Influence of an adjacent droplet on fluid convection inside an evaporating droplet of binary mixture

Accepted Manuscript Title: Influence of an adjacent droplet on fluid convection inside an evaporating droplet of binary mixture Author: Tapan Kumar Pradhan Pradipta Kumar Panigrahi PII: DOI: Reference:

S0927-7757(16)30222-9 http://dx.doi.org/doi:10.1016/j.colsurfa.2016.03.073 COLSUA 20554

To appear in:

Colloids and Surfaces A: Physicochem. Eng. Aspects

Received date: Revised date: Accepted date:

22-2-2016 29-3-2016 30-3-2016

Please cite this article as: Tapan Kumar Pradhan, Pradipta Kumar Panigrahi, Influence of an adjacent droplet on fluid convection inside an evaporating droplet of binary mixture, (2016), http://dx.doi.org/10.1016/j.colsurfa.2016.03.073 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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*Graphical Abstract (for review)

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Single droplet

Two neighboring droplets

Highlights

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We study the influence of an adjacent droplet on Rayleigh convection inside a droplet of NaCl solution using µ-PIV measurements and 3D COMSOL simulation. Evaporative flux distribution on the droplet surface, concentration field and velocity field inside the droplet get influenced by the presence of an adjacent evaporating droplet. The evaporative flux distribution, concentration field and recirculating flow pattern show asymmetric behavior for the two droplets configuration contrary to the symmetric behavior for the single droplet configuration. The influence of the adjacent droplet reduces with increase in separation distance between the two droplets leading to single droplet behavior at large separation distance.

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Tapan Kumar Pradhan, Pradipta Kumar Panigrahi∗

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Influence of an adjacent droplet on fluid convection inside an evaporating droplet of binary mixture

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Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

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Abstract

We study the fluid convection inside two adjacent evaporating droplets of binary mixture. Micro-PIV technique has been used to measure the velocity field

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inside the droplet. The experimental observation on internal convection inside droplet has been explained using the simulation results of evaporative flux distribution and concentration field inside the droplet. The interaction between

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the two adjacent evaporating droplets has been studied by varying the separa-

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tion distance between the two droplets. Evaporation from the droplet induces buoyancy driven Rayleigh convection inside the droplet. Fluid convection inside a single droplet shows a symmetrical flow pattern with two recirculating bubbles in X-Z and Y-Z wall normal planes. This behavior is attributed to the symmetric evaporative flux distribution on the droplet surface and the resulting symmetric concentration field inside the droplet. Presence of an adjacent evaporating droplet leads to asymmetric evaporative flux distribution on the droplet surface due to the influence of the neighboring droplet on the free stream mass fraction. This asymmetric evaporative flux distribution on the droplet surface results in asymmetric concentration field inside the droplet. As a result, the fluid convection in two droplets configuration shows asymmetric flow pattern with one recirculation bubble in X-Z wall normal plane and two recirculation bubbles in Y-Z wall normal plane. ∗ Corresponding

author Email addresses: [email protected] (Tapan Kumar Pradhan), [email protected] (Pradipta Kumar Panigrahi)

Preprint submitted to Journal of LATEX Templates

March 29, 2016

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Keywords: Droplet, Rayleigh Convection, Droplet interaction,

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Droplet evaporation

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1. Introduction

Evaporation driven flows are observed in droplets of various liquids and so-

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lutions [1, 2, 3, 4]. In an evaporating droplet, a surface tension gradient is created by uneven evaporation from the droplet surface. The uneven evapora5

tion leads to temperature gradient and concentration gradient along the droplet

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surface leading to generation of the surface tension gradient along the droplet surface. There is flow from the lower surface tension region to the higher surface tension region and the effect of this flow penetrates to the inner region

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of the droplet through viscous effect. Deegan et al.[4] studied the fluid flow pattern inside an evaporating droplet of pinned contact line. They observed an outward radial flow of fluid to replenish the water loss at the contact line.

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Hu and Larson [5] studied the Marangoni convection in an evaporating sessile

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droplet and observed that the surface tension gradient is generated due to the temperature gradient produced by uneven evaporation from the droplet surface.

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The non-uniform surface tension gradient leads to Marangoni stress along the air-liquid interface, which produces a thermal Marangoni convection. Highly volatile fluid like methanol causes vigorous convection inside the droplet which is disordered in nature [6]. Marangoni convection due to temperature gradients inside droplets of non evaporating fluid like silicone oil was studied by Savino and

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Fico [1] by changing the substrate temperature. Thokchom et al. [7] studied the Marangoni convection inside a droplet subjected to IR heating. Surface tension

driven Marangoni convection in binary mixtures occurs due to the concentration gradient developed on the surface of the fluid [8]. Bennacer and Safiane [9] studied the surface tension gradient driven Marangoni flow inside a drop of

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ethanol-water mixture and reported generation of surface tension gradient due to the local concentration gradient along the droplet interface. Kaneda et al. [10] numerically studied the convection inside an evaporating droplet of polymer

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solution considering both thermal and solutal Marangoni effect. Evaporation

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from a droplet of aqueous solution causes buoyancy driven Rayleigh convection

[2, 11, 12]. Kang et al. [2] numerically investigated the buoyancy driven flow

inside an evaporating droplet of aqueous NaCl solution. Savino and Monti [12]

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investigated fluid convection inside a droplet of aqueous solution containing wa-

ter, precipitating agent and protein. They observed that the flow inside the

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evaporating droplet is driven by buoyancy force caused by concentration gradient inside the droplet. Lee et al. [13] investigated both experimentally and

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numerically the flow inside a droplet of aqueous NaCl solution confined between two horizontal flat substrates. They observed that the evaporation from the surface of droplet causes buoyancy driven Rayleigh convection flow where the

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fluid moves downward along the droplet surface and moves upward at the center region of the droplet.

