Convection-enhanced water evaporation

In calm air, water vapor can be transported away from the water surface by molecular diffusion or buoyant convection because water vapor is lighter than air. Evaporation of water droplets on solid surfaces is a problem of considerable fundamental and practical importance in many areas of physics, chemistry, biology, and engineering.¹ This phenomenon has long been studied by Maxwell,² Langmuir,³ and many scientists.^4–15 Droplet evaporation plays a key role in many natural and industrial processes such as drying-mediated pattern formation^6,7 and self-assembly.¹⁶ Most liquid droplets follow diffusion-limited evaporation,^5–8 described as dV/dt ∝ –2πR in calm air (that is, evaporation rate linearly increase with contact radius). For convection-limited evaporation, a different dynamics is suggested as dV/dt ∝ –πR².^13,14 In addition to diffusion-limited evaporation, convection-limited evaporation is also actually feasible for water droplets^9,13,14 but, its direct evidence is scarce.^13,14

Droplet evaporation is typically classified into three modes^4,15 with respect to the wetting property (the contact radius R and the contact angle θ): the constant-R-mode (θ changes with time), the constant-θ-mode (R changes with time), and the mixed mode (both R and θ change with time). The constant-R-mode is mostly prevailing for microliter-scale water droplets on smooth hydrophilic surfaces.^5–8 For nanoliter water droplets, evaporation behaviors can be significantly affected by line tension;¹⁵ the constant-θ-mode emerges due to strong line tension, for instance, in a 0.8 nL water droplet.¹⁵ This suggests that the evaporation behaviors of nanoliter water droplets could be quite different from those of microliter droplets. Direct observation of evaporation for nanoliter droplets is, however, a challenging task.

In this work, we visualize evaporation of nanoliter (2 to 700 nL) water droplets resting on silicon wafer in calm air using a high-resolution dual X-ray imaging method. Temporal evolutions of contact radius and contact angle reveal that evaporation rate linearly changes with surface area, indicating convective (instead of diffusive) evaporation in nanoliter water droplets.

We describe a relation between possible evaporation mechanism and droplet geometry. For nanoliter droplets with radii from 100 to 1000 μm that are much smaller than the capillary length (∼ 2.7 mm of water), the spherical cap model¹⁴ is valid. The droplet volume is given as V = πR³f(θ) where f(θ) = (2 – 3cosθ + cos³θ)/(3sin³θ) ≈ 0.25θ at small angles (< 40°).⁵ The volume is then a function of contact radius as V ∝ R³θ ∝ R^3+β/α from θ ∝ R^β/α by assuming power-law temporal evolutions of R = R_c(t_f – t)^α and θ = θ_c(t_f – t)^β, as empirically suggested for water.^9–11 Here t_f is the complete evaporation time, α and β are the scaling exponents, and R_c and θ_c are the pre-factors. From these relations, the total evaporation rate is given as:

The evaporation mechanism is then described using the scaling factor “n = 3 + (β – 1)/α”. (A similar expression was empirically obtained without derivation.⁹) The diffusion-limited evaporation (n = 1) holds only for the specific case of 2α + β = 1.⁹ For the convection-limited evaporation (n = 2),^13,14 the possible constraint is α + β = 1. For the spreading cases (n = 3),¹⁷ the possible constraint is (β – 1)/α = 0 or α → ∞ at β ≠ 1 (equally, dR/dt ∝ –R). For the pinning cases (α = 0), we always find β = 1 (equally, a linear decrease of θ with time⁸). Finally for the shrinking cases (α > 0), either (β < 1 and n < 3) or (β > 1 and n > 3) is possible. To identify dominant mechanisms in water evaporation, we need to obtain α and β separately for each event and their correlation for many events of various cases.

