Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles

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C: Physical Processes in Nanomaterials and Nanostructures

Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles Marco Gandolfi, Aurélien Crut, Fabio Medeghini, Tatjana Stoll, Paolo Maioli, Fabrice Vallee, Francesco Banfi, and Natalia Del Fatti J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01875 • Publication Date (Web): 27 Mar 2018 Downloaded from http://pubs.acs.org on March 28, 2018

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The Journal of Physical Chemistry

Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles Marco Gandolfi1,2,3, Aurélien Crut4,*, Fabio Medeghini4, Tatjana Stoll4, Paolo Maioli4, Fabrice Vallée4, Francesco Banfi1,2 and Natalia Del Fatti4 1

Interdisciplinary Laboratories for Advanced Materials Physics (I-LAMP), Università Cattolica del Sacro Cuore, Brescia I-25121, Italy 2 Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Brescia I-25121, Italy 3 Laboratory of Soft Matter and Biophysics, Department of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Heverlee, Leuven, Belgium 4 FemtoNanoOptics group, Institut Lumière Matière, Université de Lyon, CNRS, Université Claude Bernard Lyon 1, F-69622 Villeurbanne, France * E-mail : [email protected]

Abstract Time-resolved thermoplasmonics is emerging as the go-to technique for nanoscale thermal metrology. In this context, connecting the ultrafast optical response of nano-objects to the correct thermal pathways is of paramount importance. We developed full thermo-optical models relating transient spectroscopy measurements, performed on metal nano-objects in dielectric environments, to the overall system thermal dynamics. The models are applicable to small spherical nanoparticles embedded in a homogeneous matrix, following an analytical approach, and are expanded to include the cases of arbitrarily complex geometries and sizes relying on the finite element method. These approaches are then exploited to rationalize several observations made in the context of previous time-resolved thermo-optical studies at the nanoscale. The present tools open the path for accurate retrieval of thermal parameters – notably the Kapitza resistance and the local environment thermal conductivity – from experiments. They also allow identifying the optimal parameters for selectively probing thermal dynamics of either a nano-object or its nanoscale environment.

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I. Introduction Nano-objects and nanomaterials show peculiar physical (electronic, optical, acoustic, thermal, magnetic…) properties which may considerably differ from their bulk counterpart and bear great relevance both from a fundamental and applicative standpoint. The interplay of these properties, which have been separately investigated in the past,1–7 largely remains to be understood. The present work addresses the thermo-optical response of nano-objects as arising in pump-probe optical experiments. The experimental situation involves two physics, i.e. thermal dynamics and optics. On the thermal side, light absorption by a nanoparticle (NP) induces its heating and triggers heat fluxes throughout the NP and its environment, an effect exploited in many applications among which photothermal therapy8,9 and photoacoustic imaging.10–12 On the optical side, the response of a NP is modified by heating, as it depends on the NP and environment dielectric properties which are all a function of temperature. This dependence, which represents the main focus of the present paper, is at the basis of timeresolved thermo-optical spectroscopy enabling the detection of minute thermal fluxes at the nanoscale. In the context of metal NPs, the investigation of thermo-optical coupling mechanisms is commonly referred to as thermoplasmonics, in reference to the surface plasmon resonances (SPRs) which dominate their optical response. SPRs stem from the collective electronic oscillations that are resonantly excited by an external electromagnetic field. In the spectral range of SPRs, both the absorption of a NP and its sensitivity to temperature changes are considerably enhanced. The former effect enables large heat generation from metal NPs with moderate illumination powers. The latter one is beneficial for using NPs for thermal sensing, as it may turn minute temperature changes into large modifications of their optical response.13,14 Thermo-optical coupling effects stand at the core of optical time-resolved spectroscopy, which relies on the illumination of NPs with two time-delayed pump and probe pulses (Fig. 1). In these experiments, a nanostructure is used as both heater (heat being generated in it by pump pulse absorption and subsequently propagating in its environment) and thermometer (exploiting the transient changes of probe pulse transmission/reflection induced by the pump-induced thermal dynamics). This approach enables the measurement of the ultrafast cooling (typically occurring on pico- to nanosecond timescales) of a nanostructure, following its excitation and internal thermalization.4,7,15–17 Time-resolved measurement of NP cooling dynamics started on ensembles of metal nanospheres, either in solution or embedded in a glass matrix, and addressed the effects of NP composition, size, 2 ACS Paragon Plus Environment

