Discussion of the impact of surfactant on the drag-

reduction potential of superhydrophobic surfaces

APS-DFD, 18-20 November 2018

Julien Landel University of Manchester

François J. Peaudecerf

Fernando Temprano-Coleto

Frederic Gibou

Supported by

Paolo Luzzatto-Fegiz

Julien Landel, U. Manchester UK

Superhydrophobic surfaces (SHSs)

hydrophobic surface chemistry + micro/nano patterning

500 micronsPicture by M. Reyssat

Quéré, Annu. Rev. Mater. Res. (2008)

plastron

Barthlott & Neinhuis, Planta (1997) Columbine (Aquilegia vulgaris)Lotus

Photograph by Michael Apel

Julien Landel, U. Manchester UK

Drag reduction with SHSs

slip velocity

Wikimedia Commons

J. Philip, Z. Angew. Math. Phys. (1972)

E. Lauga and H. Stone, JFM (2003)

air-water interface ~ no shear

Julien Landel, U. Manchester UK

Other applications of SHSs

emptying ketchup bottles

cooling heat transfer (e.g. cpu) self-cleaning windows

reduce pressure gradient in microfluidics

Julien Landel, U. Manchester UK

Drag reduction with SHSs

• In experiments: mixed results

• Park, Sun and Kim, JFM (2014), long gratings, up to 75%

• Gruncell (2014), meshes/amorphous, negligible DR

• McHale et al., Soft Matter (2011) • Kim and Hidrovo, PoF (2012), • Bolognesi et al.. PoF (2014), • Schäffel et al., PRL (2016)

Hypothesis: surfactants might deteriorate performance

Julien Landel, U. Manchester UK

Drag reduction with SHSs

Hypothesis: surfactants might deteriorate performance

surface tension�water

Julien Landel, U. Manchester UK

Drag reduction with SHSs

Hypothesis: surfactants might deteriorate performance

surfactants (bulk)

surfactants (adsorbed at interface)

surface concentration

surface tension

�

�(�) < �water

Julien Landel, U. Manchester UK

Drag reduction with SHSs

Hypothesis: surfactants might deteriorate performance

-> Marangoni forces resisting flow

�start < �end

�start > �end

reduced free-slip

Difficult to test experimentally:

In Schäffel et al. (2016), adding surfactant has an effect of the order of experimental uncertainty

Julien Landel, U. Manchester UK

Analogy with air bubbles rising in water

Clean bubble

airwater

UniformMarangoni force Stagnant cap Remobilisation

• Bond & Newton, Phil. Mag. (1928) • Frumkin & Levich, Zhur. Fiz. Khim. (1947) • Palaparthi et al., JFM (2006)

Julien Landel, U. Manchester UK

clean does not occur and the remobilsation reduces very slightly as well as the plastron would likely fail

Studies revealing the impact of surfactant on SHS

10�6 10�4 10�2 100

Concentration c0 (mM)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

h⌧i/

⌧ P

Free slip

No slip

Peaudecerf, Landel, Goldstein & Luzzatto-Fegiz, PNAS (2017)

Equivalent SDS concentration in fog

Increasing surfactant concentration reduces slip

101 102 103 104

Grating length g (µm)

0.0

0.2

0.4

0.6

0.8

1.0

h⌧i/

⌧ P

contaminated

clean

Increasing gap length increases slip

Julien Landel, U. Manchester UK

Studies revealing the impact of surfactant on SHSSong, Song, Hu, Du, Du, Choi, Rothstein, PRF (2018)

• Marangoni stresses significantly reduce slip on SHS and increase drag

• Presence of return flow in rectangular gratingsConvex surface Concave surface

• Large slip for circular grating

Lee, Choi & Kim, PRL (2008)Julien Landel, U. Manchester UK

Bulk

• Navier-Stokes for incompressible fluid

Steady model, periodic domain

r · u = 0Rer · (uu) = �rp+r2u

r · (uc) = 1

Per2c

@(uI �)

@x=

1

Pes

@2�

@x2+ S(cI , �)

S(cI , �) = BihkcI(1� �)� eA��

i

�k

Pe

@c

@z

����I

= �S(cI , �)

@u

@z

����I

= �Ma

✓1

1� �+A�

◆@�

@x• Marangoni stress:

• Interfacial flux:

• Modified Frumkin kinetics flux:

Top wall: air water interface

• Surfactant advection diffusion:

l/2

x

y

1

g

u

Julien Landel, U. Manchester UK

• Surfactant advection diffusion

Bulk

• Navier-Stokes for incompressible fluid

r · u = 0Rer · (uu) = �rp+r2u

r · (uc) = 1

Per2c

@(uI �)

@x=

1

Pes

@2�

@x2+ S(cI , �)