A high-resolution dual X-ray imaging method is introduced to simultaneously acquire transmission (TI) and reflection images (RI) of water droplets in real time at a grazing angle ϕ using a synchrotron X-ray phase-contrast imaging^18–22 (from the PLS 7B2 beamline, Pohang, Korea). This method allows precise measurements of droplet geometry (from TI) and complete evaporation time (from RI) for nanoliter water droplets of 2 to 700 nL, which are very small compared to microliter droplets (1–3 μL) studied in common methods.^14,23 Droplet geometry (R and θ) is monitored from TI (a spherical cap) and complete evaporation time (t_f) from RI during evaporation of a water droplet on a flat surface (e.g., Si wafer) [Figs. 1(a) and 1(b)]. The trace of the water film in RI (as marked by an arrow in 120 s image) is recorded until it eventually fades out, for instance, at 208.2 s in this event, in order to measure the complete evaporation time (t_f). The t_f measurement is not easy from TI because of limited spatial resolution.²⁴ Nanoliter-scale water droplets of 2 to 700 nL (R = 100–1000 μm) are tested [Fig. 1(c)]. The actual values of the droplet height h (= h_a + (w/2) sinϕ; w = the wafer width), the contact angle θ (= acos[1 – h/r]), and the contact radius R (= r sinθ; r = the meniscus radius) are obtained by calibrating the apparent values of h_a and R_a (measured within errors ± 0.5 μm from TI) and by compensating a shade effect from the wafer by w = 100 mm and ϕ = 0.008–0.2 rad (variable for each droplet). This collective information enables us to formulate R and θ as a function of time (t_f – t).

Spatially-coherent synchrotron X-rays (photon energy of 10-60 keV)^19–22 are useful to track the water-air meniscus [Fig. 1(b)]. After passing through the specimen, the X-ray photons are converted, using a CdWO₄ scintillator crystal, to visible ones, which are captured by a CCD camera. The scintillator-specimen distance is set at 150 mm to optimize phase contrast. The beam cross section is tuned to 1.5 × 1.2 mm² or 0.58 × 0.44 mm² according to the sample size and the microradiology spatial resolution is 0.5 μm. Sequential real-time images are taken with an acquisition time of 0.5 s (exposure time ∼ 0.01 s) per each frame. Heating or radiation effects on water evaporation during X-ray irradiation is negligible because of short exposure time and low dose.^19,22 As a substrate, a commercial silicon wafer (100 mm in diameter) is used. Ultrapure water droplets (18 MΩ, Millipore) are used and placed on the wafer. All experiments are conducted at room temperature and relative humidity (25 °C and RH = 30–50%). For each test, the silicon sample is cleaned of any impurity, by rinsing it with pure acetone and distilled water and by drying it in N₂. Nine datasets among thirty droplets tested in this study are shown to have TI and RI simultaneously in the given field of views. The initial contact angles of water droplets on cleaned silicon wafers were 20–50 degrees,²⁵ which are typical for silicon wafer with angle hysteresis (∼ 10°)^12,25 and under laboratory humidity.

Representative results of evaporation dynamics in water droplets on the silicon wafers are illustrated in Fig. 2. The temporal evolutions of R and θ are slightly different depending on the initial volumes. For large droplets (in cases of 429, 98, and 68 nL), the contact angle almost monotonically decreases with time (as the reduced time (t_f – t) decreases) while the contact radius is almost constant (the constant-R or the “pinning” mode). For intermediate droplets (in cases of 269, 13, and 10 nL), the contact angle and the radius simultaneously decrease with time (the mixed mode). For small droplets (2.5 and 3.5 nL), the contact radius decreases with time while the contact angles are invariant (the constant-θ or the “shrinking” mode). As empirically suggested,^9–11 the time-dependent relations for R and θ in each event are well fitted with power-law functions as R = R_c(t_f – t)^α and θ = θ_c(t_f – t)^β (solid lines). The exact value of t_f allows estimating the scaling exponents, α and β, more precisely. In addition, quantitative data on α and β are available separately, which would be essential to identify dominant mechanism of water droplet evaporation.

We evaluate two exponents, α and β, from the sets of R and θ for each event. We note that the scaling exponent, α or β, is significantly different per each event [Fig. 3(a)]. A pair point (square) of α and β corresponds to a single event. For all the events, our data are better fitted to α + β = 1 (solid line; convective evaporation) than 2α + β = 1 (dashed line; diffusive evaporation). The good agreement with the relation of α + β = 1 suggests that convection is more dominant than diffusion in nanoliter water evaporation.

