Mathematical Modelling of Droplet Evaporation
Stephen K. Wilson
Department of Mathematics and Statistics, University of Strathclyde, Glasgow
Email: [email protected]: @S K Wilson
IMA Workshop onDynamic Contact Lines: Progress and Opportunities
26th−28th March 2018
Joint work with Jutta Stauber, Feargus Schofield, Brian Duffyand David Pritchard (UoS), and Khellil Sefiane (Edinburgh)
Droplets are everywhere . . .
www.wonderfulphotos.com commons.wikimedia.org
www.livescience.com www.pentagonpost.com
. . . and have been extensively studied
. . . and have been extensively studied
. . . and have been extensively studied
. . . and have been extensively studied
Outline of Talk
1. Modes of evaporation
2. Diffusion-limited model
3. Lifetimes of droplets
4. Comparison with experimental results
5. Lifetimes revisited when θ0 and θ∗ are related
6. Anomalously long lifetimes due to strong thermal effects
Outline of Talk
1. Modes of evaporation
2. Diffusion-limited model
3. Lifetimes of droplets
4. Comparison with experimental results
5. Lifetimes revisited when θ0 and θ∗ are related
6. Anomalously long lifetimes due to strong thermal effects
Outline of Talk
1. Modes of evaporation
2. Diffusion-limited model
3. Lifetimes of droplets
4. Comparison with experimental results
5. Lifetimes revisited when θ0 and θ∗ are related
6. Anomalously long lifetimes due to strong thermal effects
Outline of Talk
1. Modes of evaporation
2. Diffusion-limited model
3. Lifetimes of droplets
4. Comparison with experimental results
5. Lifetimes revisited when θ0 and θ∗ are related
6. Anomalously long lifetimes due to strong thermal effects
Outline of Talk
1. Modes of evaporation
2. Diffusion-limited model
3. Lifetimes of droplets
4. Comparison with experimental results
5. Lifetimes revisited when θ0 and θ∗ are related
6. Anomalously long lifetimes due to strong thermal effects
Outline of Talk
1. Modes of evaporation
2. Diffusion-limited model
3. Lifetimes of droplets
4. Comparison with experimental results
5. Lifetimes revisited when θ0 and θ∗ are related
6. Anomalously long lifetimes due to strong thermal effects
Outline of Talk
1. Modes of evaporation
2. Diffusion-limited model
3. Lifetimes of droplets
4. Comparison with experimental results
5. Lifetimes revisited when θ0 and θ∗ are related
6. Anomalously long lifetimes due to strong thermal effects
Constant Radius (CR) Mode
• Constant contact radius R = R0
• Variable contact angle θ = θ(t)
Constant Angle (CA) Mode
• Variable contact radius R = R(t)
• Constant contact angle θ = θ0
Stick-Slide (SS) Mode
800
θ(◦)
R(×
10−3m)
R
θ
0
10
20
30
40
50
60
00 100100 200200 300300 400400 100100 600600 700700t (s)
0
0.6
1.2
CR phase CA phase
Fukai et al. (2006)
Idealised Stick-Slide (SS) Mode
R
R0
00
00
00
θ θ∗
θ0
V
V0
t
t
t
tSS
tSS
tSS
CR phase CA phase
Nguyen & Nguyen (2012), Dash & Garimella (2013),and Stauber et al. (2014)
Stick-Jump (SJ) Mode
Orejon et al. (2011)
The Diffusion-Limited Model
z
r
Free Surfacez = h
R
θ
RSubstrate
Fluid
Evaporative Flux J
Vapour
In many circumstances the evaporation of a droplet is welldescribed by the diffusion-limited model (e.g. Popov 2005).
Volume of droplet
V = V (t) =πR3
3
sin θ(2 + cos θ)
(1 + cos θ)2
The Diffusion-Limited Modelz
r
Free Surfacez = h
R
θ
RSubstrate
Fluid
Evaporative Flux J
Vapour
In many circumstances the evaporation of a droplet is welldescribed by the diffusion-limited model (e.g. Popov 2005).
