Vapor-Based Interferometric Measurement of Local Evaporation Rate

The simulation results for this arrangement are marked by a subscript Tv (“variable ... Subsequent comparison with the experimental results should c...
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Vapor-Based Interferometric Measurement of Local Evaporation Rate and Interfacial Temperature of Evaporating Droplets Sam Dehaeck,* Alexey Rednikov, and Pierre Colinet* Université Libre de Bruxelles, TIPs (Transfers, Interfaces and Processes), CP 165/67, Av. F.D. Roosevelt 50, 1050 Brussels, Belgium S Supporting Information *

ABSTRACT: The local evaporation rate and interfacial temperature are two quintessential characteristics for the study of evaporating droplets. Here, it is shown how one can extract these quantities by measuring the vapor concentration field around the droplet with digital holographic interferometry. As a concrete example, an evaporating freely receding pending droplet of 3M Novec HFE-7000 is analyzed at ambient conditions. The measured vapor cloud is shown to deviate significantly from a pure-diffusion regime calculation, but it compares favorably to a new boundary-layer theory accounting for a buoyancy-induced convection in the gas and the influence upon it of a thermal Marangoni flow. By integration of the measured local evaporation rate over the interface, the global evaporation rate is obtained and validated by a side-view measurement of the droplet shape. Advective effects are found to boost the global evaporation rate by a factor of 4 as compared to the diffusion-limited theory.

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the contact line using PIV measurements. Girard et al.18 measured interfacial temperatures using an infrared camera and, assuming that only conduction through the liquid is important (but not convection), calculated the local heat flux and thus the local evaporation rate (by dividing by the latent heat). Unfortunately, however, both of these studies did not perform an integration of the local evaporation rate and validate this against an independent global evaporation rate measurement. Ghasemi and Ward19 used microthermocouples to measure intrusively the local conductive heat flux, from both sides, toward the interface of a sessile water drop (evaporating into pure vapor). Yet in their case the validation was not possible without invoking additional nonclassical effects which are unlikely to be needed here. In the present paper, it is shown how digital holographic interferometry appears to be a better tool to extract the local evaporation rate distribution over the droplet interface, from a direct measurement of the vapor concentration gradient normal to the droplet interface. Several other techniques are in principle also capable of measuring the required vapor concentrations, such as (planar) laser induced fluorescence (or phosphorescence) (PLIF)20,21 or Fourier transform infrared spectrometry (FTIR).22 However, up to now, these techniques have not been able to yield accurate enough measurements in the vicinity of the droplet as to compute the required normal gradients. In the past, interferometry has already been used to

INTRODUCTION An evaporating droplet sitting on a flat plate is a surprisingly rich problem with many applications such as spray cooling1 or DNA handling.2 It has been under a particularly intense reexamination in the last 15 years. Motivated originally by the study of the coffee-stain effect,3,4 several papers have appeared that compute the global evaporation rate of the drop analytically3−6 or measure it experimentally.7 In addition, the effects of substrate thermal properties8−11 and convection in the gas phase12−16 have also been intensively studied. However, in our opinion, further progress has been slowed down by the absence of experimental techniques capable of testing existing models locally, as it is only the global evaporation rate averaged over a certain time period that was measured in most experiments. As such, one of the most fundamental predictions concerning this kind of configurations, namely that the local evaporation rate diverges (up to microscopic scales) toward the contact line,3,4,6 still remains without sufficient experimental evidence. In addition, the influence of other important phenomena, such as how convection or thermal effects modify the evaporation rate distribution, could also be compared to experimental results in an integral sense only. Yet, this local evaporation rate is an essential boundary condition used, for example, in simulations of particle deposition patterns3,4,6 and thermal loads on the substrate.9 Extracting local evaporation rates along the droplet interface has been performed only in a handful of papers and only through more indirect and/or model-based measurement techniques. Dhavaleswarapu et al.17 obtained local evaporation rates by measuring the mass-flux decrease when approaching © 2014 American Chemical Society

Received: October 8, 2013 Revised: February 7, 2014 Published: February 10, 2014 2002

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Figure 1. Mach−Zehnder interferometer used to measure the vapor cloud and a typical raw image. unknown local temperature T and mole fraction χ can be derived (see the Supporting Information):

study the vapor cloud qualitatively. 23−25 In addition, quantitative results were already obtained by Toker and Stricker26,27 for the global vapor mole fraction field around an evaporating droplet suspended from a millimetric wire. From the mole fraction obtained at the interface, they also deduced interfacial temperatures. In this paper, we will revisit the technique and show that it is now at a point where one can analyze the mole fraction field even further and also measure the instantaneous local evaporation rate along the droplet interface. These vapor cloud measurements also give access to two instantaneous global evaporation rate measurements, which are successfully validated by an independent side-view measurement.

