Analytical Model for Diffusive Evaporation of Sessile Droplets Coupled

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Analytical Model for Diffusive Evaporation of Sessile Droplets Coupled with Interfacial Cooling Effect Tuan Anh Huu Nguyen, Simon R. Biggs, and Anh V. Nguyen Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03862 • Publication Date (Web): 14 May 2018 Downloaded from http://pubs.acs.org on May 15, 2018

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Langmuir

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Analytical Model for Diffusive Evaporation of Sessile

2

Droplets Coupled with Interfacial Cooling Effect

3

Tuan A. H. Nguyen,* Simon R. Biggs and Anh V. Nguyen*

4

School of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072,

5

Australia

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KEYWORDS: evaporative cooling, droplet evaporation, coffee ring, contact angle, flux, toroidal

7

coordinate, Laplace equation.

8

ABSTRACT

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Current analytical models for sessile droplet evaporation do not consider the non-uniform

10

temperature field within the droplet and can over-predict the evaporation by 20%. This deviation

11

can be attributed to a significant temperature drop due to the release of the latent heat of

12

evaporation along the air-liquid interface. We report, for the first time, an analytical solution of

13

the sessile droplet evaporation coupled with this interfacial cooling effect. The two-way coupling

14

model of the quasi-steady thermal diffusion within the droplet and the quasi-steady diffusion-

15

controlled droplet evaporation is conveniently solved in the toroidal coordinate system by

16

applying the method of separation of variables. Our new analytical model for the coupled vapour

17

concentration and temperature fields is in the closed form and is applicable for a full range of

18

spherical-cap shape droplets of different contact angles and types of fluid. Our analytical results

19

are uniquely quantified by a dimensionless evaporative cooling number Eo whose magnitude is

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determined only by the thermophysical properties of the liquid and the atmosphere. Accordingly,

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the larger the magnitude of Eo , the more significant the effect of the evaporative cooling, which

22

results in stronger suppression on the evaporation rate. The classical isothermal model is

23

recovered if the temperature gradient along the air-liquid interface is negligible ( Eo = 0 ). For

24

substrates with very high thermal conductivities (isothermal substrates), our analytical model

25

predicts a reversal of temperature gradient along the droplet free surface at a contact angle of

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119°. Our findings pose interesting challenges but also guidance for experimental investigations.


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INTRODUCTION

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Diffusive evaporation of a droplet is driven by the spatial gradient of vapour between the air-

29

liquid interface and ambient air

30

solution of the Laplace equation is widely used to predict the interfacial evaporation dynamics.

31

Typically, the Laplace equation is solved either analytically or numerically together with the

32

assumption that the atmosphere just above the air-liquid interface is saturated with vapour and

33

the vapour saturation concentration is constant along the interface. Mathematically, there exists

34

an analogy between this diffusive vapour concentration fields and the electrostatic potential

35

fields around a spherical-cap conductor at constant potential since they are both governed by the

36

Laplace equation. The analytical solution for the electrostatic potential of a charged conductor

37

was first derived by Lebedev in 1965 5. Deegan et al.

38

non-uniform evaporative flux at the air-liquid interface of sessile droplets with acute contact

39

angles. This classical analytical model does not take into account the effect of interfacial heat

40

transfer; and, temperature is considered uniform across the whole droplet. Thus, the saturated

41

vapour concentration along the air-liquid interface also does not change. This model is, therefore,

42

called the “isothermal model”.

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Accurate measurements of droplet temperature, however, show a significant change in

44

temperature inside the droplpet and along the air-liquid interface

45

temperature field could be attributed to not only the non-uniform evaporative flux at the interface

46

but also to the heat transfer process within the droplet. In particular, the higher the evaporative

47

flux, the greater the latent heat of evaporation that is released, lowering the air-liquid interfacial

48

temperature locally. Furthermore, the local temperature also varies because heat conduction

49

length from the substrate to the air-liquid interface changes radially in accordance with the

1, 2, 3, 4

. For a steady-state diffusion-controlled process, the

1, 6

later used this solution to explain the

2, 7, 8

. This non-uniform


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droplet height from the centre to the edge. The combination of these heat and mass transfer

51

processes leads to a non-uniform saturated vapour concentration along the air-liquid interface,

52

different from the assumption of the “isothermal model”. Neglecting the cooling effect and

53

decoupling the thermal transport mechanism in the isothermal model may lead to considerable

54

discrepancy in predicting the droplet evaporation dynamics and associated phenomena. For

55

instance, Dash and Garimella 9 reported that evaporation rate of a 160° droplet is over-predicted

56

by ~20% when using the isothermal model. This discrepancy was attributed to a large

57

temperature drop along the air-liquid interface due to the cooling effect and was confirmed by

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the numerical simulations

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gradient due to the evaporative cooling effect along the air-liquid interface of a sessile droplet on

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a copper substrate.