Volume of droplet
V = V (t) =πR3
3
sin θ(2 + cos θ)
(1 + cos θ)2
The Diffusion-Limited Modelz
r
Free Surfacez = h
R
θ
RSubstrate
Fluid
Evaporative Flux J
Vapour
In many circumstances the evaporation of a droplet is welldescribed by the diffusion-limited model (e.g. Popov 2005).
Volume of droplet
V = V (t) =πR3
3
sin θ(2 + cos θ)
(1 + cos θ)2
Evaporative Flux
r
z
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0 −2 −1 0 1 2
θ = 10◦r
z
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0 −2 −1 0 1 2
θ = 90◦
r
z
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0 −2 0 2
θ = 170◦r
z
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0 −2 0 2
θ = 180◦
The Diffusion-Limited Model
Rate of change of V due to total evaporative flux
dV
dt=
∫J ds = −πRD(csat − c∞)
ρ
g(θ)
(1 + cos θ)2
where D is the diffusion coefficient of vapour in the air, ρ isthe density of the fluid, csat is the vapour concentration at theinterface, c∞ is the vapour concentration far from theinterface, and g = g(θ) is given by
(1 + cos θ)2{
tanθ
2+ 8
∫ ∞0
cosh2 θτ
sinh 2πτtanh [τ(π − θ)] dτ
}
Aside 1: Thermal Properties of the Substrate
• Dunn et al. (2009) showed that the thermal properties ofthe substrate can have a strong effect.
Aside 1: Thermal Properties of the Substrate
• Dunn et al. (2009) showed that the thermal properties ofthe substrate can have a strong effect.
Aside 1: Thermal Properties of the Substrate
• Dunn et al. (2009) showed that the thermal properties ofthe substrate can have a strong effect.
Aside 2: Strongly Hydrophobic Substrates
• Stauber et al. (2015) pointed out that the extreme modesbecome indistinguishable on strongly hydrophobicsubstrates, θ → π, R → 0, tπ = (41/3 log 2)−1 ' 0.9088.
• For the CR mode
θ = π −(
1− t
tπ
)−1/2(π − θ0) + O(π − θ0)3
For the CA mode
R =
(3V0
4π
)1/3(1− t
tπ
)1/2
(π − θ0) + O(π − θ0)3
Both both modes
V = V0
(1− t
tπ
)3/2
+ O(π − θ0)2
Aside 2: Strongly Hydrophobic Substrates
• Stauber et al. (2015) pointed out that the extreme modesbecome indistinguishable on strongly hydrophobicsubstrates, θ → π, R → 0, tπ = (41/3 log 2)−1 ' 0.9088.
• For the CR mode
θ = π −(
1− t
tπ
)−1/2(π − θ0) + O(π − θ0)3
For the CA mode
R =
(3V0
4π
)1/3(1− t
tπ
)1/2
(π − θ0) + O(π − θ0)3
Both both modes
V = V0
(1− t
tπ
)3/2
+ O(π − θ0)2
Aside 2: Strongly Hydrophobic Substrates
• Stauber et al. (2015) pointed out that the extreme modesbecome indistinguishable on strongly hydrophobicsubstrates, θ → π, R → 0, tπ = (41/3 log 2)−1 ' 0.9088.