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χ=

⎛ Tamb ⎞⎤ 1 T ⎡ − 1⎟⎥ ⎢Δn − (nair,amb − 1)⎜ ⎝ ⎠⎦ ⎣ nHFE,amb − nair,amb Tamb T (1)

where nHFE,amb and nair,amb are the refractive indices of the pure vapor of HFE-7000 and air, respectively, at ambient pressure and temperature Tamb. In general, this single formula is not enough to extract both concentration and temperature variations from a single measurement. However, as the impact of temperature variations is marginal, one can neglect them by setting T ≜ Tamb in the above formula which then yields: χ=

EXPERIMENTAL AND NUMERICAL METHODS

A. Vapor Cloud Concentration Measurements. A Mach− Zehnder interferometer (see Figure 1) using a red diode laser (660 nm) is used to measure the vapor cloud surrounding an evaporating pending drop of 3M Novec HFE-7000. This liquid is sold by 3M as a thermal management fluid. Before the deposition of the drop, the interferometer is set to generate a homogeneous system of vertical fringes. At this time, a first image is acquired (with a CCD camera of 1920 × 1080 pixels and a spatial resolution of 7.45 μm) and analyzed with the Fourier transform profilometry algorithm28 to yield a phase for each individual pixel corresponding to a uniform refractive index field (equal to that of pure air). Then, the drop is deposited on the lower surface of a horizontal 2" diameter silicon wafer (thickness 0.5 mm) with a syringe and left to evaporate, while shielding it from spurious air currents. Images are acquired at 30 frames/s during the complete evaporation of the droplet (∼12 s). The phase for each pixel is extracted and compared to the reference phase obtained for that pixel in the previously mentioned reference/calibration image.29 This then yields the total phase shift for that pixel. However, the information obtained in this way is “wrapped”, which means that the phase shift ranges from 0 to 2π in the image. The next step is the socalled “unwrapping” of this image, for which the algorithm described by Herraez et al.30 is used. This essentially looks for the discontinuity lines where the phase shift changes abruptly from 2π to 0 in a single pixel and adds 2π accordingly. Roughly speaking, the algorithm proceeds to investigate all of the pixels until such jumps are not present anymore. As a result, we can find a phase shift of, for example, 6π at the droplet interface, when keeping the total phase shift in the zone where neither concentration nor temperature changes occur equal to zero. Step-by-step images showing this image-processing sequence are shown in the Supporting Information. This phase shift is now proportional to the optical path length difference accumulated when the light ray traverses the axisymmetric refractive-index field surrounding the droplet. Now to extract this refractive-index field from this single projection measurement, an inverse Abel transform algorithm is needed to perform the tomographic reconstruction.31,32 The final step is then to convert this into the vapor mole-fraction field χ. Starting from the Lorentz− Lorenz equation33,34 for a mixture of gases, the following equation relating the measured refractive index difference field Δn to the

1 Δn nHFE,amb − nair,amb

(2)