Interaction between droplets plays crucial role in several applications i.e.

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droplet coalescence, digital microfluidics, dropwise condensation, protein crystal growth by vapor diffusion method and surface coating etc where droplets are

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surrounded by other droplets. The nature of the flow field inside a droplet

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has many practical implications. Nanotechnology applications using pattern formation during drying of a droplet depends on the flow inside the droplet [4, 14]. The drying pattern of a droplet is affected by the presence of another droplet placed nearer to it [14, 15]. Deegan et al. [14] observed that the nearest region of two droplets shows weakest deposits during drying of two droplets located

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side by side. Several researchers have observed different process performance

involving multiple droplets compared to that of the single droplet configuration. One example is the protein crystal growth process where, the number of drops

influences the crystal quality during lysozyme protein crystal growth by the hanging drop technique [16].

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Fluid convection inside interacting droplets significantly affects droplet coalescence. Aversana et al. [17] reported that the Marangoni convection inside droplets can prevent the coalescence of two droplets. The coalescence of two sessile droplets with different miscible liquids is delayed after forming a neck 3

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between the two droplets [18, 19, 20]. The delay occurs due to the Marangoni convection which drains out fluid from the neck region [20]. Surface tension gra-

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dient caused by the coalescence of two different miscible droplets can propel the droplets on the surface [21, 22]. Carles and Cazabat [23] reported that a droplet

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can be propelled in the presence of a neighboring volatile droplet of different fluid. They observed that this phenomenon occurs due to the surface tension gradient on the droplet surface caused by the penetration of vapor of the neigh-

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boring volatile droplet. Cira et al. [24] studied the repulsion and attraction

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between two neighboring droplets of water and propylene glycol.

Most of the previous studies in literature are limited to convection in single droplet configuration. To the best of our knowledge, influence of a neighboring droplet on the internal hydrodynamics of an evaporating droplet is not available

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in literature. In this work, we investigate the flow behavior inside a droplet of aqueous NaCl solution due to the interaction between neighboring evaporating

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droplets without any physical contact between the droplets. We experimentally demonstrate the droplet interaction effects on fluid convection inside the droplet using micro-PIV measurements. Simulation results of evaporative flux

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distribution on the droplet surface and concentration field inside the droplet using COMSOL multiphysics software are used to understand and explain the observed phenomena.

2. Experimental Details

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The experimental arrangement used for investigation of internal convection

inside two droplets using micro-PIV technique is shown in Fig. 1. The two droplets are placed on a siliconised cover slip. The volume of each droplet is equal to 0.7 µL. The contact line diameter of each droplet is equal to 1.38 mm.

The separation distance between the two droplets is equal to 260 µm ± 6 µm. 85

The siliconised cover slip acts as a hydrophobic surface. The contact angle of the droplet is equal to 910 ±40 . Contact angle is measured using DropSnake plugin in Image J [25]. The cover slip containing two droplets is enclosed by a rectangular

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X Y Scanning area

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Objective

Excitation

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Scan control Dichroic mirror

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Z

Laser

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Emission Pin hole

Display

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PMT

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Figure 1: Experimental arrangement for visualization of convection inside the droplets.

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enclosure to prevent any influence of atmospheric air convection. The working fluid used is aqueous NaCl solution with 1 M concentration. Polystyrene fluo-

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3

rescent particles (diameter 2 µm, density 1.05 g/cm ) are added to the fluid of the droplets for visualization of fluid flow pattern. These particles act as tracer particles for PIV measurement. The ambient temperature is equal to 20 0 C

and relative humidity is equal to 60 % ± 5 %. Bond number (Bo = ρgRh0 /σ)

for the droplets is equal to 6.5 × 10−2 , where ρ is the density of fluid, g is the

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acceleration due to gravity, R is the contact line radius of droplet, h0 is the droplet height and σ is the surface tension of the solution. The values of ρ and 3

σ for 1 M aqueous NaCl solution are equal to 1038 kg/m and 7.39 × 10−2 N/m respectively at 20 0 C [26, 27]. The value of R is equal to 690 µm and the value of

h0 is taken same as that of R. For this small value of Bond number, the droplets 100

form a spherical cap. Buoyancy effect due to density difference between the seeding particles and fluid may cause settling of particles. According to Stokes law, settling velocity of particles is given by, Ug =

d2p (ρp −ρ) g. 18µ

Here, dp and ρp

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are the diameter and density of the seeding particles. The value of dynamic

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viscosity (µ) of NaCl solution is equal to 1.09 × 10−3 kg/m · s [28]. The value of settling velocity is found equal to 2.4 × 10−8 m/s which is very negligible as

compared to the observed velocity in the experiment which is of the order of

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3.0 × 10−5 m/s. Hence, the effect of particle settling due to gravity on the fluid convection can be ignored.

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Images for the micro-PIV measurement are captured by a confocal microscope. The fluorescent particles present inside the droplets are illuminated by a

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laser source of wave length 488 nm. The emissions from the fluorescent particles are captured by the PMT present in the confocal microscope. Numbers of images are captured by point scanning using PMT of Leica confocal microscope.

