Evaporation-Induced Flows inside a Confined Droplet of Diluted

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Evaporation-Induced Flows inside a Confined Droplet of Diluted Saline Solution Sang-Joon Lee, Jiwoo Hong, and Yong-Seok Choi Langmuir, Just Accepted Manuscript • Publication Date (Web): 16 Jun 2014 Downloaded from http://pubs.acs.org on June 17, 2014

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Evaporation-Induced Flows inside a Confined Droplet of Diluted Saline Solution Sang Joon Lee*, Jiwoo Hong, and Yong-Seok Choi Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), San 31, Hyoja-dong, Pohang 790-784, South Korea

ABSTRACT Flow patterns inside a droplet of diluted aqueous NaCl solution confined by two flat substrates under natural evaporation were investigated both experimentally and numerically. We focused on natural convection-driven flows inside confined droplets at high Rayleigh numbers (i.e., the ratio of buoyancy to diffusion, Ra), where the convection of solutes is strongly dominant, compared to diffusion. The evaporated water at the free surface of the droplet builds up a concentration gradient inside the solution, which induces the Rayleigh convection flow. Three-dimensional trajectories of tracer particles in the droplet were tracked, and axisymmetric flow motions induced by the Rayleigh convection were experimentally measured by using a digital in-line holographic microscopy technique. In addition, the effects of the confined droplet’s aspect ratio and the liquid’s molar concentration on the evaporationinduced flows were investigated. The convection velocity is found to be increased as molar concentration increases, because Rayleigh convection becomes significant at high the molar concentration is high (i.e. high Ra). Our numerical simulation based on the Boussinesq approximation fairly well predicted the velocity profiles of evaporating confined droplets at low concentrations. Consequently, evaporation kinetics inside the confined droplets can be *

Author to whom all correspondence should be addressed. 1

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controlled with varying droplet’s aspect ratio and the liquid’s molar concentration, which provides helpful information for the design of biochemical microplating with limited resources and for tuning self-assembly micro/nanoparticle clusters.

1. INTRODUCTION The evaporation of liquids on solid surfaces is a very common, natural phenomenon that we can see everyday and everywhere. It is basic and crucial process in numerous practical applications such as ink-jet printing,1-3 thin film coating,4 spray cooling,5 and deposition of DNA/RNA micro-arrays.6 Despite of its familiarity and importance, the understanding of evaporation dynamics including internal flow pattern is considerably difficult, because hydrodynamics, contact line dynamics, mass and heat transfers are mutually coupled. Many researchers have tried to unveil the underlying physics of evaporation of sessile droplets encountered from the traditional puzzle of coffee ring strains to DNA stretching applications. Deegan et al. studied outward capillary flows inside evaporating sessile droplets with pinned contact lines.7,8 The outward flow was inducted to compensate the liquid loss at the edge of evaporating droplets. Hu and Larson numerically analyzed the time-dependent axisymmetric outward flow.9 Savino and Monti numerically investigated the natural and Marangoni convection flows inside sessile and pendant droplets.10 They found that natural convection dominates the solute transport inside the droplets. Kang et al. experimentally and numerically observed the Rayleigh convection flows inside sessile droplets of a water-alcohol mixture and a saline solution placed on a hydrophobic substrate.11,12 They demonstrated that 2


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the evaporation of a droplet is accompanied by a convective motion that is an important factor in the overall mass transport phenomenon. Kochiya and Ueno measured the trajectories of suspended particles inside drying droplets of pure water and of a water-ethanol mixture by using a three-dimensional (3D) particle-tracking velocimetry technique.13 The previous studies on the evaporation of sessile droplet were recently reviewed in detail in Ref. 14. However, these studies on the evaporation of sessile droplets have several technological difficulties, such as laborious control of evaporation kinetics and inaccurate observation due to image distortion.15,16 In addition, the study on a sessile droplet is intrinsically a 3D problem so that it is difficult to develop a numerical or mathematical model to predict the evaporation dynamics of sessile droplets.17,18 To alleviate such difficulties in the studies on the evaporation of sessile droplets, some research groups proposed to investigate the evaporation of droplet in confined configuration. In the confined configuration, evaporation proceeds by gas diffusion from the edge of the droplet toward the edge of the plates. Thus, the confined configuration naturally sets up a well-defined boundary condition.15 In addition, it facilitates direct observation of flows inside the droplet due to its thin and quasi-2D geometry.15 Clément and Leng observed the temporal volume variations of the solvent for a pure liquid and a solution of evaporating liquids in confined droplets to investigate the solvent activity.15 Leng studied the drying kinetics of a colloidal suspension with different particle sizes and volume fractions in the same confined configuration.16 Daubersies and Salmon developed a theoretical model to describe the drying phenomena of solutions and colloidal dispersions inside droplets confined between two circular plates.17 Selva et al. demonstrated experimentally and theoretically that natural convections driven by solutal density differences in a molecular binary mixture can

