Gopal Verma

Gyanendra Yadav

Chaudry Sajed Saraj

Longnan Li

Nenad Miljkovic

J P Delville

Wei Li
email: weili1@ciomp.ac.cn

A Versatile Interferometric Technique for Probing the Thermophysical Properties of Complex Fluids

Laser-induced thermocapillary deformation of liquid surfaces has emerged as a promising tool to precisely characterize the thermophysical properties of pure fluids. However, challenges arise for nanofluid (NF) and soft bio-fluid systems where the direct interaction of the laser generates an intriguing interplay between heating, momentum, and scattering forces which can even damage soft bio-fluids. Here, we report a versatile, pump-probebased, rapid, and non-contact interferometric technique that resolves interface dynamics of complex fluids with precision of approximately 1 nm in thick-film and 150 pm in thin-film regimes below the thermal limit without the use of lock-in or modulated beams. We characterize the thermophysical properties of complex NF in three exclusively different types of configurations. First, when the NF is heated from the bottom through an opaque substrate, we demonstrate that our methodology permits the measurement of thermophysical properties (viscosity, surface tension, and diffusivity) of complex NF and bio-fluids. Second, in a top illumination configuration, we show a precise characterization of NF by quantitively isolating the competing forces, taking advantage of the different time scales of these forces. Third, we show the measurement of NF confined in a metal-cavity, in which the transient thermoelastic deformation of the metal surface provides the properties of the NF as well as thermo-mechanical properties of the metal. Our results reveal how the dissipative nature of the heat wave allows us to investigate thick-film dynamics in the thin-film regime, thereby suggesting a general approach for precision measurements of complex NFs, bio-fluids, and optofluidic devices.

Introduction

Nanofluids have emerged as promising heat transport fluids with enhanced thermal conductivity in a wide range of technological applications. Nanofluids are formed by suspending metallic or nonmetallic nanoparticles in a base fluid [START_REF] Choi | Enhancing thermal conductivity of fluids with nanoparticles[END_REF][START_REF] Akilu | A review of thermophysical properties of water based composite nanofluids[END_REF][START_REF] Gupta | A review on thermophysical properties of nanofluids and heat transfer applications[END_REF][START_REF] Tawfik | Experimental studies of nanofluid thermal conductivity enhancement and applications: a review[END_REF] . Nanofluids have opened a new dimension in the enhancement of heat transfer technology merely by adding < 1% nanoparticle mass or volume fraction to the base fluids [START_REF] Choi | Enhancing thermal conductivity of fluids with nanoparticles[END_REF][START_REF] Akilu | A review of thermophysical properties of water based composite nanofluids[END_REF][START_REF] Gupta | A review on thermophysical properties of nanofluids and heat transfer applications[END_REF][START_REF] Tawfik | Experimental studies of nanofluid thermal conductivity enhancement and applications: a review[END_REF] . Since the first study published in 1995 [START_REF] Choi | Enhancing thermal conductivity of fluids with nanoparticles[END_REF] , NFs have attracted different applications such as coolants in automobile transmission systems, electronic cooling applications, solar water heating devices, nuclear reactors, radiators, low-cost spectrally selective optical filters, etc. [START_REF] Tawfik | Experimental studies of nanofluid thermal conductivity enhancement and applications: a review[END_REF][START_REF] Goodarzi | Investigation of heat transfer performance and friction factor of a counter-flow double-pipe heat exchanger using nitrogen-doped, graphenebased nanofluids[END_REF][START_REF] Eshgarf | An experimental study on rheological behavior of nonnewtonian hybrid nano-coolant for application in cooling and heating systems[END_REF][START_REF] Ray | Superior performance of nanofluids in an automotive radiator[END_REF][START_REF] Philip | Nanofluid with tunable thermal properties[END_REF][START_REF] Rasheed | Heat transfer and tribological performance of graphene nanolubricant in an internal combustion engine[END_REF][START_REF] Branson | Nanodiamond nanofluids for enhanced thermal conductivity[END_REF][START_REF] Lamhot | Optical control of thermocapillary effects in complex nanofluids[END_REF][START_REF] Ciraulo | Long-range optofluidic control with plasmon heating[END_REF][START_REF] Taylor | Nanofluid-based optical filter optimization for PV/T systems[END_REF][START_REF] Huang | Efficiency limits of concentrating spectral-splitting hybrid photovoltaic-thermal (PV-T) solar collectors and systems[END_REF] . In these applications, the thermophysical properties of the nanofluid, including its heat transfer characteristics (thermal conductivity, heat capacity) and hydrodynamic properties (surface tension, viscosity), play critical roles on performance [START_REF] Goodarzi | Investigation of heat transfer performance and friction factor of a counter-flow double-pipe heat exchanger using nitrogen-doped, graphenebased nanofluids[END_REF][START_REF] Estellé | Current trends in surface tension and wetting behavior of nanofluids[END_REF] . Therefore, the precise characterization of surface and bulk thermophysical properties of a nanofluid is indispensable in ensuring enhanced performance and predictive capability.

Various techniques have been used to explore the thermophysical properties of NFs made from nanoparticles having different shapes, sizes, the mass/volume fractions both form a theoretical and experimental approach. These includes transient hot-wire, temperature oscillation, and ω methods [START_REF] Tawfik | Experimental studies of nanofluid thermal conductivity enhancement and applications: a review[END_REF][START_REF] Goodarzi | Investigation of heat transfer performance and friction factor of a counter-flow double-pipe heat exchanger using nitrogen-doped, graphenebased nanofluids[END_REF][START_REF] Eshgarf | An experimental study on rheological behavior of nonnewtonian hybrid nano-coolant for application in cooling and heating systems[END_REF][START_REF] Ray | Superior performance of nanofluids in an automotive radiator[END_REF][START_REF] Philip | Nanofluid with tunable thermal properties[END_REF][START_REF] Rasheed | Heat transfer and tribological performance of graphene nanolubricant in an internal combustion engine[END_REF][START_REF] Branson | Nanodiamond nanofluids for enhanced thermal conductivity[END_REF][START_REF] Lamhot | Optical control of thermocapillary effects in complex nanofluids[END_REF][START_REF] Ciraulo | Long-range optofluidic control with plasmon heating[END_REF][START_REF] Taylor | Nanofluid-based optical filter optimization for PV/T systems[END_REF][START_REF] Huang | Efficiency limits of concentrating spectral-splitting hybrid photovoltaic-thermal (PV-T) solar collectors and systems[END_REF][START_REF] Estellé | Current trends in surface tension and wetting behavior of nanofluids[END_REF][START_REF] Richard | A transient hot-wire method for measuring the thermal conductivity of gases and liquids[END_REF][START_REF] Vladkov | Modeling transient absorption and thermal conductivity in a simple nanofluid[END_REF][START_REF] Yang | A new thermal conductivity model for nanorod-based nanofluids[END_REF][START_REF] Prasher | Thermal conductivity of nanoscale colloidal solutions (nanofluids)[END_REF] . Among all developed methods, laser-driven thermocapillary deformation is the most promising approach to characterize the thermophysical properties of liquids due to its non-invasive, fast, and sensitive features. Thermocapillary deformation induced from localized laser heating and its delayed thermal response have been used to measure the thermal diffusivity of and monitor the organic impurities in water [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF][START_REF] Bezuglyi | Effect of delay of the thermocapillary response of a transparent liquid layer during laser heating of the absorbing substrate[END_REF][START_REF] Da Costa | Time evolution of the caustics of a laser heated liquid film[END_REF][START_REF] Zykov | Laser-induced thermocapillary convection in thin liquid layers: effect of thermal conductivity of substrates[END_REF][START_REF] Bezuglyi | New approach to diagnostics of organic impurities in water[END_REF][START_REF] Won | Estimation of the thermocapillary force and its applications to precise droplet control on a microfluidic chip[END_REF] . However, due to its direct laser fluid interaction, thermocapillary deformation has two outstanding challenges which limit its practical application. The first is the fact that it only works for pure fluids. When implemented for nanofluids and bio-fluids, a complex interplay of radiation, thermocapillarity, and scattering forces form which can lead to inaccurate determination of thermophysical properties. The second challenge is that thermocapillary deformation does not work for applications where the pump laser can lead to damage of the fluid. These include soft biological fluids [START_REF] Siposan | Effect of low-level laser radiation on some rheological factors in human blood: an in vitro study[END_REF] or systems where the fluid is confined in a closed surface, such as heat pipes.

In this article, we address these two challenges by proposing a versatile optical technique based on pump-probe interferometry [START_REF] Verma | Time-resolved interference unveils nanoscale surface dynamics in evaporating sessile droplet[END_REF][START_REF] Verma | Universal long-range nanometric bending of water by light[END_REF][START_REF] Verma | Interferometric technique for nanoscale dynamics of fluid drops on arbitrary substrates[END_REF][START_REF] Verma | Nanomechanical effects of light unveil photons momentum in medium[END_REF] to characterize the thermophysical properties of both NFs and biological fluids. We show three exclusively different pump laser-induced heating configurations and demonstrate their wide applicability. Peculiarly, 1) for fluids that cannot be illuminated with the pump laser directly from the top such as biological fluids or NFs, we show that a bottom illumination configuration enables the characterization of the precise thermophysical properties. 2) For applications scenarios where bottom illumination is prohibited, and top illumination is required, such as the characterization of liquids on thick substrates like skin, we demonstrate the precise characterization of NFs by quantitatively isolating the competing effects of all three forces by taking advantage of their different time scales. 3) We show the direct measurement of NF thermophysical properties inside a closed cavity for applications mimicking heat pipes where the fluid cannot be exposed to light or the ambient environment. The measured results from our various configurations characterize the thermophysical quantities without introducing the competing effects of radiation pressure and scattering forces. These results are self-consistent and in good agreement with existing techniques. Our technique works for nearly all liquids and can be applied to a wide range of application scenarios for precise in-situ characterization of the thermophysical properties of complex fluids.