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As a confocal microscope uses pinhole, background noise is minimal. The image size is equal to 512 pixels × 512 pixels. The field of view of the image is equal to 1.55 mm × 1.55 mm. Images at a particular wall normal, Z-location

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from the substrate surface are captured at a time interval of 1.16 sec. The particle visualization images are processed by a PIV evaluation software called

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DynamicStudio from Dantec to obtain the velocity vector fields. Adaptive cross

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correlation with 3 step refinements for 32×32 pixels interrogation area and 25

% overlap is used during the PIV processing.

3. Numerical Modelling

We have carried out three-dimensional simulation for both single droplet and

two droplets configuration. The schematic of the droplets used in numerical

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simulation is shown in Fig. 2. The droplets are assumed hemispherical shape with 900 contact angle and 690 µm contact line radius (R). The separation distance (S) between the two droplets is equal to 260 µm. The concentration 3

of the solution is set equal to 1000 mol/m (1 M). Continuity, momentum and species transport equations are used to study 130

the flow inside the droplets. Boussinesq approximation is used in momentum equation where the density only varies in body force term. We assumed the flow

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z

z

n

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n

x

x R

(b)

S

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Figure 2: Schematic of droplets used in numerical simulation for (a) single droplet configuration and (b) two droplets configuration.

to be incompressible. The Governing equations are:

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∂u ∂v ∂w + + =0 ∂x ∂z ∂z

(1)

   2 ∂u ∂p ∂u ∂u ∂u ∂ u ∂2u ∂2u =− (2a) +u +v +w +µ + + ∂t ∂x ∂y ∂z ∂x ∂x2 ∂y 2 ∂z 2    2  ∂p ∂v ∂v ∂v ∂2v ∂2v ∂ v ∂v =− (2b) +u +v +w +µ + + ρ0 ∂t ∂x ∂y ∂z ∂y ∂x2 ∂y 2 ∂z 2     2 ∂p ∂w ∂w ∂w ∂w ∂ w ∂2w ∂2w =− + + +u +v +w − ∆ρg + µ ρ0 ∂t ∂x ∂y ∂z ∂z ∂x2 ∂y 2 ∂z 2

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ρ0

(2c)

∂c ∂c ∂c ∂c +u +v +w = Dc ∂t ∂x ∂y ∂z



∂2c ∂2c ∂2c + + ∂x2 ∂y 2 ∂z 2



(3)

Here, u, v and w represent the x, y and z component of velocities, p is the

pressure, t is the time, g is the acceleration due to gravity, ρ is the density,

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µ is the dynamic viscosity of the fluid, Dc is the diffusivity of NaCl in water and c is the solute concentration. The value of Dc is equal to 1.6 × 10−9 m2 /s [29]. The body force (∆ρg) term in the z-momentum equation is because of

the density difference arising from the concentration gradient, where ∆ρ is the density difference, ρ − ρ0 . The initial density (ρ0 ) of the solution is equal to

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3

1038 kg/m [26]. Pure water droplet does not show any convection inside the droplet at identical evaporation condition as that of NaCl solution droplet. Therefore, the con-

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tribution of thermal effect on the flow pattern observed in the present study is

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assumed negligible. Uneven evaporation from the droplet surface may cause concentration gradient along the liquid-air interface leading to solutal Marangoni

convection due to surface tension gradient along the interface. Presence of con-

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taminants on the liquid-air interface can suppress the Marangoni convection

[30, 31, 5, 1]. Even in a clean environment, the liquid-air interface gets con-

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taminated due to the adsorption of contaminants molecule on the liquid surface [5, 1]. A small trace of contaminants on the liquid surface can completely re-

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tard the surface tension driven flow at the liquid-air interface [30]. Study by Kang et al. [2] shows that the flow inside a single evaporating droplet of NaCl solution is driven by Rayleigh convection due to concentration gradient. They

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observed that the surface tension gradient has no effect on the flow observed inside the droplet. Hence, the contribution of surface tension gradient flow due to temperature and concentration gradient has been assumed negligible in the

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present study. The solute concentration inside the droplet at initial time remians uniform.

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Hence, there is no flow of fluid at initial time. The fluid velocity at the substrate

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surface is equal to zero due to the no slip boundary condition at the solid wall.

The velocity boundary conditions are summarized below:

u=v=w=0

at

u=v=w=0

t = 0 (initial condition),

(4a)

at substrate surface.

(4b)

The flow visualization image of particle movement inside a single evaporating

droplet is presented in Fig. 3. The flow visualization movie (see multimedia view) clearly shows no movement of particles at the surface of the droplet.

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Therefore, the air-liquid interface for single droplet case has been assumed to have no slip boundary condition. Kang et al. [2] also reported no movement of particles at the surface of the droplet for a single droplet of aqueous NaCl droplet. They observed that the no slip boundary condition at the liquid-air 8

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Figure 3: Visualization images of particle movement inside a single evaporating droplet (Multimedia view).