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environment and temperature on the thermal dynamics.18–23 These all-optical time-resolved methods are currently among the go-to techniques to characterize interfacial thermal transport at the nanoscale,1,3,24 i.e. determine the thermal (Kapitza) conductance (noted G in the following) at the NP/environment interface. G values were estimated for a number of interfaces, addressing both the effect of NP and environment compositions13,19,22, temperature20,23 and the effect of surfactant molecules at their interface.25–27 However, the analysis of most of the above-mentioned experimental investigations relied on the implicit assumption of a direct proportionality between the transient relative transmission/reflection variations and the metal NPs temperature. This hypothesis is a priori questionable, as subsequent heating of the local environment close to the NP also modifies its optical properties, and may thus significantly contribute to time-resolved signals. We recently demonstrated such a large environment contribution in the context of time-resolved experiments on gold NPs in liquids (water and ethanol).13 The measured transient timeresolved signals displayed a complex temporal response strongly dependent on probe beam wavelength, which could be quantitatively understood only by accounting for the contribution of solvent heating in the modeling. Signals obtained for ethanol solvent (whose refractive index strongly depends on temperature) could not be reproduced assuming their proportionality to NP temperature change only. In water, this approximation led to a 50% overestimation of interface thermal conductance of gold NPs when thermal dynamics was probed in the infrared range. The anomalously low interface conductances reported for metal NPs in toluene19 may also have been the result of toluene heating.14 The possibility of selectively probing the thermal dynamics of the solvent using a specific probe wavelength was even demonstrated. Encompassing the temperature dependence of the environment dielectric constants is thus essential for a reliable determination of thermal parameters at the interface between a metal NP and a liquid. The above discussion calls for comprehensive thermo-optical models allowing for an accurate analysis of time-resolved signals. An analytical model was developed for small spherical NPs in a homogeneous environment13 and constitutes a first step in this direction. However, it cannot address more complex situations involving large NP sizes, arbitrary shapes and/or heterogeneous environment (e.g. substrate-deposited nano-objects immersed in liquid) which have been considered in recent experiments.14,28–32 Numerical approaches are required to model the thermo-optical response of such systems, but have been previously



applied only to detailed modeling of their thermal behavior (optical aspects not being included in these models).14,30 This theoretical paper tackles thermo-optical coupling at nanoscale in two aspects. It first provides a detailed discussion of environment effects on time-resolved signals measured on spheroidal metal nano-objects embedded in a solid or liquid medium. In particular, the optimal conditions for selectively probing NP or environment thermal dynamics are identified. It then presents a numerical model for arbitrarily complex geometries, and its application to a recently investigated experimental situation (substrate-deposited gold nanodisks immersed in liquid14). In particular, the present work rationalizes the claim that nanodisk heating can be selectively addressed (i.e., that time-resolved signals proportional to the nanodisk temperature can be obtained) when the probe wavelength is very close to the SPR one. The small perturbation regime (i.e., initial lattice heating of the order of a few kelvin), experimentally accessed by high sensitivity femtosecond pump-probe spectroscopy, is considered throughout the paper. However, the presented model could be generalized to the high perturbation case by replacing the constant thermal (e.g., heat capacity and conductivity) and thermo-optical (e.g., temperature derivatives of dielectric functions) parameters by temperature-dependent functions. 23,33

II. Theory and Results II.1 Optical detection of NP and environment heating: simplified discussion A simplified situation of interest for describing the optical probing of thermal dynamics is the ideal case of a NP in a homogeneous environment with small and uniform transient temperature rises, ∆Tp(t) and ∆Tm(t) in the NP and its environment, respectively. This case differs from the actual situation encountered in time-resolved experiments and presented in Fig. 1, where an optical pump pulse initially heats the metal NP which subsequently cools via energy transfer to the surrounding medium, resulting into a nonuniform environment heating (an effect taken into account in the more complete models presented below). Nevertheless, discussing this simplified model is instructive to qualitatively appreciate the relative weights of NP and environment contributions, as well as their dependences on the laser probe wavelength. In the following, the environment medium, assumed to be transparent at optical frequencies, is characterized by a real dielectric function εm. Initial heating of a NP modifies 4 ACS Paragon Plus Environment

