Loubnan Abou-Hamdan

Claire Li

Riad Haidar

Valentina Krachmalnicoff

Patrick Bouchon
email: patrick.bouchon@onera.fr

Yannick De Wilde
email: yannick.dewilde@espci.fr

Hybrid modes in a single thermally excited asymmetric dimer antenna

The study of hybrid modes in a single dimer of neighboring antennas is an essential step to optimize the far-field electromagnetic (EM) response of large-scale metasurfaces or any complex antenna structure made up of subwavelength building blocks. Here we present far-field infrared spatial modulation spectroscopy (IR-SMS) measurements of a single thermally excited asymmetric dimer of square metal-insulator-metal (MIM) antennas separated by a nanometric gap. Through thermal fluctuations, all the EM modes of the antennas are excited and hybrid bonding and antibonding modes can be observed simultaneously. We study the latter within a plasmon hybridization model, and analyse their effect on the far-field response.

Infrared (IR) metamaterials made up of plasmonic resonators have been used in a wide range of applications, such as radiative cooling [1,2], photodetection [3][4][5][6][7], and solar cell design [8]. The building blocks of these materials are subwavelength structures that exhibit many interesting antenna effects, namely, directional thermal emission, spectral selectivity, as well as high field confinement. In this regard, MIM resonators have been studied in a variety of configurations, ranging from nanoribbons [9,10] to coated spherical nanoparticles [2], and have demonstrated such capabilities while also being spectrally tunable and angularly independent [9][10][11]. So far, these studies have mainly considered periodic arrays of antennas. However, diffraction orders as well as coupling effects between neighboring antennas take place, which affects the overall EM response of such arrays [12]. In addition, when two plasmonic resonators are in nearfield interaction, the dimer structure may exhibit hybrid plasmonic resonances [13][14][15][16][17][18][19]. Diffractive coupling between neighboring dimers placed in an array has also been shown to alter optical behavior [20]. The thermal emission of an isolated MIM antenna has been characterized in the mid-IR and has revealed interesting resonant behavior [21]. Therefore, understanding the effect of near-field interaction within an isolated asymmetric dimer on its far-field thermal emission properties is crucial to guide future efforts of optimizing the response of antennas made of the assembly of sub-wavelength building blocks separated by tiny gaps, with benefit for various applications, such as IR photodetection [4] or multispectral biosensing [22,23]. In this letter, we demonstrate simultaneous probing of the bonding and anti-bonding modes of an isolated dimer of MIM antennas by thermal excitation. This striking result, which is confirmed by finite difference time domain (FDTD) simulations and polarization analysis, shows that various coupled modes of a single nano-antenna can be simultaneously excited by thermal fluctuations, an es-sentially incoherent process arising from fluctuating thermal currents. The investigated samples consist of a homogeneous substrate supporting two coupled subwavelength gold square patch MIM antennas of different width with a silica (SiO 2 ) spacer layer, separated by a nanometric gap g, which we refer to as a BiMIM structure (see Fig. 1(a) for sample geometry). The far-field thermal radiation spectrum of the coupled MIM antennas is extracted from the overwhelming background thermal radiation via IR-SMS [21] in the spectral range between 6-13 µm. As sketched in Fig. 1 (a), the temperature of the BiMIM and the substrate is raised to 438 K by means of a hot sample stage. A lateral modulation of the sample at frequency Ω is applied, producing a small oscillatory contribution on the thermal radiation captured by an IR detector due to a light fall-off effect. Provided that the substrate is homogeneous, it coincides with the sole BiMIM contribution which is then extracted by lock-in demodulation, allowing background-free spectroscopic analysis with a Fourier transform IR (FTIR) spectrometer. A single MIM antenna with a patch of width w acts as a Fabry-Perot resonator for the gap plasmons trapped below the patch [9,24,25]. By thermally exciting the MIM antennas, all their EM modes are excited in the spectral range considered and are populated according to Bose-Einstein statistics [26][27][28]. The resonance wavelength of the modes can be roughly estimated by the phase matching condition λ = 2n ef f w, where n ef f is the effective index of refraction of the MIM mode in the insulator layer [24,25]. Fig. 1(b) shows the measured thermal emission spectrum of a BiMIM structure with a 1 µm gap (blue curve). As a reference, we also show measured spectra of the single MIM antennas making up the structure (respective widths w = 2 µm and 2.5 µm, dashed and dash-dotted curves) each consisting of two resonance peaks, which are a signature of the excitation of the MIM fundamental mode at two different wavelengths. The peak at the largest wavelength results from the peculiar dispersion of the silica spacer layer [21] and will not be discussed further here as it occurs in a region where the cavity is very lossy (see supplementary doc- The sample consists of a transparent Si substrate on top of which a 200 nm thick layer of gold is evaporated, then a 300 nm spacer layer of SiO2 is deposited, and a final top layer consisting of two subwavelength gold square patches of thickness 100 nm, and sides w1 = 2 µm and w2 = 2.5 µm, separated by a gap g. An SEM image of the investigated sample is shown in the top right corner. The scale bar is 1µm. (b) Measured IR-SMS far-field thermal emission spectrum of a BiMIM with a 1 µm gap compared to that of its constituent MIM antennas. The antennas resonate near (6.5 µm, 10.5 µm) and (7.2 µm, 11.2 µm), for w = 2 µm and 2.5 µm, respectively. See Supplementary document, section 1 for measurement details. ument, section 2 for more details). We will thus focus the remainder of our discussion on spectral features below 8.5 µm. As can be seen from Fig. 1(b), for such a large gap, the thermal emission spectrum of the BiMIM structure simply overlaps with that of the two individual antennas, meaning that the two MIM antennas behave independently. However, when the MIM antennas are separated by a smaller gap, their two gold patches form a plasmonic dimer pair that exhibits bonding and antibonding hybrid modes [14]. If an external illumination is used, the latter modes can be selectively excited by varying the polarization of the EM excitation with respect to the dimer common axis (see Fig. 2(b) inset). These hybrid modes can be studied using a plasmon hybridization model [14,29,30], in which the conduction electrons in the metal are modeled as a charged incompressable liquid, whose deformations lead to the formation of a surface charge. In our experiment, a thermal excitation of the BiMIM structure is instead produced in situ by raising the overall temperature of the sample, inducing a superposition between the longitudinal and the transverse bonding and anti-bonding modes, which modifies the observed farfield response. In particular, when the gap size is reduced to 100 nm (red curve in Fig. 2) a splitting begins to form in the resonance peak between 7 and 8 µm, while an overall red-shift in the peaks is also observed. It has been shown that the bonding hybrid modes lead to a red-shift in the spectrum, while the anti-bonding modes result in a slight blue-shift [13,16,17,31]. The observed splitting is thus, a hallmark of simultaneous excitation of the longitudinal bonding and transverse anti-bonding modes due to thermal fluctuations. FDTD calculations of the emission cross-section of the BiMIM structure (Fig. 2(b)) confirm this effect. For a larger gap size (g = 500 nm, black curve in Fig. 2) the splitting and red-shift disappear. The thermal radiation emitted by the BiMIM structure is collected by a reflective objective (N A = 0.5), which collects radiation from a solid angle between 10 and 30 o . By reciprocity this means that the BiMIM's hybrid modes can be excited by an external source at non-normal incidence. The agreement of our off-axis measurements with simulations at normal incidence suggests that the emission spectra of the hybrid modes of the considered BiMIM structure are angularly independent up to at least 30 o . As shown below, it is possible to distinguish between the various hybrid modes of the BiMIM antennas, that are thermally excited, by performing polarized thermal emission measurements. This is done by placing a wire grid polarizer the detector in order to select the E // and E ⊥ polarizations. The experimental results and the corresponding cross-section calculations are shown in Fig. 3 (a) and (b) respectively. For the parallel polarization, FDTD simulations show that the longitudinal antibonding and bonding modes are excited at the resonance positions (i) and (ii), respectively, in Fig. 3(b). At these resonance positions, charges accumulate on the left and right edges of the square patches. This produces a strong enhancement of the z component |E z | of the electric field at the edges of the squares, as shown in Fig. 3(c) (i-ii). The opposite charges, found at the edges near the gap, lead to the observed red-shift in the longitudinal bonding mode (position (ii) in Fig. 3(b)), due to attraction. On the other hand, charges of same sign are found at the edges near the gap for the longitudinal anti-bonding mode (Fig. 3(c) (i)). This latter mode appears as a small peak at position (i) in Fig. 3(b). Note that the asymmetry of the BiMIM structure is essential for the observation of this mode. If the square patches were identical in size the net dipole moment for this mode would be zero, and it would be inaccessible by far-field illumination, as it is a dark mode [29]. By reciprocity it would not produce any measurable thermal radiation to the far-field as well. For the E ⊥ polarization, charges accumulate at the and bottom edges of the patches, leading to the weakly interacting transverse bonding and anti-bonding modes at positions (iii) and (iv), respectively (Fig. 3(c) (iii-iv)). In the measured spectra (Fig. 2(a)), the thermally induced splitting is thus the result of the combined response of the longitudinal and transverse bonding and anti-bonding modes. red In contrast, the longitudinal anti-bonding and the transverse bonding modes (i and iii, respectively) are degenerate and do not produce a splitting in the unpolarized thermal emission spectrum. If a symmetric BiMIM structure is considered (where both MIM antennas of the structure resonate near 6.5 µm), Fig. 4 (black simulated curve), the spectral overlap of the resonances near 6.5 µm does not allow for the excitation of the dark mode, while a splitting due to the longitudinal bonding and transverse anti-bonding modes still occurs-as indicated by the splitting between 6 and 7.5 µm. An interesting effect is also observed in the considered BiMIM structure, when examining the simulated electric field enhancement plots, for unpolarized emission, at the resonance positions A1 and A2 (Fig. 4 (b)). The plots reveal emission hot spots from the BiMIM structure, showing that the two MIM antennas emit independently at these resonance positions, even though they are effectively in near-field interaction. On the other hand, at the resonance peak A3 the two MIM antennas act as a coupled system, and a larger field enhancement is found in the gap (Fig. 4 (b), A3). The observed behavior at positions A1 and A2 is a direct result of the nature of the weakly interacting transverse hybrid modes, and the asymmetry of the BiMIM structure under study. In fact, since the resonance wavelength of a MIM antenna scales with its width, the size difference between the two MIM antennas allows for independent emission at two different wavelengths. To illustrate this point, we also show field enhancement plots at the resonance positions S1 and S2 for the symmetric BiMIM case (Fig 4 . (b) bottom two panels). In this case, the two MIM antennas cannot emit independently, and emission hot spots are either found at the edges of the two gold patches (position S1) or at the gap (position S2).