that is, a simple proportionality relationship. Its coefficient is determined by the refractive index of the pure vapor of HFE-7000. This property still had to be measured here and yielded nHFE,amb = 1.00163 ± 3 × 10−5 (details of the procedure are given in the Supporting Information). A posteriori, it can be shown that the simplification T = Tamb leads to a bias (here an overestimation) of at most 5% in the present study. From eq 2 and other considerations, three main limitations of the technique can now be identified: (i) The refractive-index contrast of the pure vapor with the ambient gas needs to be large enough. This for instance renders a measurement of water vapor highly challenging. On the other hand, an evaporating water drop in a xenon atmosphere could be tractable. (ii) The second (related) point is the saturation concentration, which must be large enough. (iii) Finally, the size of the droplet is also important as in reality we measure an integral of the local refractive index with the distance, that is, the optical path length. Clearly micrometric droplet sizes are more difficult to measure than millimetric ones. In order to estimate the experimental uncertainty and repeatability of our measurements (with 95% confidence interval), the experiment is run 4 times (albeit with different initial drop sizes due to manual injection). Error estimates given later could then be calculated using not only the different experimental runs, but also the neighboring images (up to 6 of them) of a single run. B. Interfacial Temperature Evaluation. After the determination of the interface location, one can extract through bilinear interpolation an estimation for the refractive-index difference at each point of the interface. Assuming local thermal and chemical equilibria at the interface, an additional relation exists next to eq 1 for the determination of the two unknowns (Tσ and χσ, where the subscript σ from now on refers to quantities evaluated at the droplet interface). Namely, assuming ideal gases, χσ = Psat(Tσ)/Pamb, where Psat(T) is the saturation pressure as a function of temperature (data provided by 3M or simply the Clausius−Clapeyron relation) and Pamb = 101 500 Pa is the ambient pressure. Thus, experimental determination of the interfacial quantities is here more precise than that of the bulk-related ones in the sense that the absence of the earlier mentioned small bias from neglecting the temperature variation of the refractive index (i.e., from using eq 2 rather than eq 1). A similar treatment has already been proposed by Toker and Stricker.26 Note that the quantity in the 2003

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Psat(Tamb)(Mυ − Mair)/9 Tamb = 4.31 kg/m3, given that Psat(Tamb) = 62.2 kPa and Mair = 0.029 kg/mol (while Mυ = 0.2 kg/mol). Then, for example, for a droplet with Rc = 1.81 mm, one obtains Gr ∼ 852. This seems large indeed, hence the expectation that the boundary-layer model is here closer to the reality than the pure-diffusion one (the latter actually corresponding to Gr ≪ 1). On the other hand, notice the large gas density variations Δρg expected here, appreciably greater than the air density. As a consequence, a non-Boussinesq boundarylayer formulation is used (see the Supporting Information). While this approach should not be considered trustworthy neither too close to the contact line nor in the center of the drop, it should nevertheless give quantitative results in the remaining portion of the interface. The droplet shape used is always adopted from the experiment. Quasi-stationarity is assumed at each moment during the droplet evaporation. Three different sets of interfacial conditions are examined for this analysis. They are detailed below. First, we assume a constant interfacial temperature, equal to the ambient one, and hence χσ = Psat(Tamb)/Pamb = 0.61 along all the droplet interface, just as we did earlier in the pure-diffusion model. At the same time, the droplet interface is assumed to be stagnant; that is, the no-slip condition is applied thereat for the gas velocity. Physically, this corresponds to liquid velocities being much smaller than those due to buoyancy convection in the gas. The simulation results for this arrangement are marked by a subscript Tc (“constant temperature”). Second, we use in the boundary-layer simulations the interfacial temperature distribution Tσ(r) measured in the experiment, where r is the radial position (distance to the symmetry axis), hence χσ(r) = Psat(Tσ(r))/Pamb. However, the droplet interface is still assumed stagnant. The simulation results for this arrangement are marked by a subscript Tv (“variable temperature”). Clearly, due to evaporative cooling, the evaporation rates are expected to be lower in this case Tv than in Tc. A last approach is to take into account not only the experimentally obtained Tσ(r), but also a Marangoni flow generated by this temperature nonuniformity. Thus, instead of a stagnant interface from the gas side, we now have a nonzero tangential velocity determined by the Marangoni convection inside the droplet. The latter is computed using a Navier−Stokes formulation with prescribed Marangoni stresses (given by the experimentally determined Tσ(r), whereas viscous stresses from the gas side are neglected in view of typically small gas-to-liquid dynamic-viscosity ratios), and the details are given in the Supporting Information. The simulation results for this last arrangement are marked by a subscript Ma (“Marangoni”). To sum up, a sequence of five simulations (D, DSF, Tc, Tv, and Ma) with increasing complexity is performed, each time including more physics. Subsequent comparison with the experimental results should confirm whether the considered physical effects have an important impact on the calculated evaporation rates. Finally, the key subscripts used when representing the experimental and theoretical results are summarized in Table 1.

following subsection does contain this small bias, since extracting the normal gradient involves bulk values. C. Local Evaporation Rate Determination. From the measured field of vapor mole fraction χ, the normal gradient ∂χ/∂n at the interface can also be extracted and used in the expression for the local evaporation rate

Jl = −

M υPambDυ ∂χ 9Tamb(1 − χσ ) ∂n

(3)