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its complex dielectric function ε=ε1+iε2 and its volume V, which both affect its optical response. However, for a metal NP, the direct heating-induced volume change (described by the volume expansion coefficient αV = (∆V/V)/∆T≈ 4 10-5 K-1 for gold) is much smaller than the temperature induced perturbation of metal dielectric functions (with for instance the wavelength dependent (∆ε2/ε2)/∆T coefficient typically of the order of 10-3 K-1 for gold, dominated by increase of electronic scattering rate)4,7,17 and will be therefore neglected in the following. In the small perturbation regime, the transient change of the NP extinction crosssection σext (sum of its absorption and scattering ones) optically measured at a probe wavelength λpr then reads to first order:

 ∂σ dε ∂σ ext (λ pr ) dε 2 ∆ σ ext (λ pr , t ) =  ext (λ pr ) 1 +  ∂ε dT p ∂ε 2 dT p 1  = A p (λ pr )∆ T p (t ) + Am (λ pr )∆Tm (t )

   ∆ T p (t ) +  ∂σ ext (λ pr ) dε m  ∂ε  dTm m  

  ∆ Tm (t ) (1) 

The relative contributions of ∆Tp and ∆Tm to ∆σext thus depend on both the temperature dependences of the NP and environment optical properties (dε1,2/dTp and dεm/dTm terms) and the sensitivity of σext to dielectric function changes (∂σext/∂ε1,2,m derivatives). The dielectric function ε of a noble metal can be separated into two complex contributions associated to intraband (Drude-like) and interband (εib=1+χib) optical transitions at frequency ω=2πc/λ:34

=

− ≈ − + +

(2)

where ωp is the plasma frequency of the metal (about 9 eV for gold and silver) and γ the scattering rate at optical frequencies of the conduction electrons in the NP (with γ>|Am| except in two narrow spectral regions on the blue and red sides of the SPR, around the two wavelengths where the metal NP contribution to the signal vanishes. Conversely, |Ap|0.8 occurring in > 50 nm wide spectral windows around the two wavelengths where Ap=0, as reported in Fig. 2i). For gold nanoellipsoids in ethanol, |Ap||Am| occurring over a >5 nm window) that was inexistent for spheres in the same environment. A quantitative analysis of time-resolved experiments requires two additional steps as compared to the previous simplified discussion: (i) the evaluation of the spatio-temporal temperature variations through a thermal model and (ii) the extension of Eq. 1 to the case of a spatially non-uniform ∆Tm distribution. This can be done either analytically, in the case of small spherical NPs in a homogeneous medium, or numerically, for arbitrary NP sizes, shapes and environments, as discussed below.

II.2 Analytical dipolar thermo-optical model for nanospheres Heat dissipation from a nanosphere of radius R is described in spherical coordinates by a set of two equations ruling heat transfer through the NP surface at r = R and heat diffusion in the surrounding medium (r > R):13,20,22,23,49 dT p (t ) dt

cm

=−

3G (Tp (t ) − Tm ( R, t )) R cp

1 ∂2 ∂Tm (r, t ) (rTm (r, t )) = Λm r ∂r 2 ∂t

(5)

(6)

where cp(m) is the particle (medium) heat capacity per unit volume, Λm the thermal conductivity of the surrounding medium, G the interface thermal conductance, and r the distance to the particle centre. Eqs. 5-6 rely on two assumptions: uniform NP temperature Tp and temperature-independent cp(m). These are usually valid for describing time-resolved experiments on metal NPs, which typically involve metals with high thermal conductivity (with Λm=320 and 410 W.m-1.K-1 for gold and silver, respectively, to be compared with the much lower Λm values of the dielectric environments considered in this work listed in Table 1) and small (a few K) initial NP heating. Additionally, while this model assumes a purely diffusive (i.e., described by Fourier’s law) thermal transport in the environment, it can take into account partially ballistic heat transport in the environment as a decrease of G depending on nanosphere diameter.29,31 The NP and the non-uniform environment temperature evolutions, ∆Tp(t) and ∆Tm(r,t) respectively, following an initial particle temperature increase ∆T0 after optical pump excitation of the metal electron gas and subsequent internal thermalization with the lattice, can be computed using Laplace transform.13,20,22,23,49 The non-uniform environment temperature reflects into spatial variations of εm. To include this effect in optical calculations, 10 ACS Paragon Plus Environment

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the NP environment is decomposed into a large number N of thin concentric shells.13,50 Eq. 1 is then generalized into:

∆σ ext (λpr , t ) = Ap (λ pr )∆Tp (t ) + ∑ Am(i ) (λpr )∆Tm(i ) (t ) N

(7)

i =1

with ∆Tm the spatially averaged temperature increase throughout the ith shell (estimated from (i )