The equivalence between emission and absorption suggests that if an external electric field is incident on the considered asymmetric BiMIM structure, it will be routed to, and resonantly absorbed by each MIM antenna at the resonance positions A1 and A2. Such an attribute could be beneficial for IR photodetection applications [10].

In summary, we have shown thermally induced simultaneous probing of the various hybrid modes of a single asymmetric BiMIM structure, and measured its effect on the far-field response using IR-SMS. In contrast to other symmetric systems the considered asymmetric BiMIM structure was found to allow direct far-field excitation of the dark anti-bonding mode. Further, simulated electric field enhancement plots revealed that due to the nature of the transverse hybrid modes, and the asymmetry of the system considered, the two MIM antennas making up the structure can emit (equivalently absorb) radiation independently. This feature, along with the fact that the absorbed radiation is mainly concentrated in the insulator layer in a MIM antenna, could be of use for realizing IR photodetectors with improved responsivity at room temperature. The absorbed radiation may be harnessed by replacing the insulating SiO 2 layer considered here by a semi-conductor like HgCdTe [10,32] or a quantum well material [4]. Finally, our study may guide future efforts to optimize near-field interactions in nano-antenna arrays for applications such as bio-sensing for the detection and characterization of bio-molecules. The infrared spatial modulation spectroscopy technique (IR-SMS) used to obtain the thermal emission of the isolated BiMIM structure is based on lateral modulation of the sample along with lock-in detection. The full details of the experimental technique have already been outlined elsewhere in Ref. [1]. The experimental setup is coupled to an FTIR which allows us to obtain the thermal emission spectra shown in Fig. 2(a) of the main text. For completeness, these spectra are again plotted here over the full spectral range of measurement between 6-13 µm in Fig 2 .  During the measurements, the sample itself is the source of the thermal emission, which is brought upon by heating it to a uniform temperature. The presented measurements were performed at T sample = 438 K. The instrumental response of the FTIR was calibrated with blackbody samples at different temperatures using the method outlined in Ref. [2]. The measured thermal emission spectra, presented in the main text, were normalized to the response of a blackbody at the same temperature FIG. 2. Raw thermal emission spectra of the BiMIM structure measured using the IR-SMS technique and a mechanical chopper. The two spectra are scaled according to the sensitivity used in the lock-in detection. The inset shows a comparison of the two when normalized to 1.

