (d) Calculated ratio of the optical potential versus the thermal energy of the NP, plotted as a function of the laser irradiance. The experimental values of $I_c$ obtained for the NPs of different sizes correspond to the same ratio of ≈ 0.9.</figcaption></figure>

These observations are attributed to the existence of the gradient optical force $F_G$,⁴⁵ originating from the presence of polarizable spheres in the inhomogeneous electric field of the pump laser. This gradient force has an amplitude proportional to the dipolar moment of the NP (scaling in $R^3$) and to the gradient of the incident irradiance. It submits the NPs to a restoring force that pulls them to the center of the confocal volume, thereby delaying
their escape from the detection volume. At low pump-laser power, the optical force is too small to modify the free diffusion of the heated particles and the experimentally determined temperatures stick to the theoretical ones. At higher incident power (above $I_c$), a so-called weak regime appears, where the NPs can still rapidly escape from the optical potential due to their thermal energy, but as a consequence of $F_G$, remain in the confocal volume for an increased time. If this apparent slowing down is interpreted in terms of temperatures, this leads to artefactual lower effective temperatures, explaining the differences with the theoretical temperatures. A more quantitative analysis of the role of the optical gradient force in the observed behaviors was done by comparing the optical potential energy, from which $F_G$ derives, to the thermal energy of the particles. The optical potential writes as:

$$U_{opt} = - \int \vec{F}_G \cdot d^3\vec{r} = \frac{\epsilon_m Re(\alpha) I_p(r, z)}{2c},$$

where $Re(\alpha)$ is the real part of the polarizability of the NPs as defined in Eqn. SI-2, $\epsilon_m$ is the dielectric constant of the solvent, $c$ is the light speed in vacuum and $I_p(r, z)$ is the pump laser intensity distribution (Gaussian beam). Figure 5d shows, for different sizes of Au NPs, the amplitude of the optical potential $U_0 = \frac{\epsilon_m Re(\alpha) I_0}{2c}$, where $I_0$ is the amplitude of $I_p(r, z)$, compared to their thermal energy $U_{th} = k_B(T_0 + \Delta T)$ by plotting the ratio $U_0/U_{th}$ as a function of the irradiance of the heating laser. The experimental values of $I_c$, obtained for the three different studied AuNPs, were also plotted. From Fig. 5(d), it is possible to observe that for the three sizes of Au NPs, the $I_c$ values correspond to the same ratio value $\frac{U_0}{U_{th}} \approx 0.9$, a value which signs the beginning of the weak regime of optical trapping. Besides, as experimentally observed in Figure 4(b), NPs with smaller diameters (such as Au20) are not affected by the optical gradient force within the studied irradiance range. Finally, we highlight that owing to an optical trapping strengthened for larger NPs, their free diffusion is limited to temperature increases that become smaller as the size of the NPs increases. As is shown in Figure 5(a)-(c), the temperatures above which $F_G$ alters the free HBM-HD of the
particles are $\Delta T_{max} \sim 120$ K, $\Delta T_{max} \sim 85$ K and $\Delta T_{max} \sim 50$ K for Au30, Au40 and Au50 respectively. Extrapolating these values indicates that, for particles with diameters $d \gtrsim 65$ nm, optical forces become relevant even at very small temperature rises ($\lesssim 15$ K), which confocal DLS cannot reliably detect due to the experimental inaccuracies. Although beyond the scope of this work, such restrictions imposed by optical trapping can, in principle, be overcome by solving the Generalized Langevin Equation in the presence of a restoring force³² and by determining the corresponding effective temperatures.³³

## Temperature determination of core-shell Au@SiO₂ NPs

After validating the experimental approach on plasmonic NPs of bare Au, silicate-coated gold NPs (Au@SiO₂) were studied. These NPs are of particular interest for diverse applications, particularly those where the shell is composed of mesoporous silica since pores can act as nanoreactors⁴⁶ or as hosts for proteins³⁹ or drugs.⁴⁷ Being able to pre-determine the expected temperature at the surface of the shell is of particular interest for example, to ensure a proper release of proteins without being denatured.

