Deformation of thin viscous liquid films exposed to a transverse thermal gradient results in Bénard–Marangoni instability, which would lead to the formation of micro- and nano-sized features. Linear and nonlinear analyses are performed to investigate the thermally induced pattern formation in shear thinning and shear thickening liquid films. The so-called thin film (TF) equation is re-derived to include viscosity variations using the power-law (PL) model. The characteristic wavelength for the growth of instabilities is found using a linear stability analysis of the PL-TF equation. A finite-difference-based discretization scheme and adaptive time step solver are used to solve the PL-TF equation for the nonlinear numerical model. The results show that the rheological property affects the timescale of the process and the size and final shape of the formed features. The fastest growth pillar reaching the top substrate in a shear thickening fluid is shorter than both the shear thinning and the Newtonian fluid cases. Moreover, morphological changes between patterns of shear thinning and shear thickening fluids are correlated with local viscosity variations. The number of formed pillars considerably increases with the increasing flow behavior index. The existing model also predicts the formation of pillars and bicontinuous structures at very low and high filling ratios.
References
1.
B. D.
Gates
, Q.
Xu
, M.
Stewart
, D.
Ryan
, C. G.
Willson
, and G. M.
Whitesides
, “New approaches to nanofabrication: Molding, printing, and other techniques
,” Chem. Rev.
105
, 1171
(2005
).2.
E.
Schäffer
, T.
Thurn-Albrecht
, T. P.
Russell
, and U.
Steiner
, “Electrohydrodynamic instabilities in polymer films
,” Europhys. Lett.
53
, 518
(2001
).3.
S. Y.
Chou
, L.
Zhuang
, and L.
Guo
, “Lithographically induced self-construction of polymer microstructures for resistless patterning
,” Appl. Phys. Lett.
75
, 1004
(1999
).4.
N.
Wu
and W. B.
Russel
, “Micro- and nano-patterns created via electrohydrodynamic instabilities
,” Nano Today
4
, 180
(2009
).5.
H.
Nazaripoor
, C. R.
Koch
, M.
Sadrzadeh
, and S.
Bhattacharjee
, “Compact micro/nano electrohydrodynamic patterning: sing a thin conductive film and a patterned template
,” Soft Matter
12
, 1074
(2016
).6.
E.
Albisetti
, D.
Petti
, M.
Pancaldi
, M.
Madami
, S.
Tacchi
, J.
Curtis
, W. P.
King
, A.
Papp
, G.
Csaba
, W.
Porod
, P.
Vavassori
, E.
Riedo
, and R.
Bertacco
, “Nanopatterning reconfigurable magnetic landscapes via thermally assisted scanning probe lithography
,” Nat. Nanotechnol.
11
, 545
(2016
).7.
J. P.
Singer
, “Thermocapillary approaches to the deliberate patterning of polymers
,” J. Polym. Sci., Part B: Polym. Phys.
55
, 1649
(2017
).8.
B.
Khorshidi
, B.
Soltannia
, T.
Thundat
, and M.
Sadrzadeh
, “Synthesis of thin film composite polyamide membranes: Effect of monohydric and polyhydric alcohol additives in aqueous solution
,” J. Membr. Sci.
523
, 336
(2017
).9.
B.
Soltannia
, M. A.
Islam
, J. Y.
Cho
, F.
Mohammadtabar
, R.
Wang
, V. A.
Piunova
, Z.
Almansoori
, M.
Rastgar
, A. J.
Myles
, Y. H.
La
, and M.
Sadrzadeh
, “Thermally stable core-shell star-shaped block copolymers for antifouling enhancement of water purification membranes
,” J. Memb. Sci.
598
, 117686
(2020
).10.
B.
Khorshidi
, S.
Shabani
, and M.
Sadrzadeh
, Nanocomposite Membranes for Water and Gas Separation
(Elsevier
, 2019
), pp. 257
–297
.11.
A. M.
Koupaei
, H.
Nazaripoor
, and M.
Sadrzadeh
, “Electrohydrodynamic patterning of polyethersulfone membranes
,” Langmuir
35
, 12139
(2019
).12.