to correct for instrumental response. As such, the normalized spectra are directly comparable to the emission cross-section of the BiMIM structure (see ref. [3] for more details). To illustrate the fact that the IR-SMS technique allows one to obtain a better signal-to-noise ratio and can directly extract the thermal emission of the single BiMIM structure from the huge extended background radiation, we show in Fig. 2 raw (un-normalized) spectra of the BiMIM structure measured both using the IR-SMS technique and with a mechanical chopper. The two spectra are scaled according to the sensitivity used during the lock-in detection. As clearly indicated in Fig. 2 the signal coming from the sole BiMIM antenna, which is directly measured by the IR-SMS technique, is much smaller than that of the background emission. On the other hand, the BiMIM emission is completely lost in the overwhelming signal coming from the substrate emission and background thermal radiation when the measurement is performed with a mechanical chopper. The peak near 9.5 µm in Fig. 2 (black curve) for instance, is an absorption peak of the SiO 2 substrate. A comparison of the two spectra when normalized to 1 (Fig. 2 inset) shows that the resonance peaks of the measured BiMIM structure can be clearly resolved even in the raw untreated spectrum, showing the efficacy of the IR-SMS technique. The polarized thermal emission spectra presented in Fig. 3(a) of the main text, were obtained by adding a BaF 2 wire grid polarizer-with maximum transmission in the measured spectral range-before the detector.

II. FDTD SIMULATIONS

The simulations presented in the main text where performed using FDTD calculations (Lumerical solutions). In order to simulate an isolated BiMIM structure perfectly matched layers (PMLs) were effected for all boundaries. Unpolarized, normally incident, broadband electromagnetic radiation in the 6-13 µm range was used as the source of excitation. The simulation span was made large enough so that the distance between the edges of the patch antennas of the BiMIM structure and the PMLs was at least λ/4 in all directions, where λ is the maximum wavelength in the considered spectral range. This was done to reduce reflections by the PML boundaries and to improve the convergence of the simulations. The simulation time was also set large enough to insure proper convergence of all simulations. The emission cross-section was obtained by calculating losses due to absorption from the MIM antennas. Palik data [5] were used in the simulations for the dielectric function of the gold layers. On the other hand, dielectric data reported in Ref. [4] for SiO 2 thin films were used for the SiO 2 layer. In order to estimate the difference between the used dielectric data for SiO 2 and the real one, we measured the thermal emission from the substrate, by collecting thermal radiation from a part of the sample that does not include any gold patches, using a mechanical chopper. The measured thermal emission of the substrate can be compared to the absorption of the SiO 2 layer, represented by the imaginary part of its index of refraction. As shown in Fig. 3 (a) the peak of absorption of the imaginary part of the index of refraction agrees well with the peak of the measured thermal emission. On examining the imaginary part of the index of refraction further, we find that due to the high absorption of SiO 2 in the spectral range between 8 and 13µm, a splitting due to the hybrid modes does not occur in this region. Indeed, as illustrated in Fig. 3 (b) the imaginary part of the index of refraction is zero in the spectral range where the splitting occurs (below ∼7.5 µm), while it is non-zero for higher wavelengths.

FIG. 1 .

 1 FIG. 1. (a) Schematic illustration of the BiMIM geometry and IR-SMS technique. The sample consists of a transparentSi substrate on top of which a 200 nm thick layer of gold is evaporated, then a 300 nm spacer layer of SiO2 is deposited, and a final top layer consisting of two subwavelength gold square patches of thickness 100 nm, and sides w1 = 2 µm and w2 = 2.5 µm, separated by a gap g. An SEM image of the investigated sample is shown in the top right corner. The scale bar is 1µm. (b) Measured IR-SMS far-field thermal emission spectrum of a BiMIM with a 1 µm gap compared to that of its constituent MIM antennas. The antennas resonate near (6.5 µm, 10.5 µm) and (7.2 µm, 11.2 µm), for w = 2 µm and 2.5 µm, respectively. See Supplementary document, section 1 for measurement details.

FIG. 2 .

 2 FIG. 2. (a) Measured thermal emission spectra of the BiMIM structure for different gap size g. (b) FDTD calculations of emission cross-sections for normally incident, unpolarized illumination, corresponding to the measurements (see Supplementary document, section 2 for calculation details). The inset shows a sketch of the dimer geometry showing two of the hybrid modes of the BiMIM, for electric field polarizations parallel and perpendicular to the dimer axis. The positive and negative signs indicate the surface charge distribution.

FIG. 3 .

 3 FIG. 3. (a) Measured thermal emission and (b) calculated emission cross-section of the BiMIM structure with g = 100 nm for polarized thermal emission. (c) Plots of the z component Ez (in units of V/m) the electric field in the xy-plane at the top of the square metal patches, at the resonance peaks indicated by the markers (i), (ii), (iii), and in panel (b).arrows indicate the electric field polarization in each case.

FIG. 4 .

 4 FIG. 4. (a) Simulated emission cross-section for unpolarized light for the considered BiMIM structure with g = 100 nm compared with that of a symmetric BiMIM whose patches have widths w1 = w2 = 2 µm. (b) Electric field enhancement plots |E|/|Eo|, where |Eo| is the incident field, showing field hot spots in the xy-plane at the top of the metal patches, recorded at the resonance positions in panel (a).

FIG. 1 .

 1 FIG.1. Measured thermal emission spectra of the BiMIM structure for different gap sizes g (presented in Fig.2(a) of the main text), over the spectral range between 6-13 µm.

FIG. 3 .

 3 FIG. 3. (a)Measured thermal emission of the sample substrate, along with the imaginary part of the index of refraction of SiO2 (taken from Ref.[4]) representing the absorption due to the substrate. (b) Simulated emission cross-section of a BiMIM structure for gap size g = 100 nm (black curve) presented in the main text (red curve in Fig.2(b) of the main text), along with the imaginary part of the SiO2 index of refraction (red curve). The comparison of the two shows a high absorption due to SiO2 in the spectral region between 8 and 13 µm, which prevents the formation of a splitting in the BiMIM emission spectrum in this range.