(i ) ∆Tm(r,t) obtained before) and Am its associated weight. The latter coefficient can be

computed in the dipolar approximation by multiplying the environment optical properties temperature dependence dεm/dTm (supposed constant throughout the shells) with the numerical derivation of the σext expression for a small spheroid with multilayer shells.51 This provides the extinction cross-section sensitivity to the shell dielectric constants:

Am(i ) (λ pr ) =

∂σ ext (λ pr ) dε m (i ) dTm ∂ε m

(8)

Application of the above model to a 10 nm gold nanosphere in glass and ethanol environments is shown Fig. 3. In these calculations, a ∆T0=1K initial NP heating was considered at t = 0, and thermal interface conductance values measured for gold/ethanol13 and gold/glass22 interfaces were used (Table 1). The computed transient extinction cross-section changes ∆σext are plotted in Fig. 3a and Fig. 3d as a function of time and probe wavelength λpr. Their temporal evolution is also shown in Fig. 3b and Fig. 3e for some relevant specific λpr values, where the nanoparticle (black dashed line) and environment (red dash-dotted line) contribution to the signals (solid grey line) are highlighted. Note that, in the low perturbation regime, these weak cross-section changes induced by a 1K nanoparticle temperature rise after pump excitation, typically correspond to relative probe transmission changes measured on ensembles of nanoparticles of the order of 10-5, detectable by means of high sensitivity femtosecond pump-probe spectroscopy.13,22 The probe wavelength affects both amplitude and temporal profiles of the transient ∆σext signals. To better show the latter effect, normalized transient extinction changes ∆ λ , were computed at each wavelength by dividing

|∆σext (t)| by its maximal absolute value (Fig. 3c and Fig. 3f). The results shown in Fig. 3 follow the trends emerging from the simplified analysis based on uniform environment temperatures presented above. For both considered environments, a clear plasmonic enhancement of ∆σext signals is apparent in the spectral range around SPR. For gold nanospheres in glass, the temporal profile of ∆σext is weakly dependent 11 ACS Paragon Plus Environment


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on λpr. For most λpr values, |∆σext| monotonically decays and reflects NP cooling by propagation of the initially injected heat in its environment, the contribution of glass heating to the optical response being negligible (see Fig. 3b signals at λpr=545 nm and λpr=600 nm and Fig. 3c). The influence of glass heating in the kinetics of the optical response is manifest only in narrow λpr windows close to the SPR, such as that shown in the top panel of Fig. 3c (see also the λpr=567 nm signal in Fig. 3b), where the Ap coefficient changes sign and therefore the metal contribution is very weak. Conversely, for nanospheres in ethanol, the temporal profile of ∆σext becomes strongly wavelength−dependent and always differs from that associated to NP cooling only (Fig. 3e-f). For specific wavelengths where |Am|>> |Ap| (such as λpr=550 nm, Fig. 3e), the temporal profile of |∆σext(t) | reflects the thermal dynamics of a thin liquid layer around the NP, its optical response being sensitive to environment refractive index changes on a length scale of the order of the nanoparticle radius.34 It therefore present a rise and then a decrease of the signal, associated to heating (by heat propagation through the NP/ethanol interface) and cooling (by heat diffusion in ethanol) of the liquid on a time scale ruled by its thermal diffusivity, of the order of a few hundred picoseconds. The amplitude of this environment contribution depending on the absolute temperature increase within this interfacial layer (Eq. 7), this effect increasingly influences the total transient signal for higher thermal interface conductances and lower environment thermal conductivities. A similar analysis can be performed for silver NPs, with however more uncertainties due to the large dispersion of silver dielectric function measurements and their temperature dependencies.52–55 The optical and thermoplasmonic properties of a 10 nm diameter silver nanosphere in glass are shown in Fig. 4. Its SPR being far from interband transitions (occurring below ≈ 320 nm in silver), it has a quasi-Lorentzian shape (with λR ≈ 410 nm, Fig. 4a) and its plasmonic derivatives ∂σext/∂ε1,2,m (Fig. 4b) are therefore very similar to those of gold nanoellipsoids (Fig. 2k). Also, the silver temperature derivatives dε1,2/dTp are dominated by modifications of the Drude terms in Eq. 2. These were theoretically estimated by taking into

account

the

modification

of

plasma

frequency

due

to

dilation,

yielding

dε1 / dTP = αV ωP2 / ω 2 with αV ≈ 4.10-5 K-1 for small silver nanoparticles, 56 and the increase of electron-phonon scattering rate due to lattice temperature increase, yielding dε 2 / dTP = αγ ε 2 with αγ = (dγ e− ph / dTP ) / γ 0 ≈ 3 ⋅ 10−3 K −1 and ε 2 tabulated for bulk silver

4,7,17,54

(Fig. 4e).

