The temperature dependence of the thermal boundary resistivity is investigated in glass-embedded Ag particles of radius 4.5 nm, in the temperature range from 300 to 70 K, using all-optical time-resolved nanocalorimetry. The present results provide a benchmark for theories aiming at explaining the thermal boundary resistivity at the interface between metal nanoparticles and their environment, a topic of great relevance when tailoring thermal energy delivery from nanoparticles as for applications in nanomedicine and thermal management at the nanoscale.

With the ever decreasing size of nanodevices, investigation and modeling of heat exchange at the nanoscale has become of central technological interest. Under a fundamental standpoint, metal nanoparticles (NPs) embedded in a host matrix constitute a model system, as they can be selectively heated-up and their cooling monitored using time-resolved spectroscopy.1–3 Furthermore, the thermal dynamics occurring between an optically excited metal nanoparticle and the surrounding environment is of direct relevance for a variety of applications ranging from photothermal cancer therapy4,5 and selective drug delivery6 to thermoacoustic imaging and electromagnetic waveguiding in dielectric-embedded plasmonic devices.7 The electromagnetic energy harvested by the NPs is dissipated as thermal energy in the environment. The corresponding energy flux Jp is ruled by the thermal boundary resistivity ρbd, i.e., Kapitza resistivity, and by the temperature mismatch ΔT between the two media: Jp = ΔT/ρbd. Investigating ρbd is, therefore, crucial to tailor thermal energy delivery from the NP to the matrix or matrix-embedded target and, more generally, to analyze heat transfer at the nanoscale.

Whereas effort has been devoted to understand and model the Kapitza resistivity between two solids, both bulk and thin films,8–10 the scenario remains relatively unexplored when one of the two materials downscales to the nanometer range. Several confinement effects may modify ρbd, for instance, as the dimension of the NPs becomes comparable to the thermal diffusion length of the host material,11–13 or is reduced to a point where the continuum solid approximation to the elastic problem becomes questionable. A lack of extensive experimental evidence2,14,15 spanning the space of parameters affecting ρbd, most notably the temperature,8,9 has so far prevented a consistent account of the mechanisms ruling the Kapitza resistivity at the nanoscale. When confronted with the problem of measuring the heat transfer from a nanoscale object, the challenges stand in (a) a probe speed requirement, dictated by the fact that the time for heat exchange between the sample and the thermal reservoir decreases with the decreasing sample’s mass and (b) a non-contact probe requirement, to avoid the addendum heat capacitance contribution from the probe itself.

In this letter, we use time-resolved all-optical nanocalorimetry16 to overcome such challenges and investigate the temperature dependence of the cooling dynamics of glass-embedded Ag particles of radius R = 4.5 nm. The Kapitza resistivity is shown to increase by a factor of two with decreasing temperature from 300 to 70 K, a trend consistent with existing models.

The Ag nanospheres are embedded in a 50% BaO–50% P2O5 glass matrix of thickness L = 50 μm. The metal volume fraction is 2 · 10−4. The sample was synthesized using a fusion and heat treatment technique.17,18 The samples’ optical density (OD) shows enhanced absorption in the blue portion of the spectrum due to the localized surface plasmon resonance (LSPR) of the Ag NPs – see inset of Fig. 1.

FIG. 1.

(Color online) Measured time-resolved relative transmission change for Tcryo = 200 K. The pump and probe pulses have wavelengths, respectively, at 400 nm and 800 nm. In the cartoon, the thermal fluxes Jp and Jm are represented together with the temperature profile within the sample. Inset: measured OD of the sample outlining the Ag nanoparticles’ LSPR.

FIG. 1.

(Color online) Measured time-resolved relative transmission change for Tcryo = 200 K. The pump and probe pulses have wavelengths, respectively, at 400 nm and 800 nm. In the cartoon, the thermal fluxes Jp and Jm are represented together with the temperature profile within the sample. Inset: measured OD of the sample outlining the Ag nanoparticles’ LSPR.