Figure 4: Visualization images of particle movement inside a droplet in the presence of a neighboring droplet (Multimedia view).

interface is more realistic as compared to the slip flow boundary condition for

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a droplet of NaCl solution. Fig. 4 shows the flow visualization image of tracer

particles inside the droplet for the two droplets case. Contrary to the single

droplet case, the particles at the droplet surface shows free movement along the interface. Therefore, the boundary condition at the liquid-air interface of two droplets configuration has been taken as slip flow boundary condition. The 175

difference in the behaviour of the droplet surface between single droplet and two droplets configuration may be attributed to the asymmetry in the boundary

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condition along the droplet surface due to the assymmetric ambient water vapor

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concentration distribution. The velocity boundary conditions at the liquid-air

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interface are summarized below:

u = v = w = 0 at liquid-air interface for single droplet,

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u · n = 0 at liquid-air interface for two droplets configuration,

(5a)

(5b)

K − (K · n)n = 0 at liquid-air interface for two droplets configuration, (5c)

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K = [µ(∇u + (∇u)T )]n.

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Initial concentration (c0 ) of solute is equal to 1000 mol/m . The substrate

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surface is considered to have no solute mass flux. Evaporation from the liquidair interface increases the solute concentration at the interface. The solute from the interface diffuses to the inner fluid. The boundary condition at the liquid-air

and boundary conditions for species transport equation (Eq. 3) are summarized

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interface can be obtained by mass balance of solute at the interface. The initial

below:

c = c0

at

∂c =0 ∂z

−Dc ∇c · n = −

t = 0 (initial condition),

(6a)

at substrate surface,

(6b)

cJw ρw

at liquid-air interface.

(6c)

Here, Jw is the evaporative flux of water at the liquid-air interface, ρw is

the density of water. The density of solution as a function of solute concentration and the density is given by: ρ ≈ ρ0 [1 + β(c − c0 )], where solutal expan-

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sion coefficient (β ) is given by β = (1/ρ)(∂ρ/∂c). The value of β is equal to 3.7 × 10−5 m3 /mol [26]. Evaporative flux (Jw ) is obtained from the solution of diffusion equation for the transport of water vapor from the droplet surface to air as described below.

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Evaporation from the droplet surface is driven by the difference in vapor concentration of the droplet surface and the ambient air. Mass transport of

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water vapor due to convection outside the droplet is assumed negligible as the

droplet is isolated from the atmospheric convection because of the enclosure used

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during the experiment. Evaporation from the droplet surface can be modeled as

quasi-steady and diffusion driven [14, 32, 33]. Study by Gelderblom et al. [34] shows that the experimental data of evaporation rate from the droplet surface

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has a good quantitative agreement with the model based on quasi-steady and

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diffusion-driven evaporation. Flow inside the droplet may induce flow in the air due to velocity continuity at the liquid-air interface. The diffusion time scale for water vapor is given by, td = R2 /Dw which is of the order of 10−2 sec using the diffusivity of water vapor, Dw equal to 2.42 × 10−5 m2 /s [35]. The convective

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time scale is given by, tc = R/U which is of the order of 101 sec using the maximum velocity of fluid (U) inside the droplet equal to 35 µm/s. For these

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time scales, diffusion is much faster than advection. Therefore, the transport of water vapor due to advection can be neglected. Hence, mass transfer of water vapor from the droplet surface to the ambient air is assumed to be governed by

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the steady diffusion equation as:

∇2 Cv = 0

(7)

Here, Cv represents the concentration of water vapor in the air. Vapor

concentration at the surface of the droplet is assumed as saturated vapor concentration (Cs ). Vapor concentration at the far field is taken as ambient vapor

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concentration (C∞ ). The solid substrate surface is considered as no mass flux

boundary condition (∇Cv = 0). The boundary conditions for the solution of

vapor diffusion equation (Eq. 7) can be summarized as follows:

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(8a)

at far field ambient region,

∇Cv = 0 at substrate.

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Cv = C∞ = φCs0

at droplet surface,

(8b)

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C v = Cs

(8c)

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Where, φ is the relative humidity. Cs0 is the saturated vapor concentration corresponding to the ambient temperature (T∞ ). The saturated vapor concen3

tration (Cs0 ) is set equal to 0.96 mol/m [35] in our simulation. The vapor

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concentration at the far field (C∞ = φCs0 ) is equal to 0.58 mol/m . According to Raoult’s law, vapor concentration at the surface of a solution can be expressed

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as, Cs = Xsolvent Cs0 . Where, the mole fraction of solvent, Xsolvent is equal to 0.965. The ambient temperature (T∞ ) is equal to 20 0 C and relative humidity 225

(φ) is equal to 60 % in our experiment. The value of vapor concentration (Cs ) 3

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at the surface of droplet is equal to 0.93 mol/m and assumed constant during

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the simulation. Evaporation flux from the droplet surface is given by:

Jw = −Dw ∇Cv

(9)

The Governing equations (Eq. 1, 2, 3 and 7) are solved by FEM based COM-

SOL Multiphysics V5.0 software. The continuity (Eq. 1), momentum (Eq. 2)

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and species transport equations (Eq. 3) are solved inside the computational

domain of the droplet. The computational domain of each droplet is divided into approximately 299000 elements. The diffusion equation (Eq. 7) is solved in a computational domain of 15000 µm × 10000 µm × 6000 µm surrounding

the droplets to obtain the evaporative flux (Jw ) from the droplet surface. The

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computational domain for the solution of diffusion equation is divided into approximately 770000 elements. First, the steady diffusion equation (Eq. 7) is solved to get the evaporative flux on the droplet surface and then the transient continuity, momentum and species transport equations are solved inside the computational domain of the droplet. The simulation is performed for a

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period of 200 second with time step of 0.1 second. Due to the quasi-steady

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assumption of the process, we assumed fixed droplet shape for our simulation.

The simulation results provide a better understanding of the physics of flow pattern observed in our experiment. Close match between our experimental

assumptions used in our numerical modeling.