3


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enhance the transport of colloids.18 Recently, Giorgiutti et al. directly observed the concentration profiles of particles inside a confined droplet using a fluorescence microscopy.19 Studies on practical applications based on the evaporation of confined droplets have also been reported. Bensimon et al. proposed a simple method to stretch DNA by evaporating a liquid droplet placed between a silanated surface and an untreated glass.20 To study wetting dynamics of samples and to mitigate evaporation in biochemical microplating process, Ng and his coworkers suggested a convenient method to measure contact angle of small volumes using capillary well microplating and microscopic imaging methods.21,22 Yunker et al. measured the bending rigidity of a colloidal monolayer membrane inside a confined geometry.23 To enhance the performance of abovementioned practical applications, the evaporation-driven flows inside confined droplets should be clearly understood. In this study, internal flows and concentration distributions of saline droplets confined between two plates under evaporation are investigated. The 3D trajectories of tracer particles inside the droplet are measured by using digital holographic particle tracking velocimetry (DHPTV) technique, after which the velocity field information of the flow are deduced.24 DHPTV technique is capable of recording 3D volumetric field information on a single hologram. The effects of the confined droplet’s aspect ratio and the liquid’s molar concentrations on the evaporationinduced flows are also investigated by using simple numerical simulation based on the Boussinesq approximation.

2. EXPERIMENTAL SETUP 4


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The experimental setup used in this study is similar to that for evaporation of confined droplets (Figure. 1).15,16 A NaCl solution diluted with deionized (D.I.) water was used to form a liquid droplet. The molar concentration (C) of the diluted solution was varied to 0.1, 0.5, and 1 M. Polystyrene micro particles with a mean diameter of 4 µm to 7 µm were seeded in the solution as flow tracers. The kinematic viscosity ν of the tested fluid was approximately 0.94 mm2/s. Using a micropipette, a small amount of the solution was placed between two slide glasses that are separated by a spacer to form a confined droplet. The heights (H) of the confined droplet varied between 0.5 and 2.0 mm. The initial volume of the confined droplets tested in this study ranged from 1 µL to 20 µL. The slide glasses were spincoated with polydimethylsiloxane (PDMS) to make the surface hydrophobic and to increase the contact angle. This surface treatment enabled the confined droplet to have an approximately 90° contact angle at the edges, which resulted in an almost rectangular cylindrical shape, as shown in Figure. 1(b). However, given the hydrostatic pressure, the diameters at the bottom were slightly larger than those at the top. Therefore, the mean of the two diameters was used as the nominal diameter of the confined droplet. The temperature and relative humidity are kept in the ranges of 25 °C ± 0.5 °C and 45% ± 5%, respectively. The volume of the evaporating droplet was evaluated quantitatively from the optical images obtained from the lateral direction, as shown in Figure 1(b). The droplet images were consecutively recorded, and the transient volume variation of the droplet was estimated with the assumption that the droplet has an axisymmetric geometry. As the evaporation proceeded, the diameter of the confined droplet slowly shrunk while the contact angle was maintained. The receding velocity ( u0 ) of the evaporating surface was measured by tracing the wall position from the images. 5