Direct Laser Interactions with Complex NF:

The schematic diagram shown in Fig. 1(a) represents the fundamental processes that are established during direct laser interaction with a complex NF. These include optical momentum induced deformation [START_REF] Verma | Time-resolved interference unveils nanoscale surface dynamics in evaporating sessile droplet[END_REF][START_REF] Verma | Universal long-range nanometric bending of water by light[END_REF] , scattering with nanoparticles [START_REF] Schroll | Liquid transport due to light scattering[END_REF][START_REF] Wunenburger | Fluid flows driven by light scattering[END_REF] in the fluid, heating induced deformation [START_REF] Chaudhary | Picometer resolved nanoscale optomechanics of micro-droplet[END_REF][START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF][START_REF] Chraïbi | Thermocapillary flows and interface deformations produced by localized laser heating in confined environment[END_REF] , as well as potential damage that may be induced by the laser, especially for soft bio-fluids. The surface deformation at the interface occurs as a result of these competing processes. The complex interplay of these processes makes it challenging to investigate the transient surface deformation for precise thermophysical characterization of NFs. Here, we address these challenges and demonstrate the characterization of a NF using three exclusively different types of pump laser-induced heating configurations as shown in Fig. 1(b). Below we discuss each configuration and demonstrate their wide applicability.

Results

Pump laser illumination from the bottom of the substrate: For our first approach, we introduce a configuration where the pump laser is illuminated from the bottom of the substrate. Instead of direct laser liquid interaction, this configuration allows us to precisely characterize the thermophysical properties of complex NFs and biological fluids by eliminating scattering and radiation pressure effects. It also isolates the potential damage from the pump laser for applications when characterizing biological fluids is required [START_REF] Siposan | Effect of low-level laser radiation on some rheological factors in human blood: an in vitro study[END_REF] . In the bottom illumination setup (Fig. 2), we used a solid-state green laser. This laser serves as a heating beam for the generation of thermocapillary convection in the nanofluid by substrate heating. The power of the heating beam was varied depending on experimental requirements. The probe laser beam (He-Ne laser, 𝜆𝜆 = 632 𝑛𝑛𝑛𝑛, beam waist 1 mm) was focused on the liquid surface with a lens (focal length 200 mm) to form a Newton Ring type interference with fringe pattern reported in Refs. 20, 27-30. The focused probe beam has a beam waist of 110 𝜇𝜇 m (see Fig. S1). The experimental procedure includes the following steps: a water-based nanofluid made form 𝐴𝐴𝑙𝑙 2 𝑂𝑂 3 or a hybrid NF (𝐴𝐴𝑙𝑙 2 𝑂𝑂 3 +Graphene-oxide (GO) nanosheet) having initial volume fraction (𝜙𝜙 = 0.05%) were placed in a Copper (Cu) substrate having a large radius of curvature (see Materials and methods). The thickness of the nanofluid layer (ℎ 0 ) was chosen to exclude the influence of thermo-gravitational convection in the process of thermocapillary deformation, ℎ 𝑐𝑐 [START_REF] Zykov | Laser-induced thermocapillary convection in thin liquid layers: effect of thermal conductivity of substrates[END_REF] . We used a transparent glass enclosure to reduce the evaporation rate of the NF so that the concentration variation of the NF during measurement is minimized.

The evaporation of NF produced oscillations in 𝐼𝐼(𝑡𝑡) (blue curve in Fig. 2b) which served as a reference to determine the direction (dimple/bump) of the deformation [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF][START_REF] Verma | Time-resolved interference unveils nanoscale surface dynamics in evaporating sessile droplet[END_REF][START_REF] Verma | Universal long-range nanometric bending of water by light[END_REF][START_REF] Verma | Interferometric technique for nanoscale dynamics of fluid drops on arbitrary substrates[END_REF][START_REF] Verma | Nanomechanical effects of light unveil photons momentum in medium[END_REF] . The static evaporation rate also enabled us to have different concentrations of NF for our experiment. Tracking the central intensity of the fringe 𝐼𝐼(𝑡𝑡) allowed self-calibrating timeresolved measurement of optical thickness variation. One fringe collapse resulted in (𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚 -𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚 ) ≡ 𝜆𝜆/4𝑛𝑛 (for 𝜆𝜆 = 632 𝑛𝑛𝑛𝑛, 𝑛𝑛 = 1.33) = 118 𝑛𝑛𝑛𝑛 change in the optical path length in the NF drop. By further resolving the intensity levels between the maxima and the minima, we achieved remarkable precision of approximately 5 𝑛𝑛𝑛𝑛. We have also shown the connection of the intensity signal in the center of the photodetector with the complex temporal and radial dependence of the laser-induced surface deformation in the fluid (Eqns. S1, S2), and probe beam waist dependent phase shift in and corresponding thermophysical properties change (Figs. S2,S3).

The base fluid, nanomaterials, and human saliva initial properties are listed in Table 1 When a Gaussian laser beam is incident on an absorbing Cu substrate covered with a transparent NF droplet, a non-uniform temperature distribution 𝛥𝛥𝛥𝛥(𝑟𝑟, 𝑡𝑡, 𝑧𝑧) is produced at the solid-NF interface [START_REF] Chaudhary | Picometer resolved nanoscale optomechanics of micro-droplet[END_REF][START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF][START_REF] Chraïbi | Thermocapillary flows and interface deformations produced by localized laser heating in confined environment[END_REF] . The heat from this source distributes both into the depth of the substrate and the bulk of the NF layer. As a result, the isotherm will reach the free surface of the NF after a lag time [START_REF] Bezuglyi | Effect of delay of the thermocapillary response of a transparent liquid layer during laser heating of the absorbing substrate[END_REF][START_REF] Zykov | Laser-induced thermocapillary convection in thin liquid layers: effect of thermal conductivity of substrates[END_REF] of 𝜏𝜏 𝑑𝑑 = ℎ 0 2 /4𝐷𝐷 𝑓𝑓 , where 𝐷𝐷 𝑓𝑓 = 𝑘𝑘/𝜌𝜌𝑐𝑐 𝑝𝑝 denotes the thermal diffusivity of the fluid medium having thermal conductivity 𝑘𝑘 and specific heat 𝑐𝑐 𝑝𝑝 . For small 𝛥𝛥𝛥𝛥(𝑟𝑟, 𝑡𝑡, 𝑧𝑧), all thermal properties are assumed to be constant except for surface tension. The interface deformation ℎ is allowed to vary linearly (Eq. S3) with the liquid temperature [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF][START_REF] Chaudhary | Picometer resolved nanoscale optomechanics of micro-droplet[END_REF] . Temperature variation of the free liquid surface leads to a local decrease in surface tension, 𝛾𝛾 = 𝛾𝛾 0 + 𝛾𝛾 𝑇𝑇 (𝛥𝛥 -𝛥𝛥 0 ) where, 𝛾𝛾 0 is the surface tension of the liquid at a reference temperature 𝛥𝛥 0 , 𝛾𝛾 𝑇𝑇 = 𝑑𝑑𝛾𝛾/𝑑𝑑𝛥𝛥 is surface tension coefficient, which is negative for the majority of pure liquids [START_REF] Bezuglyi | Effect of delay of the thermocapillary response of a transparent liquid layer during laser heating of the absorbing substrate[END_REF][START_REF] Zykov | Laser-induced thermocapillary convection in thin liquid layers: effect of thermal conductivity of substrates[END_REF] . Consequently, a radially outward surface tension gradient forms along the free liquid surface, 𝜕𝜕𝛾𝛾/𝜕𝜕𝑟𝑟 = 𝛾𝛾 𝑇𝑇 (𝜕𝜕𝛥𝛥/𝜕𝜕𝑟𝑟) > 0 that triggers tangential thermocapillary stress along the interface directed toward the regions of highest surface tension. This force is balanced by the velocity gradient (viscus stress) in the bulk layer. This process induces a radially outward thermocapillary flow, which leads to a beamcentered dimple on top of the NF droplet interface [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF] . For most fluids, including 𝐴𝐴𝑙𝑙 2 𝑂𝑂 3 -NF, 𝛾𝛾(𝛥𝛥) decreases with higher 𝛥𝛥 ( 𝛾𝛾 𝑇𝑇 > 0 ) which leads to a thermocapillary dip [START_REF] Chaudhary | Picometer resolved nanoscale optomechanics of micro-droplet[END_REF][START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF] . To measure the transient height of the thermocapillary dimple ℎ(𝑡𝑡) , we detected local interference intensity 𝐼𝐼(𝑡𝑡) of the central fringe with a photo-diode (PD) and a 2 mm aperture opening as shown in Fig. 2(a).

Just after switching on the pump laser beam, the probe signal did not change for a certain time interval, herein referred to as the thermocapillary delay time 𝜏𝜏 𝑑𝑑 of the response. The maximum deformation ℎ(𝑟𝑟 = 0) rapidly increases in time after the delay time 𝜏𝜏 𝑑𝑑 and approaches steady state thereafter. Note that the time to reach the temperature isotherm at the Cu-NF interface for a Cu substrate thickness ℎ 𝐶𝐶𝐶𝐶 = 300 𝜇𝜇𝑛𝑛 is 𝜏𝜏 𝐶𝐶𝐶𝐶 = ℎ 𝐶𝐶𝐶𝐶 2 /4𝐷𝐷 𝐶𝐶𝐶𝐶 = 9 𝑛𝑛𝑚𝑚 which is very small compared to 𝑡𝑡 𝑑𝑑 = 160 𝑛𝑛𝑚𝑚. We demonstrate three different concentrations of NF particle volume fractions of 𝜙𝜙 = 0.1, 0.3 and 0.5% in water to show the experimental signal in Fig. 3(a). We found that for the same experimental parameters (laser power and beam waist), the deformation height for 𝜙𝜙 = 0.1,0.3 and 0.5% with water to have a decreasing order because 𝛾𝛾(𝛥𝛥) decreases with an increase in 𝜙𝜙 . The transient signal in Fig. 3a gives the 𝜏𝜏 𝑑𝑑 and time constant for the thermocapillary deformation 𝜏𝜏 𝑡𝑡ℎ which are related to the diffusivity and viscosity of the NF (as shown in Fig. 3b) and water using the relation 𝜏𝜏 = ℎ 0 2 /(4𝐷𝐷 𝑓𝑓 ) and τ th = 3𝜂𝜂𝑤𝑤 𝑡𝑡ℎ 4 /(𝛾𝛾ℎ 0 3 ) . For example, water gives 𝜏𝜏 𝑡𝑡ℎ = 0.82 𝑚𝑚. This is very large for a thick sample (ℎ 0 = 300 𝜇𝜇𝑛𝑛).