Close modal

The time-resolved measurements were performed using a Ti:Sapphire cavity dumped oscillator – 800 nm wavelength, 120 fs pulse temporal width at full width half maximum. The Ag NPs are selectively excited by the frequency-doubled pulse at 400 nm wavelength close to the LSPR in order to maximize energy absorption in the particle. This leads to a fast heating of the electrons of the NPs that thermalize with the lattice on a few picoseconds time-scale, the thermal energy being subsequently delivered to the matrix. Care was taken to minimize average heating19 of the glass matrix by keeping the energy per pulse as low as possible while granting a detectable transmission variation. To this end, the laser repetition rate was tuned to 540 kHz by means of a cavity dumper. Accounting for transmission losses along the optical path, the energy density per pump pulse on the sample surface was I0 ∼ 0.5 J/m2, while the energy density absorbed per particle per pulse was UV ∼ 4 · 107 J/m3. The cooling dynamics of the hot NPs to the glass matrix is then followed by measuring the relative change in transmission across the sample ΔTr/Tr of a time-delayed 800 nm probe pulse. Probing out of the LSPR grants proportionality between the experimental signal and the NPs temperature rise2 (at the expense of the signal amplitude).

A typical experimental trace is reported in Fig. 1 for a cryostat temperature Tcryo = 200 K. After excitation by the pump pulse and internal electron-lattice thermalization, i.e., after 6 ps (this step has been extensively investigated in these systems1,20 and will not be discussed here), the signal decay reflects cooling of the hot NP to the matrix. This is governed by the thermal flux Jp and Jm from the NP to the matrix and from the matrix portion adjacent to the NP to the rest of the matrix, respectively. Considering these two processes, the energy balance is governed by

CptTp(t)=-3Rρbd[Tp(t)-Tm(R,t)],
(1)
CmtTm(r,t)=Λmr-1r2[rTm(r,t)],
(2)

where Tp is the NPs temperature, assumed as constant throughout the particle volume, Tm the matrix’ temperature, Cp and Cm the particle’s and matrix’s specific heat per unit volume, respectively, and Λm the matrix’ thermal conductivity. For the case of constant thermal parameters, the temperature increase for the NP, ΔTp(t), and for the matrix portion in contact with it, ΔTm(R,t), are analytically accessible working in Laplace space21 and read

ΔTp(t)=0duf(u,t),
(3)
ΔTm(R,t)=0du[1-u2kgR]f(u,t),
(4)

where

f(u,t)=2k(Rg)2ΔT0πu2exp(-κu2t/R2)[u2(1+Rg)-kRg]2+(u3-kRgu)2,
(5)

where ΔT0 is the initial temperature increase22 and κ = Λm/Cm, k = 3 Cm/Cp, g = 1/Λmρbd. In our analysis, the thermal resistivity ρbd is set as a fit parameter, together with Λm, which is not precisely known for our glass material.

As low temperatures are investigated (in particular, around and below the NPs Debye temperature, TD ∼ 215 K for Ag), Cp and Cm (Ref. 23) cannot be set to their Tcryo value and regarded as constant over the particle and matrix’s temperature excursion taking place during the experiment. The solution of Eqs. (1) and (2), taking into account the temperature dependent specific heats, is then retrieved iteratively. Fitting is performed starting at a time-delay of 6 ps and setting Tm,0 = Tcryo, and Tp,0 = Tcryo + ΔT0. The corresponding values Cp(Tp,0) and Cm(Tm,0) are inserted in Eqs. (3) and (4), and the new temperatures Tp,1 = Tcryo + ΔTp,1 and Tm,1 = Tcryo + ΔTm,1 are calculated and adopted in the subsequent time step. The procedure is iterated to reach the maximum experimental time-delay of 320 ps. Values for ρbd and Λm are obtained maximizing the likelihood between the theoretical Tp(t;ρbdbd)/ΔT0 and experimental −ΔTr/Tr traces.