Conversely to a liquid environment, the thermoplasmonic coefficient associated to the metal 12 ACS Paragon Plus Environment

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is much stronger than the glass one (|Ap|>> |Am|, see Fig. 4c). As a consequence, for most λpr values, the dynamics of the optical transient signals induced by metal heating reflects the cooling of the silver nanoparticle, at the exception of two spectral windows located on both sides of the SPR (Fig. 4d). In this spectral range, the metal contribution is reduced and changes sign, the measured optical dynamics being therefore affected by the thermal response of glass, as for λpr ≈ 420 nm (Fig. 4f). II.3 Numerical model for arbitrary NP morphology and environment In order to extend the previous thermo-optical analysis, limited to dipolar approximation and spherical geometry, and encompass arbitrary NP size, shape and environments (including inhomogeneous ones), a numerical model was developed using a finite-element method (FEM) commercial software. This extension is particularly relevant to address for instance the impulsive thermal dynamics involving nano-objects patterned on a solid substrate. Such designs have been recently adopted as model systems to investigate several aspects of nanoscale heat transport ranging from nanosecond decoupling of electron and phonon temperatures57 to non-Fourier heat transport effects.29,31 In view of thermal nanometrology applications involving characterization of solid material thermal properties, surface nanopatterning, as opposed to spherical nanoparticles embedded within the solid volume, allows greater geometry flexibility and enhanced control of the solid-solid interface between the nano-object and the environment, a prerequisite for a thorough understanding of the Kapitza resistance.58 The numerical thermo-optical model computes the out-of-equilibrium optical response of metal NPs in two steps, corresponding to thermal and optical calculations. Thermal calculations, i.e. determination of T(r,t) following sudden NP temperature modification from T0 to T0+∆T0, are based on the numerical resolution of Fourier law in the NP and in its environment, with Kapitza resistances incorporated at the interface between the NP and its environment. The thermal domain was chosen large enough so that the temperature perturbation does not reach the boundaries in the time interval of interest. Then on the external boundaries the temperature is kept fixed at the environment temperature T0. These calculations can be numerically performed in 2D geometry for systems presenting revolution symmetry, such as nanospheres and nanodisks, considerably reducing memory and computational time requirements as compared to 3D systems.



Optical calculations are performed as a function of time, determining at each time t the spatial modifications of dielectric functions defined locally as ε(r,t)=ε(T0)+(dε/dT) (T(r,t)T0), and then computing the out-of-equilibrium optical response as a function of probe wavelength λpr. The optical simulation domain is enclosed by a spherical perfectly matched layer to avoid spurious reflections at its border. Calculations are performed in the scattered field formulation, i.e. defining an incident electromagnetic wave (here a plane wave) and calculating the scattered one. The absorption cross-section σabs is deduced from the computed electromagnetic field by integration over the NP volume of the Joule dissipated power density, while the scattering one σscat is obtained by integration of the scattered intensity over a closed surface surrounding the NP. ∆σext(λpr, t) is obtained as the difference between the out-of-equilibrium σext values obtained from calculations with heating-modified dielectric functions and those corresponding to the equilibrium T0 temperature. Details concerning this numerical model are discussed in section S1 of the Supporting Information. The validity of the developed FEM model was checked by comparing its results with those of the analytical dipolar model presented above (section S2 of Supporting Information). Fig. 5 presents FEM computations on a complex geometry corresponding to relevant experimental situations: gold nanodisks (120 nm diameter and 20 nm height) deposited on a substrate (fused silica) and immersed in a liquid environment (ethanol).14 This configuration requires going beyond the analytical model presented above (i.e. nano-object with small size, spherical shape and homogenous environment), calling for a numerical modeling approach. Optical calculations performed for nanodisk illumination by a plane light wave propagating orthogonally to the substrate surface lead to an extinction spectrum with a SPR peak centered at 790 nm wavelength (Fig. 5a). Results of the complete thermo-optical calculations following an initial heating of the nanodisk by ∆T0=1K yield the transient extinction changes ∆σext(λpr, t) as a function of the probe wavelength close to the SPR (Fig. 5b). To highlight the temporal dynamics, these are also plotted with normalized amplitudes (Fig. 5c) and at specific wavelengths of interest (Fig. 5d).