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The in-line DHPTV setup depicted in Figure 1(a) was employed to obtain 3D, threecomponent velocity vector field information. The setup comprised a continuous diodepumped solid-state Nd:YAG laser (λ = 532 nm, 100 mW, CrystaLaser, USA), a beam expander (20×, Newport, USA), a microscope objective lens (4×, NA=0.1, Nikon, Japan), and a digital camera (PCO.2000, PCO, Germany). The focal plane of the objective lens was positioned above the top substrate to generate holographic images and to obtain the overlap of the scattered and unscattered waves. The holographic images magnified by the objective lens were captured by using the digital camera and then stored in a computer hard disk. The 3D motion of the flow tracers inside the confined droplet was analyzed by employing the digital image processing techniques described in our previous work.24 Figure 1(c) shows a typical holographic image processed in this study. The flow motion was immediately initiated after the liquid-bridge was formed, and a stable flow pattern was established after approximately 1 min. The holograms were recorded consecutively for 10 min after the stable flow patterns settled. Every experiment was repeated at least three times, and the experimental errors are typically within ±5% of the data in each figure.

Figure 1. (a) Schematic diagram of experimental setup and the coordinate system used for 6


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numerical analysis. (b) Side view of the confined droplet. The contact angle is approximately 90° after coating the substrates with PDMS. (c) A typical hologram captured in this study.

3. NUMERICAL ANALYSIS We conducted a numerical analysis on the flow field and concentration distribution inside a slowly evaporating confined droplet. The stretching coordinate method12 was employed to account for the effect of boundary shrinkage during evaporation. The geometric boundary and concentration distribution in the evaporating droplet were assumed to be cylindrical and axisymmetric, respectively. We assumed several hypotheses to develop a numerical model that does not require complex numerical techniques. Similar to the assumptions employed in the Kang et al.’s numerical model,12 the thermal convection and Marangoni effects were disregarded and the local density variation caused by concentration gradient of the solute was considered only via the body-force term in the momentum equation (i.e., Boussinesq approximation). The flow inside the confined droplet then has to satisfy the following momentum and species equations:

ρ 0 ( ∂v ∂t + v ⋅ ∇v ) = −∇p + µ∇ 2 v + ρ g, ρ 0 ( ∂c ∂t + v ⋅∇c ) = ρ D∇ 2 c

(1)

where v and p denote the velocity and the pressure of the evaporating droplet, respectively. ρ0 is the initial liquid density, ρ is the local liquid density, µ is the dynamic viscosity of the fluid, g is the constant of gravitational acceleration, c is the concentration of species in mass fraction (wt%), and D is the molecular diffusivity. The local density is approximated by the 7


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equation ρ ≅ ρ 0 1 + β ( c − c0 )  ,

where

c0

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is

the

initial

concentration,

and

β ≡ (1 ρ )( ∂ρ ∂c ) = 7×10-3/wt% is the solutal expansion coefficient. On the evaporating surface, the following species boundary condition should hold: ρ D∇c ⋅ n = ρ cu0 , where n is the unit normal vector, and u0 is the receding velocity of the evaporating droplet surface. The characteristic length a(t) and characteristic time t = a0 uc were used to formulate the nondimensional governing equations. In this work, a(t) is the nominal radius of the liquid-bridge, and a0 is its initial value. The characteristic velocity uc is defined as uc = D a0 . Other nondimensional variables include V = v uc , C = c c0 , R = r a(t ) , Z = z H , P = p ( ρ 0uc 2 ) , and λ = a(t ) a0 = 1 − u0t a0 . Using these variables, Equation (1) is converted into the following non-dimensional equations:

λ ( ∂V ∂τ ) + V ⋅∇% V = −∇% P + ( Sc λ ) ∇% 2 V − RaScλ ( C − 1) k − Pe ( R P ⋅∇% V ) , λ ( ∂C ∂τ ) + V ⋅∇% C = (1 λ ) ∇% 2C − Pe ( R P ⋅∇% C ) .