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An improved model for the droplet evaporation, therefore, needs to allow the change in

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saturation vapour concentration along the air-liquid interface as a function of temperature. This

63

“non-isothermal model” has also to be coupled with the equation of heat transfer inside the

64

droplet and solved in an iterative manner. This coupled problem of diffusive vapour and heat

65

conduction has been solved numerically by Dunn et al 12, Xu and Ma 8. Pan et al 10, 11 also carried

66

out a numerical modelling to study the competing effects of external natural convection and the

67

evaporative cooling. In another attempt to explain the discrepancy between experimental and

68

theoretical results, Gleason and Putnam

69

with replacing the vapour concentration along the air-liquid interface by a function of

70

temperature. Hu and Larson

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inside evaporating droplets whose boundary condition at the air-liquid interface was the

72

analytical expression for the evaporative flux from the isothermal model. This one-way coupling

8, 10, 11

2

. Chandramohan et al.

13

7

recently reported a large temperature

used the analytical solution of the isothermal model

used a finite-element method to compute the temperature field


4

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model also shows that a non-uniform evaporation leads to a non-uniform distribution of

74

temperature along the air-liquid interface. Despite these efforts, the complete theory for the

75

effects of the interface cooling on the evaporation of liquid droplets is still lacking.

z α=0

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Figure 1. Schematic of a sessile droplet (spherical cap) evaporating from a flat substrate in

78

rotationally symmetric cylindrical ( r , z ) and toroidal (α , β ) coordinates. These coordinates are

79

z + ir iR coth (α + i β 2 ) , where R is the droplet base radius linked through complex mapping =

80

and i=

81

thermal conduction within the internal domain of π − θ ≤ β ≤ π , where θ is the contact angle of

82

the liquid phase. Vapour diffuses from the droplet free surface through the ambient air confined

83

within the external domain of 2π ≤ β ≤ 3π − θ .

−1 . The substrate is heated at constant temperature Tsub ≥ T∞ , which triggers the


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THEORETICAL ANALYSIS

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In this paper, we report, for the first time, closed-form, analytical solutions of vapour diffusion

86

from a sessile droplet into the surrounding air by taking into account the interfacial cooling. The

87

analytical solutions can be used for a full range of small spherical cap droplets of different

88

contact angles and types of fluid, resting on a flat substrate heated at constant temperature Tsub .

89

This assumption of an isothermal surface is applicable for substrates made of materials having

90

high thermal conductivity, such as copper (398 W/m K), gold (315 W/m K), aluminium (247

91

W/m K) 14, etc. For simplicity, we ignore the effects of varying the substrate temperature, natural

92

vapour convection, and convective heat transfer within the droplet. That said, the present model

93

is a quasi-steady diffusion-controlled droplet evaporation model. As discussed by Larson 15, the

94

validity of the quasi-steady-state assumption for vapor-phase mass-transfer is governed by the

95

ratio of the characteristic time for the vapor field to adjust to the change in droplet shape to the

96

−3 characteristic time of droplet shrinkage (which is just the drying time), tvap t f ≤ 5 ×10 . This

97

ratio can be approximated as five times the ratio of mass density of vapour and mass density of

98

droplet liquid (  5 ρvap ρ L ) which is quite small, thus the vapour concentration above the

99

surface of the droplet reaches a quasi-steady state well before the completion of drying. While

100

the vapour phase quasi-steady-state assumption is normally very well justified, thermal

101

equilibration throughout the liquid droplet is much less rapid than equilibration of the vapor

102

concentration field. Using similar arguments for the vapour phase, the criterion for attaining a

103

quasi-steady state of heat transfer within a droplet is the ratio of heat equilibrium time in a

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droplet and the total evaporation time (droplet lifetime) theat t f ≤ 0.1 (detail discussion can be