Similarly to the case of gold NPs in ethanol, both the amplitudes and temporal profiles of ∆σext(t) strongly depend on probe wavelength. Numerical simulations demonstrate that specific probing of thermal dynamics is possible for both the environment (when probing at λpr=735 nm) and the gold nanodisk (at λpr=790 nm), as shown in Fig. 5d. In order to better


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understand the influence of the inhomogeneous environment on the out-of-equilibrium optical response, and separate the sensitivity of σext to ethanol or silica heating, two distinct thermoplasmonic coefficients (playing the role of Am for homogeneous environments) can also be defined and numerically computed by FEM for the investigated situation, i.e.

Aethanol =

∂σ ext (λ pr ) dε ethanol and Asilica = ∂σ ext (λ pr ) dε silica (Fig. 5e). As expected, σext is ∂ε ethanol dTethanol ∂ε silica dTsilica

much more sensitive to temperature changes in ethanol than in silica. With this respect, the relative amplitudes and λpr dependences of Ap and Aethanol show large similarities with those obtained in the case of small gold nanoellipsoids in ethanol treated in Fig. 2j-m. In agreement with complete thermo-optical calculations, cancelations of either of these coefficients, ruling the metal (Ap) and environment (Aethanol) contributions respectively, occur on the two flanks (at 735 and 835 nm wavelengths) and close to the center (790 nm wavelength) of the SPR.

III. Discussion As illustrated above in several situations, the temporal response of out-of-equilibrium metal NPs in a dielectric environment measured by time-resolved spectroscopy generally reflects a combination of both the thermal dynamics of the NPs and of their environment. The absolute and relative weights of their contributions depend on several parameters, which can be classified as optical, thermal and thermo-optical ones. - Optical parameters are summarized by the coefficients ∂σext/∂ε1,2,m quantifying the sensitivity of the optical response (e.g., its extinction cross-section, the relevant quantity in transmission pump-probe spectroscopy) of a NP to changes of its dielectric function and of that of its environment. Due to plasmonic effects, these λpr-dependent coefficients all are enhanced when λpr is chosen in the SPR domain. Their different spectral variations in this range induce large variations of NP and environment relative weights in time-resolved signals (Figs. 2, 4). - Thermal parameters are those appearing in Eqs. 5 and 6, i.e. heat capacity, thermal conductivity and interface thermal conductance. They control the efficiency with which the heat generated in the NP propagates in its proximal environment and diffuses further away from it, where its influence on the NP optical response becomes negligible. - Thermo-optical parameters are those allowing NP and environment heating to leave an optical signature, i.e. the temperature-dependence of dielectric functions dε1,2,m/dT and the 15 ACS Paragon Plus Environment


resulting Ap,m coefficients. They constitute a major source of variability between experimental configurations, both because dε1,2/dT display large spectral variations for noble metals depending on the considered spectral range (within or far from interband transitions, Fig. 2a and 4e) and because |dεm/dT| greatly varies from an environment to another, with values of the order of 10-3, 10-4 and 10-5 K-1 for ethanol, water and glass, respectively. As a result, assuming that time-resolved signals solely reflect the evolution of NP temperature is often valid for crystalline or amorphous solid environments at the exception of some specific wavelengths (Figs. 2e, 3b, 4d,f), but almost never valid for liquid ones (Figs. 2i,m, 4d, f, 5d). These effects are all included in the complete analytical and numerical thermo-optical models presented in this work, which thus have the potential for a quantitative analysis of experimental time-resolved signals on metal NPs, and a reliable extraction of unknown thermal parameters (provided their number is not too large) such as thermal interface resistances. Moreover, a major outcome of these models is to clarify whether and how the temperature evolutions in the NP, or in their environment, can be selectively probed. For NPs in glassy dielectric environments, |dεm/dT| is typically much smaller than |dε1,2/dT|. In this case, specific probing of NP temperature evolution is approximately achieved for most λpr values, except around those where Ap vanishes (Figs. 2d,e and 4c,d). For these specific λpr ranges, the dynamics of the out-of-equilibrium optical response is affected by the thermal responses of both the metal and the close environment of the NPs. In the case of silver NPs in glass, previous experimental reports have demonstrated specific probing of NP heating in particular when probing in the infrared range (inset of Fig. 4c). A totally different temporal dynamics was obtained for λpr on the red side of SPR, which indicated a sensitivity to the NP environment, in excellent agreement with the computed vanishing of Ap in this spectral region (Fig. 4f).22 For liquid environments, for which |dεm/dT| is of the order of |dε1,2/dT|, a strong probe wavelength dependence of the amplitude and temporal profiles of time-resolved signals is predicted. In this case, the thermal response of the liquid environment has a strong impact on the optical signal, in addition to the metal one (Figs. 3e and 5d). The spectral windows for specific probing of environment temperature changes become broader (Figs. 2i,m), as already demonstrated for gold NPs in ethanol.13 Conversely, the possibility of specifically probing NP cooling does not always exist, and depends on whether environment contribution may vanish (i.e. Am=0). This condition is not fulfilled for gold nanospheres in ethanol (Figs. 2i), but occurs for elongated nano-object shapes such as nanoellipsoids (as the prolate case presented 16 ACS Paragon Plus Environment