• For the CR mode
θ = π −(
1− t
tπ
)−1/2(π − θ0) + O(π − θ0)3
For the CA mode
R =
(3V0
4π
)1/3(1− t
tπ
)1/2
(π − θ0) + O(π − θ0)3
Both both modes
V = V0
(1− t
tπ
)3/2
+ O(π − θ0)2
Aside 2: Strongly Hydrophobic Substrates
t/T
θ
CA mode (θ0 = 10◦)CR mode (θ0 = 10◦)
CA mode (θ0 = 90◦)CR mode (θ0 = 90◦)
CA mode (θ0 = θcrit)CR mode (θ0 = θcrit)
CA mode (θ0 = 170◦)CR mode (θ0 = 170◦)
0.0 0.2 0.4 0.6 0.8 1.00
π8
π4
3π8
π2
5π8
3π4
7π8
π
t/T
R/V1/30
CA mode (θ0 = 10◦)
CR mode (θ0 = 10◦)
CA mode (θ0 = 90◦)CR mode (θ0 = 90◦)
CA mode (θ0 = θcrit)CR mode (θ0 = θcrit)
CA mode (θ0 = 170◦)
CR mode (θ0 = 170◦)
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
Aside 2: Strongly Hydrophobic Substrates
t/T
V/V0
CA mode (θ0 = 10◦)
CR mode (θ0 = 10◦)
CA mode (θ0 = 90◦)
CR mode (θ0 = 90◦)
CA mode (θ0 = θcrit)
CR mode (θ0 = θcrit)
CA mode (θ0 = 170◦)
CR mode (θ0 = 170◦)
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Droplet Lifetimes
• Reference Timescale
ρ
2D(csat − c∞)
(3V0
2π
)2/3
• Constant Radius (CR) Mode
tCR =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 ∫ θ0
0
2 dθ
g(θ)
• Constant Angle (CA) Mode
tCA =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3sin θ0(cos θ0 + 2)
g(θ0)
• Stick-Slide (SS) Mode (θ0 > θ∗)
tSS =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 [∫ θ0
θ∗
2 dθ
g(θ)+
sin θ∗(2 + cos θ∗)g(θ∗)
]
Droplet Lifetimes• Reference Timescale
ρ
2D(csat − c∞)
(3V0
2π
)2/3
• Constant Radius (CR) Mode
tCR =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 ∫ θ0
0
2 dθ
g(θ)
• Constant Angle (CA) Mode
tCA =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3sin θ0(cos θ0 + 2)
g(θ0)
• Stick-Slide (SS) Mode (θ0 > θ∗)
tSS =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 [∫ θ0
θ∗
2 dθ
g(θ)+
sin θ∗(2 + cos θ∗)g(θ∗)
]
Droplet Lifetimes• Reference Timescale
ρ
2D(csat − c∞)
(3V0
2π
)2/3
• Constant Radius (CR) Mode
tCR =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 ∫ θ0
0
2 dθ
g(θ)
• Constant Angle (CA) Mode
tCA =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3sin θ0(cos θ0 + 2)
g(θ0)
• Stick-Slide (SS) Mode (θ0 > θ∗)
tSS =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 [∫ θ0
θ∗
2 dθ
g(θ)+
sin θ∗(2 + cos θ∗)g(θ∗)
]
Droplet Lifetimes• Reference Timescale
ρ
2D(csat − c∞)
(3V0
2π
)2/3
• Constant Radius (CR) Mode
tCR =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 ∫ θ0
0
2 dθ
g(θ)
• Constant Angle (CA) Mode
tCA =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3sin θ0(cos θ0 + 2)
g(θ0)
• Stick-Slide (SS) Mode (θ0 > θ∗)
tSS =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 [∫ θ0
θ∗
2 dθ
g(θ)+
sin θ∗(2 + cos θ∗)g(θ∗)
]