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and simulation results reported in the following sections confirm the validity of

4. Results and discussion

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The influence of an adjacent evaporating droplet on internal convection of a droplet of binary mixture has been studied using both experimental and simulation results. Single droplet convection has been studied as a reference case and the separation distance between the two droplets has been varied systematically

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to study the influence of the adjacent droplet. The complete three dimensional flow field inside the droplet has been obtained by reconstruction of experimental

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velocity fields in several X-Y planes. The velocity vector field from numerical

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simulation has been compared and validated with the experimental velocity 255

vector field. The flow physics on interaction between evaporating droplets is

explained using the evaporative flux distribution and concentration field results from the numerical simulation. Both the experimental results and numerical simulation results are discussed in the following sections. 4.1. Experimental Results

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The experimental results from the present study have been discussed below

in the following sequence: (a) convection inside a single droplet, (b) convection inside a droplet in the presence of a neighboring droplet and (c) Velocity vector field in wall normal planes.

4.1.1. Convection inside a single droplet 265

Fig. 5 shows the velocity vector field at Z=50 µm and 350 µm from the substrate surface for a single droplet configuration. The direction of the flow field is opposite near the substrate region (Z=50 µm) compared to the location 13

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(a)

(b) 500

500 (µm/s)

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(µm/s)

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16

14

12 01

0

08

Y (µm)

Y (µm)

14

12 01

0

08 06

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06

04

04

02

02

-500

-500

0

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-500

X (µm)

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0

500

X (µm)

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Figure 5: Velocity vector field in X-Y plane inside a single droplet (a) at Z = 50 µm from the substrate surface and (b) at Z = 350 µm from the substrate surface.

away from the substrate surface (Z=350 µm). Flow inside the droplet is due

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to the buoyancy driven natural convection caused by the variation of NaCl concentration inside the droplet [2]. Evaporation from the droplet causes higher solute concentration on the droplet surface. The lighter fluid from the centre of

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the droplet moves upward along the center of the droplet and fluid with higher

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concentration flows downwards along the surface. The flow pattern observed here for a single droplet case matches with the result of Kang et al. [2]. Both

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our results and the results of Kang et al. [2] show an inward movement of fluid

near the substrate surface and an outward flow at the apex region of the droplet. Overall, the convection inside the droplet shows symmetric behavior about the vertical axis (Z-axis).

4.1.2. Convection inside a droplet in the presence of a neighboring droplet

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Fig. 6(a) and (b) show the velocity vector field inside the two neighboring

droplets close to the substrate surface (Z=50 µm) and away from the substrate

surface (Z=350 µm) respectively. Fig. 6(c) and (d) show the flow field inside

the left droplet at Z=50 µm and 350 µm respectively. There is an inward flow towards the adjacent region of the two neighboring droplets along the contact 285

line at Z=50 µm (Fig. 6(c)) and an outward flow at Z=350 µm (Fig. 6(d)) from the adjacent region of the neighboring droplets. The radial flow pattern

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(a)

(b) 500

500

(µm/s)

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(µm/s)

16

16

14

12

Y (µm)

Y (µm)

14

01

0

08

12 01

0

08 06

cr

06

04

04

02

02

-500

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-500

-500

-500

500

0

500

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(d)

(c )

16 14 12 01

0

(µm/s) 16 14 12 01

0

08

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08

Y (µm)

500

(µm/s)

Y (µm)

500

0

X (µm)

X (µm)

06

06

04

04

02

-500

02

d

-500

-500

-1000

0

-500

-1000

0

X (µm)

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X (µm)

Figure 6: Velocity vector field in X-Y plane inside droplets in the presence of a neighboring droplet for different distance (Z) from the substrate surface: (a) at Z=50 µm inside two adjacent droplets; (b) at Z=350 µm inside two adjacent droplets; (c) at Z=50 µm inside the left droplet; (d) at Z=350 µm inside the left droplet. Bright field image of the droplet has been superposed on the vector field.

observed in Fig. 5 for single droplet configuration is transformed to source and sink like flow in Fig. 6 (c) and (d) for the two droplets configuration. The

presence of neighboring droplet generates an asymmetric flow behavior in each

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droplet along the vertical axis (Z-axis). However, when both the droplets are

considered, the flow patterns of both the droplets are mirror image to each other.

4.1.3. Velocity vector field in wall normal planes PIV measurements were acquired at eleven X-Y planes for different Z-locations. 295

There is negligible change in the droplet height during the measurement period of all eleven planes. Therefore, it may be assumed that there is no appreciable 15

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Single droplet

Two droplets (b) 600

400

400

Z (µm)

200

-500

0

0

500

-500

0

Y (µm)

(d)

200

400

200

-500

0

0

500

-1000

-500

0

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X (µm)

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400

600

Z (µm)

Z (µm)

20 µm/s

20 µm/s

600

0

500

Y (µm)

( c)

2nd Droplet

0

200

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Z (µm)

20 µm/s

20 µm/s

600

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(a)

X (µm)

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Figure 7: Velocity vector field in Y-Z and X-Z planes for single droplet configuration (a and c) and inside left droplet for two droplets configuration (b and d).

change in the flow field inside the droplet during this time period. The flow is quasi-steady and the measurements taken at different planes show spatial vari-

have used the steady continuity equation to obtain the w component of velocity

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ation in flow field only with no time dependence. Based on this assumption we

from the 2D velocity fields at these planes. ∂w ∂u ∂v + + =0 ∂x ∂y ∂z

(10)

The complete 3D velocity field for both single droplet and two droplets

configuration has been reconstructed using the continuity equation (Eq. 10) [3, 36]. Finite difference scheme is used for the reconstruction of 3D velocity

305

field.