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in Fig. 2m, the oblate one yielding similar results) and nanodisks (Fig. 5d,e) when λpr is chosen very close (slightly on the blue side) of their SPR. These results thus confirm the validity of the experimental strategy chosen in previous experiments on immersed substratedeposited gold nanorods30 and nanodisks,14 where λpr was fixed at the SPR wavelength to probe nano-object cooling and time-resolved data were confronted to a thermal model to extract interface resistances.

IV. Conclusions In conclusion, we developed comprehensive analytical and numerical thermo-optical models linking the optical signals measured by ultrafast transient spectroscopy on metal nano-objects in dielectric environments to their thermal dynamics. These models show that the usual assumption of proportionality between the time-resolved optical trace and the metal NP temperature change may not hold. Furthermore, they provide a tool allowing to discern at which probe wavelengths the optical signal entails the NP or embedding environment thermal dynamics. These tools have been here exploited to rationalize a large number of observations made in the context of previous time-resolved studies. In perspectives, these models constitute a valuable tool for a reliable extraction of thermal parameters from time-resolved experiments, most notably the Kapitza resistance at the nanoscale and modification of the thermal conductivity of the NP embedding environment in close proximity to the nanometric interface. A correct assessment of these parameters is the prerequisite to enhance heat dissipation in next-generation micro- and nanodevices, low thermal dissipation across nanoscale interfaces being among the main impediments towards further circuits downscaling. Supporting Information. Numerical thermo-optical models: nanospheres in a homogeneous environment and immersed nanodisks on a substrate; benchmarking of the numerical approach.

Acknowledgements F.B. and M.G. acknowledge financial support from the MIUR Futuro in ricerca 2013 Grant in the frame of the ULTRANANO Project (Project No. RBFR13NEA4), and support from Università Cattolica del Sacro Cuore through D.2.2 and D.3.1 grants. M.G acknowledges financial support from Academische Stichting Leuven and FWO. This work was supported by



the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Authors Informations ORCID Marco Gandolfi: 0000-0001-7700-9255 Francesco Banfi: 0000-0002-7465-8417 Natalia Del Fatti: 0000-0002-8074-256X


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Related figures Heat capacity cm (106 J.m-3.K-1) Thermal conductivity Λm (W.m-1.K-1) Interface conductance with gold G (MW.m-2.K-1) Dielectric function εm dεm/dT (K-1)

Glass 2,3,4 1.8

Ethanol 2,3,5 1.92

Fused silica 5 1.63

0.8

0.18

1.3

102

35

65

2.56 1 10-5

1.86 -1.1 10-3

2.1 1.2 10-5

Table 1. Environment Parameters. Thermal interface conductances are taken from refs. 13, 14, 22.



Figure 1. Thermo-optical response of a metal nanoparticle in a dielectric environment in the context of optical pump-probe experiments (schematic representation). Pump pulse absorption by a metal NP leads to its selective heating. NP cooling occurs via heat transfer at the interface and in the surrounding medium. These processes lead to transient modifications of the dielectric function of the NP (∆ε), and of the environment (∆εm), the latter being strongly non-uniform. This results in time-dependent modifications of the NP optical extinction crosssection, and thus of the transmission of the probe beam.