Droplet Lifetimes• Reference Timescale
ρ
2D(csat − c∞)
(3V0
2π
)2/3
• Constant Radius (CR) Mode
tCR =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 ∫ θ0
0
2 dθ
g(θ)
• Constant Angle (CA) Mode
tCA =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3sin θ0(cos θ0 + 2)
g(θ0)
• Stick-Slide (SS) Mode (θ0 > θ∗)
tSS =
(2(1 + cos θ0)2
sin θ0(cos θ0 + 2)
)2/3 [∫ θ0
θ∗
2 dθ
g(θ)+
sin θ∗(2 + cos θ∗)g(θ∗)
]
Droplet Lifetimes for 0 ≤ θ∗ ≤ π/2
tCA
tCR
θ∗ = π/64
θ∗ = π/16
θ∗ = π/8
θ∗ = 3π/16
θ∗ = π/4θ∗ = 5π/16
θ∗ = 3π/8 θ∗ = 7π/16 θ∗ = π/2
0.2
0.4
0.6
0.8
1.0
tSS
θ0π8
π4
3π8
π2
5π8
3π4
7π8
π0
ææ
tCAtCR
θ∗ = π/64
θ∗ = π/16θ∗ = π/8
θ∗ = 3π/16θ∗ = π/4
θ∗ = 5π/16θ∗ = 3π/8
θ∗ = 7π/16θ∗ = π/2
0.85
0.90
0.95
1.00
tSS
θ0π2
9π16
5π8
11π16
3π4
13π16
7π8
15π16
π
θcrit ≃ 2.5830
0.9354
Droplet Lifetimes for π/2 ≤ θ∗ ≤ π
ææ
tCAtCR
θ∗ = π/2θ∗ = 9π/16
θ∗ = 5π/8
θ∗ = 11π/16
θ∗ = 3π/4
θ∗ = 13π/16
θ∗ = 7π/8θ∗ = 15π/16
0.85
0.90
0.95
1.00
tSS
θ0π2
9π16
5π8
11π16
3π4
13π16
7π8
15π16
π
θcrit ≃ 2.5830
0.9354
Comparing the Lifetimes of the Modes
I
VI
II
V
θ∗
θ0
0
π8
π4
3π8
π2
5π8
3π4
7π8
π
0 π8
π4
3π8
π2
5π8
3π4
7π8
π
tSS = tCA
tSS = tCA
tCR = tCA
tSS =
θcrit
θcrit
tCR
III
IV • Region I: tCR < tSS < tCA
• Region II: tCR < tCA < tSS
• Region III: tCA < tCR < tSS
• Region IV: tCA < tSS < tCR
• Region V: tSS = tCA < tCR
• Region VI: tCR < tSS = tCA
Experimental Data for the Stick-Slide Mode
• 29 sets of experimental data from 9 different authors
• Liquids used:• water• deionised water• xylene-polystyrene• diethylene glycol with coffee particles• latex dispersions
• Substrates used:• polished epoxy resin• monolayers on glass• monolayers on silicon• monolayers on gold-covered mica• platinum• Teflon• pyrex glass
Comparison with Experimental Results
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.050.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05tSS
texp
−10%
10%
−5%
5%
Comparing the Lifetimes of the Modes
I
VI
II
V
θ∗
θ0
0
π8
π4
3π8
π2
5π8
3π4
7π8
π
0 π8
π4
3π8
π2
5π8
3π4
7π8
π
tSS = tCA
tSS = tCA
tCR = tCA
tSS =
θcrit
θcrit
tCR
III
IV • Region I: tCR < tSS < tCA
• Region II: tCR < tCA < tSS
• Region III: tCA < tCR < tSS
• Region IV: tCA < tSS < tCR
• Region V: tSS = tCA < tCR
• Region VI: tCR < tSS = tCA
Comparing the Lifetimes of the Modes
I
VI
II
V
θ∗
θ0
0
π8
π4
3π8
π2
5π8
3π4
7π8
π
0 π8
π4
3π8
π2
5π8
3π4
7π8
π
tSS = tCA
tSS = tCA
tCR = tCA
tSS =
θcrit
θcrit
tCR
III
IV • Region I: tCR < tSS < tCA
• Region II: tCR < tCA < tSS
• Region III: tCA < tCR < tSS
• Region IV: tCA < tSS < tCR
• Region V: tSS = tCA < tCR
• Region VI: tCR < tSS = tCA
The Relationship Between θ0 and θ∗?