We plot the evaporation rate (–dV/dt) as a function of R [Fig. 3(b)]. The evaporation rate is taken from the derivative of a power-law form of V(t) for each event. The rates are in the range of 0.1–3.0 nL s⁻¹ for R = 100–1000 μm. By fitting all the evaporation rate data for all events with (–dV/dt) ∝ Rⁿ, we obtain n ≈ 1.44 (not integer), similar to previous observation (n ≈ 1.5).⁹ Meanwhile, by fitting the evaporation rate for each event with n = 2, we find a good agreement by adjusting the coefficient A from (–dV/dt) = AR². This result shows that the evaporation rate linearly changes with the contact area (πR²) for each event, consistently suggesting the convective evaporation for nanoliter water droplets.^9,13,14 Here the coefficient A seems to increase from 2.4 × 10⁻⁶ to 12 × 10⁻⁶ nL s⁻¹ μm⁻² [1 nL s⁻¹ μm⁻² = 10⁸ μg s⁻¹ cm⁻²] as R₀ decreases from 1000 to 100 μm. This trend is reasonable because the evaporation flux j₀ inversely changes with the initial contact radius as j₀ ∝ R₀⁻¹ (Ref. 7) and the coefficient linearly increases with the evaporative flux as A ∝ j₀.⁹ Actually in our data, we find an inverse proportionality as A ≈ 0.0037R₀^{−1.12 ± 0.11} [inset of Fig. 3(b)].

To compare diffusive and convective evaporation rates, we estimate the diffusive mass loss of a pinned droplet using a general form as (–dM/dt) = πRD(1 – H)c_v(0.27θ² + 1.30).⁸ If we take typical values⁸ for water such as the vapor diffusivity in air (D = 26.1 mm² s⁻¹), the saturation vapor concentration (c_v = 2.32 × 10⁻⁸ g mm⁻³), the liquid density (ρ = M/V = 998 g mm⁻³), the relative humidity (H = 0.4), and the contact angle (θ = 0.5 rad), then we obtain B = (–dV/dt)/R = 15.7 μg s⁻¹ cm⁻¹, similar to previous results of B = 15.4 μg s⁻¹ cm⁻¹⁵ or B = 16.1 μg s⁻¹ cm⁻¹.⁷ By comparing the measured A (270–1200 μg s⁻¹ cm⁻²) with the estimated BR⁻¹ (160–940 μg s⁻¹ cm⁻²) for the given R₀ values, we find that A (convective evaporation) is larger than BR⁻¹ (diffusive evaporation) at R < 1000 μm, showing the crossover between the two regimes at R ∼ 1000 μm. This indicates that convection of water vapor would play a dominant role in evaporation kinetics at nanoliter scales and enhance the transport rate of water vapor away from water droplets.

The significance of buoyant convection of water vapor in the atmosphere should not be ignored at small scales. The upward velocity of the air/vapor mixture is given as υ = (gΔρ/ρL)^1/2 where g is the gravitational acceleration (9.8 m s⁻²), Δρ/ρ is the reduced density difference (∼ 1% for water vapor), and L is the length scale of the quasi-steady buoyant convection flow in the atmosphere.¹³ Taking L = 0.1 m (as a realistic condition),¹⁴ we obtain υ = 0.1 m s⁻¹. For the given L, the buoyant convection would become more significant at small scales than at large scales. The natural buoyancy at the velocity of υ ∼ 0.1 m s⁻¹ can trigger the convective mass transfer in calm air.¹⁴ The natural convective evaporation, (–dV/dt) ∝ R², is analogous to the forced convective evaporation, (–dV/dt) ∝ R^1.8 under air flow of 0.09 to 4.05 m s⁻¹ (by a fan).²⁶ The competition of convection and diffusion is important at small scales.²⁷ Our finding shows that convection of water vapor indeed enhances the evaporation rate of nanoliter water droplets, as suggested.^13,14

In conclusion, we present a direct visualization of evaporating nanoliter water droplets of 2 to 700 nL resting on silicon wafer with a high-resolution dual X-ray imaging method. All evidence supports that convection of water vapor would be important in evaporation behaviors of nanoliter water droplets. For each evaporation event, contact radius and contact angle quite well follow power-law functions, as empirically suggested. From the power-law temporal evolutions, we find that a sum of two exponents becomes one and the evaporation rate linearly changes with surface area, suggesting the convective (instead of diffusive) evaporation for nanoliter water droplets. These findings would contribute to better understanding of the role of vapor convection in enhancing water evaporation at nanoliter scales, for instance, on microdroplets or inside nanochannels, which are common in microfludics and lab-on-chip systems.