S(cI , �) = BihkcI(1� �)� eA��

i

�k

Pe

@c

@z

����I

= �S(cI , �)

@u

@z

����I

= �Ma

✓1

1� �+A�

◆@�

@x• Marangoni stress:

• Interfacial flux:

• Modified Frumkin kinetics flux:

Top wall: air water interface

• Surfactant advection diffusion:

l/2

x

y

1

g

u

Julien Landel, U. Manchester UK

• Surfactant advection diffusion

gU

D

gU

Ds

RT�m

µU

gd

�ma

gd

Uc0a

d

Steady model, periodic domain

Bulk

• Navier-Stokes for incompressible fluid

r · u = 0Rer · (uu) = �rp+r2u

r · (uc) = 1

Per2c

@(uI �)

@x=

1

Pes

@2�

@x2+ S(cI , �)

S(cI , �) = BihkcI(1� �)� eA��

i

�k

Pe

@c

@z

����I

= �S(cI , �)

@u

@z

����I

= �Ma

✓1

1� �+A�

◆@�

@x• Marangoni stress:

• Interfacial flux:

• Modified Frumkin kinetics flux:

Top wall: air water interface

• Surfactant advection diffusion:

l/2

x

y

1

g

u

Julien Landel, U. Manchester UK

• Surfactant advection diffusion

gU

D

gU

Ds

RT�m

µU

gd

�ma

gd

Uc0a

d

aspect ratio

gas fraction

Steady model, periodic domain

Numerical simulations137 finite-element Comsol simulations

Julien Landel, U. Manchester UK

4⇥ 10�4 Re 105

10�7 k 102

10�6 Pe 106

2 Pes 2⇥ 107

1.25⇥ 10�5 Bi 2⇥ 2.5

3 Ma 1012

2⇥ 10�3 � 2⇥ 102

Parameter variations:

0 20 40 60 80 100 120 140Simu #

10-10

10-5

100

105

1010DKkPesPe

Diffusive flux to surface Surface convection

Adsorption flux Surface convection

4⇥ 10�6 D =�(1 + k)

Pe1/2 7⇥ 102

10�5 K = Bi(1 + k) 50

2 extra parameters:

Palaparthi et al., JFM (2006)

Uniform Stagnant cap

Uniform regime vs stagnant cap regime

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5x

10-5

100

γM

a

uniform

stagnant cap

transition

Palaparthi et al., JFM (2006)