The first analyzed particles were synthesized in the lab following the protocol described in the Experimental Procedures Section, obtaining CS15@50 NPs with a Au core of 15 nm in diameter and an overall diameter $d_s = 50$ nm. More statistical information is detailed in Table 2 and Figure 3-SI. The experimental approach remains the same as previously described, while the calculation of the increase of temperature at the surface of the NPs writes as:

$$ \Delta T = T_s - T_0 = \frac{\sigma_{abs}I}{4\pi\kappa R_s}, \qquad (7) $$

where $R_s = d_s/2$ is the overall radius of the NP. The absorption cross-section $\sigma_{abs}$ was calculated using the Equation SI-3 of the polarizability of core-shell NPs.⁴⁴

Figure 6(a) shows the evolution of $\tau_D$ obtained for core-shell particles in an aqueous
solvent, where a fivefold decrease is observed after fitting each ACF according to Eq. 3. As in the previous section, the theoretical EM surface temperature elevations $\Delta T$ were confronted to the experimental ones, considering the HBM model in the high density regime. Indeed, with a mass density of 2.2 g/cm$^3$ for the silicate shell, $\rho_{NP}/\rho_s = 2.66 > 1$, in the case of the present CS15@50 NPs. However, as can be seen in Figure 6(b), using this model leads to experimentally determined temperatures very different than the theoretical ones. Moreover, considering the inadequate classical Stokes-Einstein relation with a solvent at the same temperature $T = T_s$ than the surface of the NPs, provides a better agreement with the expected temperature values (Fig. 6(b)). These observations may appear unexpected, but it should be recalled that HBM low density regime was predicted theoretically,$^{33}$ for which the effective temperature are very close to that of the surface of the NPs. Fig. 6(b) also exhibits the temperatures considering HBM-LD model where $T_{eff} = T_0 + 0.86\Delta T$, confirming a very good agreement.
<figure><img src="image_7.png" /><figcaption>Figure 6: (a) $\tau_D$ decay as a function of the pump laser irradiance, measured on CS15@50 in water. Inset: SEM of the NPs exhibiting in white the Au core and in gray the silica shell. (b) Experimentally determined temperature increase obtained considering a HBM-HD (red dots), HBM-LD (grey rombos) and the isothermal case (black squares). HBM-LD results are very close to the case of an isothermal solvent at the same temperature as that of the surface of the NP. The theoretical EM temperatures obtained by Eq. 7 are represented by the blue solid line.</figcaption></figure>

According to Falasco et al.,³³ a continuous transition from HBM-HD regime to HBM-LD regime occurs by varying the density ratio $\rho_{NP}/\rho_s$. The latter is claimed to play a role in the characteristic times of the vorticity of the solvent and consequently in the motion of the particles. In HBM-HD regime, described by effective temperatures given in Eq. 5, $\rho_{NP}/\rho_s \gg 1$, the vorticity spreads out far from the particle which experiences a diffusion
in a solvent at a temperature averaged over a gradient of decreasing temperatures, thus
it is influenced by cooler regions of the fluid.³³ By contrast, at small ratios ρ<sub>NP</sub>/ρ<sub>s</sub> ≪ 1,
the fluid cannot diffuse heat significantly from the particle surface resulting in an effective
temperature of the solvent around the particle surface of T ~ T<sub>0</sub> + 0.86ΔT, i.e almost that
of the particle. For all the Au NPs analyzed in Figures 4 and 5, ρ<sub>NP</sub>/ρ<sub>s</sub> = 19.3 ≫ 1, a result
that reinforces the good accordance of our data with high density HBM regime as already
observed. For the core-shell particles analyzed in Figure 6, ρ<sub>NP</sub>/ρ<sub>s</sub> = 2.66, which shows that
the HBM of the core-shell NPs tends to the low density regime at significantly higher density
ratio values than those predicted theoretically.