D.
Bhattacharjee
, H.
Nazaripoor
, B.
Soltannia
, M. F.
Ismail
, and M.
Sadrzadeh
, “An experimental and numerical study of droplet spreading and imbibition on microporous membranes
,” Colloids Surf., A
615
, 126191
(2021
).13.
S.
Shabani
, B.
Khorshidi
, and M.
Sadrzadeh
, Nanocomposite Membranes for Water and Gas Separation
(Elsevier
, 2019
), pp. 219
–236
.14.
T.
Ambjörnsson
, M. A.
Lomholt
, and P. L.
Hansen
, “Applying a potential across a biomembrane: Electrostatic contribution to the bending rigidity and membrane instability
,” Phys. Rev. E
75
, 051916
(2007
).15.
J. T.
Schwalbe
, P. M.
Vlahovska
, and M. J.
Miksis
, “Lipid membrane instability driven by capacitive charging
,” Phys. Fluids
23
, 041701
(2011
).16.
W.
Huang
, X.
Yu
, Y.
Liu
, W.
Qiao
, and L.
Chen
, “A review of the scalable nano-manufacturing technology for flexible devices
,” Front. Mech. Eng.
12
, 99
(2017
).17.
J. M.
Davis
and S. M.
Troian
, “Influence of attractive van der Waals interactions on the optimal excitations in thermocapillary-driven spreading,” Phys. Rev. E
67
, 016308
(2003
).18.
M.
Dietzel
and S. M.
Troian
, Thermocapillary Patterning of Nanoscale Polymer Films
(Cambridge University Press
, 2009
), p. 1179
-BB08-02
.19.
H.
Nazaripoor
, C. R.
Koch
, M.
Sadrzadeh
, and S.
Bhattacharjee
, “Electrohydrodynamic patterning of ultra-thin ionic liquid films
,” Soft Matter
11
, 2193
(2015
).20.
S. Y.
Chou
and L.
Zhuang
, “Lithographically induced self-assembly of periodic polymer micropillar arrays
,” J. Vac. Sci. Technol. B
17
, 3197
(1999
).21.
E.
Schäffer
, S.
Harkema
, M.
Roerdink
, R.
Blossey
, and U.
Steiner
, “Morphological instability of a confined polymer film in a thermal gradient
,” Macromolecules
36
, 1645
(2003
).22.
E.
Schäffer
, S.
Harkema
, M.
Roerdink
, R.
Blossey
, and U.
Steiner
, “Thermomechanical lithography: Pattern replication using a temperature gradient driven instability
,” Adv. Mater.
15
, 514
(2003
).23.
E.
McLeod
, Y.
Liu
, and S. M.
Troian
, “Experimental verification of the formation mechanism for pillar arrays in nanofilms subject to large thermal gradients
,” Phys. Rev. Lett.
106
, 175501
(2011
).24.
H.
Nazaripoor
, M. R.
Flynn
, C. R.
Koch
, and M.
Sadrzadeh
, “Thermally induced interfacial instabilities and pattern formation in confined liquid nanofilms
,” Phys. Rev. E
98
, 043106
(2018
).25.
R. V.
Craster
and O. K.
Matar
, “Dynamics and stability of thin liquid films
,” Rev. Mod. Phys.
81
, 1131
(2009
).26.
K. R.
Fiedler
and S. M.
Troian
, “Early time instability in nanofilms exposed to a large transverse thermal gradient: Improved image and thermal analysis
,” J. Appl. Phys.
120
, 205303
(2016
).27.
M.
Dietzel
and S. M.
Troian
, “Mechanism for spontaneous growth of nanopillar arrays in ultrathin films subject to a thermal gradient
,” J. Appl. Phys.
108
, 074308
(2010
).28.
A.
Oron
, S. H.
Davis
, and S. G.
Bankoff
, “Long-scale evolution of thin liquid films
,” Rev. Mod. Phys.
69
, 931
(1997
).29.
A.
Oron
and P.
Rosenau
, “On a nonlinear thermocapillary effect in thin liquid layers
,” J. Fluid Mech.
273
, 361
(1994
).30.