The reconstructed velocity vector fields in X-Z and Y-Z planes for single

droplet and two droplets configurations are shown in Fig. 7. The reconstructed velocity vector fields in vertical planes provide better visualization of the flow pattern. Fig. 7(a) and (c) show symmetric nature of velocity vector field in

310

X-Z and Y-Z planes for the single droplet case. Two recirculation bubbles are observed in both X-Z and Y-Z plane. Fig. 7(b) shows symmetric flow pattern in Y-Z plane for two droplets case similar to that of the single droplet

16

Page 18 of 32

case in Fig. 7(a). However, Fig. 7(d) shows asymmetric flow field in X-Z

315

ip t

plane for two droplets case contrary to the single droplet case (Fig. 7(c)). One recirculation bubble is observed for the two droplets configuration in comparison to two recirculation bubbles for the single droplet configuration. This behavior

cr

can be attributed to the presence of a neighboring droplet in the X-direction.

The recirculation bubble near the adjacent region between two droplets gets

320

us

disturbed due to the presence of the neighboring droplet. This behavior can be attributed to the interaction between the two droplets. Evaporative flux at the

an

nearest region of the two droplets is less which causes low solute concentration as compared to the rest of the droplet. Fluid with lower concentration at the nearest region of the two droplets moves upward along the droplet surface. The

325

and move downward.

d

4.2. Simulation Results

M

higher concentration fluid element at the apex region of droplet gets displaced

The flow phenomenon observed in the experiment has been studied in details

Ac ce pt e

using evaporative flux, concentration field and velocity vector field results from the numerical simulation. The simulation results are discussed below in the

330

following sequence: (a) evaporative flux distribution, (b) concentration field, (c) velocity vector field and (d) effect of separation distance. 4.2.1. Evaporatice flux distribution Evaporative flux from the droplet surface is obtained by solving the diffusion

equation (Eq. 7). The evaporative flux from the droplet surface with and

335

without a neighboring droplet is shown in Fig. 8. The evaporative flux from the single droplet without a neighboring droplet (Fig. 8(a)) is symmetric about the Z axis. The evaporative flux distribution on the droplet surface for two droplets configuration is presented in Fig. 8(b). It shows the evaporative flux is lowest at the

340

adjacent region between the two droplets. The evaporative flux on the droplet surface increases from the adjacent region between two droplets to the opposite

17

Page 19 of 32

2

1.0E-02 mol/m s

800 600 400

1.0E-02 mol/m2s

800 600 400 200

200 0

1000

-1000

-500

0

0 -2000

1000

500

-1500

-1000

-500

0

X (µm)

500

1000

1500

2000

cr

X (µm)

ip t

(b)

1000

Z (µm)

Z (µm)

(a)

us

Figure 8: Evaporative flux from the droplet surface for (a) single droplet and (b) two droplets configuration.

end of the droplets. The presence of another droplet close to an evaporating droplet influences the vapor concentration distribution of the diffusion boundary

345

an

layer due to the increase in water vapor content because of evaporation from the neighboring droplet. Therefore, there is reduction in evaporation flux in this region due to decrease in the driving force for evaporation. Non-uniform

M

evaporative flux on the droplet surface in the presence of a neighboring droplet generate asymmetric flow behavior inside the droplet.

350

d

4.2.2. Concentration field Evaporation from the droplet surface increases the solute concentration at

Ac ce pt e

the liquid-air interface region creating a concentration gradient inside the droplet. The concentration gradient induces buoyancy driven natural convection. The

concentration and velocity field inside the droplet gradually changes with time after the initiation of the evaporation process and attains a saturation state

355

after some time duration. The temporal evolution of concentration field inside

the droplet with and without a neighboring droplet is shown in Fig. 9. The concentration field gradually changes with time and attains a constant distribution pattern after some time though the absolute value of concentration increases with time. Concentration fields at time 100 and 150 sec show similar distribu-

360

tion pattern. The concentration field for single droplet shown in Fig. 9 shows symmetric concentration distribution about the Z-axis. Symmetric concentration field inside the droplet is attributed to symmetric evaporation from the droplet surface for single droplet configuration (Fig. 8(a)). For single droplet, the salt concentration (Fig. 9(e)) is lower in the central region of the droplet

365

and higher on the droplet surface region . The salt concentration is lower on 18

Page 20 of 32

the apex region of the droplet surface compared to the region near the substrate

ip t

surface. The concentration field in two droplets configuration shows asymmet-

ric distribution about the Z-axis. The asymmetric concentration field is due to

the non uniform evaporative flux from the droplet surface in the presence of a neighboring droplet (Fig. 8(b)). The salt concentration (Fig. 9(f)) is lower in

cr

370

the proximity region compared to that in the opposite side of the neighboring

Ac ce pt e

d

M

an

us

droplet.

Figure 9: Concentration field inside an evaporating droplet of NaCl solution in single droplet and two droplets configuration at different time instants: (a,b) t= 10 sec, (c,d) t= 100 sec and (e,f) t=150 sec.