σ

(with Mυ = 0.2 kg/mol the molar mass, Pamb = 101500 Pa the ambient pressure, Dυ = 8.11 × 10−6 m2/s the diffusion coefficient measured by Stefan-tube experiments, 9 = 8.31 J/(mol K) the universal gas constant, and Tamb = 297.15 K the ambient temperature) to yield the experimental values Jl,Exp thereof. The subscript Exp underlines that this is an experimentally measured quantity, in contrast with other subscripts to be used with Jl later on. D. Global Evaporation Rate Determination. Three ways to obtain the global evaporation rate Jg are used. A reference measurement Jg,side comes from the extraction of the contour of the drop shadow over time enabling calculation of its volume evolution. A second one is obtained by integration of the local evaporation rate (eq 3) over the entire droplet interface as follows:

Jg =

∫σ Jl dS

(4)

which when used with the measured local flux Jl = Jl,Exp yields the quantity hereafter designated as Jg,surf. Finally, a similar integration (see the Supporting Information for its derivation) can be performed over any iso-concentration surface encompassing the drop and terminating on the substrate (assuming the quasi-stationarity of the concentration field): Jg = −

MP D

∫χ =const 9T υ (1amb−υχ ) amb

∂χ dS ∂n

(5)

which, when used with the measured concentration field, yields the quantity hereafter designated as Jg,iso. E. Comparison with Available Basic Models. It is instructive to compare the measured vapor cloud shape with the reference case of pure dif f usion, formally valid in the limit of sufficiently small vapor concentrations χ ≪ 1 and negligible buoyancy. To this purpose, a simulation is performed with OpenFOAM 2.1 for the axisymmetric Laplace equation ∇2χ = 0 with χσ given by the saturation concentration at the ambient temperature (i.e., χσ = Psat(Tamb)/Pamb = 0.61), a nonpenetration condition ∂χ/∂n = 0 at the gas−solid surface, and χ → 0 far away from the droplet. For the corresponding local and global evaporation rates, analytical formulas found in the literature6 for flattened droplets are rather used for simplicity. In terms of eq 3, they correspond to χσ ≪ 1 neglected in the denominator (negligible Stefan flow), consistent with the overall approach. These results are marked by the subscript D in the following. As the mole fraction at the interface is here nonetheless large (∼60%), a semiheuristic Stefan−Fuchs correction35 is also considered, which consists in formally leaving the denominator intact even though χ is still determined from ∇2χ = 0, that is, neglecting the convective transport. It is denoted by a subscript DSF. See the Supporting Information for further details. On the other hand, as a buoyancy-induced convective flow of gas along the droplet interface is anticipated in the present situation, a boundary-layer approach (large-Grashof-number limit)36 is also tested, while a full simulation still awaits its realization. The Grashof number Gr can here be defined as Gr =

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RESULTS AND DISCUSSION A. Global Vapor Cloud Configuration. In Figure 2a, a contour map of the measured vapor mole-fraction field is shown, containing just a few contours for the sake of clarity. It is clear that a strong convective plume falling along the axis of the droplet is present, which is a result of the gas mixture being heavier than ambient air by a factor of 4. The continuous black lines in this figure are the iso-concentration lines obtained from the numerical simulations. The left-hand-side contours correspond to the pure-diffusion case (D), the right-hand-side ones to the boundary-layer approximation including the Marangoni-driven flow (Ma). Clearly the pure-diffusion simulation is not representative of the reality, whereas the boundary-layer simulation yields a much better agreement, especially when not too close to the center or edge of the drop. Concerning the different boundary-layer simulations performed, it is instructive to look at the effect of the Marangoni

Δρg gR c 3 μair νair

(6)

where Δρg is the scale of gas density variations, g is the gravity acceleration, Rc is the contact radius of the pendant drop, μair = 1.85 × 10−5 Pa s is the dynamic viscosity of pure air, and νair = 1.58 × 10−5 m2/s its kinematic viscosity. Assuming ideal gases, Δρg is estimated at 2004