To provide further support to these results, we conducted experiments in various con-
ditions to vary parameters likely to influence the effective temperatures and the motion of
the particles. First, the ratio ρ<sub>NP</sub>/ρ<sub>s</sub> was modified, while keeping water as a solvent, by
considering (i) core-shell particles with a slightly increased Au-core diameter CS20@65 for
which ρ<sub>NP</sub>/ρ<sub>s</sub> = 2.75, and (ii) considering core-shell particles CSm15@85, where *m* indicates
that the shell particle is porous. Here the porosity of the shell is 70% and ρ<sub>shell</sub> = 1.4 g/cm<sup>3</sup>,
such that ρ<sub>NP</sub>/ρ<sub>s</sub> = 1.5. In both cases, shown in Fig. 7(a) and (b), the HBM-HD is far from
the expected theoretical EM values, while HBM-LD regime fits accordingly, being consistent
with the results presented in Figure 6. These results reinforce the fact that the HBM-LD
regime exists at density ratio close to unity.
<figure><img src="image_8.png" /><figcaption>Figure 7: Experimentally determined temperature increases obtained considering a HBM model in the high density regime (red dots) and in the low density regime (gray rombos) (a) for CS20@60 in water and (b) for CS15@85 in water. The theoretical EM temperature obtained by Eq. 7 are represented by the blue line. In (a) the vertical dashed line indicates the critical irradiance above which a weak optical trapping occurs, as detailed for the Au NPs.</figcaption></figure>

The same particles were then analyzed with changing the solvent to modify the viscosity and the thermal conductivity of the media, which are important factors for the motion of the particles (viscosity) and their surface temperature (heat dissipation). Dense CS20@65 NPs were suspended in an Ethanol:Water (80:20 v:v) mixture whose viscosity is only about twice that of pure water at 298 K, as it is shown in Figure 8(c), while its thermal conductivity decreases more significantly to $\kappa_{Et:W} \approx \frac{1}{3}\kappa_W$ (see Figure 11-SI(b)). This reduction in heat dissipation is expected to increase the solvent temperature in the vicinity of the NP. Figure 8(a) exhibits the experimental temperatures obtained for both LD and HD-HBM regimes, re-obtaining an adequate response for the HBM-LD model for the case of core-shell particles. As expected, the temperature increases $\Delta T$ are higher than the ones obtained using water as a solvent 7(a). Remarkably, the system remains within the same LD effective temperatures regime, at that even for temperature increase approaching 200 K.
<figure><img src="image_9.png" /><figcaption>Figure 8: Experimentally determined temperature increases obtained for HBM-HD regime (red circles) and HBM-LD regime (gray rhombos) for (a) CS20@65 in an ethanol:water mixture (80:20 in v:v) and (b) CS15@85 in a glycerol:water mixture (50:50 in wt). The blue line represents the theoretical EM temperature obtained by Eq. 7. In (a) the vertical dashed line indicates the critical irradiance above which a weak optical trapping occurs, as detailed for the Au NPs. (c) Temperature behavior of the dynamic viscosity of the different studied solvents.</figcaption></figure>

By contrast, mesoporous CS NPs (CSm15@85) were suspended in a Glycerol:Water (50:50 v:v) mixture, where the viscosity is strongly enhanced to $\eta_{G:W} \approx 6\eta_W$ at T = 298 K, significantly affecting the drag force and diffusion timescale. Here again, the thermal response of the nanoparticles stays in the same effective temperature regime, i.e. with $T_{eff} \sim T_0 + 0.86\Delta T$ as it is shown in Figure 8(b). The thermal conductivity of the glycerol-water mixture, used in the EM calculations, is $\kappa_{G:W} \approx \frac{2}{3}\kappa_W$ as observed in Figure 11-SI(b). In both cases of Figure 8, the solvent changes keep the density ratio $\rho_{NP}/\rho_s$ in the order of magnitude of unity (3.2 for CS20@65 and 1.3 for CSm15@85). Despite noticeable changes in viscosity and thermal conductivity, the diffusion of the NPs remains in the HBM-LD regime, indicating that the density ratio governs the transition between regimes. Consistently, all investigated core-shell NPs with a low density ratio showed experimental temper-
ature increase $\Delta T$ in agreement with theoretical predictions of the HBM-LD regime, while the HBM-HD systematically yielded incorrect, higher values. Interestingly, the HBM-LD regime was observed at $\rho_{NP}/\rho_m \approx 1$, i.e at higher values than those theoretically predicted, a result supported by the robustness and reproducibility of the experiments. This difference from theory could maybe stem from the assumption of a temperature-independent solvent viscosity in the model.