We expect this because heating is a dissipative process; hence the thermal spot size increases at the liquid surface. Therefore, the dynamics of the interface deformation induced by heating start to belong to the thin-film regime, which has a time scale [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF] of 𝜏𝜏 𝑡𝑡ℎ = 3𝜂𝜂𝑤𝑤 𝑡𝑡ℎ 4 /𝛾𝛾ℎ 0 3 . Since we know other parameters in 𝜏𝜏 𝑡𝑡ℎ we found 𝑤𝑤 𝑡𝑡ℎ = 4.8 𝑛𝑛𝑛𝑛. Hence, ℎ 0 /𝑤𝑤 𝑡𝑡ℎ << 1 is satisfied.

Figure 3(c) shows 𝛥𝛥𝛥𝛥(𝑟𝑟, 𝑡𝑡, 𝑧𝑧) and its time derivative at the air-NF interface. The peak in the plot indicates that the present temperature gradient has become sufficiently large enough to initiate observable surface deformation at the NF-air interface. This peak also enables us to calculate the value of 𝜏𝜏 𝑑𝑑 . To obtain 𝜏𝜏 𝑑𝑑 with higher precision, we plot the semi-log of this curve (inset of Fig. 3c), where the maxima verified our approach for this calculation. In Fig. 3(d), one could easily notice that the thermal spot size at the Cu substrate and NF-air interface was different as it was broadened at the air-water interface (also at the air-NF interface, not shown) due to the diffusive nature of heat. This peculiar nature of thermal spot width variation allowed us to investigate the thick NF sample under a thin-film regime and provide reliability in our calculation of thermophysical quantities. To validate our experimental results, we performed a numerical simulation using COMSOL Multiphysics (see Materials and methods). We used the data from Table 1 for the base fluid, different nanoparticles, and saliva as a reference for calculating the physical properties of the NF and saliva using the mixture model [START_REF] Jiang | Numerical investigation of nanofluid thermocapillary convection based on two-phase mixture model[END_REF] .

Hybrid-nanofluids and complex biological fluids (human saliva):

To further demonstrate the wide applicability of our technique, we investigated fluids with more complex components. As a proof-of-concept demonstration, we choose a hybrid-NF and a bio-fluid (human saliva) as two examples and investigated their thermophysical properties. A hybrid nanofluid is a solid-liquid mixture formed by dispersing two or more different nanoparticles suspended in a liquid, which displays enhanced thermophysical properties and rheological characteristics compared to nanofluids made from a single nanoparticle component [START_REF] Lu | Surface tension, viscosity, and rheology of water-based nanofluids: a microscopic interpretation on the molecular level[END_REF][START_REF] Chinnam | Measurements of the surface tension of nanofluids and development of a new correlation[END_REF][START_REF] Richardson | Experimental and theoretical studies of light-to-heat conversion and collective heating effects in metal nanoparticle solutions[END_REF] . We prepared an aqueous 𝐴𝐴𝑙𝑙 2 𝑂𝑂 3 and graphene oxide (𝐺𝐺𝑂𝑂 ) nano-sheet hybrid NF for the characterization of thermal properties. Figure 4(a) presents the timeresolved deformation height and numerical results for the hybrid NF and human saliva samples. Figure 4(b) shows the influence of the NP volume fraction on the dynamic viscosity and thermal diffusivity of the hybrid-NF, which is extracted from Fig. 4(a). The results indicate that both properties of the hybrid-NF increase with increased NP volume fraction in the range from 0% to 0.5%. The results show that the nanoparticle volume fraction had a significant influence on the hybrid nanofluid thermophysical properties because the thermal conductivity of 𝐺𝐺𝑂𝑂 and 𝐴𝐴𝑙𝑙 2 𝑂𝑂 3 nanoparticles were larger than that of deionized water as well as the 𝐴𝐴𝑙𝑙 2 𝑂𝑂 3 NF. The nanoscale sensitivity of our non-invasive optical technique to surface deformation measurement allowed us to characterize the complex biological fluid (human saliva) in a similar non-contact manner without damaging it. A lack of damage is ensured due to the micro-Kelvin scale temperature rise required to achieve a few nanometers deformation of the sample surface. Figures 4(c,d) show that the transient signal of the deformed surface and extracted viscosity and diffusivity of fresh human saliva is consistent with previously reported values [START_REF] Gittings | Characterisation of human saliva as a platform for oral dissolution medium development[END_REF] . It is noticeable that the thermal diffusivity of saliva decreases as water content decrease. Human saliva was collected from a healthy donor, aged 25-35 years, between 9:00 and 11:00 AM. The donor refrained from eating or drinking for 2 h prior to collection. Human saliva is comprised of 99.5 % water and contains many essential substances, including electrolytes, mucus, antibacterial compounds, and various enzymes. Hence for our volume concentration of saliva calculation, we considered the solid content as spherical particles (𝑟𝑟 0 ).

Top pump laser illumination on the NF surface:

We now present our second configuration using top pump laser illumination. This configuration can be used in application scenarios where bottom illumination with the laser is not easily accessible (e.g., semi-infinite substrate, structured surfaces like cloths, skin, etc…). Unlike previous work 20- 25 where the complex interplay of radiation, thermocapillarity, and scattering forces were not considered, here we show that we can precisely characterize NFs by quantitively isolating the competing effects of all three forces by taking advantage of their different time scales. For optically transparent fluids (water) and substrates that do not absorb light at the pump laser wavelength, thermal effects are negligible on the water surface. Light momentum discontinuity results in a radiation pressure 𝛱𝛱 0 = where, 𝐽𝐽 0 is the zeroth-order Bessel function [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF][START_REF] Verma | Universal long-range nanometric bending of water by light[END_REF] . For dilute 𝐴𝐴𝑙𝑙 2 𝑂𝑂 3 NF and bio-fluids, the absorption coefficient (𝐴𝐴 𝑒𝑒 ) is higher than that of water. Therefore, in this case, due to weak heating, we observed a complex interplay between three forces (Fig. 5a) comprised of thermocapillary stress, radiation pressure, and volumetric force due to light scattered 𝑓𝑓 𝑠𝑠𝑐𝑐𝑚𝑚𝑡𝑡𝑡𝑡 by the nanoparticles [START_REF] Schroll | Liquid transport due to light scattering[END_REF][START_REF] Wunenburger | Fluid flows driven by light scattering[END_REF] . In the case of a spherical particle of radius 𝑟𝑟 0 such that 𝑟𝑟 0 << 𝜆𝜆, the order of magnitude of the scattering force density involved in our experiments is inferred from the relationship 𝑓𝑓 𝑠𝑠𝑐𝑐𝑚𝑚𝑡𝑡𝑡𝑡 ≃ 𝜏𝜏𝐼𝐼𝑛𝑛/𝑐𝑐. [START_REF] Schroll | Liquid transport due to light scattering[END_REF][START_REF] Wunenburger | Fluid flows driven by light scattering[END_REF] The turbidity (𝜏𝜏) of the NF has been found to be of the order of 50 -100 𝑛𝑛 -1 . Notice that 𝜏𝜏 ≫ 𝛼𝛼 𝐴𝐴 𝑏𝑏 , confirming that the light-induced force density originates mainly from light scattering. Using 𝑛𝑛 = 1.33 as a characteristic value for the refractive index and 𝐼𝐼 ∼ 2𝑃𝑃/𝜋𝜋𝑤𝑤 𝑒𝑒 2 with 𝑃𝑃 0 = 0.1 -1 𝑊𝑊, we obtained 𝑓𝑓 𝑠𝑠𝑐𝑐𝑚𝑚𝑡𝑡𝑡𝑡 ≃ 1 × 10 2 -2 × 10 3 𝑁𝑁 𝑛𝑛 -3 . These forces are exerted perpendicular to the NF surface, and their competing effect is shown in Figs. 5(b,c). We implemented the𝑓𝑓 𝑠𝑠𝑐𝑐𝑚𝑚𝑡𝑡𝑡𝑡 as a volumetric force in Eq. (2) (see Materials and methods) and radiation pressure and heating induced surface stress at the air-water/NF interface as a boundary condition. Radiation pressure-induced fluid deformation has a fast time scale ( 𝜏𝜏 𝑟𝑟 = 2𝜂𝜂𝑤𝑤 𝑒𝑒 /𝛾𝛾) as one can see in Figs. 5(b,c). The other two forces are related to the bulk flow of liquid. Therefore, radiation pressure fluid deformation appears on a different time scale and becomes distinguishable after some time of laser exposure. Figure 5(b) shows this expected longterm dynamics of interface deformation. Differences in time scales are even more noticeable when the pump laser is turned off. At first, the interface relaxes rapidly. Radiation pressure induces a bulge following the pump beam with steps near the rising/falling edges, as previously observed in transparent fluids [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF][START_REF] Verma | Universal long-range nanometric bending of water by light[END_REF][START_REF] Verma | Nanomechanical effects of light unveil photons momentum in medium[END_REF] . Thermocapillary and scattering force-induced dimple formation exhibited slow dynamics, which were isolated as shown in Fig. 5(b). The scattering force depends on the direction of incident laser light. Hence it could induce a bulge (as the radiation pressure case) on the NF surface when we shine a laser from the bottom of the glass substrate. We also performed an experiment in a thin-film NF sample for sample height (ℎ 0 ) about 70 𝜇𝜇𝑛𝑛 and 𝜙𝜙 = 0.05, which allowed us to measure picometer scale sensitivity of the NF surface deformation. Figure 5(c) shows the power dependent transient probe signal which linearly depends on laser power as shown in Fig. 5(d). The precision in surface deformation measurement is restricted due to the thermal limit, which is associated with the NF's interfacial fluctuations and given by �𝑘𝑘 𝐵𝐵 𝛥𝛥/𝛾𝛾, where 𝑘𝑘 𝐵𝐵 is the Boltzmann's constant. The value for this system is approximately ℎ = 260 pm. Noticeably, we have measured the deformation height of approximately 150 pm, which is beyond the thermal limit. This displacement is measured without any lock-in or modulation as contrary to Ref. 33 as for thin-films as viscous forces dominate and suppress other fluctuations. These different degrees of precision in thick and thin films of the sample are due to bottom frictions of substrate and gravity-capillary waves. For thick films, bottom friction vanishes and gravity-capillary waves are not damped by viscous dissipation [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF] allowing us approximately a 1 nm precision in deformation height measurement. However, bottom substrate friction becomes dominant in the thin-film regime, and gravity-capillary waves are damped by viscous dissipation, allowing us to reach picometer-scale precision.