The resulting dynamics of temperatures and specific heats is exemplified in Figs. 2(a) and 2(b), respectively, for the lowest studied temperature. The internal thermalization of the NP is achieved at Tp = 93 K. As time evolves, the NP cools down, increasing Tm(R); the maximum value of Tm(R) being attained at t ∼ 15 ps. For longer time-delays, both Tp and Tm(R) decay toward the asymptotic value Tcryo. The same trend applies to the specific heats Cp and Cm(R), showing maximum relative changes during the experiment in the 15%-20% range. A similar behavior is obtained for measurements performed at higher temperatures, although with a smaller excursion amplitude (as indicated by the arrows in Fig. 3). These variations stress the importance of taking into account the temperature dependence of the specific heat when measuring at cryogenic temperatures and make difficult the extraction of ρbd at values of Tcryo < 70 K. The Kapitza resistivity ρbd(Tcryo) increases by a factor of two, spanning values from 3.2 to 6.5 m2 K/GW, as the cryostat’s temperature decreases from 300 K to 77 K (see Fig. 3). The extracted value corresponds to a mean value of ρbd over the NPs temperature excursion during the experiment (indicated by arrows in Fig. 3). Values for Λm were found in the range 0.2-0.7 W/mK, comparable to the ones reported for thermal conductivity of glasses with similar compositions.2 

FIG. 2.

(Color online) Time evolution of temperature and specific heat for Tcryo = 70 K. Panel (a): relative temperature variation (left axis) and absolute temperature (right axis) of the NP (full line) and of the adjacent matrix (dashed line). Inset: experimental transmission change normalized to the value at 6 ps (red curve) and its best fit (black curve). Panel (b): relative specific heat variation (left axis) and absolute specific heat (right axis) of the NP (full line) and of the adjacent matrix (dashed line).

FIG. 2.

(Color online) Time evolution of temperature and specific heat for Tcryo = 70 K. Panel (a): relative temperature variation (left axis) and absolute temperature (right axis) of the NP (full line) and of the adjacent matrix (dashed line). Inset: experimental transmission change normalized to the value at 6 ps (red curve) and its best fit (black curve). Panel (b): relative specific heat variation (left axis) and absolute specific heat (right axis) of the NP (full line) and of the adjacent matrix (dashed line).

Close modal
FIG. 3.

(Color online) Kapitza resistivity ρbd vs Tcryo (black circles). The horizontal arrows indicate the temperatures spanned by the NPs during the thermalization process. Plot of the function ACp-1, A being a multiplication constant with dimensions ms−1 (red curve). Inset: normalized transmission change (red curve) and its best fit (black curve) for the case Tcryo = 200 K.

FIG. 3.

(Color online) Kapitza resistivity ρbd vs Tcryo (black circles). The horizontal arrows indicate the temperatures spanned by the NPs during the thermalization process. Plot of the function ACp-1, A being a multiplication constant with dimensions ms−1 (red curve). Inset: normalized transmission change (red curve) and its best fit (black curve) for the case Tcryo = 200 K.

Close modal

Starting from the general expression for the Kapitza resistivity,8 and assuming both a frequency independent and/or frequency-averaged phonon transmission coefficient t̃, and group velocity vg̃, one finds ρbd(t̃vg̃Cp)-1. This trend is experimentally retrieved in our data where ρbd is found to roughly follow the temperature dependence of Cp-1, see Fig. 3.

In conclusion, we measured via time-resolved all-optical nanocalorimetry the Kapitza resistivity of a 4.5 nm radius glass-embedded Ag nanoparticle in the temperature range from 300 K to 70 K. ρbd increases monotonically from an ambient temperature value of 3.2 m2 K/GW to 6.5 m2 K/GW. The present findings constitute a benchmark for theories aiming at explaining the Kapitza resistivity in nanosystems, a fundamental issue for applications in nanomedicine and thermal management at the nanoscale.

We acknowledge Dr. Aurelién Crut and Dr. Paolo Maioli for enlightening discussions and useful suggestions. This work was partially funded by grant D.2.2-2011 of Università Cattolica and the Opthermal grant of the Agence Nationale de la Recherche. N.D.F. and F.B. acknowledge the support from the Institut Universitaire de France and CNRS, respectively. Open Access publication was sponsored in the frame of Scientific Dissemination Grant D.3.1-2011 of Università Cattolica.

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