## Conclusion

Photothermal properties of Au and Au@SiO₂ core–shell NPs were investigated by an unprecedented confocal DLS method. This setup allows the heating of individual NPs as they diffuse through a small observation volume (~ 1 fL), while simultaneously probing their diffusion coefficient with a second laser. From the dependence of the measured diffusion coefficient to the pump laser irradiance, the temperature at the NPs surface was determined. Moreover, owing to the use of tightly focused lasers, confocal DLS permits to have pump irradiance sufficiently high to trigger significant temperature increases (> 10 K) of individual NPs, while collective effects are needed to obtain equivalent temperature increases with collimated, or weakly focused, lasers.

Measurements on bare Au NPs (20–50 nm in diameter) confirmed the HBM-HD regime, which accounts for the solvent temperature gradient generated by the NP heat dissipation. Experimentally determined temperature increases at the surface of the NPs agreed well with theoretical predictions over a wide range of heating powers, without any adjustable parameters. The study was then extended to Au@SiO₂ core–shell NPs, with absorbing cores of only 15 nm in diameter. A change in the diffusion regime was observed, with behavior corresponding to the HBM-LD regime where the solvent temperature probed by the NPs is almost equals to that of the NPs surface. This transition is attributed to the reduced density ratio of core–shell particles as compared to that of the bare Au NPs. Temperature determi-
nation was successfully achieved for different core-shell NPs and for solvents with markedly different viscosity or thermal conductivity, demonstrating the robustness and versatility of the method. Importantly, we highlight that, to our knowledge, this result corresponds to the first observation of a HBM regime change. This finding is significant, since assuming a HBM in the high-density limit for core-shell NPs would lead to temperature estimates off by several tens of percent.

These results establish confocal DLS as a powerful technique for probing the photothermal properties of individual nanoparticles. The method enables accurate temperature measurements for nano-heaters as small as 15 nm—below the size limitations of conventional characterization techniques—and provides reliable temperature increases within the application-relevant range of 15–150 K. Confocal DLS thus opens new perspectives for investigating the temperature of individual nano-heaters, including mesoporous core-shell NPs and non-ideal geometries, paving the way for a variety of temperature-dependent applications such as controlled catalysis and drug release.

## Acknowledgments

M.L.M.R. thanks the University Grenoble Alpes for an invited professor grant. A. M.-O. thanks CONICET and University Grenoble Alpes for a PhD fellowship. The authors are indebted to the NanoAndes Network and La Puya Internationale as well as to CNRS (IRN NanoAndes) for their continuous cooperative support. This work is supported by the French National Research Agency in the framework of the “Investissements d’avenir” program (ANR-15-IDEX-02), by Universidad de Buenos Aires, Argentina (UBACyT 20020190100299BA, and 2002022 0200115BA), by CONICET (PIP 11220210100205), and by ANPCyT PICT 2021 I-A-0045 (without disbursements since December 2023). The TEM facility JEOL NEOARM at CNRS Institut Néel was co-financed by the European Union under the European Regional Development Fund (ERDF, contract no. RA0023813).

Alice Déchelette and Tomás Mautino are specially acknowledged for the ACF analysis
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# Supporting Information Available