(2)

where Pe = u0 a0 D , λ = 1 − Peτ , Ra = g β c0 a0 3 (ν D) , and Sc = ν D . Pe, Ra, Sc, and ν

denote the Peclet number (i.e., the ratio of convection to diffusion), saline Rayleigh number (i.e., the ratio of buoyancy to diffusion), Schmidt number (i.e., the ratio of momentum diffusivity to mass diffusivity), and kinematic viscosity, respectively. k is the z-directional unit vector, and Rp is the non-dimensional position vector. The values ∇% and ∇% 2 denote the non-dimensional derivative operators; initial conditions are V = 0 and C = 1. For the boundary conditions of the momentum equation, a slip condition is assumed on the evaporating surface, whereas a no-slip condition is applied on the substrates. A free-shear rate (∂uz/∂r = 0) is applied at the boundary of the slip boundary condition. The non-dimensional 8


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% C ⋅ n = ( Peλ ) C . concentration condition at the evaporating surface is ∇ FLUENT (VERSION 6.3) was employed for the numerical calculations, and the 2D axisymmetric unsteady model with an implicit scheme was used. User-defined functions were used to solve the non-dimensional equations. The non-dimensional constants (i.e., Pe, Ra, and Sc) obtained from the experimental values are presented in Table 1. The Ra number exhibited the largest change because of the geometrical features. In addition, the Ra number approximately increased ten times because the diameter of the confined droplet was doubled.

Table 1. Calculated non-dimensional constants

Pe

Ra

Sc

H = 2 mm, V = 5 µL, C = 1 M

3.31×10-2

1.70×105

5.84×102

H = 2 mm, V = 20 µL, C = 1 M

4.32×10-2

1.39×106

5.84×102

4. RESULTS AND DISCUSSION

The temporal volume variations of the confined droplets of C = 0, 0.1, and 1 M and H = 0.5 and 1 mm were consecutively measured (Figure 2). The evaporation rate was derived

from inclination of the volume profiles. The evaporation rates are nearly constant except for C= 1 M , because the diameter of the confined droplet constantly shrinks without contact-line

pinning during droplet evaporation. When C = 1 M, the evaporation rate is not only slower, it also slows further down. The evaporation rate is approximately proportional to vapor pressure, which is decreased by the presence of NaCl. This is because the NaCl ions interact with the water molecules and make it difficult for them to escape into the air. The evaporation 9


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rates are also proportional to the droplet height. The evaporation rates for H = 1 mm are nearly two times of those for H = 0.5 mm. This result is attributed to the fact that the evaporation of a confined droplet is governed by the vapor diffusion along the gap between the two substrates.

Figure 2. Temporal volume variations of slowly evaporating confined droplets.

The 3D trajectories of the tracer particles suspended in the confined droplet with a height of H = 2 mm, an initial volume of V = 5 µL, and a molar concentration of C = 1 M are illustrated in Figure 3. The axial velocities are coded in color, with red and blue indicating the upward and downward motions, respectively. When water evaporates from the droplet surface, the surface concentration of NaCl becomes higher and the concentration gradient is generated along the radial direction. The density variation due to the concentration gradient of NaCl induces the natural convection (i.e. buoyancy-induced flow) that rises along the symmetry axis and flows down along the droplet surface.12 In addition, this flow can influences the spatial distributions of NaCl concentration inside the droplet and the concentration is increased as it approaches to the lower parts of the droplet surface. This 10


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result will be confirmed by the numerical simulation shown in Figure 5. On the contrary, any noticeable convective motion was not observed when we tested the confined droplet of D.I. water. The temperature gradient induced by the latent heat of evaporation is negligible, because the surface evaporation rate of the diluted aqueous NaCl solution is almost the same as that of pure water. The outward capillary flows reported by Deegan et al.7,8 and other researchers9,25 are not observed in this study, because the contact lines are not pinned on the hydrophobic surface. From previous results for evaporation-induced flows inside sessile droplets under similar experimental conditions,11,12 the Marangoni effect attributed to the temperature and concentration gradients in the internal flow of confined droplets seems to be negligible.

Figure 3. 3D trajectories of tracer particles in the confined droplet: H = 2 mm, V = 5 µL, and

C = 1 M.

3D trajectories of tracer particles in the confined droplet of large volume (20 µL) under same conditions, and C = 1 M are shown in Figure 4. The convectional flow motion is similar to the previous results shown in Figure 3. The trajectories are more tightly packed near the center region of the bottom surface, compared to those near the top surface. 11


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Accordingly, the region of fast upward motion is adjacent to the bottom surface. A slight asymmetry is observed in the x-y plane. It may result from uncontrollable experimental conditions such as the slightly different evaporation rates around the surface and a very small misalignment of two substrates.