105

found in reference

15

). Generally, the quasi-steady assumption is satisfied for a droplet if the


6

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averaged droplet temperature is smaller than ~75 °C, 38 °C, 36 °C, and 29 °C for water,

107

isopropanol, ethanol, and methanol, respectively

108

transfer can be justified by the Péclet number (Pe), which is defined as the ratio of the rate of

109

advection to thermal diffusion 15, 17. It is reported that heat convection only becomes appreciable

110

once Pe exceeds approximately 10 or so

111

(Marangoni) convection inside a water droplet is valid for low Péclet number, say Pe < 10 . Once

112

this happens, the temperature field throughout the liquid droplet is determined by pure heat

113

conduction.

114

With above assumptions, mass conservation for vapour evaporation by diffusion is described by

115

second Fick’s law. The scale analysis indicates that the evaporation of sessile droplets can be

116

described by the quasi-steady state with neglecting the transient term in Fick’s law1, 4, 5, 18, giving

117

∇ 2C = 0 . Heat transfer in a drying droplet can also be considered as quasi-steady state

118

dominated by the heat conduction over the convection term

119

within a slowly evaporating droplet can be described by the Laplace equation, ∇ 2T = 0 . These

120

Laplace equations for C and T can be conveniently solved for the vapour concentration and

121

temperature fields by applying the method of separation of variables in the toroidal coordinate

122

system (α , β , ϕ )

123

the physical domain: ∞ > α ≥ 0 and 3π − θ ≥ β ≥ 2π , and satisfies the boundary condition

124

0 at the C ( 0, 2π ) = C∞ at infinity of ambient air at temperature T∞ ( T∞ ≤ Tsub ), ( ∂C / ∂α )a =0 =

125

0 at the solid-vapour axis of symmetry and the zero flux of vapour diffusion ( ∂C / ∂β ) β = 2π =

126

interface. On the other hand, the solution for temperature is bounded by ∞ > α ≥ 0 and

4, 5

15

16

. The relative importance of convection heat

, thus the assumption of neglecting thermal

19

2

and is

. Therefore, the temperature field

(Fig. 1). Accordingly, the solution for vapour concentration is bounded by


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π ≥ β ≥ π − θ , and satisfies the boundary conditions of T (α , π ) = Tsub at the droplet base and

128

0 at the axis of symmetry, giving: ( ∂T / ∂α )a =0 =

129

 (α , β ) ≡ C (α , β ) − C∞ = C 2 cosh α − 2 cos β ∫ EC (τ ) Piτ −1/2 ( cosh α ) cosh ( 2π − β )τ  dτ Ce − C∞ 0

(1)

130

T (α , β ) − TSub 2 cosh α − 2 cos β T (α , β ) ≡ = TSub − T∞

(2)

∞

∞

∫ E (τ ) Pτ ( cosh α ) sinh (π − β )τ  dτ T

i −1/2

0

131

where Ce = Csat (Te ) is the saturated vapour concentration of the liquid at temperature Te of the

132

droplet edge ( α → ∞ and Te = Tsub ), τ is the integration dummy, Piτ −1/2 ( cosh α ) is the toroidal

133

function (i.e., the first-kind Legendre function of the complex half-integral degree and the

134

argument of the hyperbolic cosine function). These solutions are independent of ϕ because of the

135

rotational symmetry. EC (τ ) and ET (τ ) in Eq. (1) and (2), respectively, are functions of the

136

integration dummy (independent of the toroidal coordinates α and β ), which can be determined

137

from the boundary conditions at the air-liquid interface as follows.

138

The net of the heat and mass fluxes at the droplet surface depend on the toroidal coordinate and

139

 are D∇C ⋅ iβ determined by j (α ) =

= j (α )

140

Ce − C∞ R D

2 ( cosh α + cos θ )



( D∇C ⋅ i ) β

32

β= 3π −θ

 and q (α ) = k ∇T ⋅ iβ

(

)

β= π −θ

, yielding

×

 cosh (θ − π )τ  sin θ  α τ θ π τ cosh sinh E P − −   ( ) ( )    dτ ∫0 C iτ −1 2  2 ( cosh α + cos θ ) 

∞

(3)


8

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Langmuir

 sinh θτ sin θ  cosh cosh α τ θτ − E P ( )  dτ − T i τ 1 2 ∫0  2 ( cosh α + cos θ ) 

∞

141

Tsub − T∞ 32 2 ( cosh α + cos θ ) q (α ) = Rk

142

where k is the liquid thermal conductivity and D the vapour diffusion coefficient. The local

143

energy balance on the air-liquid interface establishes a relationship between the evaporative mass

144

flux and the heat flux by q (α ) = Lj (α ) , where L is the liquid latent heat of vaporization.