4.2.3. Velocity vector field The concentration gradient inside the droplet as seen in Fig. 9 induces 375

Rayleigh convection. The velocity vector fields inside the droplet with and without a neighboring droplet are shown in Fig. 10 at different time instants. 19

Page 21 of 32

Two Droplets (b)

800

800 20 µm/s

600 400

2nd Droplet

Z (µm)

200

200

-500

0

0

500

-1500

-500

-1000

X (µm)

X (µm)

t = 10 sec (d)

800

800

20 µm/s

20 µm/s

600

Z (µm)

400 200

400 200

-500

0

an

Z (µm)

600

0

0

us

(c)

cr

Z (µm)

400

0

ip t

20 µm/s

600

0

500

-1500

-1000

-500

2nd Droplet

Single Droplet (a)

0

X (µm)

X (µm)

t = 100 sec (f)

800

800

M

(e)

20 µm/s

20 µm/s

600

400 200

200

-500

0

0

500

Ac ce pt e

X (µm)

d

0

400

2nd Droplet

Z (µm)

Z (µm)

600

t = 150 sec

-1500

-500

-1000

0

X (µm)

Figure 10: Velocity vector field inside an evaporating droplet of NaCl solution in single droplet and two droplets configuration at different time instants: (a,b) t= 10 sec, (c,d) t= 100 sec and (e,f) t=150 sec.

The flow field gradually changes with time and attains a constant saturation state after some time. The flow field for the single droplet configuration in Fig. 10 shows symmetric flow pattern, which correlates with the symmetric

380

concentration field inside the droplet (Fig. 9). The low density fluid element

in the central region of the droplet moves upward. There is an inward flow from the contact line region along the substrate surface to replenish the fluid in the central region of the droplet resulting in formation of two recirculation bubbles. The flow pattern for single droplet in Fig. 10(e) from simulation is

385

similar to the experimental results in Fig. 7(c). Velocity vector field inside the droplet in the presence of a neighboring droplet shows asymmetric pattern. The concentration in the proximity region of the two droplets is less as compared

20

Page 22 of 32

to the region opposite to the neighboring droplet (Fig. 9(f)). Fluid with low

390

ip t

density (lesser concentration) rises up along the droplet surface at the proximity

region of two droplets (10(f)). Higher density fluid (higher concentration) from the opposite end of the droplet (Fig. 9(f)) moves along the substrate surface

cr

to replenish the low density fluid. The asymmetric flow pattern incase of two droplets correlates with the asymmetric concentration field inside the droplet

395

us

and the asymmetric evaporative flux distribution from the droplet surface. The simulation results in Fig. 10(f) compares well with the experimental vector field

an

in Fig. 7(d). One recirculation bubble is observed in two droplets configuration compared to two recirculation bubbles in single droplet configuration. The nature of velocity field inside the droplet can be described using Rayleigh

400

gβ∆cR3 Dc ν ,

where, ∆c is the

M

number values. Rayleigh number is given by Ra =

difference of maximum and minimum value of solute concentration inside the droplet. The value of Rayleigh number is equal to 4118 and 4134 for single

d

droplet and two droplets case for concentration fields shown in Fig. 9(e) and Fig. 9(f) respectively. Both the values of Rayleigh number are above the critical

Ac ce pt e

Rayleigh number of 1702 [37]. The role of diffusion and convection on solute

405

transport inside the droplet is governed by Peclet number. The Peclet number which is the ratio of advective mass transport to diffusive mass transport is given by P e = RU/Dc , where, U is the velocity of fluid and its maximum value is

approximately equal to 35 µm/s and 36 µm/s for single droplet and two droplets

configuration respectively at time 150 sec. The value of Peclet number is equal

410

to 15 and 15.5 for single droplet and two droplets configuration respectively. For

these values of Peclet number, the solute transport is dominated by advection. The velocity vector field in X-Y plane parallel to the substrate surface is

presented in Fig. 11 for both single droplet and two droplets configurations at time 150 sec. The velocity vector field for a single droplet is symmetric about

415

the center near the substrate surface and in the apex region of the droplet as shown in Fig. 11(a) and (c). However, the direction of the radial flow is opposite to each other. The inward radial flow near the substrate surface and the outward radial flow at the apex region (Fig. 11(a) and (c)) corresponding 21

Page 23 of 32

Single droplet

(a)

Two droplets

(b)

20 µm/s

500

-500

0

A

-500

-500

0

500

-1500

X (µm)

-500

-1000

0

X (µm)

(d) 20 µm/s

an

(c )

20 µm/s

500

Y (µm)

M

500

0

d

Y (µm)

A

cr

A

A

us

Y (µm)

Y (µm)

500

0

ip t

20 µm/s

-500

Ac ce pt e

-500

-500

0

0

500

-1500

X (µm)

-500

-1000

0

X (µm)

Figure 11: Velocity vector field in X-Y plane inside the evaporating droplet at different distance (Z) from the substrate surface (a) Z = 50 µm for single droplet, (b) Z = 50 µm inside left droplet for two droplets configuration, (c) Z = 350 µm for single droplet and (d) Z = 350 µm inside left droplet for two droplets configuration.

to the lower and upper part of the circulation bubbles shown in Fig. 10(e). The

420

velocity vector field results in Fig. 11(a) and (c) from simulation is similar to the experimental results shown in Fig. 5. Fig. 11(b) and (d) show the velocity

vector field in X-Y plane for two droplets configuration. The flow field is no more radially symmetric as that of single droplet case in Fig. 11(a) and (c). The direction of the velocity vector field is towards the adjacent region near the

425

substrate surface (Fig. 11(b)) and away from the adjacent region (Fig. 11(d)), demonstrating the sink and source like behavior. Correlation of Fig. 11(b) and Fig. 11(d) with Fig. 10(f) provides complete picture of the interaction between two evaporating droplets. The flow pattern in Fig. 11(b) and (d) is similar to 22

Page 24 of 32

the experimental velocity vector field in Fig. 6(c) and (d).