• In any particular experiment, θ0 and θ∗ are (empiricallydetermined) numbers.
• This begs the natural questions “What is the functionalrelationship between θ0 and θ∗, and how does it effect theway the droplet lifetimes vary with θ0?”
• We analyse the consequences of one physically crediblerelationship based on the unbalanced Young force at thecontact line.
The Relationship Between θ0 and θ∗?
• In any particular experiment, θ0 and θ∗ are (empiricallydetermined) numbers.
• This begs the natural questions “What is the functionalrelationship between θ0 and θ∗, and how does it effect theway the droplet lifetimes vary with θ0?”
• We analyse the consequences of one physically crediblerelationship based on the unbalanced Young force at thecontact line.
The Relationship Between θ0 and θ∗?
• In any particular experiment, θ0 and θ∗ are (empiricallydetermined) numbers.
• This begs the natural questions “What is the functionalrelationship between θ0 and θ∗, and how does it effect theway the droplet lifetimes vary with θ0?”
• We analyse the consequences of one physically crediblerelationship based on the unbalanced Young force at thecontact line.
The Relationship Between θ0 and θ∗?
• In any particular experiment, θ0 and θ∗ are (empiricallydetermined) numbers.
• This begs the natural questions “What is the functionalrelationship between θ0 and θ∗, and how does it effect theway the droplet lifetimes vary with θ0?”
• We analyse the consequences of one physically crediblerelationship based on the unbalanced Young force at thecontact line.
Behaviour at the Contact Line
Behaviour at the Contact Line: Droplet is Deposited
vapour
liquid
substrateγSV γSL
γ
Fp(0) θ0
• Droplet is deposited on• rough substrate• chemically inhomogeneous substrate
• Force balance at the contact line:
γSV + Fp(0) = γSL + γ cos θ0
Behaviour at the Contact Line: Droplet is Evaporating
vapour
liquid
substrateγSV γSL
γ
Fp(t) θ(t)
γSV + Fp(t) = γSL + γ cos θ(t)
Behaviour at the Contact Line: Contact Line Depins
vapour
liquid
substrateγSV γSL
γ
Fmax θ∗
• Force balance at the contact line:
γSV + Fmax = γSL + γ cos θ∗
• Relationship between θ0 and θ∗
fp = (Fmax − Fp(0))/γ = cos θ∗ − cos θ0
Relationship Between θ0 and θ∗
θ∗ = arccos(fp + cos θ0) (0 ≤ fp ≤ 2)
fp = 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
fp = 2.0
θ∗
0
π8
π4
3π8
π2
5π8
3π4
7π8
π
θ0
0π8
π4
3π8
π2
5π8
3π4
7π8
π
Droplet Lifetimes
tCA
tCR
c = 0.05
0.2
0.40.6
0.81.01.2
1.4 1.6 1.81.9
0.2
0.4
0.6
0.8
1.0
tSS
θ0π8
π4
3π8
π2
5π8
3π4
7π8
π
tCA
tCR
c = 0.05
0.2 0.4 0.60.8
1.0
1.2
1.4
1.61.8
1.9
0.85
0.90
0.95
1.00
θ0π2
9π16
5π8
11π16
3π4
13π16
7π8
15π16
π
tSS
0
[Note that c = fp!]
