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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
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