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Table 1. Meaning of Key Subscripts l g σ Exp surf iso side D DSF Tc Tv Ma

as in Jl, local evaporation rate as in Jg, global evaporation rate interfacial value as in Jl,Exp, experimental value as in Jg,surf, measured at interface as in Jg,iso, measured at isoconcentration lines as in Jg,side, based on side view measurement based on pure-diffusion model with constant interfacial temperature (equal to Tamb) pure-diffusion model with Stefan−Fuchs correction boundary-layer model with constant interfacial temperature (equal to Tamb) and stagnant droplet interface boundary-layer model with nonuniform interfacial temperature (adopted from experiment) but still stagnant droplet interface boundary-layer model with nonuniform interfacial temperature (adopted from experiment) and thereby induced Marangoni flow inside the droplet

Figure 3. Normal profiles at different dimensionless radial positions r along the interface for a droplet with Rc = 1.81 mm and y being the normal distance to the interface.

flow on what is going on in the gas phase. As the Marangoniinduced velocity at the droplet interface can attain ∼10 cm/s and this is of the same order of magnitude as the predicted buoyancy-induced velocities in the vapor, there is indeed an effect. Figure 2b shows the simulations with and without this effect and it is clearly seen how the interfacial velocity leads to a substantial thinning of the vapor concentration boundary layer close to the contact line as both flows are in the same direction. This results in higher local evaporation rates, which agrees better with the measurements (see also below). Another important representation of the measured vapor cloud is given by mole-fraction profiles normal to the detected droplet interface. Six such profiles are shown in Figure 3 for different starting positions r (where r = 0 corresponds to the center and r = 1 to the contact line). B. Interfacial Temperature Evaluation. The measured interfacial temperature profile from the symmetry axis (r = 0) to the contact line (r = 1) is represented in Figure 4 (open circles). This shows that the temperature in the center of the drop is considerably smaller (by nearly 11 °C) than at the contact line, which is supposed to be at the ambient temperature Tamb = 24 °C due to the highly conductive

substrate used here. For estimations showing this is appropriate, see the Supporting Information. Note however that the plot does not indicate that this is the case. The difficulty is attributed to excessively large concentration gradient values expected when approaching the contact line. Yet, as the algorithm for the inverse Abel transform is optimized to limit the noise level by smoothing out high frequency changes (and hence large gradients), it is clear that a zone with too large gradients cannot be reconstructed correctly and the temperature will always be underestimated there. This limitation is demonstrated in the Supporting Information. A tentative fit of the interfacial temperature which does reach the ambient temperature at the contact line and which is used in the computations presented above, is shown by the solid line (its mathematical form is given in the Supporting Information). Note the large temperature gradients near the contact line, which generate a substantial Marangoni driving force. Finally, remark that such a “quasi-stationary” profile is established in a

Figure 2. (a) Measured mole-fraction contour map together with the isocontours obtained in the (left) pure-diffusion and (right) boundary-layer approach. (b) Calculated gas velocity and mole-fraction fields in the case of (left) stagnant droplet interface and (right) Marangoni flow at the droplet interface. The coordinates are nondimensionalized with the contact radius Rc. The results are here shown for Rc = 1.81 mm. 2005

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interfacial temperature variation and the Marangoni flow, Jl,Ma, yields the best overall correspondence, while Jl,Tv (temperature variation but no Marangoni flow) and Jl,Tc (constant temperature of 24 °C) clearly under- and overpredict Jl,Exp respectively. Note also that the boundary layer thinning due to the Marangoni flow visible in Figure 2b does indeed lead to an important increase in local evaporation rate (Jl,Ma versus Jl,Tv) as anticipated. D. Global Evaporation Rate Determination. As detailed in the description of the experimental methods, the global evaporation rate is measured in three different ways: Jg,side based on the droplet volume variation, Jg,surf based on the integration of Jl,Exp over the droplet interface and Jg,iso based on the integration over isoconcentration contours. The results are shown in Figure 6. The first unexpected feature is that Jg,side is

Figure 4. Measured interfacial temperature versus the dimensionless radial position r for Rc = 1.81 mm, the solid line being the fit thereof (see the Supporting Information) used in the simulations.

few seconds following the deposition of the drop and that the experimental uncertainty is estimated at ±0.9 °C with larger values toward both the center and the contact line. C. Local Evaporation Rate Determination. The extracted local evaporation rates Jl,Exp along the droplet interface are shown in Figure 5.37 The uncertainty is estimated at 10%,

Figure 6. Global evaporation rate versus the droplet contact radius.