Nanofluid characterization inside a metallic cavity:

We now introduce our third configuration where we show that the nanofluid inside a metallic cavity can be characterized using our approach. This configuration has its unique advantages in applications where the fluid is confined within a closed surface [START_REF] Chraïbi | Thermocapillary flows and interface deformations produced by localized laser heating in confined environment[END_REF] such as for a heat pipe, or where a fluid cannot be exposed to light or the ambient environment. This is particularly important for applications where volatile organic compounds are present and can be absorbed by the working fluid [START_REF] Oh | The apparent surface free energy of rare earth oxides is governed by hydrocarbon adsorption[END_REF][START_REF] Yan | Atmosphere-Mediated Scalable and Durable Biphilicity on Rationally Designed Structured Surfaces[END_REF][START_REF] Yan | Atmosphere-mediated superhydrophobicity of rationally designed micro/nanostructured surfaces[END_REF][START_REF] Preston | Effect of hydrocarbon adsorption on the wettability of rare earth oxide ceramics[END_REF][START_REF] Cha | Nanoscale-agglomerate-mediated heterogeneous nucleatio[END_REF] . It has been shown that the effect of surface heat transfer from the sample to the surroundings could be meaningful in the determination of physical parameters by laser-excited photothermal lens spectroscopy [START_REF] Lukasievicz | A theoretical and experimental study of timeresolved thermal mirror with non-absorbing heat-coupling fluids[END_REF] . Here we extracted the thermophysical properties of a NF by measuring the thermoelastic response of the laserheated metal (Cu) cavity. The nanometric sensitivity of our technique allowed us to measure the laser heating-induced thermoelastic deformation (see Materials and methods). Figure 6(a) shows the schematic of the set-up and surface thermoelastic response of a metal cavity filled with air, water, and two different concentrations of NFs. We observed (Fig. 6b) that the thermoelastic deformations in the cavity are higher when there is air when compared to water or NF. This is because of the increasingly higher thermal conductivity when water or NF is introduced as a fluid, compared to air. Our results could be used to obtain the physical properties of fluids having low optical absorption by using a reference solid sample in both thermal lens and thermal mirror experiments. This configuration could be useful for the non-invasive measurement of blood or any other fluid inside the viscoelastic material tube (e.g., Nylon, vein/artery). Because the flow of blood or external stresses produce pressure on the vessel wall, wall deformation may occur [START_REF] Gupta | A review on thermophysical properties of nanofluids and heat transfer applications[END_REF] . The thermal, optical, and mechanical properties of the cavity listed below (Table 3) associated with characteristics values found from experimental data (shaded gray) and Ref. 48 are used for the numerical simulations. We now discuss the self-consistency and robustness of our technique. Table 4 summarizes the measured viscosity and surface tension coefficient of a NF, hybrid NF, human saliva, and compares them with literature values. Good agreements are found for all fluids. It is also noteworthy to mention that the measured quantities under all heating configurations are consistent (see Table S1). For example, the measured viscosity of Al2O3 NF (with φ = 0.1) in the bottom (Fig. 3a), top (Fig. 5a) and cavity (Fig. 6b) configurations are 1.14 1.12 and 1.15 mPa•s respectively. The measured results are consistent and very close to the theoretical value of 1.13 mPa•s. Furthermore, the measured quantities are independent of apparatus parameters such as substrate thickness and laser spot size, depicting the robustness of our technique.

Discussion

Four prime capabilities of our technique that prove its versatility in measuring thermophysical and mechanical properties of fluids. The first capability is the nanometric sensitivity (picometer for thin-film) and self-calibrating nature of the technique. This capability allowed us to measure the time-resolved thermophysical properties of NFs, biological fluids in various configurations. The delayed thermocapillary signal directly gives the physical properties (NF and saliva) for the substrate used. An important consequence of heating the substrate from the bottom and NF inside the metal cavity is that it works for nearly all kinds of liquid as it eliminates the effects of scattering and radiation forces, as well as the potential damage induced by the laser. Additionally, measuring the surface deformation beyond the thermal limit without the use of any electrical modulation distinguishes our work from past approaches. The second capability is the ability to measure weakly absorbing NFs thermal properties for a thick substrate by isolating the competing effects of thermal, scattering, and momentum transfer. The fluid dynamics can be obtained by analyzing different time scales in the surface deformation signal. This analysis can also be applicable to structured surfaces, e.g., skin, cloth, where it is difficult to shine laser from the bottom for heating [START_REF] Verma | Interferometric technique for nanoscale dynamics of fluid drops on arbitrary substrates[END_REF] . The third capability is that the transient thermo-elastic signal is very useful for NFs in confined environments, where heat coupling between the solid and NF provides thermophysical properties of the nanofluid only if a solid reference sample is used. The fourth capability is the noninvasive nature of the technique which is crucial for investigating the properties of complex hybrid-NFs and biological fluids [START_REF] Siposan | Effect of low-level laser radiation on some rheological factors in human blood: an in vitro study[END_REF][START_REF] Lu | Surface tension, viscosity, and rheology of water-based nanofluids: a microscopic interpretation on the molecular level[END_REF] . We have demonstrated the measurement of thermophysical properties of human saliva by merely changing the temperature of the sample by a few µK and in a single measurement. Apart from this, our approach is also applicable for high concentration NFs where non-Newtonian behavior (visco-elastic) is expected [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF][START_REF] Lu | Surface tension, viscosity, and rheology of water-based nanofluids: a microscopic interpretation on the molecular level[END_REF][START_REF] Vajjha | Measurements of specific heat and density of Al2O3 nanofluid[END_REF][START_REF] Shah | Temperature-dependent thermal conductivity and viscosity of synthesized 𝛼𝛼-alumina nanofluids[END_REF][START_REF] Dietzel | Laser-induced motion in nanoparticle suspension droplets on a surface[END_REF] . Nanofluid films having thicknesses of a few nanometers inside the cavity would be interesting because the change in Brownian motion of the nanoparticles would play an important role in enhancing the thermal conductivity [START_REF] Prasher | Thermal conductivity of nanoscale colloidal solutions (nanofluids)[END_REF] .

In conclusion, our results open the possibility of applying an interferometric method for the measurement of thermophysical properties of NFs, hybrid-NFs, and biological fluids in three different types of pump laser-induced heating configurations. Future work is needed to extend the precision of our technique below 100 pm by using a twisted light beam as the probe beam [START_REF] Verma | Compact picometer-scale interferometer using twisted light[END_REF] , that can be used to detect femto-Newton-range forces with sensitivity close to the thermal limit [START_REF] Stoev | Highly sensitive force measurements in an optically generated, harmonic hydrodynamic trap[END_REF] . It would also be helpful in laser-induced cooling of solids [START_REF] Clark | Laser cooling in the condensed phase by frequency up-conversion[END_REF] to find the bump on NF/complex fluid surface. Our results and method have the potential make notable contributions to opto-rheology [START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF] , thin-film nanometrology [START_REF] Schmidt | Hybrid nanoparticle-microcavity-based plasmonic nanosensors with improved detection resolution and extended remote-sensing bility[END_REF] , and designing tailored optical propulsion forces for controlled transport of resonant opticalforce-guided microfluidics [START_REF] Rodrigo | Tailored optical propulsion forces for controlled transport of resonant gold nanoparticles and associated thermal convective fluid flows[END_REF] .

𝑘𝑘 𝑛𝑛𝑛𝑛 𝑘𝑘 𝑛𝑛 = 1-𝜑𝜑+2𝜑𝜑 𝑘𝑘 𝑛𝑛 𝑘𝑘 𝑛𝑛 -𝑘𝑘 𝑛𝑛 𝑙𝑙𝑚𝑚� 𝑘𝑘 𝑛𝑛 +𝑘𝑘 𝑛𝑛 2𝑘𝑘 𝑛𝑛 � 1-𝜑𝜑+2𝜑𝜑 𝑘𝑘 𝑛𝑛 𝑘𝑘 𝑛𝑛 -𝑘𝑘 𝑛𝑛 𝑙𝑙𝑚𝑚� 𝑘𝑘 𝑛𝑛 +𝑘𝑘 𝑛𝑛 2𝑘𝑘 𝑛𝑛 � (1)
Temperature Fields: We used the Heat Transfer in Solid Fluids module to find temperature rise distribution within the sample and substrate. This is given by the solution of the coupled heat conduction differential equation [START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF] ,𝜕𝜕 𝑡𝑡 𝛥𝛥 𝑚𝑚 (𝑟𝑟, 𝑧𝑧, 𝑡𝑡) -𝐷𝐷 𝑚𝑚 𝛻𝛻 2 𝛥𝛥 𝑚𝑚 (𝑟𝑟, 𝑧𝑧, 𝑡𝑡) = 𝑞𝑞 𝑚𝑚 𝑞𝑞(𝑟𝑟, 𝑧𝑧) with proper boundary and initial conditions as given in Ref. 34. Where, 𝐷𝐷 𝑚𝑚 = 𝑘𝑘 𝑚𝑚 /𝜌𝜌 𝑚𝑚 𝑐𝑐 𝑝𝑝𝑚𝑚 is the thermal diffusivity, 𝑘𝑘 𝑚𝑚 is the thermal conductivity, 𝜌𝜌 𝑚𝑚 is the mass density, and 𝑐𝑐 𝑝𝑝𝑚𝑚 is the specific heat for the solid substrate (𝑖𝑖 = 𝑚𝑚) and fluid (𝑖𝑖 = 𝑓𝑓). The laser as a heat source is defined as