Julien Landel, U. Manchester UK

Stagnant cap regime:K 5⇥ 10�4

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D 5⇥ 10�2<latexit sha1_base64="Z2L8jDbJBggcFCju41lRFa/gD/g=">AAACCXicbVDLSsNAFJ34rPUVdelmsAhuLElRdFnUhcsK9gFNLJPpTTt08nBmIpSQrRt/xY0LRdz6B+78GydtFtp64MLhnHu59x4v5kwqy/o2FhaXlldWS2vl9Y3NrW1zZ7clo0RQaNKIR6LjEQmchdBUTHHoxAJI4HFoe6PL3G8/gJAsCm/VOAY3IIOQ+YwSpaWeiZ2AqCElPL3KHA73+BQ7igUgsW3dpce1rGdWrKo1AZ4ndkEqqECjZ345/YgmAYSKciJl17Zi5aZEKEY5ZGUnkRATOiID6GoaEr3MTSefZPhQK33sR0JXqPBE/T2RkkDKceDpzvxuOevl4n9eN1H+uZuyME4UhHS6yE84VhHOY8F9JoAqPtaEUMH0rZgOiSBU6fDKOgR79uV50qpVbatq35xU6hdFHCW0jw7QEbLRGaqja9RATUTRI3pGr+jNeDJejHfjY9q6YBQze+gPjM8f4YCZJA==</latexit><latexit sha1_base64="Z2L8jDbJBggcFCju41lRFa/gD/g=">AAACCXicbVDLSsNAFJ34rPUVdelmsAhuLElRdFnUhcsK9gFNLJPpTTt08nBmIpSQrRt/xY0LRdz6B+78GydtFtp64MLhnHu59x4v5kwqy/o2FhaXlldWS2vl9Y3NrW1zZ7clo0RQaNKIR6LjEQmchdBUTHHoxAJI4HFoe6PL3G8/gJAsCm/VOAY3IIOQ+YwSpaWeiZ2AqCElPL3KHA73+BQ7igUgsW3dpce1rGdWrKo1AZ4ndkEqqECjZ345/YgmAYSKciJl17Zi5aZEKEY5ZGUnkRATOiID6GoaEr3MTSefZPhQK33sR0JXqPBE/T2RkkDKceDpzvxuOevl4n9eN1H+uZuyME4UhHS6yE84VhHOY8F9JoAqPtaEUMH0rZgOiSBU6fDKOgR79uV50qpVbatq35xU6hdFHCW0jw7QEbLRGaqja9RATUTRI3pGr+jNeDJejHfjY9q6YBQze+gPjM8f4YCZJA==</latexit><latexit sha1_base64="Z2L8jDbJBggcFCju41lRFa/gD/g=">AAACCXicbVDLSsNAFJ34rPUVdelmsAhuLElRdFnUhcsK9gFNLJPpTTt08nBmIpSQrRt/xY0LRdz6B+78GydtFtp64MLhnHu59x4v5kwqy/o2FhaXlldWS2vl9Y3NrW1zZ7clo0RQaNKIR6LjEQmchdBUTHHoxAJI4HFoe6PL3G8/gJAsCm/VOAY3IIOQ+YwSpaWeiZ2AqCElPL3KHA73+BQ7igUgsW3dpce1rGdWrKo1AZ4ndkEqqECjZ345/YgmAYSKciJl17Zi5aZEKEY5ZGUnkRATOiID6GoaEr3MTSefZPhQK33sR0JXqPBE/T2RkkDKceDpzvxuOevl4n9eN1H+uZuyME4UhHS6yE84VhHOY8F9JoAqPtaEUMH0rZgOiSBU6fDKOgR79uV50qpVbatq35xU6hdFHCW0jw7QEbLRGaqja9RATUTRI3pGr+jNeDJejHfjY9q6YBQze+gPjM8f4YCZJA==</latexit><latexit sha1_base64="Z2L8jDbJBggcFCju41lRFa/gD/g=">AAACCXicbVDLSsNAFJ34rPUVdelmsAhuLElRdFnUhcsK9gFNLJPpTTt08nBmIpSQrRt/xY0LRdz6B+78GydtFtp64MLhnHu59x4v5kwqy/o2FhaXlldWS2vl9Y3NrW1zZ7clo0RQaNKIR6LjEQmchdBUTHHoxAJI4HFoe6PL3G8/gJAsCm/VOAY3IIOQ+YwSpaWeiZ2AqCElPL3KHA73+BQ7igUgsW3dpce1rGdWrKo1AZ4ndkEqqECjZ345/YgmAYSKciJl17Zi5aZEKEY5ZGUnkRATOiID6GoaEr3MTSefZPhQK33sR0JXqPBE/T2RkkDKceDpzvxuOevl4n9eN1H+uZuyME4UhHS6yE84VhHOY8F9JoAqPtaEUMH0rZgOiSBU6fDKOgR79uV50qpVbatq35xU6hdFHCW0jw7QEbLRGaqja9RATUTRI3pGr+jNeDJejHfjY9q6YBQze+gPjM8f4YCZJA==</latexit>