## Literature overview

<table><caption>Table SI-1: Selected overview of reported experimental photothermal studies, highlighting in particular the NP architecture and size of metal NPs used, the employed method, and whether the heating was studied at the collective or individual level, and in colloidal suspension or immobilized configurations. Remarkably, no technique has reported the individual study of ~ 15 nm Au nanospheres in colloidal suspension.</caption><thead><tr><th>Ref.</th><th>Method</th><th>NP architecture</th><th>NP Size</th><th>Collective or Individual</th><th>Colloidal or Immobilized</th></tr></thead><tbody><tr><td>7</td><td>IR thermographic analysis</td><td>Au@SiO<sub>2</sub></td><td>D = 15@30 - 70 nm</td><td>Collective</td><td>Colloidal</td></tr><tr><td>8</td><td>Kinetics of organic reactions</td><td>Au nanospheres</td><td>D = 13 nm</td><td>Collective</td><td>Colloidal</td></tr><tr><td>13</td><td>UCNP nanothermometry</td><td>UCNPSiO<sub>2</sub>@Au</td><td>D<sub>Au</sub> = 15 nm<br>D<sub>tot</sub> = 282 nm</td><td>Undefined</td><td>Colloidal</td></tr><tr><td>14</td><td>Anti-Stokes luminescence</td><td>Au nanorods</td><td>D = (25 × 50) nm</td><td>Individual</td><td>Immobilized</td></tr><tr><td>15</td><td>Transient Thermal Infrared Emission</td><td>Au nanospheres</td><td>D = 13 to 89 nm</td><td>Individual</td><td>Immobilized</td></tr><tr><td>16</td><td>Hyperspectral Imaging<br>Anti-Stokes Thermometry</td><td>Au nanospheres</td><td>D = 48 to 103 nm</td><td>Individual</td><td>Immobilized</td></tr><tr><td>18</td><td>Wavefront Microscopy - Phase Analysis</td><td>Au nanodisks</td><td>D = 300 nm<br>× h = 40 nm</td><td>Individual</td><td>Immobilized</td></tr><tr><td>19</td><td>Hyperspectral Imaging<br>Anti-Stokes Thermometry</td><td>Au and Au@Pd nanospheres</td><td>D = 67nm<br>D = 67@2,67@4 nm</td><td>Individual</td><td>Immobilized</td></tr><tr><td>22</td><td>Photothermal Correlation Spectroscopy</td><td>Au nanospheres</td><td>D = 40, 60 nm</td><td>Individual</td><td>Colloidal</td></tr><tr><td>48</td><td>Polymerization reaction</td><td>Au nanodisks</td><td>D = 23 nm<br>× h = 50 nm</td><td>Collective</td><td>Immobilized</td></tr><tr><td>49</td><td>Dynamic light scattering</td><td>Au@SiO<sub>2</sub>@PNIPAM</td><td>D = 150 nm</td><td>Collective</td><td>Colloidal</td></tr><tr><td>50</td><td>Melting of lipid bilayers<br>Optical trapping</td><td>Au nanospheres</td><td>D = 80 to 200 nm</td><td>Individual</td><td>Immobilized</td></tr><tr><td>51</td><td>Optical trapping</td><td>Au nanorods</td><td>D = (60 × 25) nm</td><td>Individual</td><td>Immobilized</td></tr></tbody></table>
Confocal DLS autocorrelation function

<figure><img src="image_10.png" /><figcaption>Figure 1-SI: Standard autocorrelation function (ACF) for confocal DLS technique for polystyrene NPs of 100 nm. Arrows indicate the regions where it is possible to observe the coherence time $τ_C$ and the diffusion time $τ_D$.</figcaption></figure>

Au NPs morphology

TEM images were recorded using a JEOL NEOARM microscope operating at 200 kV and equipped with a high dynamic and high resolution Gatan Oneview camera (CMOS, 4k × 4k). Before imaging, 3 µL of a the NPs suspension were dried on a carbon-coated copper grid wafer.
<figure><img src="image_11.png" /><figcaption>Figure 2-SI: TEM images of the different studied Au NPs and histogram of the size distribution obtained from the analysis of several images for each sample.</figcaption></figure>

CSAu@SiO₂ NPs morphology

SEM images were taken using a Zeiss Gemini ULTRA plus electron microscope operating at 20.0 kV at a working distance of 8 mm. 3 µL of the CSAu@SiO₂ NPs suspension were dried on a piece of doped silicon wafer before observation.
## Calibration of the in-plane waist $\omega_0$ of the confocal volume, for different pinhole apertures

The calibration was realized by measuring DLS ACFs of polystyrene NPs of diameter $d = 100$ nm, at a known room temperature $T_0$, for the different pinhole sizes. The value of $\omega_0$ is obtained from the diffusion time $\tau_D$ using the relationship $\omega_0 = \sqrt{4k_B T_0 \tau_D / 3\pi\eta(T_0)d_H}$

<figure><img src="image_12.png" /><figcaption>Figure 4-SI: $\omega_0$ values for different pinhole apertures. Dotted line serves as reference, does not correspond to a model.</figcaption></figure>