ACKNOWLEDGMENTS

The authors would like to thank Christophe Dupuis for the SEM image. This work received financial support from LABEX WIFI under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*, and ANR Project CarISOVERRE ANR-16-CE09-0012.

Supplementary information

Loubnan Abou-Hamdan,


Joris Doumouro

Elodie Perros

Alix Dodu

Nancy Rahbany

Dominique Leprat

Valentina Krachmalnicoff

Rémi Carminati

Wilfrid Poirier

Yannick De Wilde

Quantitative measurement of the thermal contact resistance between a glass microsphere and a plate

Accurate measurements of the thermal resistance between micro-objects made of insulating materials are complex because of their small size, low conductivity, and the presence of various ill-defined gaps. We address this issue using a modified scanning thermal microscope operating in vacuum and in air. The sphere-plate geometry is considered. Under controlled heating power, we measure the temperature on top of a glass microsphere glued to the probe as it approaches a glass plate at room temperature with nanometer accuracy. In vacuum, a jump is observed at contact. From this jump in temperature and the modeling of the thermal resistance of a sphere, the sphere-plate contact resistance RK = (1.4 ± 0.18) × 10 7 K.W -1 and effective radius r = (36 ± 4) nm are obtained. In air, the temperature on top of the sphere shows a decrease starting from a sphere-plate distance of 200 µm. A jump is also observed at contact, with a reduced amplitude. The sphere-plate coupling out of contact can be described by the resistance shape factor of a sphere in front of a plate in air, placed in a circuit involving a series and a parallel resistance that are determined by fitting the approach curve. The contact resistance in air R * K = (1.2 ± 0.46) × 10 7 K.W -1 is then estimated from the temperature jump. The method is quantitative without requiring any tedious multiple-scale numerical simulation, and is versatile to describe the coupling between micro-objects from large distances to contact in various environments.

I. INTRODUCTION

The effective thermal conductivity of complex insulating materials drastically depends on their micro-structure [START_REF] Kwon | Effective thermal conductivity of various filling materials for vacuum insulation panels[END_REF][START_REF] Langlais | Isolation thermique à température ambiante. bases physiques[END_REF]. In general, they are made of low thermal conductivity micrometer-sized elements in contact or near to contact. Enhanced insulation results from a high resistance between them that hampers heat flow through the material. It is of crucial importance to have a detailed knowledge of the thermal resistance between the constitutive micro-elements of insulating materials, both in air and in vacuum, but accurate measurements are complex to perform because of their small size and very low thermal conductivity. To describe the heat transfer processes at a microscopic scale, a distinction must be made between situations in which micro-elements are close to contact or in contact. Different heat transfer channels are expected to contribute between very close micro-elements. If they are separated by a vacuum gap smaller than the thermal wavelength λ BB ≈ 10 µm at room temperature, the radiative heat transfer should be enhanced by the contribution of evanescent waves [START_REF] Wilde | Thermal radiation scanning tunnelling microscopy[END_REF][START_REF] Kittel | Near-field heat transfer in a scanning thermal microscope[END_REF][START_REF] Wischnath | The near-field scanning thermal microscope[END_REF][START_REF] Rousseau | Radiative heat transfer at the nanoscale[END_REF][START_REF] Shen | Surface phonon polaritons mediated energy transfer between nanoscale gaps[END_REF][START_REF] Ghashami | Precision measurement of phonon-polaritonic near-field energy transfer between macroscale planar structures under large thermal gradients[END_REF][START_REF] Cui | Study of radiative heat transfer in Ångströmand nanometre-sized gaps[END_REF], but under ambient conditions this radiative contribution is outplayed by conductive and convective transfer. If the micro-elements are in contact, an additional heat transfer occurs due to conduction through the solid phase and a water meniscus potentially surrounding the contact area [START_REF] Assy | Temperature-dependent capillary forces at nano-contacts for estimating the heat conduction through a water meniscus[END_REF]. * yannick.dewilde@espci.fr

In insulating materials made of fibers or grains, microelements are separated by a large variety of gaps. A wellknown example is glass wool, a biphasic material made of randomly distributed glass fibers with a diameter on the order of tens of micrometers, interacting at various distances, from hundreds of micrometers to contact. From a theoretical point of view, handling this variety of scales is a challenge for numerical simulations. Therefore an analytical model would be versatile to estimate the heat flux as a function of the gap. Yet, there has been little effort to analytically model heat transfer between two bodies separated by small gaps or in contact under ambient conditions [START_REF] Batchelor | Thermal or electrical conduction through a granular material[END_REF].

On the experimental side, scanning thermal microscopy (SThM) is ideally suited to perform local thermal investigations which require a control of the position of a small thermal sensor with nanometer accuracy. It consists of an atomic force microscope probe including a very small thermometer on its tip, whose position with respect to the sample surface is adjusted by means of piezoelectric translations. Such a sensor is generally employed for thermal characterization with nanometer spatial resolution of nanomaterials, nanostructures, and nanoscale devices [START_REF] Gomès | Scanning thermal microscopy: A review[END_REF][START_REF] Menges | Temperature mapping of operating nanoscale devices by scanning probe thermometry[END_REF]. It has also been used to investigate the thermal contact resistance between the probe tip and a planar substrate [START_REF] Jarzembski | Feedback control of local hotspot temperature using resistive on-substrate nanoheater/thermometer[END_REF][START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] Park | Experimental investigation on the heat transfer between a heated microcantilever and a substrate[END_REF][START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF][START_REF] Zhang | A noncontact thermal microprobe for local thermal conductivity measurement[END_REF] and to measure the thermal resistance due to a water meniscus formed around the probe-substrate contact region [START_REF] Assy | Heat transfer at nanoscale contacts investigated with scanning thermal microscopy[END_REF]. Most SThM studies use a thermal sensor placed in direct contact with the sample, which raises subtle metrological issues [START_REF] Gomès | Scanning thermal microscopy: A review[END_REF][START_REF] Ramiandrisoa | A dark mode in scanning thermal microscopy[END_REF]. In recent studies a microsphere was glued on the probe of a SThM to study the near-field radia-tive heat transfer in a sphere-plate geometry on a phase transition material [START_REF] Menges | Thermal radiative near field transport between vanadium dioxide and silicon oxide across the metal insulator transition[END_REF] or on a photovoltaic cell [START_REF] Lucchesi | Harnessing near-field thermal photons with efficient photovoltaic conversion[END_REF].