Pes � 4⇥ 105<latexit sha1_base64="WAmECsaWfBXGPS6M4bM0/LOZDQo=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWARXJVEKrosunFZwT6giWEyvWmHTiZxZiKUUDf+ihsXirj1L9z5N04fC209cOFwzr3ce0+Ycqa043xbhaXlldW14nppY3Nre8fe3WuqJJMUGjThiWyHRAFnAhqaaQ7tVAKJQw6tcHA19lsPIBVLxK0epuDHpCdYxCjRRgrsgzoEyuvBPa5iT7MYFHadu/xsFNhlp+JMgBeJOyNlNEM9sL+8bkKzGISmnCjVcZ1U+zmRmlEOo5KXKUgJHZAedAwVxOzy88kHI3xslC6OEmlKaDxRf0/kJFZqGIemMya6r+a9sfif18l0dOHnTKSZBkGni6KMY53gcRy4yyRQzYeGECqZuRXTPpGEahNayYTgzr+8SJqnFdepuDfVcu1yFkcRHaIjdIJcdI5q6BrVUQNR9Iie0St6s56sF+vd+pi2FqzZzD76A+vzB6kZlbk=</latexit><latexit sha1_base64="WAmECsaWfBXGPS6M4bM0/LOZDQo=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWARXJVEKrosunFZwT6giWEyvWmHTiZxZiKUUDf+ihsXirj1L9z5N04fC209cOFwzr3ce0+Ycqa043xbhaXlldW14nppY3Nre8fe3WuqJJMUGjThiWyHRAFnAhqaaQ7tVAKJQw6tcHA19lsPIBVLxK0epuDHpCdYxCjRRgrsgzoEyuvBPa5iT7MYFHadu/xsFNhlp+JMgBeJOyNlNEM9sL+8bkKzGISmnCjVcZ1U+zmRmlEOo5KXKUgJHZAedAwVxOzy88kHI3xslC6OEmlKaDxRf0/kJFZqGIemMya6r+a9sfif18l0dOHnTKSZBkGni6KMY53gcRy4yyRQzYeGECqZuRXTPpGEahNayYTgzr+8SJqnFdepuDfVcu1yFkcRHaIjdIJcdI5q6BrVUQNR9Iie0St6s56sF+vd+pi2FqzZzD76A+vzB6kZlbk=</latexit><latexit sha1_base64="WAmECsaWfBXGPS6M4bM0/LOZDQo=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWARXJVEKrosunFZwT6giWEyvWmHTiZxZiKUUDf+ihsXirj1L9z5N04fC209cOFwzr3ce0+Ycqa043xbhaXlldW14nppY3Nre8fe3WuqJJMUGjThiWyHRAFnAhqaaQ7tVAKJQw6tcHA19lsPIBVLxK0epuDHpCdYxCjRRgrsgzoEyuvBPa5iT7MYFHadu/xsFNhlp+JMgBeJOyNlNEM9sL+8bkKzGISmnCjVcZ1U+zmRmlEOo5KXKUgJHZAedAwVxOzy88kHI3xslC6OEmlKaDxRf0/kJFZqGIemMya6r+a9sfif18l0dOHnTKSZBkGni6KMY53gcRy4yyRQzYeGECqZuRXTPpGEahNayYTgzr+8SJqnFdepuDfVcu1yFkcRHaIjdIJcdI5q6BrVUQNR9Iie0St6s56sF+vd+pi2FqzZzD76A+vzB6kZlbk=</latexit><latexit sha1_base64="WAmECsaWfBXGPS6M4bM0/LOZDQo=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWARXJVEKrosunFZwT6giWEyvWmHTiZxZiKUUDf+ihsXirj1L9z5N04fC209cOFwzr3ce0+Ycqa043xbhaXlldW14nppY3Nre8fe3WuqJJMUGjThiWyHRAFnAhqaaQ7tVAKJQw6tcHA19lsPIBVLxK0epuDHpCdYxCjRRgrsgzoEyuvBPa5iT7MYFHadu/xsFNhlp+JMgBeJOyNlNEM9sL+8bkKzGISmnCjVcZ1U+zmRmlEOo5KXKUgJHZAedAwVxOzy88kHI3xslC6OEmlKaDxRf0/kJFZqGIemMya6r+a9sfif18l0dOHnTKSZBkGni6KMY53gcRy4yyRQzYeGECqZuRXTPpGEahNayYTgzr+8SJqnFdepuDfVcu1yFkcRHaIjdIJcdI5q6BrVUQNR9Iie0St6s56sF+vd+pi2FqzZzD76A+vzB6kZlbk=</latexit>

Modelling the uniform regime: velocity field

Bulk

• Navier-Stokes for incompressible fluid

Assumptions

Expansion solution of the surfactant-free Stokes problem,coefficients computed numerically (Lauga & Stone JFM, 2003)

Solution

l/2

x

y

1

g

u

Julien Landel, U. Manchester UK

• Stokes flow:

• Uniform Marangoni shear at interface:

Modelling the uniform regime: surfactant Marangoni shear

Full transport problem coupled with the Stokes flow solution and the surfactant kinetics

l/2

x

y

1

g

u

Julien Landel, U. Manchester UK

Assumptions

� ⌧ 1• small surfactant concentration:

• linearisation of transport equations

Solution

�Ma ⇡ �2F(g, s)k⇤

g

2F(g, s)k⇤

g+

1

Pes+

1

2

✓1

2g+

1

2

◆2 Bi

1 + BiPe�g �

!�1

from Stokes flow Note: k⇤ = kMaMarangoni weighted normalised bulk concentration

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1|γMa|

10-5

100

105

|λtheory−λda

ta|/|λ

data| With Surfactant

No surfactant

Slip length in uniform regime

Julien Landel, U. Manchester UK

Free slip interface

No slip interface

10% error

�Ma ⇡ �2F(g, s)k⇤

g

2F(g, s)k⇤

g+

1

Pes+

1

2

✓1

2g+

1

2

◆2 Bi

1 + BiPe�g �

!�1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1|γMa|

10-5

100

105

|λtheory−λda

ta|/|λ

data| With Surfactant

No surfactant

Slip length in uniform regime

Julien Landel, U. Manchester UK

Free slip interface

No slip interface

10% error

�Ma ⇡ �2F(g, s)k⇤

g

2F(g, s)k⇤

g+

1

Pes+

1

2

✓1

2g+

1

2

◆2 Bi

1 + BiPe�g �

!�1

Conclusion

Julien Landel, U. Manchester UK

•Surfactants can reduce significantly slip and drag reduction for SHS

•In 2D, 2 regimes observed: •uniform •stagnant cap

•How to reduce surfactant Marangoni effects: • increase grating length •(reduce surfactant concentration?) •no stagnation point: e.g. circular lanes •3D geometries: pillars, rough surfaces?

scaling model