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New Concepts at the Interface: Novel Viewpoints and Interpretations, Theory and Computations
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
1
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
6
KEYWORDS: evaporative cooling, droplet evaporation, coffee ring, contact angle, flux, toroidal
7
coordinate, Laplace equation.
8
ABSTRACT
9
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,
21
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
26
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”.
43
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
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by ~20% when using the isothermal model. This discrepancy was attributed to a large
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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
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“non-isothermal model” has also to be coupled with the equation of heat transfer inside the
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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
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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
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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
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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
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−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
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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.
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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
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described by the quasi-steady state with neglecting the transient term in Fick’s law1, 4, 5, 18, giving
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∇ 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
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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 =
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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
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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
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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
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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.
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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)
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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
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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.
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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
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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
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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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Langmuir
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
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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
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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
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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
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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
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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