In this paper we propose a SThM-based method to investigate the thermal resistance between a glass microsphere and a glass plate separated by a gap varying from hundreds of micrometers to contact, both under vacuum and ambient conditions. The strategy relies on the use of a SThM probe working in active mode, i.e. both as thermometer and heater, with a glass microsphere glued on its tip. The transfer between the glass sphere and the plate is disentangled from the transfer between the heater and the environment through other paths, by monitoring the temperature at the top of the sphere as it approaches the plate and enters in contact with it. The study intends to provide a better insight regarding how heat is transferred between micrometer-sized glass elements within complex insulating materials like glass wool.

First, we describe the experimental setup. Then, we solve the problem of heat conduction in vacuum based on measurements combined with an analytical expression of the thermal resistance of a sphere, which leads to an accurate estimate of the sphere-plate thermal contact resistance and of the contact effective diameter. Finally, the heat conduction problem in ambient conditions is also tackled theoretically using a phenomenological approach based on a shape factor which properly describes the approach curve of the sphere towards the plate. A good estimate of the sphere-plate contact resistance is also obtained in these more complex conditions.

II. EXPERIMENTAL SETUP

To investigate the sphere-plate geometry a borosilicate glass sphere of calibrated diameter D = 20.7 ± 1µm from Duke Standards is attached with epoxy glue to the truncated pyramidal tip of a SThM probe fabricated by Kelvin Nanotechnology (KNT), as shown in Fig. 1. The probe is made of silicon nitride (Si 3 N 4 ) and carries a thermoresistive thin palladium strip patterned on the flat portion of the pyramidal tip. Its temperature coefficient of 0.74 Ω/K is obtained by placing the entire SThM probe in an isolated oven whose temperature can be accurately controlled while the strip electrical resistance R is measured. To perform the thermal measurements, we developed a setup which can operate either under vacuum or ambient conditions. It combines a SmarAct linear piezoelectric stepper-motor and a Piezosystem Jena linear continuous piezoelectric positionner. This allows us to perform displacements for measurements requiring a sphere-plate separation up to several hundreds of micrometers, and fine approach measurements when the sphere is close to contact with the plate. The latter is a borosilicate glass microscope coverslip. A secondary vacuum of approximately 10 -6 mbar is achieved by connecting a turbo pump to the chamber. The top of the glass microsphere is glued to the palladium strip on the SThM probe. The latter is used both for local temperature mea- surements and heating. The heated microsphere is held above the glass plate at a room temperature, T Room . The gap between the base of the sphere and the plate surface is precisely adjusted with the piezoelectric stages while the temperature at the sphere top is monitored. The SThM probes from KNT used in these experiments have a typical room temperature resistance of 350 Ω. The temperature at the top of the microsphere T T op is obtained by measuring the electrical resistance of the thermoresistive palladium strip glued to it. This electrical resistance R(T ) is accurately measured by integrating it in a highly-sensitive Wheatstone resistance bridge built at LNE (see Appendix). The bridge is biased with an AC voltage superimposed to a DC voltage which are controlled by the internal oscillator of a lock-in detector and a Yokogawa voltage source respectively. An AC polarization current I AC (≈ 30 µA at 873 Hz, typically) circulates through the palladium strip. The resistance R(T ) is obtained from the three other reference resistances constituting the bridge and the AC voltage unbalance detected by the lock-in detector. T T op is then easily obtained based on a fine calibration of the temperatureresistance relationship carried out prior to measurements. We are able to resolve temperature variations on the order of 10 mK. To use the probe in active mode, a DC current I DC of a few hundreds ( 400 -500 µA, typically) is superimposed on the AC current. This produces a well-defined and well-controlled local Joule heating of the SThM tip and of the glass microsphere attached to it. This injected power, P J = RI DC 2 , was adjusted around 70 µW. All measurements were performed without a deflection laser nor tip illumination, in order to prevent uncontrolled heat injection and to reduce measurement noise [START_REF] Ramiandrisoa | A dark mode in scanning thermal microscopy[END_REF][START_REF] Spiece | Improving accuracy of nanothermal measurements via spatially distributed scanning thermal microscope probes[END_REF].

In the experiments, the dissipated heat power P J is applied at the top of the glass microsphere and results in a temperature increase of the SThM probe above room temperature T Room (≈ 30 • C in the experiments discussed in this paper) whose magnitude ∆T T op = (T T op -T Room ) depends on the environmental conditions of the microsphere. This heat flux is dissipated through several channels including radiation between bodies, conduction in the air, the glass sphere, the SThM cantilever, through the sphere-plate contact, and convection. In vacuum, conduction and convection in air are suppressed.

To characterize these thermal conductance channels, we measured the temperature T T op at the top of the sphere as a function of the separation d between its base and the glass plate from large distances to contact, first in vacuum, and then in air. The result of these measurements are presented and discussed in Sections III and IV, respectively.

III. HEAT TRANSPORT UNDER VACUUM CONDITIONS

The first measurements were performed in high vacuum at a pressure of 10 -6 mbar such that the mean free path of air molecules ≈ 100 m is much larger than the vacuum chamber. Fig. 2 (a) presents the results of temperature measurements recorded when the microsphere is approached towards the plate. In the experiment, a heat power P J = 71µW produces a temperature increase ∆T T op = 15.30 K in the absence of contact between the microsphere and the plate. The total thermal conductance between the thermoresistive palladium strip and the environment before contact is thus G T otal (d > 0) = 4.6 µW.K -1 . The temperature at the top of the sphere as function of the distance is mostly constant and only exhibits an abrupt drop in amplitude δT T op 175 mK at d = 0. This drop corresponds to contact: a new conductive channel opens and substantially increases heat leakage. Note that besides the step-like variation, slow fluctuations are superimposed on the approach curve as the vacuum configuration requires the use of an electrical wiring which is not as well optimized as in air in terms of length, intermediate connections, and thermal shielding.