Figure 4. 3D trajectories of tracer particles in the confined droplet: H = 2 mm, V = 20 µL,

and C = 1 M.

The spatial distributions of concentration and velocity field obtained from the numerical simulation and the experiment for the case of H = 2 mm and C = 1 M are compared in Figure 5. The temporal variations of the concentration and the axial velocity profiles are described in Supporting Information. The experimental velocity field was obtained by ensemble averaging of the particle velocities at each grid position (r, z) which 12


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were acquired for 10 min with intervals of 1 sec after the stable flow patterns was settled. The streamlines obtained by numerical simulation show good agreement with the flow patterns obtained from the experiment. The flow speed of a 20 µL droplet is faster than that of a 5 µL droplet. This is contributed to the fact that buoyancy effect is more dominant for a larger droplet. In addition, the stagnation point at which the axial velocity is zero shifts toward the evaporation surface as the diameter of the confined droplet increases. A plateau is also observed in the concentration profiles along the midline of the 20 µL droplet. For V = 5 µL, the flow patterns in the upper and lower parts are nearly symmetric. However, for V = 20 µL, the upward flow near the center axis decelerates after passing the midline. The different flow patterns are primarily attributed to the wide region of radially quasi-equivalent concentrations above the midline.

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Figure 5. Spatial distributions of concentration (a and b) and flow speed (c and d) obtained

from the numerical simulation. (e and f) Velocity vector fields obtained from the experiment. Here, the amplitude of velocity vector field was indirectly estimated from spatial distributions of the simulated flow speeds; (a, c, e) H = 2 mm, V = 5 µL, and C = 1 M; (b, d, f) H = 2 mm, V = 20 µL, and C = 1 M.

The axial velocity profiles numerically extracted from the midline z/Z = 0.5 of the confined droplets at different ion concentrations are compared with experimental results (Figure 6). Droplets having the same geometry of V = 1 µL and H = 0.5 mm were tested with varying molar concentration of C = 0.1, 0.5, and 1 M. As the molar concentration increases, 14


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the convection velocity is also increased. The simulated velocity profiles are in good agreement with the experimented results when the molar concentration is low. As the concentration increases, however, both profiles show large deviations, especially in the region near the outer edge of the confined droplet. The flow velocity near the droplet surface obtained from the numerical simulation is faster than the experimental results. This large deviation seems to be attributed to the slip boundary condition applied to the momentum equation on the free surface. In our previous study on Rayleigh convection inside an evaporating sessile droplet, the flow velocity near the droplet surface for the case of the noslip condition is slower than that for the case of the slip condition.12 Thus, we deduced that the boundary condition shifts midway between the slip and no-slip conditions at higher molar concentrations.

Figure 6. Variations of axial velocity profiles along the midline z/Z = 0.5 at different ion

concentrations. The convection velocity is increased, as the molar concentration increases. The experimental data and simulation results exhibit larger deviations at higher concentrations, especially in the region near the free surface of the confined droplet. H = 0.5 15


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mm and V = 1 µL.

5. CONCLUSIONS

The evaporation-induced convection flows in confined droplets of diluted aqueous NaCl solutions were both experimentally and numerically. The 3D trajectories of tracer particles in confined droplets were found to exhibit axisymmetric flow motions by using digital holographic microscopy technique. In addition, the evaporation rates of the confined droplets were also measured. The evaporation rates are nearly constant, except for high concentration (C = 1 M). They are proportional to the droplet height. The decrease of the evaporation rate at high concentration (C = 1 M) is evident as the confined droplet evaporates. In addition, the convection velocity of internal flow is increased as the molar concentration increases. The numerical simulation based on the Boussinesq approximation predicted fairly well the velocity profiles of evaporating confined droplets at low concentrations. However, the comparison between numerical predictions and experimental results at high concentrations show large deviations. This implies that the slip boundary condition used for the momentum equation at the free surface is inappropriate at high concentrations. The present study would be useful in the design of biochemical microplating and fabrication of micro/nanostructures by using evaporation-driven flows.