very noisy even though a substantial averaging is involved in the volume differentiation with time (which involves a fitting step). This is believed to be due to a slight stick−slip behavior of the drop. The vapor-based interferometric methods fall within this reference global evaporation rate measurement and its uncertainty. More important however is that these measurements are not only instantaneous (no differentiation with time is involved) but also much less noisy. The uncertainty is estimated at 5%, increasing for smaller drops. We also notice that Jg,surf seems to be consistently smaller than Jg,iso (on average by 10%). This can be associated with the fact that the local evaporation rate at the contact line is apparently underpredicted, as discussed earlier. The global evaporation rates calculated from the purediffusion models Jg,D and Jg,DSF are also shown in Figure 6, assuming an isothermal interface at 24 °C. This shows that neither the variation with the radius nor the absolute values are well captured (Jg,D underestimates the experimental results by a factor 4 for Rc = 2 mm even if calculated at the maximum temperature available in the system). While correct trends are anticipated based on the results of Kelly-Zion et al.,15 the correlation proposed there for the effect of convection on the evaporation rate of sessile droplets also underestimates by a factor of 2 the global evaporation rate when applied to the present pending droplets. This could perhaps be attributed to the different configuration (pending instead of sessile), which here leads to a more intense convection in the vapor cloud.

Figure 5. Local evaporation rate versus the dimensionless radial position r for a droplet with Rc = 1.81 mm.

increasing above this value toward both the center and the contact line. We immediately note that, even though a clear increase in the local evaporation rate is apparent, no divergence is observed when approaching the contact line. As above, this can be attributed to limitations in the tomographic algorithm in this zone. Therefore, no conclusive answer is here given on the precise behavior close to the contact line. In Figure 5, the local evaporation rate predicted from purediffusion calculations, Jl,D, and the one with the Stefan−Fuchs correction, Jl,DSF, are also shown. None of them yield a good agreement, especially given that they are obtained using an isothermal interface at 24 °C. Indeed, the measured temperature decrease should actually make their values even smaller by up to 32%. The boundary-layer approach including the 2006

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Langmuir Nevertheless, for smaller drops the discrepancy becomes smaller. This is expected as the role of convection diminishes at smaller scales (the Grashof number scaling as the diameter cubed, eq 6). The estimations resulting from the boundarylayer theory yield the correct trend and straddle the experimental results. As the theory is expected to yield valid quantitative results only in a limited region of the drop interface, it is not surprising that the integration over the entire surface is slightly off, however. Of particular importance is the fact that the simulations including the internal Marangoni flow lead to an evaporation rate that can be 30% larger than for a droplet with the same interfacial temperature but without internal flow. While this brings it closer to the isothermal evaporation rate Jg,Tc, there is however still an important gap. Thus, in order to perform a full simulation of the present setup, independent from any experimental input, it is not only the buoyancy-induced convection in the gas phase that should be considered, but also the heat transfer and Marangoni flow inside the droplet, leading to a highly coupled and challenging problem.