The radiative conductance between a 20 µm diameter glass microsphere and a glass plate is only of a few nW.K -1 at maximum in the far field: G f f (T ) = 2πR 2 4σε(T Sphere )T 3 , with R the radius of the sphere, σ the Stefan constant and ε(T Sphere ) the emissivity of silicon dioxide at the sphere temperature T Sphere [START_REF] Palik | Handbook of Optical Constants of Solids, Five-Volume Set: Handbook of Thermo-Optic Coefficients of Optical Materials with Applications[END_REF]. For a sphere heated up to 45 • C the conductance is G f f (T ) = 1.6 nW.K -1 . This value increases by a few nW.K -1 in the near field. The radiative conductance has only been observed so far with SThM techniques provided that the temperature difference between the sphere and the plate was on the order of hundreds of Kelvin [START_REF] Menges | Thermal radiative near field transport between vanadium dioxide and silicon oxide across the metal insulator transition[END_REF][START_REF] Lucchesi | Harnessing near-field thermal photons with efficient photovoltaic conversion[END_REF]. Its measurement with smaller temperature differences was possible with ultrasensitive fluxmeters [START_REF] Rousseau | Radiative heat transfer at the nanoscale[END_REF][START_REF] Cui | Study of radiative heat transfer in Ångströmand nanometre-sized gaps[END_REF]. The sensitivity of our SThM based setup is deduced from the noise on the temperature measurement before contact which is found to be 10 mK, corresponding to a conductance sensitivity of 3 nW.K -1 , and having the same order as the near-field contribution observed in [START_REF] Rousseau | Radiative heat transfer at the nanoscale[END_REF] for 22 µm glass sphere. Near-field effects are not observable at this stage with our setup for such a sphere diameter.

Neglecting this contribution compared to conduction, the cantilever conductance can be directly inferred from measurements out of contact (d > 0). Fig. 2(b) sketches the thermal resistance circuit to be considered in the experiment. It involves three thermal resistances associated with the cantilever, R C , the sphere, R S , and the sphereplate contact, R K , and the corresponding conductances G C,S,K = 1/R C,S,K . Since before contact R K ∞, the sphere is isothermal and all the heat dissipated at the palladium strip flows through the cantilever whose anchor point is at room temperature. The cantilever conductance G C is equal to the total thermal conductance before contact:

G C = G T otal (d > 0) = 4.6 µW.K -1 .
Figure 2(a) shows that the temperature measured at the top of the sphere drops abruptly by an amount δT T op at d = 0 when the base of the sphere enters in contact with the plate. This drop corresponds to the opening of a new conduction channel for heat at contact, as illustrated in Fig. 2(b). Note that ∆T T op still decreases by ∼ 25 mK between 0 and -10 nm as the sphere is further pressed against the plate, which is likely due to a small deformation of the contact region. With the sphere in contact with the plate held at T Room , heat can flow from the heated palladium strip through the sphere and the contact. The sphere is no longer isothermal. Its resistance is in series with the sphere-plate contact resistance, and the total conductance increases by an additional term (1/G S + 1/G K )

-1 compared to its value G C before contact.

G T otal (d = 0) = P J ∆T T op (0) = 1 R S + R K + G C (1)
where ∆T T op (0) is the temperature difference at contact. This provides a way to calculate the contact resistance R K from the measured values of G T otal in contact and before contact. The calculation of R K from the measurement of G T otal (d = 0) thus requires to determine the sphere thermal resistance R S . The latter can be calculated in analogy to the electrical resistance of a sphere because the poles are considered as heat source and sink, and the surface of the sphere is isolated everywhere else due to a negligible radiative heat transfer. The problem of a conducting sphere was addressed analytically by Landau and Lifshitz [START_REF] Landau | Electrodynamics of continuous media[END_REF] and Solivérez [START_REF] Solivérez | The notion of electrical resistance[END_REF]. The resistance of a sphere of radius R and conductivity k whose poles are connected with equal-size contacts of radius r R is R S (r) = (1/πk) [(1/r) + (1/2R) (ln (4R/r) -1)] [START_REF] Solivérez | The notion of electrical resistance[END_REF]. In our experiment, k = (1 ± 0.1) W.m -1 .K -1 is the conductivity of glass. Heat is injected via the palladium strip which forms a contact of radius a = 1 µm with the sphere top, and then sinks via the sphere-plate contact of radius r. We thus have two hemispheres with different contact radii in series. The sphere resistance is then R S (a, r) = 1 2 R S (a)+ 1 2 R S (r), which yields the expression of the resistance of a sphere of radius R and conductivity k with different contact radii a, r at the north and south poles respectively:

R S (a, r) = 1 2πk 1 a + 1 r + 1 2R ln 16R 2 a r -2 . (2)
According to Eq.( 2), the sphere resistance is expected to strongly depend on r which is a priori not known. However, since the phonon mean free path of glass near room temperature is less than 1 nm we assume that the transport of heat across the contact is purely diffusive [START_REF] Kittel | Interpretation of the thermal conductivity of glasses[END_REF]. In this regime, the resistance is directly related to the constriction size: R K = 1/(2kr) [START_REF] Prasher | Microscopic and macroscopic thermal contact resistances of pressed mechanical contacts[END_REF]. r can thus be expressed in terms of R K in Eq. ( 2) and using Eq. ( 1) and G T otal (d > O) = G C it is possible to determine both R K and R S from the measurement of the temperature jump at contact δT T op = ∆T T op (d > 0) -∆T T op (d ≤ 0).

The thermal resistance of the glass microsphere-plate contact in vacuum is obtained to be R K = (1.4 ± 0.18) × 10 7 K.W -1 , or equivalently G K = 72 nW.K -1 . This corresponds to an effective contact radius r = (36 ± 4) nm. It follows that the sphere resistance is R S ∼ 4.6 × 10 6 K.W -1 . With the values of the resistances found, we can thus calculate that only 1.2 % of the generated power flows through the sphere and the contact, while the majority of it flows through the cantilever. A series of measurements were performed for several temperature differences (∆T T op = 2, 12, 15 K) and at different locations of the glass plate. We find the same value of R K ≈ 1.4 10 7 K.W -1 . Note that a contact resistance of a few 10 7 K.W -1 was also measured between two crossing glass micro-fibers [START_REF] Nguyen | Thermal exchange of glass micro-fibers measured by the 3omega technique[END_REF]. Let's also mention that the contribution of the glue between the SThM probe and the sphere has been neglected in this analysis. Its thickness is ≈100 nm, which corresponds to a resistance of about 10 5 K.W -1 given the conductivity k G ≈ 0.3 W.m -1 .K -1 of the epoxy glue in use in the experiment.