## Calibration of the out-of plane waist $z_0$ of the confocal volume

In order to determine the value of $z_0$, a drop of water is used as a sample and its Raman signal is recorded as the focus point is scanned along the propagation direction of the laser (z-axis). The obtained signal is given by the cumulative distribution function:

$$I(z) = \frac{I_{max}}{2} \left( 1 + \text{erf} \left( \frac{(z - z_c)\sqrt{2}}{z_0} \right) \right) + I_B$$

where $I_B$ is the background signal, and $z_c$ is an arbitrary position of the center of the focus point. Figure 5-SI shows the Raman intensities measured with a pinhole of 50 µm pinhole.
The best fit of $I(t)$ was obtained for $z_0=3.8$ µm

<figure><img src="image_13.png" /><figcaption>Figure 5-SI: Raman intensity of the solvent measured as a function of the position of the focus point along the z-direction (dot) and fitting with $z_0=3.8$ µm (line).</figcaption></figure>

The shape factor is thus $w = \frac{z_0}{\omega_0} \approx 13 \gg 1$. Its values stays much larger than one for all the different pinhole sizes.

## Calibration of the pump-laser volume waist $\omega_H$

The determination of $\omega_H$ was done in the same way as for $\omega_0$ without any pinhole and using Au40 NPs. Its value was found to be $\omega_H = 575$ nm.
## UV-Vis Spectra

<figure><img src="image_14.png" /><figcaption>Figure 6-SI: Normalized extinction spectrum for all studied particles. (a) bare Au and (b) Au@SiO₂ NPs in water. For all cases, the resonance wavelength is λ<sub>LSPR</sub> ≈ (525 – 533) nm, which are close to the pump laser wavelength λ = 532 nm.</figcaption></figure>

## Blanks

The confocal DLS setup was tested on 3 different non-resonant particles (blanks), mesoporous SiO₂ NPs, dense SiO₂ NPs synthesized by Stöber method and commercial polystyrene particles (all three types of particles are of diameter 100 – 120 nm). It is possible to observe in Figure 7-SI that τ<sub>D</sub> is constant within the error bars and particularly compared to the decay observed for Au NPs and Au@SiO₂ NPs.
<figure><img src="image_15.png" /></figure>

Figure 7-SI: τ<sub>D</sub> values for different heating laser powers for non-plasmonic nanoparticles of different materials: polystyrene of nominal 100 nm (blue dots) and SiO<sub>2</sub> (Stöber method) synthesized at the lab of ≈ 100 nm (black dots). Also SiO<sub>2</sub> mesoporous particles of porosity 70% (red dots) were evaluated.

The confocal DLS setup was also tested on AuNP40 excited at a non-resonant wavelength,
namely λ = 671 nm.
<figure><img src="image_16.png" /></figure>

Figure 8-SI: τD values for different heating laser powers for Au40 NP excited (a) near their plasmonic resonance (λ = 532 nm) and (b) far from their plasmonic resonance (λ = 671 nm.)

Effect of dilution

The confocal DLS setup was tested on 2 different concentrations of Au40 NP that led to the
same behaviour of the NP diffusion time.
<figure><img src="image_17.png" /><figcaption>Figure 9-SI: $\tau_D$ evolution as a function of power for Au40 NPs in water for two concentrations. Red dots correspond to a not diluted sample (OD=1), black squares correspond to sample diluted to OD=0.5. Both measurements were done with pinhole 200µm</figcaption></figure>

## Role of the pinhole

The incident net irradiance $I$ of the pump laser is defined as the ratio between the total incident power transmitted through the waist $\omega_0$ of the detection volume of the probe laser and the transmitted surface. For a pump-laser of Gaussian shape and waist $\omega_H$, and an incident power $P_{inc}$, $I$ writes as Eq. SI-1 :

$$I = \frac{2P_{inc}(1 - e^{\frac{2\omega_0^2}{\omega_H^2}})}{\pi\omega_0^2} \qquad (\text{SI-1})$$
<figure><img src="image_18.png" /><figcaption>Figure 10-SI: (a) $\tau_D$ evolution as a function of incident power for different pinhole apertures $\omega_0 = (25, 50, 100 \text{ and } 200\mu\text{m})$ for Au40 NPs. (b) Normalized $\tau_D$ as a function of net irradiance according to the confocal volume. Inset scheme denotes the confocal volume with $\omega_0$ in-plain waist within the $\omega_H$ pump waist. In all cases, lines are visual references not models.</figcaption></figure>

