Mathematical Modelling of Droplet Evaporation...1.Modes of evaporation 2.Di usion-limited model...

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Mathematical Modelling of Droplet Evaporation

Stephen K. Wilson

Department of Mathematics and Statistics, University of Strathclyde, Glasgow

Email: s.k.wilson@strath.ac.ukTwitter: @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

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

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

θ(◦)

10−3m)

00 100100 200200 300300 400400 100100 600600 700700t (s)

CR phase CA phase

Fukai et al. (2006)

Idealised Stick-Slide (SS) Mode

θ θ∗

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

Free Surfacez = h

RSubstrate

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

sin θ(2 + cos θ)

(1 + cos θ)2

The Diffusion-Limited Modelz

Free Surfacez = h

RSubstrate

Evaporative Flux J

Vapour

Volume of droplet

V = V (t) =πR3

sin θ(2 + cos θ)

(1 + cos θ)2

The Diffusion-Limited Modelz

Free Surfacez = h

RSubstrate

Evaporative Flux J

Vapour

Volume of droplet

V = V (t) =πR3

sin θ(2 + cos θ)

(1 + cos θ)2

Evaporative Flux

0.0 −2 −1 0 1 2

θ = 10◦r

0.0 −2 −1 0 1 2

θ = 90◦

0.0 −2 0 2

θ = 170◦r

0.0 −2 0 2

θ = 180◦

The Diffusion-Limited Model

Rate of change of V due to total evaporative flux

∫J ds = −πRD(csat − c∞)

(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{

∫ ∞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 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

)−1/2(π − θ0) + O(π − θ0)3

For the CA mode

)1/3(1− t

(π − θ0) + O(π − θ0)3

Both both modes

V = V0

(1− t

+ O(π − θ0)2

• For the CR mode

θ = π −(

1− t

)−1/2(π − θ0) + O(π − θ0)3

For the CA mode

)1/3(1− t

(π − θ0) + O(π − θ0)3

Both both modes

V = V0

(1− t

+ O(π − θ0)2

• For the CR mode

θ = π −(

1− t

)−1/2(π − θ0) + O(π − θ0)3

For the CA mode

)1/3(1− t

(π − θ0) + O(π − θ0)3

Both both modes

V = V0

(1− t

+ O(π − θ0)2

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

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

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

Droplet Lifetimes

• Reference Timescale

2D(csat − c∞)

• Constant Radius (CR) Mode

(2(1 + cos θ0)2

sin θ0(cos θ0 + 2)

)2/3 ∫ θ0

• Constant Angle (CA) Mode

(2(1 + cos θ0)2

)2/3sin θ0(cos θ0 + 2)

g(θ0)

• Stick-Slide (SS) Mode (θ0 > θ∗)

(2(1 + cos θ0)2

)2/3 [∫ θ0

g(θ)+

sin θ∗(2 + cos θ∗)g(θ∗)

Droplet Lifetimes• Reference Timescale

2D(csat − c∞)

(2(1 + cos θ0)2

)2/3 ∫ θ0

(2(1 + cos θ0)2

)2/3sin θ0(cos θ0 + 2)

g(θ0)

(2(1 + cos θ0)2

)2/3 [∫ θ0

g(θ)+

2D(csat − c∞)

(2(1 + cos θ0)2

)2/3 ∫ θ0

(2(1 + cos θ0)2

)2/3sin θ0(cos θ0 + 2)

g(θ0)

(2(1 + cos θ0)2

)2/3 [∫ θ0

g(θ)+

2D(csat − c∞)

(2(1 + cos θ0)2

)2/3 ∫ θ0

(2(1 + cos θ0)2

)2/3sin θ0(cos θ0 + 2)

g(θ0)

(2(1 + cos θ0)2

)2/3 [∫ θ0

g(θ)+

2D(csat − c∞)

(2(1 + cos θ0)2

)2/3 ∫ θ0

(2(1 + cos θ0)2

)2/3sin θ0(cos θ0 + 2)

g(θ0)