IV. HEAT TRANSPORT UNDER AMBIENT CONDITIONS

Similar experiments with a SThM probe from a different batch and a glass sphere with D = 20.7 ± 1µm glued on its tip were also carried out under ambient conditions, i.e. in air under atmospheric pressure with a relative humidity of (70 ± 10)%. These conditions are more complex to deal with since several heat loss channels coexist even when the sphere and the plate are out of contact. With separations between the sphere and the plate reaching up to hundreds of micrometers, heat can still be transferred through air whose thermal conductivity k a = (26.6 ± 0.1) × 10 -3 W.m -1 .K -1 is small but not enough to be neglected. Other heat transfer channels may also contribute as both the sphere and cantilever constitute fins. While the thermal problem could probably be treated exactly by solving the heat equation with appropriate boundary conditions, we adopt in what follows a phenomenological model based on thermal resistance circuitry to describe the temperature variations recorded at the top of the sphere when it approaches the plate from hundreds of micrometers to contact. This procedure is sufficient to obtain a good estimate of the thermal-contact resistance between the sphere and the plate in air.

Figure 3 (a) shows the variation of ∆T T op as a function of the sphere-plate separation d recorded in air from 200 µm to contact with P J = 68.4 µW. High accuracy measurements of the temperature drop occurring at contact are presented in Fig. 3 (b). At large distances (Fig. 3 (a)) , the curve first exhibits a slow decay as d is reduced, which increases below 100 µm. As in vacuum, a jump in temperature is observed at contact, but with a reduced amplitude of (28 ± 3) mK.

The underlying idea of the model is to separate the thermal resistance due to the sphere-plate coupling from other heat-loss channels which do not depend on the distance d. When approaching the sphere to the plate in air, the contact at d = 0 produces a small additional contribution to the sphere-plate coupling by conduction through air R * SP (d), whose thermal resistance R * K can be evaluated by measuring the temperature jump. The thermal resistance circuit on which the model relies is sketched in the inset of Fig. 3(a). We approximate the various heat-loss channels by a set of three thermal resistances: a series resistance R * s and a parallel resistance R * p , both representing the distance independent losses, and a resistance

R * SP ||K (d) = R * SP (d)||R * K = (1/R * SP (d) + 1/R * K ) -1
which describes the loss channel between the sphere and the plate. The latter includes two resistances placed in parallel: * SP (d) which depends on d and R * K the contact resistance which takes a finite value only at contact

(d ≤ 0). Since R * K = ∞ out of contact, we have R * SP ||K (d) = R * SP (d) for d > 0.
In contrast with the vacuum case (section III), a fitting procedure is needed to find the values of R * s and R * p . Regarding the sphere-plate coupling resistance out of contact R * SP (d), we found that the conduction shape factor [START_REF] Kreith | Principles of Heat Transfer[END_REF] of an isothermal sphere placed close to a semi-infinite isothermal plane S(d) = (2πD)/ 1 -D/4 d + D/2 describes the coupling very well. This suggests that a large portion of the glass sphere is quasi-isothermal. We thus set R * SP (d) = 1/ S(d) k a in the fit. In Fig. 3 (a) we show the fit of the approach curve obtained in this way, superimposed on the experimental curve. The fit corresponds to R * s = (2.84 ± 0.13) × 10 5 K.W -1 and R * p = (4.22 ± 0.08) × 10 5 K.W -1 . Note that these resistances are the only fitting parameters of the model. They mimic the losses through the solid phases as well as through air, provided that these losses are independent of the sphere-plate distance. Once the values of R * s and R * p have been determined, we use the experimental values of ∆T T op measured at d = 5 nm and d = 0 (contact), i.e. on both sides of the jump in Fig.

* K = 1/R * SP ||K (d = 0) -1/R * SP ||K (d = 5nm
), which yields the value of the contact resistance in air R * K = (1.2 ± 0.46) × 10 7 K.W -1 . This value is close to that found in vacuum, which suggests that if a water meniscus is present within the contact area between the sphere and the plate, it only has a minor influence on the value of the thermal contact resistance. This is consistent with reported results obtained with SThM probes [START_REF] Assy | Temperature-dependent capillary forces at nano-contacts for estimating the heat conduction through a water meniscus[END_REF]. Finally, the resistance circuit shown in the inset of Figure 3 (a) enables one to calculate the relation which links R * K and ∆T Rel = δT T op /∆T T op (d = 0), the temperature jump occurring at contact measured at the top of the sphere relative to the temperature difference with respect to T Room .

R * K = ∆T -1 Rel R * 2 SP R * p (R * s + R * SP ) -1 R * s + R * SP + R * p -1 .
The minimum jump of temperature which can be resolved by our SThM setup in air is ≈ 5 mK around ∆T T op =14.5 K, i.e. ∆T Rel = 3.4 10 -4 K.K -1 . The highest value of R * K which could thus be measured is ≈ 6.8 10 7 K.W -1 in air at atmospheric pressure, while we estimate it to be ≈ 3.3 10 8 K.W -1 in vacuum. An improvement of the sensitivity by one order of magnitude should still be possible by modifying the measurement electronics to reduce the uncertainty on ∆T T op below 1 mK.

V. CONCLUSION

A quantitative characterization method of the thermal resistance between a glass microsphere and a glass plate in vacuum or in air has been demonstrated. Using a SThM tip glued on the sphere as both a heater and a temperature sensor, it is possible to vary the sphereplate separation with nanometer precision from hundreds of micrometers to contact. Both in vacuum and air, a jump of temperature is observed at contact due to the opening of a new conduction channel. In spite of the low thermal conductivity of glass and of the complex geometry of the experiment (the contact is opposite to the temperature sensor on the glass sphere), it was possible to disentangle the value of the thermal contact resistance from the height of the jump. In vacuum, we could measure the thermal resistance of the SThM cantilever in use. With an analytical expression of the thermal resistance of a sphere with different contact radii at the poles, and assuming diffusive heat transport, we were able to determine the value of the sphere-plate thermal contact resistance and the contact size, which we found to be of tens of nanometers. In air, the measurements showed that the distance dependence of the approach curve is governed by conduction through the gas phase. Based on a phenomenological model involving the conduction shape factor and a fitting procedure it was then possible to estimate the sphere-plate thermal contact resistance in air. We found a value close to that measured in vacuum.