𝑞𝑞 𝑠𝑠 = 2𝑃𝑃(1 -𝑅𝑅)𝜙𝜙 𝑙𝑙ℎ /𝜋𝜋𝑐𝑐 𝑠𝑠 𝜌𝜌 𝑠𝑠 𝑤𝑤 𝑒𝑒 2 and 𝑞𝑞(𝑟𝑟, 𝑧𝑧) = 𝑒𝑒𝑒𝑒𝑒𝑒 � -2𝑟𝑟 2
𝑤𝑤 𝑒𝑒 (𝑧𝑧) 2 � 𝑒𝑒𝑒𝑒𝑒𝑒(-𝐴𝐴 𝑏𝑏 𝑧𝑧). Since the fluid is assumed to be non-absorbing, 𝑞𝑞 𝑓𝑓 = 0. The portion of the absorbed power converted into heat is represented by 𝜙𝜙 𝑙𝑙ℎ , which in this case is ≃ 1. We used boundary heat source in COMSOL as a boundary condition, where 𝑃𝑃 0 is the excitation power, 𝑅𝑅 is the sample surface reflectivity. The heat flux or losses from boundaries is specified by 𝑄𝑄 𝑧𝑧 = -𝑛𝑛 �⃗ ⋅ 𝛻𝛻𝛥𝛥 = ℎ 𝑁𝑁 (𝛥𝛥 -𝛥𝛥 ∞ ) + 𝜀𝜀𝜎𝜎 𝐵𝐵 (𝛥𝛥 4 -𝛥𝛥 ∞ 4 ) where 𝑛𝑛 �⃗ is the outward-oriented unit normal vector. ℎ 𝑁𝑁 is the convective heat transfer coefficient of the substrate, 𝜎𝜎 𝐵𝐵 is the Stefan-Boltzmann constant, 𝜀𝜀 is the emissivity of the substrate. 𝑣𝑣 describes the flow velocity, 𝑃𝑃 is the pressure, 𝜌𝜌 is the fluid density, 𝜇𝜇 is the dynamic viscosity and 𝐹𝐹 𝑣𝑣 is the volume force. The ratio between the convective and diffusive mechanisms is given by the thermal Peclet number, which is defined as 𝑃𝑃𝑒𝑒 𝑡𝑡ℎ = 𝐶𝐶 𝑝𝑝 𝑈𝑈𝑈𝑈/𝑘𝑘, where 𝑈𝑈 is a characteristic distance, the height ℎ 0 of the liquid layer in this case and 𝑈𝑈 is a characteristic velocity. The Peclet number attains a maximum of 100 at the interface near the laser spot and values below 20 elsewhere. In consequence, the convective transfer mechanism is dominant in the system, meaning that the flow velocity field will have a significant effect on the temperature field. Maximum attained Reynolds number 𝑅𝑅𝑒𝑒 = 𝑈𝑈𝑈𝑈/𝜇𝜇 is obtained at the interface near the laser spot with a value of ≃ 10 -4 and values below this elsewhere. This means that the convective transfer mechanism is the dominant one and the flow is considered to be laminar. We used our extracted experimental parameters as input parameters for simulation e.g., viscosity, and surface tension coefficient (see Table 1), and other thermophysical properties (density, heat capacity, and thermal conductivity) of the nanofluid using Table 2 as a function of volume fraction using the relation given in Refs. 39,50.

Thermoelastic Displacement: The Theory of Elasticity can be used to calculate the surface displacement caused by a laser-induced, non-uniform temperature distribution. In the quasi-static approximation, the thermoelastic equation [START_REF] Lukasievicz | A theoretical and experimental study of timeresolved thermal mirror with non-absorbing heat-coupling fluids[END_REF] is given by:

(1 -2𝑣𝑣)𝛻𝛻 2 𝑢𝑢(𝑟𝑟, 𝑧𝑧, 𝑡𝑡) + 𝛻𝛻[𝛻𝛻 ⋅ 𝑢𝑢(𝑟𝑟, 𝑧𝑧, 𝑡𝑡)] = 2(1 + 𝑣𝑣)𝛼𝛼 𝑇𝑇 𝛻𝛻𝛥𝛥(𝑟𝑟, 𝑧𝑧, 𝑡𝑡)

Here, 𝑢𝑢(𝑟𝑟, 𝑧𝑧, 𝑡𝑡) is the displacement vector, 𝑣𝑣 is Poisson's ratio, and 𝛼𝛼 𝑇𝑇 is the linear thermal expansion coefficient. We applied laser heat source as a boundary condition (in Heat Transfer module) at the free surface of a Cu and "Structural Mechanics Module" to solve the above equation for thermoelastic displacement. We applied a no-slip boundary condition at the solid and fluid interface and a fixed boundary condition at the edges. As shown in Fig. 6, the experimental and FEA results are in good agreement. Initially, we experimented air-filled cavity for a given laser power 2 𝑊𝑊 and 𝑤𝑤 𝑒𝑒 = 105 𝜇𝜇𝑛𝑛 in crosscheck the value of Cu properties. Then, we experimented with water and NFs.

Figure 1 .

 1 Figure 1. Direct laser interactions with a complex NF. (a) Schematic depicting the fundamental processes that can exist during direct laser interactions with complex NFs. The three square insets beneath the schematic illustrate the direction of interface deformation (orange arrows) induced by optical momentum (left), heating (middle), and scattering (right). Schematics not to scale. Green arrows represent the laser incident direction. The complex interplay of these processes makes it challenging to precisely characterize complex NFs using laser-based techniques. (b) Schematics of the proposed configurations including bottom illumination, top illumination, and cavity configurations.

Figure. 2

 2 Figure.2 Schematic of the pump-probe set-up for the bottom illumination configuration. a. The schematic of the experimental setup. A solid-state green laser (532 nm, 1-5 W) serves as a heating beam, and a red laser (632 nm, 20 mW) probes the thermocapillary induced deformation. BS: 50:50 beam splitter, M: mirror, L: lens (focal length 200 mm), T = 295 K, and relative humidity 50%. Inset: Photograph of an NF during probing from the bottom. b. An exemplary transient signal on a photodiode.

Figure 3 .

 3 Figure 3. Time-resolved deformation height and numerical results for NF in a bottom illumination configuration. a. Time-resolved surface deformation height h and shutter signal for four liquids including water (blue curve) and three Al2O3 NFs with volume fractions of 0.1% (brown curve), 0.3% (green curve), and 0.5% (pink curve). Experimental conditions: P0 = 0.2 W and we =105 µm. The solid red lines which fit on each height curve represent finite element analysis (FEA) results. b. Experimentally measured dynamic viscosity as a function of nanoparticle concentrations calculated from the thin film dynamics time scale 𝜏𝜏 𝑇𝑇ℎ = 3𝜂𝜂𝑤𝑤 𝑇𝑇ℎ 4 /𝛾𝛾ℎ 0 3 . c. FEA plot of temperature variation at the fluid surface and its derivative. For clarity, only water and 𝜑𝜑 = 0.5% is shown. Inset: semi-log plot of the rising temperature profile. d. Radial temperature profile at the air-NF interface. Inset: simulated surface plot of the temperature distribution.

Figure. 4

 4 Figure. 4 Time-resolved deformation height and numerical simulations for a hybrid NF and human saliva in the bottom illumination configuration. (a, c) Time-resolved deformation height and shutter signal for hybrid-NF and human saliva fluids for P0 = 0.2 W and we =105 µm. Thermal diffusivity was extracted directly by measuring the delay time (τd): Dnf = h0 2 /4τd. (b, d) Viscosity and diffusivity (right y-axis) of Al2O3 (green), hybrid NF (orange) and human saliva (blue) as a function of nanoparticle concentration. Solid lines represent lines of best fit with, 𝜂𝜂 = 𝜂𝜂 0 (1 + 2.5 𝜙𝜙).

  where 𝐼𝐼 0 = 2𝑃𝑃 0 /𝜋𝜋𝑤𝑤 𝑒𝑒 2 . The radiation pressure produces a transient nanoscale bulge having a time scale[START_REF] Verma | Interferometric technique for nanoscale dynamics of fluid drops on arbitrary substrates[END_REF][START_REF] Verma | Contactless thin-film rheology unveiled by laser-induced nanoscale interface dynamics[END_REF][START_REF] Wunenburger | Fluid flows driven by light scattering[END_REF] 𝜏𝜏 𝑟𝑟 = 2𝜂𝜂𝑤𝑤 𝑒𝑒 /𝛾𝛾 and an attained stationary height, ℎ 𝑚𝑚 (𝑟𝑟) = 𝑃𝑃 0 (𝑚𝑚-1) 𝜋𝜋𝑐𝑐(𝑚𝑚+1) ∫ 𝑘𝑘𝐽𝐽 0 (𝑘𝑘𝑟𝑟)𝑒𝑒 -𝑤𝑤 𝑒𝑒 2 𝑘𝑘 2 /8

Figure. 5

 5 Figure. 5 Time-resolved deformation height and numerical results for a NF with picometer sensitivity in the thin-film regime in the top illumination configuration. a. Transient probe laser signal. Inset: schematic of the corresponding NF surface deformation induced by the pump laser. Schematic not to scale. b. Transient surface deformation, h(t) measurement for water, and two different NFs having different Al2O3 NP concentrations. The solid red lines represent FEA results. c. Time-resolved deformation height and numerical results for an NF demonstrating picometer sensitivity in the thin-film regime, for a variety of input laser powers. d. Variation of h(t) and ∇T(r,z,t) (right axis) with pump power using an NF. The left inset demonstrates the time-resolved curve signal used to obtain the surface deformation and the magnification of the curve in the right inset shows the 150 pm resolutions of the measurement.

Figure 6 .

 6 Figure 6. Schematic of the metal cavity setup and measured time-resolved deformation height. a. Schematic of the experimental set-up. Schematic not to scale. b. Measured thermoelastic deformation height ℎ(𝑡𝑡) of the cavity top surface (Cu-Fluid-Cu) for an air-filled cavity, water filled cavity, and a cavity filled with two different NFs having different concentrations of NPs. Solid lines represent FEA results calculated from Eq. (3). The inset represents a reduced range view of ℎ(𝑡𝑡) near the saturation regime. Numerically calculated temperature and thermoelastic bulge profile of the cavity surface are also shown in Fig. S4.

Thermocapillary Deformation:

  The Laminar Two-Phase Flow, Moving Mesh module was used to solve the Navier-Stokes equation for incompressible flow. 𝜌𝜌 𝜕𝜕𝑣𝑣 𝜕𝜕𝑡𝑡 + 𝜌𝜌(𝑣𝑣 ⋅ 𝛻𝛻)𝑣𝑣 = -𝛻𝛻𝑃𝑃 + 𝜇𝜇𝛻𝛻 2 𝑣𝑣 + ∑𝐹𝐹 𝑣𝑣 (2)

Table 1 .

 1 , which have been obtained from Refs. 36,37,38. Base fluid, nanomaterials, and human saliva properties at T = 295 K.

	Parameters	Water	𝐀𝐀𝐀𝐀 𝟐𝟐 𝐎𝐎 𝟑𝟑	GO	Saliva
	𝝆𝝆 (𝐤𝐤𝐤𝐤 𝐦𝐦 -𝟑𝟑 )	997	3987	1800	1002
	𝒌𝒌 (𝐖𝐖 𝐦𝐦 -𝟏𝟏 𝐊𝐊 -𝟏𝟏 )	0.6	40	3000	0.58
	𝐜𝐜 𝐩𝐩 (𝐉𝐉 𝐤𝐤𝐤𝐤 -𝟏𝟏 𝐊𝐊 -𝟏𝟏 )	4180	773	790	3760
	𝜼𝜼 (𝐦𝐦𝐦𝐦𝐦𝐦 𝐒𝐒)	0.89	-	-	1.12
			.