145

Table 1. Magnitude of the dimensionless number ( Eo ) characterizing the interfacial cooling

146

effect for different liquids at T = 295 K and p = 99.8 kPa. D is the coefficient of vapour diffusion

147

in air, L the liquid latent heat of vaporization, k the liquid thermal conductivity, and b the

148

thermal gradient of vapour saturation concentration at T. D

L

k

b

Eo

[m2 s-1]

[m2 s-2]

[kg m s-3 K-1]

[kg m-3 K-1]

-

Water

2.44×10-5

2.45×106

0.604

1.11×10-3

0.11

Methanol

1.50×10-5

1.20×106

0.203

9.47×10-3

0.84

Acetone

1.06×10-5

5.49×105

0.161

2.84×10-2

1.03

(4)

Liquid

149

Additionally, the atmosphere just above the air-liquid interface of the droplet is considered

150

saturated with vapour, whose temperature dependence can be linearized locally as follows:8

151

dCsat dT = a Ce − bTsub . C (α ,3π − θ ) =Cs (T ) =a + bT , where b = ( Ce − C∞ ) (Tsub − T∞ ) and =

152

 (α , β ) = 1 + T (α , β ) along the droplet surface. Given these conditions, E (τ ) and Thus, C C

153

ET (τ ) can now be determined as follows (with detailed derivations in Supporting Information):


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1 dF   τ F coth τθ −  cosh τθ 3 dθ   EC = cosh τπ cosh τ (θ − π )   1 dF  1 dF    τ F coth τθ −  − Eo τ F tanh τ (θ − π )  −  3 dθ  3 dθ    

(5)

155

1 dF   τ F tanh τ (θ − π )  −  Eo cosh τθ 3 dθ   ET = cosh τπ sinh τθ  1 dF  1 dF    τ F coth τθ −  − Eo τ F tanh τ (θ − π )  −  3 dθ  3 dθ    

(6)

156

where

= dF dθ 157

F (θ ,τ ) = sinh τθ ( sinh τπ sin θ )

(τ cosh τθ sin θ − sinh τθ cos θ )

( sinh τπ sin θ ) . 2

and

Eo = bLD k is an evaporative cooling

158

number and comes from the relationship between the fluxes at the droplet surface q (α ) = Lj (α ) ,

159

  ⋅i which can be represented in the non-dimensional form as Eo ∇C β

160

 = 1 + T along the droplet surface, the left hand side of the above With the given mapping C

161

 ) field across the droplet surface with a equation can be considered as a “quasi-temperature” ( C

162

“thermal” conductivity Eo , while the right hand side is an interfacial diffusive heat transfer with

163

a thermal conductivity of 1. Hence, the evaporative cooling number Eo could be interpreted as

164

the ratio between the interfacial diffusive and conductive heat transfer rates across the droplet

165

surface, and in a broad sense, Eo could possibly be assigned as the Nusselt number. Generally,

166

Eo characterizes the strength of the evaporative cooling effect on droplet’s evaporation. Its

167

magnitude is determined by the ratio of the interfacial diffusive to conductive heat transfer rates

168

and depends on diffusion coefficient of liquid vapour in the atmosphere, the temperature

169

dependence of the vapour saturation concentration, and the thermal properties of the fluid12, 20, 21,

(

)

 = ∇T ⋅ iβ

(

β= 3π −θ

)

.

β= π −θ


10

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22

171

cooling. When the effect of evaporative cooling can be neglected (i.e., Eo = 0 ) the problem of

172

vapour diffusion in the atmosphere is decoupled from the problem of the heat transfer in the

173

droplet and the present model reduces to the classical “isothermal model” 23, 24:

174

= C iso (α , β )

(Table 1). The larger the magnitude of Eo , the more significant the effect of evaporative

∞

2 cosh α − 2 cos β ∫ Piτ −1/2 ( cosh α ) 0

175

jiso (α= )

Cs − C∞ R D

cosh θτ cosh ( 2π − β )τ  cosh πτ cosh (π − θ )τ 

dτ

(7)

∞ τ cosh θτ tanh (π − θ )τ   32  sin θ dτ  (8) 2 cosh cos α θ + + ( ) ∫ Piτ −1 2 ( cosh α )  2 cosh πτ 0  

176

and T iso = 0 . In these equations, the subscript “iso” stands for the isothermal condition at the air-

177

liquid interface.