The proposed methods have the advantage of being very generic and based on analytical modelling. With little effort, they could be adapted to estimate the thermal contact resistance between micrometer sized objects made of materials other than glass or with different shapes, or to study the impact of the environment on the transport of heat at small scales. This opens large perspectives for future investigations of heat transfer processes at a microscopic scale between the elements of complex insulation materials used in the building sector. For instance, to determine the contact resistance between glass fibers crossing with an arbitrary angle like in glass wool, it should be possible to calculate numerically only a few points of the curve R * SP (d) in the simplified geometry of two crossing cylinders without cantilever, and to determine R * s and R * p by a fit of the experimental data, which would provide enough parameters to calculate R * K .

Appendix: Thermistance measurement instrument

The thermal sensor of the SThM probe is based on a palladium resistive component with a nominal resistance of about 320 Ω at room temperature and a typical temperature dependence of 0.7 Ω/K. The measurement of the temperature therefore consists of measuring a resistance value which varies as a function of the temperature with a law that can be calibrated. Extracting quantitative information about the thermal exchanges in the experiment requires the knowledge not only of the temperature but also of the joule power which can be set by supplying the resistance sensor with a known current. To achieve this double objective, the measurement instrument, described in fig. 4, is based on a Wheatstone bridge allowing the combination of an AC (alternating current) measurement of the temperature and of a DC (direct current) heating of the sensor. The resistance value, R, of the thermal sensor (thermistance) is obtained from the three other reference resistors and the voltage unbalance of the bridge according to the relationship:

R ≈ 360 × (400 -160α) -377600γ 360 + 800γ (A.1)
where γ = V AB V AC , V AB is the amplitude of the ac unbalance voltage measured by the lock-in detector, V AC is the ac applied voltage to the Wheatstone bridge and α ∈ [0..1] is the potentiometer setting. The DC and AC currents, I DC and I AC that circulate through the thermal sensor, are given by: I DC,AC ≈ 720V DC,AC 377600 + 800R (A.

2)

The setting α of the 20 kΩ potentiometer in parallel with the 80 Ω resistor is used to balance the bridge (V AB ≈ 0) in order to accommodate for sensor resistance values away from 320 ohms. Detection of a low voltage signal allows choosing the highest sensitivity range of the lock-in detector. The Wheatstone bridge being equipped FIG. 4. Scheme of thermistance bridge with a Wagner arm with a mid-point at ground, permits both V A and V B detection voltages to be close to the ground potential. This allows the absence of a commonmode voltage which is favorable for having better sensitivity and accuracy.

To obtain good reproducibility and stability of measurements, several techniques were implemented [START_REF] Schopfer | Testing universality of the quantum hall effect by means of the wheatstone bridge[END_REF][START_REF] Schopfer | Quantum resistance standard accuracy close to the zero-dissipation state[END_REF][START_REF] Poirier | A resistance bridge based on a cryogenic current comparator achieving sub-10-9 measurement uncertainties[END_REF]. First, the bridge is made of S102CT Vishay resistors having a relative drift below 10 -5 /year and a low temperature coefficient below 0.6 ppm/K. The electronic circuits delivering the biasing AC and DC voltages are based on precision operational amplifiers supplied by stabilized voltage sources powered by rechargeable batteries. They are strongly isolated from the ground with PTFE material. Second, the thermal sensor is connected with 4 wires in a way (a triangle of two 80 Ω resistors on one side and one 80 Ω resistor and the potentiometer on the other side) that makes the bridge balance more immune to any change (for example caused by temperature) of the wire resistances. This implementation is nevertheless fully effective for a thermal sensor resistance close to 320 Ω. Finally, pairs of high and low potential wires are independently twisted and placed in a shield at ground as the case of the bridge instrument. Because of the midpoint at ground of the Wagner arm, current leakages between wires due to capacitance and insulating resistance are redirected towards the ground without changing the balance of the bridge. This avoids measurement errors, and circulation of high-frequency electrical noise, and al-lows increasing the measurement bandwidth. The efficiency of the electric guarding system is also optimum for R ≈ 320 Ω.

Close to equilibrium, i.e. α is set so that γ ≈ 0, the resistance sensitivity is given by: δR ≈ - and an AC polarization current I AC of 20 µA, a resistance sensitivity of 3 × 10 -4 Ω/Hz 1/2 can be achieved which corresponds to a temperature sensitivity of about 1 mK/Hz 1/2 at best.

FIG. 2 .

 2 FIG. 2. (a) Vacuum measurement with PJ = 71µW of the temperature difference ∆TT op = (TT op -TRoom) as a function of the distance d between the base of the microsphere and the glass plate. d = 0 corresponds to the occurence of the sphereplate contact where a sudden temperature jump of amplitude δTT op = 175 mK occurs. (b) Scheme showing the various resistances through which heat can flow before and after the contact is established between the sphere and the plate. RC is the thermal cantilever resistance, RS is the sphere resistance, and RK is the sphere-plate contact resistance. (c) Isotherms of an isolated sphere heated on a pole and cooled on the other. a is the size of the palladium strip and r is the equivalent size of contact.

FIG. 3 .

 3 FIG. 3. (a) (blue dots) Experimental approach curve showing the temperature of the palladium sensor glued on the north pole of the sphere as a function of the distance d between the south pole and the plate. d = 0 corresponds to the sphereplate contact. The red curve is the result of a fit based on the conduction shape factor and the resistance circuit shown in the inset. (b) Approach curve at short distance showing the jump of temperature occurring at contact due to the opening of a new conduction channel.

  3(b), to calculate the corresponding values of R * SP ||K (d). The occurrence of the sphere-plate contact coincides with the opening of a new conduction channel besides the coupling through air. We thus have 1/R

2 ×

 2 δV AB /I AC (A.3) Considering a voltage detection sensitivity of 5 nV/Hz 1/2

ACKNOWLEDGMENTS

This research was supported by LABEX WIFI (Laboratory of Excellence within the French Program Investments for the Future) under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*, and Agence Nationale de la Recherche (ANR), Project CarISOVERRE, ANR-16-CE09-0012. The authors thank K. Joulain, H. Kallel (Inst. Pprime), O. Bourgeois (Inst. Néel), and J.-Y. Laluet, V. Grigorova-Moutiers, J. Meulemans (Saint-Gobain) for their constructive comments and encouraging support.