Table 2 .

 2 Additional physical parameters obtained from the experimental data of Figures3 and 4.

			Parameters	Water	𝐀𝐀𝐀𝐀 𝟐𝟐 𝐎𝐎 𝟑𝟑	𝐀𝐀𝐀𝐀 𝟐𝟐 𝐎𝐎 𝟑𝟑 + GO	Saliva
			𝝉𝝉 𝒅𝒅 (𝒎𝒎𝒎𝒎)	156	154, 148, 144	145, 143, 140	140, 155, 160
	-	𝒅𝒅𝒅𝒅 𝒅𝒅𝒅𝒅	(𝟏𝟏𝟏𝟏 -𝟑𝟑 𝑵𝑵 𝒎𝒎 -𝟏𝟏 𝑲𝑲 -𝟏𝟏 )	0.18	0.28, 0.26, 0.23 0.16, 0.15, 0.13 0.18, 0.17, 0.16

Table 3 .

 3 Base fluid, nanomaterial, and initial substrate properties at room temperature.

	Parameters	Units	Values
	𝒌𝒌 𝐬𝐬,𝐦𝐦𝐚𝐚𝐚𝐚,𝐰𝐰𝐦𝐦𝐰𝐰𝐰𝐰𝐚𝐚	( W m -1 K -1 )	201, 0.026, 0.6
	𝑫𝑫 𝐬𝐬,𝐦𝐦𝐚𝐚𝐚𝐚,𝐰𝐰𝐦𝐦𝐰𝐰𝐰𝐰𝐚𝐚	(10 -6 m 2 s -1 )	6.94, 21.9, 0.14
	𝑫𝑫 𝐧𝐧𝐧𝐧 𝝓𝝓=𝟏𝟏.𝟏𝟏,𝟏𝟏.𝟓𝟓	(10 -6 m 2 s -1 )	0.14, 0.17
	𝝆𝝆 𝐬𝐬,𝐦𝐦𝐚𝐚𝐚𝐚,𝐰𝐰𝐦𝐦𝐰𝐰𝐰𝐰𝐚𝐚	( kg m -3 )	8960, 1.17, 998
	𝒄𝒄 𝐬𝐬,𝐦𝐦𝐚𝐚𝐚𝐚,𝐰𝐰𝐦𝐦𝐰𝐰𝐰𝐰𝐚𝐚	( J kg -1 K -1 )	384, 1005, 4180
	𝑹𝑹	-	0.52
	𝜶𝜶 𝐓𝐓	(10 -6 K -1 )	10.3
	𝒗𝒗 𝐬𝐬	-	0.31
	𝑬𝑬 𝒎𝒎	(10 9 Pa)	120
	𝒉𝒉 𝐂𝐂𝐂𝐂 (𝐓𝐓𝐓𝐓𝐩𝐩)	μm	300
	𝒉𝒉 𝐂𝐂𝐂𝐂 (𝐁𝐁𝐓𝐓𝐰𝐰𝐰𝐰𝐓𝐓𝐦𝐦)	μm	500

Table 4 .

 4 Measured dynamic viscosity and surface tension coefficient of a NF (grey shaded rows) and literature reported values for Al 2 O 3 (Ref. 36), Al 2 O 3 + GO (Ref. 37) and saliva (Ref. 38). The error bar is shown in figure 3.

			Parameter	Water	𝐀𝐀𝐀𝐀 𝟐𝟐 𝐎𝐎 𝟑𝟑	𝐀𝐀𝐀𝐀 𝟐𝟐 𝐎𝐎 𝟑𝟑 +GO	Saliva
					(ϕ=0.1,0.3 &	(ϕ=0.1,0.3 &0.5)	(ϕ=0.5, 0.7 &
					0.5)		0.9)
			𝛈𝛈 (𝐦𝐦𝐦𝐦𝐦𝐦 • 𝐬𝐬)	1.03	1.14,1.19, 1.27 1.18,1.25, 1.38 1.74,3.48, 9.20
				0.95	1.13,1.17, 1.26 1.17,1.27, 1.39 1.75,3.40, 9.10
	-	𝐅𝐅𝐝𝐝 𝐅𝐅𝐓𝐓	(𝟏𝟏𝟏𝟏 -𝟑𝟑 𝐍𝐍 𝐦𝐦 -𝟏𝟏 𝐊𝐊 -𝟏𝟏 )	0.18 0.19	0.28,0.26,0.23 0.16,0.15,0.13 0.29,0.27,0.21 0.17,0.16,0.12	0.17, 0.15, 0.11 0.18,0.16,0.13

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant no. 62134009, 62121005), the Innovation Grant of Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP), Jilin Provincial Science and Technology Development Project [grant no. YDZJ202102CXJD002], Bill & Melinda Gates Foundation (INV-009181).

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Materials and methods

Nanofluid preparation:

The average diameter of Al2O3 nanoparticles (provided by Nachen, Beijing, China) is 40 nm. Morphologies of these particles are basically spherical or near-spherical. Al2O3-water nanofluids with 0.01 vol.% ~ 0.5 vol.% were produced by a two-step method without any surfactant. Ultrasonic vibration with an ultrasound generator (20 kHz, 100 W) was used to disperse Al2O3 nanoparticles in DI water for three hours. Figure 7 shows photographs of Al2O3-water.

We quantified the concentration of the evaporating NF during experiments by measuring the drop height as shown in Fig. 7. Author contributions G.V., J. P. and W.L. jointly conceived the idea. G.V. and C. S. S performed the experiments. G. V., G. Y., L. L., N. M., J. D. and W. L. performed simulations and analyses. G.V., N. M. and W. L wrote the manuscript with input from all authors. W. L. supervised the project.

Conflict of interest

The authors declare no competing interests.


N Dietrich
email: dietrich@ensic.inpl-nancy.fr

S Poncin

Huai Z Li

Experimental Investigation of Bubble Formation in Micro-devices

Keywords: bubble formation, micromixer, flow focusing, correlation, µ-PIV

The multiphase flows (gas-liquid-solid) in Newtonian and non-Newtonian fluids are involved in many industrial processes. Among these processes we can mention polymer devolatization, bubble columns, cosmetics, petrochemical, water treatment and bioprocesses. These systems are generally accompanied by heat and mass transfer as well as chemical reactions. In this context, the effectiveness of the heat and mass transfers is a main parameter that depends primarily on the nature of the gas-liquid or solid-liquid interfaces. In addition, some reactions, such as the fluorination of toluene are very dangerous and difficult to achieve at a large scale. Thus the realization of this kind of reactions in microsystems is much more safe and effective.

 focused on the application of these processes at smaller scales, ranging from one hundred micrometers to a millimeter, leading to the emergence of gas-liquid micro-reactors. In the past, several studies have been focused on certain aspects of these flows, particularly the exchange of mass and pressure differences, but most of these studies are numerical and very few experimental studies have been conducted yet.

. A correlation is proposed to estimate the bubble size for the different micromixers. This paper contributes to an original improvement in the domain where a few flow field characterisation and micro mixers comparisons have been published in the literature.

Introduction

Multiphase flows in microfluidic devices have received more attention because of the foreseeable advantages that unique microscale properties have to offer with regard to enhanced heat and mass transfer efficiency, reduced axial dispersion and smaller sample volumes. In order to realize these benefits, a good understanding of the complex multiphase flow behavior in microfluidic devices must be developed. Formation of bubbles serves various applications in industrial processes such as the generation of biogas bubbles by anaerobic sludge granules in a bioreactor [START_REF] Wu | Disruption of granules by hydrodynamic force in internal circulation anaerobic reactors[END_REF], bubble nucleation in polymer devolatization processes [START_REF] Frank | Bubble nucleation and growth in fluids[END_REF], two-phase micro-mixing [START_REF] Garstecki | Formation of droplets and bubbles in a microfluidic T-junctionscaling and mechanism of break up[END_REF], fluorinations [START_REF] Chambers | Elemental fluorine -Part 13. Gas-liquid thin film microreactors for selective direct fluorination[END_REF], hydrogenations [START_REF] Kobayashi | A Microfluidic Device for Conducting Gas-Liquid-Solid Hydrogenation Reactions[END_REF], biochemical reactions such as DNA analysis [START_REF] Burns | An Integrated Nanoliter DNA Analysis Device[END_REF], micro-channel heat exchangers [START_REF] Qu | Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink[END_REF], materials synthesis [START_REF] Yen | A microfabricated gas-liquid segmented flow reactor for high-temperature synthesis: The case of CdSe quantum dots[END_REF][START_REF] Zhang | Microfluidic production of biopolymer microcapsules with controlled morphology[END_REF], drug discovery [START_REF] Dittrich | Lab-on-a-chip: microfluidics in drug discovery[END_REF].

The two-phase flow patterns in microchannels are determined by the flow conditions, the channel geometry and the properties of both fluids [START_REF] Waelchli | Two-phase flow characteristics in gas-liquid micro reactors[END_REF]. The bubbly flow is characterized by the formation of single spherical bubbles with bubble lengths smaller than, or equal to the channel width. Increasing the gas flow rate causes the coalescence of small bubbles leading to cylindrical bubbles (separated from the wall by a thin film), this regime is known as slug. Generally, bubbly flow appears at high liquid velocities and low gas velocities and slug flow occurs at intermediate gas and liquid velocities while annular flow appears usually at high gas velocities. In this study experiments are mainly focused on the segmented gas-liquid flows, i.e., bubbly and slug regimes. The most popular geometries for the production of dispersed phases are T-junctions [START_REF] Garstecki | Formation of droplets and bubbles in a microfluidic T-junctionscaling and mechanism of break up[END_REF][START_REF] Van Der Graaf | Lattice Boltzmann Simulations of Droplet Formation in a T-Shaped Microchannel[END_REF][START_REF] Guillot | Stability of parallel flows in a microchannel after a T junction[END_REF][START_REF] Nisisako | Droplet formation in a microchannel network[END_REF][START_REF] Thorsen | Dynamic pattern formation in a vesicle-generating microfluidic device[END_REF][START_REF] Tice | Formation of droplets and mixing in multiphase microfluidics at low values of the Reynolds and the capillary number[END_REF][START_REF] Xu | Controllable gas-liquid phase flow patterns and monodisperse microbubbles in a microfluidic T-junction device[END_REF] and flow-focusing devices [START_REF] Anna | Formation of dispersions using ''flow focusing'' in microchannels[END_REF][START_REF] Cubaud | Bubble dispenser in microfluidic devices[END_REF][START_REF] Ganan-Calvo | Perfectly monodisperse microbubbling by capillary flow focusing[END_REF][START_REF] Garstecki | Mechanism for flowrate controlled breakup in confined geometries[END_REF]. [START_REF] Garstecki | Mechanism for flowrate controlled breakup in confined geometries[END_REF] used the flow focusing device with a small orifice to generate monodispersed bubbles. They found that the bubble formation was due to the pressure gradient and the breakup could be controlled by the flow rate of the continuous liquid phase. Cubaud and Ho (2004) studied the formation of bubbles in square micro-channels. The bubbles were generated in a cross-shaped mixing section. Theses authors reported that the breakup mechanism in their device could be understood as the competition between the pressure drops respectively in the gas and liquid phases.