(2(1 + cos θ0)2

)2/3 [∫ θ0

g(θ)+

Droplet Lifetimes for 0 ≤ θ∗ ≤ π/2

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

θ0π8

tCAtCR

θ∗ = π/64

θ∗ = π/16θ∗ = π/8

θ∗ = 3π/16θ∗ = π/4

θ∗ = 5π/16θ∗ = 3π/8

θ∗ = 7π/16θ∗ = π/2

θ0π2

11π16

13π16

15π16

θcrit ≃ 2.5830

0.9354

Droplet Lifetimes for π/2 ≤ θ∗ ≤ π

tCAtCR

θ∗ = π/2θ∗ = 9π/16

θ∗ = 5π/8

θ∗ = 11π/16

θ∗ = 3π/4

θ∗ = 13π/16

θ∗ = 7π/8θ∗ = 15π/16

θ0π2

11π16

13π16

15π16

θcrit ≃ 2.5830

0.9354

Comparing the Lifetimes of the Modes

tSS = tCA

tCR = tCA

θcrit

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

1.05tSS

−10%

tSS = tCA

tCR = tCA

θcrit

tSS = tCA

tCR = tCA

θcrit

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

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

Fp(t) θ(t)

γSV + Fp(t) = γSL + γ cos θ(t)

Behaviour at the Contact Line: Contact Line Depins

vapour

liquid

substrateγSV γSL

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

fp = 2.0

Droplet Lifetimes

c = 0.05

0.40.6

0.81.01.2

1.4 1.6 1.81.9

θ0π8

c = 0.05

0.2 0.4 0.60.8

1.61.8

θ0π2

11π16

13π16

15π16

[Note that c = fp!]

IIIIIIVIII

(θcrit, fp crit)

(π, fpπ)

θcrit

• Region I: tCR < tSS < tCA

• Region VII: tCA < tSS = tCR

• Region VIII: tSS = tCR < tCA

θ∗ = arccos(0.2005 + cos θ0)

(0.6443, 0)

(π, 2.4973)

The theory curve is plotted for fp = 0.2005 (R2 = 0.9676)

tSS tCR

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,

∂z= 0 on z = 0 for r > R,

andc → 0 as (r2 + z2)1/2 →∞.

The total evaporation rate is given by

dt= −π

dt(θR3) = 2π

0J(r , t) rdr ,

J(r , t) = −∂c∂z

∣∣∣∣z=0

∆C =θ(0)LDc ′sat(Ta)

k, S =

θ(0)R(0)ks,

E =θ(0)LD(1− H)csat(Ta)

• 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

dt̃(θ0R

30 ) = −4R2

dt̃(θ0R

30 ) = −4R2

dt̃(θ0R

30 ) = −4R2

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 =

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

for t̃? < t̃ < t̃SS, where

t̃? =∆C

4(1− θ?) and t̃SS =

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).

θ⋆ = 1 θ⋆ = 0

θ⋆ = 1

θ⋆ = 0

t̃⋆

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

t̃n =∆C

[1− θmax + (θmax − θmin)

1− (θmin/θmax)n/3

1− (θmin/θmax)1/3

Taking the limit n→∞ yields

t̃SJ =∆C

[1− θmax + (θmax − θmin)

θmax1/3

θmax1/3 − θmin

Evolution of a Droplet Evaporating in the SJ Mode

θmin = 0

θmin = 1

θmin = 0

θmin = 1

Evolution of a Droplet Evaporating in the SJ Mode

(0, 0.25)

(1, 0.75)

An Extreme Example: Water on Aerogel

V [nl]

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).

ρθ(0)R(0)2

D(1− H)csat(Ta),

ks(1− H)csat(Ta).

ρθ(0)R(0)2

D(1− H)csat(Ta),

ks(1− H)csat(Ta).

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

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: s.k.wilson@strath.ac.uk, twitter: @S K Wilson

Mathematical Modelling of Droplet Evaporation...1.Modes of evaporation 2.Di usion-limited model...

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