Comparing the Lifetimes of the Modes
0.0IV
IIIIIIVIII
VII
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
fp
θ0
0π8
π4
3π8
π2
5π8
3π4
7π8
π
(θcrit, fp crit)
(π, fpπ)
θcrit
• Region I: tCR < tSS < tCA
• Region II: tCR < tCA < tSS
• Region III: tCA < tCR < tSS
• Region IV: tCA < tSS < tCR
• Region VII: tCA < tSS = tCR
• Region VIII: tSS = tCR < tCA
Comparison with Experimental Results
θ∗ = arccos(0.2005 + cos θ0)
(0.6443, 0)
(π, 2.4973)
θ∗
0
π8
π4
3π8
π2
5π8
3π4
7π8
π
θ0
0π8
π4
3π8
π2
5π8
3π4
7π8
π
The theory curve is plotted for fp = 0.2005 (R2 = 0.9676)
Comparison with Experimental Results
tSS
tSS tCR
tCA
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
θ00
π8
π4
3π8
π2
5π8
3π4
7π8
π
The theory curve is plotted for fp = 0.2005 (R2 = 0.9676)
Droplet Lifetimes
Qualitative similarities with the behaviour hypothesised byShanahan et al. (2011).
Anomalously long lifetimes due to strong thermal effects
Scaling using the standard timescale for a thin droplet,
ρθ(0)R(0)2
D(1− H)csat(Ta),
the vapour concentration c(r , z , t) satisfies
∇2c = 0 in z > 0,
c = 1 + ∆C (h + S)∂c
∂zon z = 0 for 0 ≤ r ≤ R,
∂c
∂z= 0 on z = 0 for r > R,
andc → 0 as (r2 + z2)1/2 →∞.
Anomalously long lifetimes due to strong thermal effects
The total evaporation rate is given by
−dV
dt= −π
4
d
dt(θR3) = 2π
∫ R
0J(r , t) rdr ,
where
J(r , t) = −∂c∂z
∣∣∣∣z=0
and
∆C =θ(0)LDc ′sat(Ta)
k, S =
khs
θ(0)R(0)ks,
E =θ(0)LD(1− H)csat(Ta)
kTa.
Anomalously long lifetimes due to strong thermal effects
• Consider the situation of a droplet on a substrate with highthermal resistance (corresponding to the limit S →∞).
• In this limit, the droplet evolves on a long timescale ofO(S)� 1 relative to the basic timescale.
• In suitably rescaled variables, the leading-order evolution ofthe droplet satisfies
d
dt̃(θ0R
30 ) = −4R2
0
∆C.
Anomalously long lifetimes due to strong thermal effects
• Consider the situation of a droplet on a substrate with highthermal resistance (corresponding to the limit S →∞).
• In this limit, the droplet evolves on a long timescale ofO(S)� 1 relative to the basic timescale.
• In suitably rescaled variables, the leading-order evolution ofthe droplet satisfies
d
dt̃(θ0R
30 ) = −4R2
0
∆C.
Anomalously long lifetimes due to strong thermal effects
• Consider the situation of a droplet on a substrate with highthermal resistance (corresponding to the limit S →∞).
• In this limit, the droplet evolves on a long timescale ofO(S)� 1 relative to the basic timescale.
• In suitably rescaled variables, the leading-order evolution ofthe droplet satisfies
d
dt̃(θ0R
30 ) = −4R2
0
∆C.
Evolution of a Droplet Evaporating in the CR Mode
For a droplet evaporating in the CR mode
R0 ≡ 1, θ0 = 1− 4
∆Ct̃, V0 =
πθ04,
and hence t̃CR = ∆C/4.
Evolution of a Droplet Evaporating in the CA Mode
For a droplet evaporating in the CA mode
R0 = 1− 4
3∆Ct̃, θ0 ≡ 1, V0 =
πR30
4,
and hence t̃CA = 3∆C/4 = 3t̃CR.
Evolution of a Droplet Evaporating in the SS Mode
For a droplet evaporating in the SS mode, the solution for the CRmode holds for 0 < t̃ < t̃?, and
R0 =(1 + 2θ?)∆C − 4t̃
3θ?∆C, θ0 ≡ θ?, V0 =
πθ0R30
4,
for t̃? < t̃ < t̃SS, where
t̃? =∆C
4(1− θ?) and t̃SS =
∆C
4(1 + 2θ?).