In order to gain more insight into the bubble breakup in such microfluidics devices, liquid flow field velocities were investigated using µ-PIV techniques. The µ-PIV technique was successfully used to characterize single-phase flow in microfluidic T-junctions [START_REF] Lindken | Stereoscopic micro particle image velocimetry[END_REF] and segmented multiphase flows in labs-on-chips. The µ-PIV studies in literature mainly deal with the flow or the formation in one channel and very few papers are devoted to the velocity flow field at a bubble formation. Velocity fields obtained by µ-PIV at bubble formation were reported for a T-junction [START_REF] Van Steijn | µ-PIV study of the formation of segmented flow in microfluidic T-junctions[END_REF] and for a Y-junction [START_REF] Dietrich | Dynamique des bulles dans un micro-réacteur[END_REF] but it is not the case for the flow focusing micro-mixer to our best knowledge.

For many industrial applications, it is essential that one can predict the geometric features of the flow pattern, such as bubble and slug length. In several studies, correlations have been proposed to determine the characteristics of the bubble (length or volume). Ganan-Calvo (1998) proposed a correlation for jet flows; Ganan-Calvo and Gordillo (2001) studied a cross flow focusing mixer and proposed a correlation between the bubble length and flow rate ratios without including physical properties. [START_REF] Cubaud | Bubble dispenser in microfluidic devices[END_REF] showed that the length of the produced confined bubbles follows a law based on the channel size and the liquid volume fraction. [START_REF] Garstecki | Formation of droplets and bubbles in a microfluidic T-junctionscaling and mechanism of break up[END_REF] studied a cross flow micro-mixer and also developed a simple correlation between flow rates ratios and the length of the bubble. To our knowledge, no correlation based on the bubble volume or on the physical properties of the liquid phase has been proposed and the existing correlations have always been based on only one geometry of micro-mixer.

The effect of the micromixer geometry was investigated by [START_REF] Haverkamp | Hydrodynamics and Mixer-Induced Bubble Formation in Micro Bubble Columns with Single and Multiple-Channels[END_REF]. The authors designed two mixing geometries, namely, the "T-type" and the "smooth" mixers and studied the flow of gas-water in both mixers. They reported that the breakup by pressure gradient was only observed in the T-type mixer, while the jet instability was the only mechanism for bubble formation in the smooth mixer. [START_REF] Fan | Experiment and lattice Boltzmann simulation of two phase gas-liquid flows in microchannels[END_REF] studied two types of mixer geometries including the cross-shape and the converging shape channels. The bubble shape and size and the formation mechanism were investigated for different flow rates and different mixer geometries. Theses authors compared simulated results with experimental data based on dimensionless numbers. A good agreement was found in general. Different flow regimes with different bubble shapes were found depending on the capillary number of the flow. The simulation confirmed that the breakup was induced by the pressure difference in both phases. The geometry of the mixing section was also observed to have an impact on the size of the gas and liquid slugs, but no experimental quantification of bubble size and velocity flow field has been done. The present paper is focused on the formation of bubbles in flowfocusing micro-reactors of different geometries. Since the most important forces in multiphase flows are the inertial force, viscous force and interfacial force, the goal of our study is to understand the importance of theses forces for different reactor geometries in order to give more insight into then bubble formation in flow focusing micro-mixers. A µ-PIV system and a high speed microscopic camera were employed for these reactors leading to new features in this domain.

Experimental setup

The different geometries of microfluidics devices used in this study are reported in Figure 1. The micro-channels were fabricated in polymethyl methacrylate (PMMA). This flow-focusing geometry had a central channel for the gas dispersed phase flow and two side channels for the inlet of the continuous liquid phase. Two sizes of liquid inlet and outlet channels (600 µm and 1000 µm) and of gas inlet channels (200 µm and 500µm) were used. The section of the gas inlet was circular in order to avoid wetting problems, the other sections being square to assure a better view of the flow field. A pressurized tank of 10 -3 m 3 was used to maintain a constant pressure and to push the liquid and air streams into the micro-channel with a regular flow rate of both phases. A gas flowmeter is used to determine the flow rate with precision. Images of bubbles were captured at steady state on the high speed camera CamRecord600 from Optronis GMBH (Kehl, Germany), capable of 500 frames per second at full resolution (1280×1024) with a microscopic objective ranging from × 100 to × 600. The length of the bubble was determined by an image analysis software and the volume of the bubble was calculated from the gas flow rate and the bubble formation frequency determined by the high speed camera as follows:

formation g b f Q v = (1)

Gas flow

Liquid flow

The Instantaneous liquid velocity flow fields were measured by a µ-PIV of DANTEC Dynamics (Skovlunde, Danemark). The system consisted of a Flowsense Dantec Camera with a 2048×2048 pixel array and a 7 Hz frequency. The microscope was equipped with different objectives ranging from ×5 to ×100. The micro-device under investigation faced the microscope and was illuminated from the back by a micro-strobe emitting at 530×10 -9 m. The liquid velocity was measured by imaging (and evaluating) the motion of seeding particles suspended in the fluid. The principle of PIV is reported in Figure 2. The camera takes two successive images at the maximum intensity of the micro stroboscope impulse. The acquired images of the flow are evaluated by dividing the images into a few thousands of small interrogation areas of 16×16 pixels. A cross-correlation is then applied on the interrogation areas in consecutive images with a 50% window overlap. For each interrogation area, the displacement vector is calculated from the location of the correlation peak. The velocity field is then calculated from the displacement vectors and the time chosen between the images. The depth of field of the microscope is between 10 and 40 µm, which means that measurements are not realized on a plane but on a slice of the micromixer. However, this value is negligible in comparison with the thickness of the microchannel. To obtain velocity flow fields, the tracer particles must perfectly follow the flow. Seeding should be uniform and at a reasonable concentration. The density should be similar to the studied fluid. If the tracer used is made of too small particles, Brownian motion (Robert Brown, 1927) can influence the measurements. This random motion of particles sets a lower limit to achieve µ-PIV measurements. Starting from the equation of Einstein-Sutherland (1905), one can estimate the influence of this Brownian motion. Equation 2 establishes the relationship between the diffusion coefficients of a colloid D and colloid properties. (2)

T is the temperature, µ the viscosity of the fluid, r is the radius of the particle and k the Boltzmann constant. This equation shows that more the radius of the particle is low more the Brownian motion is great. The standard deviation of the random movement of a particle is given by equation 3:

t D 2 p x p δ ≈ ∆ = Σ (3)
Thus, for a camera acquisition frequency of 4 s -1 and a particle diameter of 200×10 -9 m, the random movement is estimated at 3×10 -6 m.s -1 . The values of the liquid velocity used in this study being of about a few hundred µm per second, one gets an error of several percent on the velocity estimation. We must therefore focus on larger particles to avoid problems due to Brownian motion. But bigger the particle is, bigger is its settling velocity. The sedimentation velocity of a particle diameter d is estimated as follows:

(

) µ ρ - ρ = 18 p g g 2 d p V (4)
For seeding particles of 3×10 -6 m, the value of the sedimentation velocity becomes of the order of 1.5×10 -6 m per second. Finally to contain the contribution of both Brownian motion and settling velocity less than 1% of the measured velocity, the optimal size of seeding particle has to be in the range [0.5-3×10 -6 m]. Furthermore, the size of the geometry, viewing windows and objectives also will help to refine the value of the particle diameter. In this study, hydrophilic latex microspheres (Merck Estapor, Fontenay sous Bois, France) with a density of 1056 kg.m -3 and a mean diameter calibrated of 0.88×10 -6 m were used as seeding particles. These particles were small enough to follow the fluid and large enough to avoid the Brownian motion effects. When the flow is correctly inseminated, the measurement errors of the measured velocities are less than 5%.

In this work, the experiments were performed using air as gas phase and three different liquids (pure water, viscous Newtonian Emkarox HV45 10% and 20% wt dilute solutions in demineralised water). A Rheometric Fluid Spectrometer RFS II (Rheometric Scientific) was employed to measure the rheological properties of the liquids (Table 1). The surface tensions and the contact angle of the liquid on PMMA surface were measured using a tensiometer, by the pending drop technique on a Tracker apparatus from I.T. Concept (Longessaigne, France). Sodium Dodecyl Sulphate surfactant (SDS, Amersco, USA) was also used to control the tension force, which allows to compare separately the effect of tension force and viscosity.

Experimental results

In this work, experimental results of the bubble formation obtained in three different micromixer geometries (180, 90 and 60°) are presented. In order to understand the key parameter of the bubble formation process, a preliminary study was realized to understand the influence of each factor. The gas and liquid flow rates ranged from 10 -12 m 3 s -1 to 10 -6 m 3 s -1 , which corresponded to bubbly and slug flow regimes and bubble diameters between 50×10 -6 m and 5×10 -3 m. Our first contribution in this work is to characterize the bubble formation using µ-PIV and high speed camera images. The evolution of the bubble shape at different time during the formation process of a 10 -9 m 3 bubble growth up to its detachment is shown in Fig. 3. The flow was highly periodic with a bubble frequency of 34 s -1 . Figure 3 shows the growth process during one period. The bubble shapes were obtained from a high speed camera images. One can distinguish three steps during the formation of the bubble: a first rapid expansion step of the growing bubble, followed by a linear increase of the bubble during the second step, and finally the last step corresponds to a very quick elongation of the bubble which is stretched from the orifice and elongated due to the lateral flow leading to its rupture.