ASSOCIATED CONTENT Supporting Information

16


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Temporal evolutions of the concentration, effective concentration distributions, and the velocity field are described in Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Authors *

Phone: +82-54-279-2169. Fax: +82-54-279-3199. E-mail: [email protected].

ACKNOWLEDGMENT

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2008-0061991).

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021405. (17) Daubersies, L; Salmon, J. -B. Evaporation of solutions and colloidal dispersions in confined droplets. Phys. Rev. E 2011, 84, 031406. (18) Selva, B.; Daubersies, L; Salmon, J. -B. Solutal convection in confined geometries: Enhancement of colloidal transport. Phys. Rev. Lett. 2012, 108, 198303. (19) Bensimon, D.; Simon, A. J.; Croquette, V.; Bensimon, Stretching DNA with a receding meniscus: Experiments and models. A. Phys. Rev. Lett. 1995, 74, 4754. (20) Giorgiutti-Dauphiné, F.; Pauchard, L. Direct observation of concentration profiles induced by drying of a 2D colloidal dispersion drop. J. Colloid Interf. Sci. 2013, 395, 263– 268. (21) Cheong, B. H. -P.; Ng, T. W.; Yu, Y.; Liew, O. W. Using the meniscus in a capillary for small volume contact angle measurement in biochemical applications. Langmuir 2011, 27, 11925–11929. (22) Hunyh, T.; Muradoglu, M.; Liew, O. W.; Ng, T. W. Contact angle and volume retention effects from capillary bridge evaporation in biochemical microplating. Colloids Surf. A: Physicochem. Eng. Aspects 2013, 436, 647–655.

(23) Yunker, P. J.; Gratale, M.; Lohr, M. A.; Still, T.; Lubensky, T. C.; Yodh, A. G. Influence of particle shape on bending rigidity of colloidal monolayer membranes and particle deposition during droplet evaporation in confined geometries. Phys. Rev. Lett. 2012, 108, 228303. (24) Choi, Y. S.; Lee, S. J. Three-dimensional volumetric measurement of red blood cell motion using digital holographic microscopy. Appl. Opt. 2009, 48, 2983–2990. (25) Abramchuk, S. S.; Khokhlov, A. R.; Iwataki, T.; Oana, H.; Yoshikawa, K. Direct

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observation of DNA molecules in a convection flow of a drying droplet. Europhys. Lett. 2001 55, 294–330.

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Figure 1. (a) Schematic diagram of experimental setup and the coordinate system used for numerical analysis. (b) Side view of the confined droplet. The contact angle is approximately 90° after coating the substrates with PDMS. (c) A typical hologram captured in this study. 80x49mm (300 x 300 DPI)


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Figure 2. Temporal volume variations of slowly evaporating confined droplets. 80x59mm (300 x 300 DPI)


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Figure 3. 3D trajectories of tracer particles in the confined droplet: H = 2 mm, V = 5 µL, and C = 1 M. 160x53mm (300 x 300 DPI)


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Figure 4. 3D trajectories of tracer particles in the confined droplet: H = 2 mm, V = 20 µL, and C = 1 M. 160x116mm (300 x 300 DPI)


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Figure 5. Spatial distributions of concentration (a and b) and flow speed (c and d) obtained from the numerical simulation. (e and f) Velocity vector fields obtained from the experiment. Here, the amplitude of velocity vector field was indirectly estimated from spatial distributions of the simulated flow speeds; ; (a, c, e) H = 2 mm, V = 5 µL, and C = 1 M; (b, d, f) H = 2 mm, V = 20 µL, and C = 1 M. 80x141mm (300 x 300 DPI)


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Figure 6. Variations of axial velocity profiles along the midline z/Z = 0.5 at different ion concentrations. The convection velocity is increased, as the molar concentration increases. The experimental data and simulation results exhibit larger deviations at higher concentrations, especially in the region near the free surface of the confined droplet. H = 0.5 mm and V = 1 µL. 80x74mm (300 x 300 DPI)


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Evaporation-Induced Flows inside a Confined Droplet of Diluted

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