Note that t̃? = t̃SS = t̃CR when θ? = 0 (i.e. the CR mode), andt̃? = 0 and t̃SS = t̃CA = 3t̃CR when θ? = 1 (i.e. the CA mode).
Evolution of a Droplet Evaporating in the SS Mode
t̃
θ⋆ = 1 θ⋆ = 0
R0
θ⋆
t̃
θ0
θ⋆ = 1
θ⋆ = 0
Evolution of a Droplet Evaporating in the SS Mode
t̃
V0
θ⋆
t̃
V0
∆C
Evolution of a Droplet Evaporating in the SS Mode
V0
t̃SS
t̃⋆
θ⋆
Evolution of a Droplet Evaporating in the SJ ModeFor a droplet evaporating in the SJ mode, during the 1st stickphase from t = t0 = 0 to t = t1 = ∆C (1− θmin)/4 the solutionfor the CR mode holds with R0 = R1, and during the nth stickphase (n = 2, 3, 4, . . .) from t = tn−1 to t = tn,
R0 ≡ Rn, θ0 = θmax −4
∆CRn(t̃ − t̃n−1), V0 =
πθ0R3n
4,
where
t̃n =∆C
4
[1− θmax + (θmax − θmin)
1− (θmin/θmax)n/3
1− (θmin/θmax)1/3
].
Taking the limit n→∞ yields
t̃SJ =∆C
4
[1− θmax + (θmax − θmin)
θmax1/3
θmax1/3 − θmin
1/3
].
Evolution of a Droplet Evaporating in the SJ Mode
t̃
R0
θmin = 0
θmin = 1
t̃
θ0
θmin = 0
θmin = 1
Evolution of a Droplet Evaporating in the SJ Mode
t̃
V0
θmin
t̃
V0
∆C
Evolution of a Droplet Evaporating in the SJ Mode
θmin
t̃SJ
(0, 0.25)
θmax
θmax
t̃SJ
θmin
(1, 0.75)
An Extreme Example: Water on Aerogel
t[s]
V [nl]
CR CA
CR CA
On a metal substrate evaporates completely in 69 to 103 seconds.On an aerogel substrate with ks = 0.015 kg m s−3 K−1 evaporatescompletely in 194 to 583 seconds.
• Qualitatively similar (but more mathematically complicated)results hold in the limit ∆C →∞.
• The key result is that when thermal effects are strong thelifetime of the droplet is much longer than the basic timescale,
ρθ(0)R(0)2
D(1− H)csat(Ta),
by a factor of size S∆C � 1, i.e. it is actually on the muchlonger timescale
ρθ0R0Lhsc ′sat(Ta)
ks(1− H)csat(Ta).
• Evaporation is limited by diffusion of heat through the dropletand the substrate (rather than by diffusion of vapour in theatmosphere).
• Qualitatively similar (but more mathematically complicated)results hold in the limit ∆C →∞.
• The key result is that when thermal effects are strong thelifetime of the droplet is much longer than the basic timescale,
ρθ(0)R(0)2
D(1− H)csat(Ta),
by a factor of size S∆C � 1, i.e. it is actually on the muchlonger timescale
ρθ0R0Lhsc ′sat(Ta)
ks(1− H)csat(Ta).
• Evaporation is limited by diffusion of heat through the dropletand the substrate (rather than by diffusion of vapour in theatmosphere).
• Qualitatively similar (but more mathematically complicated)results hold in the limit ∆C →∞.
• The key result is that when thermal effects are strong thelifetime of the droplet is much longer than the basic timescale,
ρθ(0)R(0)2
D(1− H)csat(Ta),
by a factor of size S∆C � 1, i.e. it is actually on the muchlonger timescale
ρθ0R0Lhsc ′sat(Ta)
ks(1− H)csat(Ta).
• Evaporation is limited by diffusion of heat through the dropletand the substrate (rather than by diffusion of vapour in theatmosphere).
Conclusions
• We used the diffusion-limited evaporation model to give acomplete description of the CR, CA and SS modes.