Figure 3. Formation of micro bubbles in water. On the left: formation of a bubble in a cross flow focusing microreactor (channel size inlet 200 µm and outlet 1000 µm) using a high speed camera (1024*1280 pixels, 1000 Hz). On the right, the velocity field obtained with a µ-PIV system (2048*2048 pixels, 7 Hz) using seeding particles of 0.88 µm. V bubble = 1.14 10 -9 m 3 . Q liquide = 10 -7 m 3 /s . Q gas = 10 -8 m 3 /s. t formation = 26 ms

Obviously, bubbles expand spherically at the beginning of the bubble formation, then evolved from spherical shape to symmetrical teardrop shape. In this stage the bubble grows without significant movement into the stream direction and the liquid flows easily around the growing bubble. Subsequently, the effect of flow inertia on the bubble becomes gradually important during the growth of the bubble, the interface moves and the bubble is stretched forming a neck that is elongated to an obvious teardrop shape up to the rupture.

The µ-PIV technique is a powerful tool to evaluate the main forces acting on the bubble interface during the formation. Fig. 3 represents the velocity fields around a forming bubble in a cross shape flow focusing micro-mixer.. As shown in this figure, the neck interface undergoes shearing which can be estimated from the parallel liquid velocities at the interface of the bubble. In examples presented in Figure 3, the shear rate is about 500 s -1 . But, by evaluating the others pressure drop force acting on the bubble shows that, the shear is not the key force. The main mechanism depends on the competition between several different forces.

Formation is due to a balance between static pressures forces and tension force. The measurement of the flow field around a bubble in formation in a flow focusing microfluidic systems comforts the literature results [START_REF] Cubaud | Bubble dispenser in microfluidic devices[END_REF]Garstescki et al., 2006) about the physical formation mechanism. Moreover, as shown in Figure 3, the bubble shape is not so deformed. This can be explained from the values of the pressure drop at the bubble formation and that of the interfacial pressure defined by the Laplace-Young law. The interfacial pressure estimated for the case presented in Figure 3 is quite high aroundt 483 Pa, which results in a very small deformation of the forming bubble. In addition to the measurement of local flow properties by means of µ-PIV, the global properties such as bubble length or volume were measured using a high speed camera and a flowmeter. It is therefore possible to determine the physical key factors involved in the formation mechanism.

The flow rate ratio seems to have an important influence on the formation of micro-bubbles. Figure 4a presents the evolution of the dimensionless ratio L/W of the bubble length to the channel width with the gas flow rate (when liquid flow is constant) and inversely. These results show that the detached bubble length is affected by both gas and liquid flow rates. The bubble volume gradually increases with the gas flow rate, and decreases with the liquid flow rate. This slow evolution may be explained by the increase of shear stress and elongation when the liquid flow rate increases leading to the formation of smaller bubbles. The evolution of L/W ratio as a function of the Q gas /Q liquid ratio for various sizes of gas inlet and gas-liquid outlet (Figure 4b) may be represented by the following correlation:

α β         = liquid Q gas Q W L (5)
where α = 0.25 is the power index and constant β value depends on the liquid physical properties and the geometry of the micro-mixers. This correlation describes well the influence of both flow rate and also the size of the micro-mixer channels. Furthermore, Eq. 5 is in agreement with that proposed by [START_REF] Ganan-Calvo | Perfectly monodisperse microbubbling by capillary flow focusing[END_REF], who found a power factor α of 0.37 instead of 0.25 for this study. This is probably due to the wall effects occurring in their micro-channel of which the inlet channel is bigger. In order to improve this equation, the liquid physical properties such as the surface tension σ, the viscosity µ and also the geometry of the mixer were varied to investigate their influence. Figure 5 shows the evolution of L/W ratio as a function of the Q g /Q l flow rate ratio. One can observe for all the studied geometries, the same tendency of curves reported in Figures 5a,5b and 5c with the same power index α = 0.25. The effect of the mixing section geometry on the bubble length was studied for three types of mixers as shown in Figure 5a: a cross-shaped mixer for which the liquid inlet channel was perpendicular to the outlet channel.

The two other types were the converging mixers, with angles of 45° and 30° between both liquid inlet channels and the central gas-liquid outlet channel.

The width of the channels in these geometries was 200 µm for the gas inlet and 1000 µm for the liquid inlet. Experiments were carried out for each geometry using different operating conditions. For the same geometry, increasing the ratio of gas flow rate to liquid flow rate yielded an increase in the bubble length. However, the bubble size increases when the angle θ decreases. The effect of surface tension was investigated using the SDS surfactant in water in one micromixers geometry (180°).

As expected the bubble length is affected by the surface tension and increases with the surface tension (Figure 5b). Finally the effect of the liquid viscosity was investigated in the 180° angle micro-mixer using three liquids of the same surface tension, which was obtained by SDS surfactant addition, as showed in Figure 6c. Once again, the bubble length normally increases with the liquid viscosity which is due to the flattened liquid velocity profiles and to the enhanced shear obtained with higher liquid viscosities.

The β factor presented in Eq. 5 seems to be linked to these three parameters. A dimensional analysis was applied to obtain a dimensionless correlation equation of bubble length for all operating conditions and mixer geometries investigated in this study. For this purpose, 150 data were used to determine such correlation. Table 3 presents the correlations obtained for each mixer geometry. Dimensionless numbers based on viscosity and surface tension were introduced using the physical properties of water µ ref and σ ref as reference.

(b) (a) (c)

A good agreement is obtained between measured and correlated values with an average relative error below 3% and a maximum error of less than 13%.

3 . 10                 = ref ref µ µ σ σ β (6) ε = 2.64% error (max 10%) 10 1 9                 = ref ref µ µ σ σ β (7) ε = 3% error (max 12%) 8 1 4 3 08 . 7                 = ref ref µ µ σ σ β (8) ε = 2.75% error (max 13%)
Two conclusions can be therefore drawn. Firstly, the influence of surface tension is more important than the viscosity, as shear stress is smaller than surface tension pressure. Secondly, the influence of surface tension fincreases when θ decreases (i.e. elongation effect increases) contrarily to the effect of viscosity which increases with θ (i.e. with increasing shear stress). According to Eq. 6, 7 and 8, the viscosity has a bigger influence (1/8 > 1/5) on high angle geometries (for which higher shear stress is obtained) and surface tension has a bigger influence (6/5 >3/4) on small angle geometries (for which higher elongation and formation angle are observed). The effect of surface tension is bigger in smaller angle geometries which confirms once again the effect of interfacial force obtained in correlation of Table 2. Thus µ-PIV measurements can provide useful information in order to validate the correlations established for the studied mixers. In order to take into account the influence of the mixer angle on the behavior induced by the effect of the liquid viscosity and surface tension, a θ/θ max parameter was added in the correlation of Table 3 where θ max is the maximal angle (i.e. 180°). The following dimensionless correlation (Eq 9.) was thus obtained which was based on about 150 experimental data. The average relative error is about 6.5% with a maximum error of 18.7%. Quite good results are obtained from this correlation for bubble formation in liquid of viscosity ranging from 1 to 30 mPa.s, surface tension of 40 to 72.5 mN/m, and different geometries and size of micromixers ( angle of 180, 90 and 60° and W = 200, 500 and 1000 µm). It could be a useful tool for micro-reactor design to take into account the liquid physical properties as well as the geometry of the micro-mixers.

                                µ µ σ σ - θ θ = (9)

Conclusion

In this work, experiments were carried out to study the gas-liquid flows in microchannels. The segmented flow of a train of bubbles is important in many applications of multiphase flow in microfluidic devices. The bubble shape, size and formation mechanism were investigated under different operating conditions. The mechanism of bubble formation in a flow focusing micro-mixer was described previously [START_REF] Garstecki | Formation of droplets and bubbles in a microfluidic T-junctionscaling and mechanism of break up[END_REF][START_REF] Cubaud | Bubble dispenser in microfluidic devices[END_REF]; the bubbles were usually pinched off by the pressure difference in both phases. The current work confirms the general mechanism of bubble formation by the mean of µ-PIV measurements. The liquid flow field of the continuous phase during the formation of bubbles provides important quantitative details and more insight into the bubble length which was never reported in literature for this type of micro-mixer. Moreover, bubble size was shown to depend on the liquid physical properties (viscosity and surface tension) and the geometries of the mixing section. In regard to a different mixer, the viscosity and the surface tension have variable influence. Finally, the correlations were proposed to predict the bubble volume and length for all gas-liquid systems and micro-mixer geometries investigated and a good agreement was observed between experimental results and predicted values.

Figure 1 .

 1 Figure 1. Illustration of the micro fluidic chip used to generate micro bubble. From the left to the right, cross shape (180°), Convergence shape (90° and 60°).

Figure 2 .

 2 Figure 2. Micro Particle Image Velocimetry principle (µ-PIV). 50×50 µm windows.

Figure 4 .

 4 Figure 4. (a) Influence of gas or liquid flow rate on the bubble length to channel width ratio (L/W). (×) versus Q g at constant Q l ( ) versus Q l at constant Q g .Inlet channel size 200µm and outlet channel size 1000 µm.(b) Effect of inlet and outlet channel sizes on the evolution of L/W as function of the flow rate ratios in water.

Figure 5 .

 5 Figure 5. Influence of the liquid physical properties and micro-mixer geometry. Inlet of 200 µm and outlet of 1000 µm. (a) Influence of junction angle mixer θ mixer . (b) Influence of Tension force. (c) Influence of viscosity.

Figure 6

 6 Figure6shows the good agreement obtained between experimental L/W values and those estimated from equation 14.

Figure 6 .

 6 Figure 6. Parity diagram between correlated data and experimental results.

ρ

  Density, kg.m -3 µ Viscosity of the liquid , Pa.s σ Tension force, N.m -1 α, β Power law coefficient Σ Standard deviation, m

Table 1 .

 1 Table 1 lists all properties of the various liquids used. All experiments were carried out at a constant temperature of 293K. Properties of the different liquids used in this study at 293K.

	Fluid	ρ (kg.m -3 ) µ (×10 -3 Pa.s) σ (×10 -3 N.m -1 )
	Water	1000	1	72
	Water + 0.10% SDS (wt)	1000	1	50
	Water + 0.15% SDS (wt)	1000	1	40
	HV45 10% (wt)	1030	10	50
	HV45 10% (wt) + 0.05% SDS (wt)	1030	10	40
	HV45 25% (wt)	1050	30	40

Table 2 .

 2 Comparison of correlations proposed for the studied micro mixers

	θ 60	°=	°= θ 90	θ 180	°=
	6	1
	5	15