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REFERENCES

(1) di Marzo, M.; Tinker, S. Evaporative cooling due to a sparse spray. Fire Saf. J. 1996, 27, 289−303. (2) Dugas, V.; Broutin, J.; Souteyrand, E. Droplet evaporation study applied to DNA chip manufacturing. Langmuir 2005, 21, 9130−9136. (3) Deegan, R.; Bakajin, O.; Dupont, T.; Huber, G.; Nagel, S.; Witten, T. Capillary flow as the cause of ring stains from dried liquid drops. Nature 1997, 389, 827−829. (4) Deegan, R.; Bakajin, O.; Dupont, T.; Huber, G.; Nagel, S.; Witten, T. Contact line deposits in an evaporating drop. Phys. Rev. E 2000, 62, 756−765. (5) Hu, H.; Larson, R. Evaporation of a sessile droplet on a substrate. J. Phys. Chem. B 2002, 106, 1334−1344. (6) Popov, Y. Evaporative deposition patterns: Spatial dimensions of the deposit. Phys. Rev. E 2005, 71, 036313. (7) Cachile, M.; Bénichou, O.; Poulard, C.; Cazabat, A. M. Evaporating droplets. Langmuir 2002, 18, 8070−8078. (8) David, S.; Sefiane, K.; Tadrist, L. Experimental investigation of the effect of thermal properties of the substrate in the wetting and evaporation of sessile drops. Colloids Surf., A 2007, 298, 108−114. (9) Ristenpart, W. D.; Kim, P. G.; Domingues, C.; Wan, J.; Stone, H. A. Influence of substrate conductivity on circulation reversal in evaporating drops. Phys. Rev. Lett. 2007, 99, 234502. (10) Sobac, B.; Brutin, D. Thermal effects of the substrate on water droplet evaporation. Phys. Rev. E 2012, 86, 021602. (11) Lopes, M. C.; Bonaccurso, E.; Gambaryan-Roisman, T.; Stephan, P. Influence of the substrate thermal properties on sessile droplet evaporation: Effect of transient heat transport. Colloids Surf., A 2013, 432, 64−70. (12) Shahidzadeh-Bonn, N.; Rafaï, S.; Azouni, A.; Bonn, D. Evaporating droplets. J. Fluid Mech. 2006, 549, 307−313. (13) Kelly-Zion, P. L.; Pursell, C.; Booth, R.; Tilburg, A. V. Evaporation rates of pure hydrocarbon liquids under the influences of natural convection and diffusion. Int. J. Heat Mass Transfer 2009, 52, 3305−3313. (14) Saada, M.; Chikh, S.; Tadrist, L. Numerical investigation of heat and mass transfer of an evaporating sessile drop on a horizontal surface. Phys. Fluids 2010, 22, 112115. (15) Kelly-Zion, P.; Pursell, C.; Vaidya, S.; Batra, J. Evaporation of sessile drops under combined diffusion and natural convection. Colloids Surf., A 2011, 381, 31−36. (16) Carle, F.; Sobac, B.; Brutin, D. Experimental evidence of the atmospheric convective transport contribution to sessile droplet evaporation. Appl. Phys. Lett. 2013, 102, 061603. (17) Dhavaleswarapu, H.; Migliaccio, C.; Garimella, S.; Murthy, J. Experimental investigation of evaporation from low-contact-angle sessile droplets. Langmuir 2010, 26, 880−888. (18) Girard, F.; Antoni, M.; Sefiane, K. Use of IR thermography to investigate heated droplet evaporation and contact line dynamics. Langmuir 2011, 27, 6744−6752. (19) Ghasemi, H.; Ward, C. Energy transport by thermocapillary convection during sessile-water-droplet evaporation. Phys. Rev. Lett. 2010, 105, 136102. (20) Bazile, R.; Stepowski, D. Measurements of vaporized and liquid fuel concentration fields in a burning spray jet of acetone using planar laser induced fluorescence. Exp. Fluids 1995, 20, 1−9.

OUTLOOK This paper has demonstrated the wide potential of digital holographic interferometry for vapor concentration measurements. This is actually the case not only for pendant/sessile droplets of HFEs or other volatile liquids, but also likely for many other configurations (acoustically levitated droplets, spray drying/combustion, condensation, etc.). Using several interferometers, nonaxisymetric concentration fields could also be tomographically reconstructed. From the fluid-physics point of view, our results quantitatively show that the fast evaporation dynamics of high-molecular-weight liquids is a rather complex nonisothermal process (even for a highly conducting substrate), involving intricate couplings between boundary-layer-like solutal buoyancy-driven flows in the gas and thermal Marangoni flows inside the droplet (with an essential convective influence outside), in addition to the classical Stefan flow. We hope that experimental and theoretical ideas proposed here and in the Supporting Information will stimulate further research in this challenging area. ASSOCIATED CONTENT

* Supporting Information S

Experimental details are given on (i) the specific image treatment algorithms, (ii) the refractive index to mole fraction conversion, (iii) the temperature influence on mole fraction measurements, and (iv) a theoretical study on the capability of an Abel inverse transform to recreate a function with singularities. Additionally, the full formulation used in the numerical analysis is also given as well as estimations on the expected temperature decrease in the wafer (global and local). This material is available free of charge via the Internet at http://pubs.acs.org.

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ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support from EVAPORATION and HEAT TRANSFER projects funded by the European Space Agency and the Belgian Science Policy Office PRODEX Programme, and from the Interuniversity Attraction Poles Programme (IAP 7/38 MicroMAST) initiated by the Belgian Science Policy Office. P.C. thankfully acknowledges financial support from the Fonds de la Recherche Scientifique - FNRS. This work was also performed under the umbrella of COST Action MP1106.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected] Notes

The authors declare no competing financial interest. 2007

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2008

dx.doi.org/10.1021/la404999z | Langmuir 2014, 30, 2002−2008