178

RESULTS AND DISCUSSION

179

Model validation

180

Fig. 2 shows the comparison between our analytical results and the published numerical results8

181

for evaporative mass flux and temperature profile along the droplet surface. The comparison

182

shows good agreement for different levels of cooling effect, i.e. Eo = 0.5, 2.0, and 10 .

183

Additionally, the classical isothermal model is recovered from our new model if the temperature

184

gradient along the air-liquid interface is negligible, Eo = 0 (Eqs. (7) and (8), red lines in Fig. 2).

185

These numerical results were already validated against available experimental evaporation data

186

of sessile droplets resting on a highly thermal conductive aluminium substrate. Thus, these

187

agreements verify the validity of our analytical model for the sessile droplet evaporation coupled


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188

with the interfacial cooling effect. It is worth noting that the advancement of our analytical

189

coupling model is its applicability for a full range of droplets with different contact angles and

190

types of liquid.

191

0.0

2.0 Evaporative Flux

-0.1

(Eqs. 3 & 5)

0.0 0.5 2.0 10

1.5

-0.2 -0.3

1.0

-0.4

0.5

Interfacial temperature

-0.5 -0.6

Eo

Normalized evaporative flux, J

Normalized interfacial temperature

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(Eqs. 2 & 6)

-1.2

-0.8

θ=20° -0.4 0.0 0.4 0.8 Normalized radial coordinate, r/R

1.2

0.0

Xu & Ma, 2015 Flux Temp.

Present model

192

Figure 2. Comparison between the analytical solution obtained from this study and the

193

numerical results presented in reference.8 Evaporative flux along the air-liquid interface is

194

normalized by dividing by D ( Ce − C∞ ) / R ); the normalized interfacial temperature T (α , π − θ )

195

is given by Eqs. 2 & 6. The contact angle between the droplet and the substrate is 20°.

196

Spatial distribution of vapour and heat fields

197

The close-form analytical solutions in Eqs. (1) and (2) now can be used to predict and investigate

198

the effect of evaporative cooling on the vapour concentration and temperature fields. For

199

example, Fig. 3 shows the theoretical predictions for the concentration contours of the vapour

200

field above the air-liquid interface of a droplet with a contact angle of 60° using Eq. (1) and (5).

201

When Eo = 0 , the current model reduces to the classical model of isothermal, thus the saturated

202

vapour concentration just above the free surface of the droplet is constant. Closed contour lines


12

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203

are concentrated near the free interface of the droplet and gradually diluted outwards in a

204

uniform pattern. However, when the interfacial cooling effect is coupled (i.e. Eo > 0 ), the

205

vapour concentration near the free interface gradually decrease from the droplet apex towards the

206

contact line. Thus, the isoconcentrations lines near the free surface are not in closed contours.

Normalized vertical distance, z/R

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207

Eo= 0.0

Eo = 0.5

Eo = 2.0

Eo = 5.0

Normalized radial distance, r/R

208

Figure 3. Contour plots of vapour distribution above an evaporating droplet, θ = 60o , with (

209

Eo = 0.5, 2 and 5 ) and without ( Eo = 0 ) cooling effect. The colour bar represents the

210

normalized vapour concentration ( C − C∞ ) ( Ce − C∞ ) .

211

The temperature field inside a drying droplet is presented in Fig. 4 in terms of isotherms

212

calculated from Eq. (2). A cooler zone near the interface at the droplet apex is predicted in

213

comparison with the zone near the droplet edge. This cold zone is corresponded to the low

214

saturated vapour concentration zone observed in Fig. 3. This can be explained by heat

215

conduction distance between the heated substrate and the air-liquid interface which becomes

216

greater from the droplet edge towards the centre (with acute contact angles). In other words, a


13

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217

relatively tall droplet (large contact angle) has a large effective thermal resistance between the

218

substrate and the air-liquid interface. The higher the value of Eo used in Eq. (2), the stronger the

219

cooling effect.