Figure 10-SIa shows for Au40 NPs the mentioned differences of behavior for the diffusion times measured at different pinhole sizes and plotted as a function of the total incident power $P_{\text{inc}}$. As expected, $\tau_D$ values are larger for larger confocal volumes. However, Fig.10-SIb shows that measurements result independent of the pinhole selection if $\tau_D$ is normalized by $\tau_{D0}$ (value at $P_{\text{inc}} = 0$ which corresponds to no heating)) and $P_{\text{inc}}$ is converted into irradiance $I$ according Eq. SI-1. The required $\omega_0$ values for the different pinholes and $\omega_H = 575$ nm were obtained from calibration measurements (Figure 4-SI).
Polarizability and cross section expressions

Polarizability equations

Au NPs

$$ \alpha = 4\pi R^3 ((\varepsilon_{Au} - \varepsilon_m)/(\varepsilon_{Au} + 2\varepsilon_m)) \qquad (\text{SI-2}) $$

Au@SiO₂ core-shell NPs

$$ \alpha = 4\pi R_s^3 \frac{(\varepsilon_s - \varepsilon_m)(\varepsilon_{Au} + 2\varepsilon_s) + f(\varepsilon_{Au} - \varepsilon_s)(\varepsilon_m + 2\varepsilon_s)}{(\varepsilon_s + 2\varepsilon_m)(\varepsilon_{Au} + 2\varepsilon_s) + f(2\varepsilon_s - 2\varepsilon_m)(\varepsilon_{Au} - \varepsilon_s)} \qquad (\text{SI-3}) $$

where *R* corresponds to the radius of the NP for bare Au NP, while *R*<sub>c</sub> and *R*<sub>s</sub> correspond to the radius of the core and of the total size of the NP for the Au@SiO₂ case, respectively; and *f* = (R<sub>c</sub>/R<sub>s</sub>)³ is the volume filling fraction. ε<sub>Au</sub> is the Au complex dielectric permittivity taken from⁵² and ε<sub>m</sub> and ε<sub>s</sub> correspond the dielectric permittivity of the media where the NP is immersed (solvent) and of the SiO₂ shell, respectively. For the cases where the SiO₂ shell is mesoporous, the effective dielectric permittivity is calculated as ε<sub>s</sub><sup>eff</sup> = ε<sub>m</sub> * p + ε<sub>s</sub> * (1 - p), where *p* indicates the degree of porosity of the shell.

Absorption cross-section equation

$$ \sigma_{abs} \sim \sigma_{ext} = k_0 n_m \mathfrak{J}\mathfrak{m}(\alpha) \qquad (\text{SI-4}) $$

where k₀ = ω/c, n_m = √ε_m is the refractive index of the solvent and α is the corresponding polarizability for the Au (Eq. SI-2) or the Au@SiO₂ NP (Eqn. SI-3), depending on the case of study.
Viscosity and Thermal conductivity parameters

<figure><img src="image_19.png" /><figcaption>Figure 11-SI: (a) viscosity η and (b) thermal conductivity, κ as a function of temperature for the 3 solvents used in this work: water, ethanol:water (80:20), glycerol:water (50:50)</figcaption></figure>

<table><caption>Table SI-2: Values of the viscosity η(T) and thermal conductivity κ for different solvents. Numbers in brackets indicate the reference used for each parameter.</caption><thead><tr><th>Solvent</th><th>Water</th><th>Ethanol:Water (80:20)</th><th>Glycerol:Water (50:50)</th></tr></thead><tbody><tr><td>η(T) (kg/s m)</td><td>0.029 e<sup>496.9/(T-152)</sup> [53]</td><td>1.89<sup>-5</sup> e<sup>772/(T-126.7)</sup> [54,55]</td><td>4.5<sup>-5</sup> e<sup>624.13/(T-165.65)</sup> [56]</td></tr><tr><td>κ (nW/nm K)</td><td>0.598 [57]</td><td>0.253 [55]</td><td>0.44 [58]</td></tr></tbody></table>
TOC Graphic

<figure><img src="image_20.png" /></figure>

Table of Contents figure - Scheme for confocal
DLS for bare Au and CS NPs corresponding to
High Density (HD) and Low Density (LD) Hot
Brownian Motion (HBM) regimes