20

20

Experimental Simulation

0

-10

-10

-20

-20

0 X (µm)

500

-30 -1500

-1000

an

-500

cr

0

us

u (µm/s)

u (µm/s)

Experimental Simulation

10

10

-30

ip t

(b) 30

(a) 30

X (µm)

-500

0

430

M

Figure 12: Comparison of velocity profile (u) along the central line (Y=0, Z=50 µm) for (a) single droplet and (b) two droplets configurations.

The X-component velocity (u) along a central line at Y = 0 and Z = 50 µm (along A-A line of Fig. 11 (a) and Fig. 11 (b)) inside the droplet with

d

and without a neighboring droplet is presented in Fig. 12. The nature of

Ac ce pt e

experimental and simulation velocity profile for both single droplet (Fig. 12(a)) and two droplets (Fig. 12(b)) shows close match with each other. The small

435

deviation in the magnitude and position of the peak may be attributed to the location error in reporting the experimental velocity profile. 4.3. Effect of separation distance

The interaction between adjacent evaporating droplets is expected to be a

function of the separation distance between the two droplets. Fig. 13 com-

440

pare the evaporative flux, concentration field and velocity field as a function

of separation distance (S = 100 µm, 600 µm and 1200 µm) for two droplets configuration. The increase in separation distance between the two droplets leads to reduction in the asymmetry of evaporative flux distribution. This may be attributed to the reduced interaction between the diffusion boundary layer

445

of the neighboring droplets with increase in separation distance. For smaller separation distance, evaporative flux from the droplet surface at the adjacent region of the two droplets is less as the neighboring droplet suppresses the rate 23

Page 25 of 32

of evaporation. When the separation distance increases, the evaporative flux at

450

ip t

the adjacent region of the two droplets increases due to reduction in the influence of the neighboring droplet on the evaporation rate. The higher asymmetry in evaporative flux at lower separation distance also leads to higher asymmetric salt

cr

concentration distribution inside the droplet. The strength of the recirculation

bubble also reduces with increase in separation distance between the droplets.

455

us

At higher separation distance, a small second recirculation bubble develops indicating that flow field approaches to single droplet flow behavior with increase

an

in separation distance. With further increase in separation distance, there will be no interaction between the droplets resulting in single droplet behavior. 1000

1000

1000

1.0E-02 mol/m2s

2

1.0E-02 mol/m s

800

200

600 400 200

0

-1500

-1000

-500

0 -2000

0

-1000

X (µm)

600

400 200 0

-500

-2000

-1500

X (µm)

-1000

-500

Ac ce pt e

d

X (µm)

-1500

Z (µm)

600 400

2

1.0E-02 mol/m s

800

M

Z (µm)

Z (µm)

800

800

800

20 µm/s

800 20 µm/s

400

400

200

200

0 -1500

-1000

X (µm)

-500

0

0

-1500

20 µm/s

600

Z (µm)

600

Z (µm)

Z (µm)

600

400 200

-1000

X (µm)

(a)

(b)

-500

0 -2000

-1500

X (µm)

-1000

( c)

Figure 13: Evaporative flux (top row), concentration field (middle row) and velocity field (bottom row) distribution inside droplet with a neighboring droplet placed at right hand side for separation distance of (a) S = 100 µm, (b) S = 600 µm and (c) S = 1200 µm between the two droplets.

5. Conclusion We have studied the change in internal convection behavior of a binary mix460

ture (aqueous NaCl solution) droplet in the presence of a neighboring droplet. The concentration of NaCl is set equal to 1 M and the radius of the droplet 24

Page 26 of 32

is equal to 690 µm. The separation distance between two adjacent droplets

ip t

has been varied between 100 µm and 1200 µm. Confocal micro-PIV technique

has been used for measuring the velocity field inside the droplets. Three dimen465

sional velocity vector field inside the droplet has been reconstructed from the 2D

cr

experimental velocity distribution using continuity equation. Numerical simula-

tion using COMSOL multiphysics software has been carried out to estimate the

us

evaporation flux distribution around the droplets, concentration field inside the droplet and velocity field inside the droplet. The velocity vector field obtained from the numerical simulation has been validated with the experimental results.

an

470

Some of the important observations from the present study are summarized as follows.

M

(a) The presence of an adjacent evaporating droplet influences the evaporative flux on the droplet surface, concentration field inside the droplet and the inter475

nal convection pattern inside a droplet.

d

(b) There is a reduction in the evaporative flux from the droplet surface leading to lower salt concentration inside the droplet in the proximity region between

Ac ce pt e

the two droplets compared to the remaining region. (c) The symmetric Rayleigh convection flow pattern in the single droplet evap-

480

oration case becomes asymmetric in nature due to the presence of a neighboring droplet. One recirculating bubble is observed in wall normal X-Z plane for the two droplets configuration contrary to presence of two recirculating bubbles for the single droplet configuration.

(d) The influence of adjacent evaporating droplet on evaporative flux distribu-

485

tion, salt concentration and the velocity vector field reduces with increase in separation distance between the droplets leading to single droplet behavior at larger separation distance.

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