Normalized vertical distance, z/R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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220

Eo = 0.0

Eo = 0.5

Eo = 2.0

Eo = 5.0

Normalized radial distance, r/R

221

Figure 4. Contour plots of temperature field within an evaporating droplet, θ = 60o , with (

222

Eo = 0.5, 2 and 5 ) and without ( Eo = 0 ) cooling effect. The colour bar represents the

223

normalized temperature (T − Tsub ) (Tsub − T∞ ) .

224

Temperature profile along the air-liquid interface due to the cooling effect

225

To visualize the cooling effect in more detail, the temperature profile along the air-liquid

226

interface is plotted in Fig. 5 for droplets of different contact angles and evaporative cooling

227

numbers. The interfacial temperature profile is found to be dependent on contact angle of the

228

droplet and on the evaporative cooling number Eo . When the contact angle is smaller than 90°

229

(Fig. 5A), the interfacial temperature decreases monotonically from the droplet edge to the

230

droplet center. The greatest interfacial temperature drop observed at the droplet center is due to

231

the longest heat conduction distance between the heated substrate and the air-liquid interface


14

Page 15 of 26

there. Therefore, the larger the contact angle, the greater the temperature drop can be. Together

233

with the droplet geometry, fluid type also has significant influence on the interfacial temperature

234

profile: a higher evaporative cooling number Eo causes a greater temperature drop.

235

1.0

θ=

30o

-0.40 -0.50

0.5

2.0 1.0

-0.30

θ = 60o 0.0

-0.48 -0.58

-0.64 -0.66

-0.63

-0.70 -0.72

-0.73

-0.74

-0.78

-0.76

-0.88

120°

119.5° 121°

0 1 2 3 4 5 6 7

α

-0.83 1.2

Eo=1.0

-0.68

-0.68

Eo=2.0

0.2 0.4 0.6 0.8 1.0 Normalized radial coordinate, r/R

B

-0.53

θ = 119o

0.5

-0.10 -0.20

0.1

0.0

0.2 0.4 0.6 0.8 1.0 Normalized radial coordinate, r/R

θ = 120o

0.1

Norm. Interfacial Temp.

A

Classical model (Eo=0.0)

0.00

θ = 115o

232

Normalized interfacial temperature

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

1.2

236

Figure 5. Temperature distribution along the air-liquid interface of evaporating sessile droplets

237

of different contact angles and evaporative cooling effect. Temperature are calculated by Eqs. (2)

238

and (6), and normalized as (T − Tsub ) (Tsub − T∞ ) . The inserted graph in B is T (α , π − θ ) plotted

239

at Eo = 1.0 with contact angles near the critical value ( θ  119o ).

240

For droplets with large contact angles (on hydrophobic surfaces, Fig. 5B), on the other hand,

241

the evaporative cooling effect leads to more complex profiles of the interfacial temperature.

242

Especially, near the droplet waist and the contact line region, there is a counteracting/combining

243

effect of the local geometric confinement for mass transfer and the short distance for heat

244

conduction from the substrate. Similar to the case of droplets with acute contact angles, the fluid

245

near the contact line region experiences high local temperature due to short heat conduction path

246

from the substrate, thus supports high local saturated vapour concentration at the air-liquid

247

interface. However, due to the geometry of obtuse contact angle droplets, the confined space


15

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 26

248

between the waist and the substrate could lead to a non-uniform gradient of vapour concentration

249

(i.e., non-closed isoconcentration contours) near the droplet surface (similar to Fig. 3 with

250

Eo ≠ 0 ). The combination of these two effects leads to interesting predictions as shown in Fig.

251

5B: (i) there is a narrow range of contact angle θ  119o (red lines in the insert of Fig. 5B and

252

Fig. S2) in which temperature near the contact line and the droplet apex are almost similar, (ii)

253

for θ < 119o the trend of interfacial temperature profile is similar the one of acute droplets (the

254

contact line region has the highest local temperature), (iii) for θ ≥ 120o the droplet surface near

255

the contact line is colder than at the droplet apex. For substrates with very high thermal

256

conductivities (close to the isothermal condition used in our study), previous studies shown that

257

there is no inversion of the temperature gradient along the air-liquid interface in the range of 0

258

Analytical Model for Diffusive Evaporation of Sessile Droplets Coupled

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