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Figure 2. Thermoplasmonic properties of gold nano-objects. a) Measured temperature dependence of the real dε1/dT (blue line) and imaginary dε2/dT (green dashed line) parts of the dielectric function of gold.39 b-e) calculations for a 10 nm diameter gold nanosphere in glass, using Johnson and Christy dielectric tables modified to include quantum confinement effects: 37,38,45

b) computed extinction cross-section σext using Mie theory c) plasmonic derivatives

∂σext/∂ε1 (blue line), ∂σext/∂ε2 (green dashed line) and ∂σext/∂εm (red dash-dotted line), d)



thermoplasmonic coefficients Ap (black line) and Am (red dash-dotted line) (see eq. 1) and e) their relative weights |Ap|/(|Ap|+|Am|) (black line) and |Am|/(|Ap|+|Am|) (red dash-dotted line). fi) Same information for a 10 nm diameter gold nanosphere in ethanol. j-m) Same information for a prolate gold ellipsoid with 20 nm length and 8 nm diameter in ethanol illuminated with light polarized along its long axis (same trends are obtained for oblate ellipsoids). The central SPR position is indicated by a dashed vertical line in each case (λR= 540, 525 and 640 nm from left to right).


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Figure 3. Dipolar modeling of the thermo-optical dynamics of a small gold nanosphere. The cases of 10 nm diameter gold nanospheres in glass (a-c, same situation as in Fig. 2b-e) and ethanol (d-f, same situation as in Fig. 2f-i) are considered. Their SPR central position is shown by dashed horizontal lines (λR= 540 and 525 nm from left to right). a) Transient extinction change ∆σext(λpr, t) following the initial heating of the sphere (∆T0=1K). b) ∆σext(t) 23 ACS Paragon Plus Environment


dynamics at three specific probe central wavelengths (grey lines). To take into account the experimental spectral content of femtosecond pulses, these transient signals are computed by averaging Eq. 7 over a 10 nm probe spectral width. The specific contributions of nanoparticle and environment heating are indicated as black dashed and red dash-dotted lines, respectively. c) ∆σext(t) from panel (a) divided by its maximum value, for each probe wavelength. A zoom in the 565-568 nm λpr range is shown on top. d-f) Same information for gold nanospheres in ethanol.


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Figure 4. Thermoplasmonic properties of a small silver nanosphere in glass. a-d) Calculations for a 10 nm diameter silver nanosphere in glass, using Johnson and Christy dielectric tables modified to include quantum confinement effects:37,45 a) Computed extinction cross-section σext using Mie theory. b) plasmonic derivatives ∂σext/∂ε1 (blue line), ∂σext/∂ε2 (green dashed line) and ∂σext/∂εm (red dash-dotted line), c) thermoplasmonic coefficients Ap (black line) and Am (red dash-dotted line) in the blue and infrared (inset) spectral ranges (see eq. 1) and d) their relative weights |Ap|/(|Ap|+|Am|) (black line) and |Am|/(|Ap|+|Am|) (red dash-dotted line).



The central SPR position is indicated by a dashed vertical line (λR≈ 410 nm). e) Computed temperature dependence of the real dε1/dT (blue line) and imaginary dε2/dT (green dashed line) parts of the dielectric function of silver.4 f) Transient extinction change dynamics ∆σext(t) following the initial heating of the sphere (∆T0=1K) at three specific probe central wavelengths (grey lines). To take into account the experimental spectral content of femtosecond pulses, these transient signals are computed by averaging Eq. 7 over a 10 nm probe spectral width. The specific contributions of nanoparticle and environment heating are indicated as black dashed and red dash-dotted lines, respectively.


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Figure 5. Finite-element modeling of the thermo-optical dynamics of a gold nanodisk on silica substrate in ethanol. Nanodisks with 120 nm diameter and 20 nm height were considered, as investigated in recent experiments.14 a) Nanodisk extinction cross-section σext. The SPR central position is shown by the dashed vertical line (λR= 790 nm) and horizontal ones in the next panels. b) Transient extinction change ∆σext(t, λpr) following the initial heating of the nanodisk by ∆T0=1K. c) ∆σext(t) from panel (b) divided by its maximum value, for each probe wavelength. d) ∆σext(t) dynamics for λpr=735 nm and λpr=790 nm probe central wavelengths (grey lines). The contributions of nanoparticle and environment heating are indicated as black dashed and red dash-dotted lines, respectively. e) Numerically computed thermoplasmonic coefficients Ap (black line), Aethanol (red dash-dotted line) and Asilica (blue line). 27 ACS Paragon Plus Environment


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Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles

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