• The lifetime of a droplet in the SS mode is not alwaysconstrained by the lifetimes of droplets in the extreme modes.
• Theoretical results are in surprisingly good agreement withavailable experimental data.
• The consequences of a physically credible relationship betweenθ0 and θ∗ were explored.
• Strong thermal effects lead to anomalously long lifetimes.
• There are still many other open questions to address!
Conclusions
• We used the diffusion-limited evaporation model to give acomplete description of the CR, CA and SS modes.
• The lifetime of a droplet in the SS mode is not alwaysconstrained by the lifetimes of droplets in the extreme modes.
• Theoretical results are in surprisingly good agreement withavailable experimental data.
• The consequences of a physically credible relationship betweenθ0 and θ∗ were explored.
• Strong thermal effects lead to anomalously long lifetimes.
• There are still many other open questions to address!
Conclusions
• We used the diffusion-limited evaporation model to give acomplete description of the CR, CA and SS modes.
• The lifetime of a droplet in the SS mode is not alwaysconstrained by the lifetimes of droplets in the extreme modes.
• Theoretical results are in surprisingly good agreement withavailable experimental data.
• The consequences of a physically credible relationship betweenθ0 and θ∗ were explored.
• Strong thermal effects lead to anomalously long lifetimes.
• There are still many other open questions to address!
Conclusions
• We used the diffusion-limited evaporation model to give acomplete description of the CR, CA and SS modes.
• The lifetime of a droplet in the SS mode is not alwaysconstrained by the lifetimes of droplets in the extreme modes.
• Theoretical results are in surprisingly good agreement withavailable experimental data.
• The consequences of a physically credible relationship betweenθ0 and θ∗ were explored.
• Strong thermal effects lead to anomalously long lifetimes.
• There are still many other open questions to address!
Conclusions
• We used the diffusion-limited evaporation model to give acomplete description of the CR, CA and SS modes.
• The lifetime of a droplet in the SS mode is not alwaysconstrained by the lifetimes of droplets in the extreme modes.
• Theoretical results are in surprisingly good agreement withavailable experimental data.
• The consequences of a physically credible relationship betweenθ0 and θ∗ were explored.
• Strong thermal effects lead to anomalously long lifetimes.
• There are still many other open questions to address!
Conclusions
• We used the diffusion-limited evaporation model to give acomplete description of the CR, CA and SS modes.
• The lifetime of a droplet in the SS mode is not alwaysconstrained by the lifetimes of droplets in the extreme modes.
• Theoretical results are in surprisingly good agreement withavailable experimental data.
• The consequences of a physically credible relationship betweenθ0 and θ∗ were explored.
• Strong thermal effects lead to anomalously long lifetimes.
• There are still many other open questions to address!
Conclusions
• We used the diffusion-limited evaporation model to give acomplete description of the CR, CA and SS modes.
• The lifetime of a droplet in the SS mode is not alwaysconstrained by the lifetimes of droplets in the extreme modes.
• Theoretical results are in surprisingly good agreement withavailable experimental data.
• The consequences of a physically credible relationship betweenθ0 and θ∗ were explored.
• Strong thermal effects lead to anomalously long lifetimes.
• There are still many other open questions to address!
Some References
Dunn et al. J. Fluid Mech. 623, 329–351 (2009)
Dunn et al. Phys. Fluids 21, 052101 (2009)
Sefiane et al. Phys. Fluids 21, 062101 (2009)
Stauber et al. J. Fluid Mech. 744, R2 (2014)
Stauber et al. Langmuir 31, 3653–3660 (2015)
Stauber et al. Phys. Fluids 27, 122101 (2015)
Schofield et al. submitted for publication (2017)
Any questions?
Professor Stephen K. WilsonDepartment of Mathematics and Statistics
University of Strathclyde, Glasgowe-mail: [email protected], twitter: @S K Wilson