H.
Nazaripoor
, C. R.
Koch
, and M.
Sadrzadeh
, “Ordered high aspect ratio nanopillar formation based on electrical and thermal reflowing of prepatterned thin films
,” J. Colloid Interface Sci.
530
, 312
(2018
).31.
A.
Mohammadtabar
, H.
Nazaripoor
, A.
Riad
, A.
Hemmati
, and M.
Sadrzadeh
, “A numerical study for thermocapillary induced patterning of thin liquid films
,” Phys. Fluids
32
, 024106
(2020
).32.
Y.
Alhendal
and A.
Turan
, “Thermocapillary bubble dynamics in a 2D axis swirl domain
,” Heat Mass Transfer
51
, 529
(2015
).33.
G.
Karapetsas
, N. T.
Chamakos
, and A. G.
Papathanasiou
, “Thermocapillary droplet actuation: Effect of solid structure and wettability
,” Langmuir
33
, 10838
(2017
).34.
A.
Mohammadtabar
, H.
Nazaripoor
, A.
Riad
, A.
Hemmati
, and M.
Sadrzadeh
, “Two-layer modeling of thermally induced Bénard convection in thin liquid films: Volume of fluid approach vs thin-film model
,” AIP Adv.
11
, 045317
(2021
).35.
36.
S.
Miladinova
, G.
Lebon
, and E.
Toshev
, “Thin-film flow of a power-law liquid falling down an inclined plate
,” J. Non-Newtonian Fluid Mech.
122
, 69
(2004
).37.
B. S.
Dandapat
and A.
Mukhopadhyay
, “Waves on the surface of a falling power-law fluid film
,” Int. J. Non Linear Mech.
38
, 21
(2003
).38.
R. S. R.
Gorla
, “Rupture of thin power-law liquid film on a cylinder
,” J. Appl. Mech. Trans. ASME
68
, 294
(2001
).39.
S. M.
Troian
, X. L.
Wu
, and S. A.
Safran
, “Fingering instability in thin wetting films
,” Phys. Rev. Lett.
62
, 1496
(1989
).40.
V.
Garg
, P. M.
Kamat
, C. R.
Anthony
, S. S.
Thete
, and O. A.
Basaran
, “Self-similar rupture of thin films of power-law fluids on a substrate
,” J. Fluid Mech.
826
, 455
(2017
).41.
R.
Sarma
and P. K.
Mondal
, “Marangoni instability in a heated viscoelastic liquid film: Long-wave versus short-wave perturbations
,” Phys. Rev. E
100
, 13103
(2019
).42.
K. X.
Hu
, M.
He
, and Q. S.
Chen
, “Instability of thermocapillary liquid layers for Oldroyd-B fluid,” Phys. Fluids
28
, 033105
(2016
).43.
K. X.
Hu
, M.
He
, Q. S.
Chen
, and R.
Liu
, “Linear stability of thermocapillary liquid layers of a shear-thinning fluid
,” Phys. Fluids
29
, 073101
(2017
).44.
M.
Iervolino
, J. P.
Pascal
, and A.
Vacca
, “Thermocapillary instabilities of a shear–thinning fluid falling over a porous layer
,” J. Non-Newtonian Fluid Mech.
270
, 36
(2019
).45.
M.
Naïmi
, M.
Hasnaoui
, and J. K.
Platten
, “Marangoni convection of non‐Newtonian power law fluids in a shallow rectangular cavity
,” Eng. Comput.
17
, 638
(2000
).46.
Z.
Alloui
and P.
Vasseur
, “Onset of Marangoni convection and multiple solutions in a power-law fluid layer under a zero gravity environment
,” Int. J. Heat Mass Transfer
58
, 43
(2013
).47.
C. H.
Chen
, “Marangoni effects on forced convection of power-law liquids in a thin film over a stretching surface,” Phys. Lett. A
370
, 51
(2007
).48.
I. M. R.
Sadiq
and R.
Usha
, “Long-wave instabilities in a non-Newtonian film on a nonuniformly heated inclined plane,” J. Fluids Eng. Trans. ASME
131
, 0312021
(2009
).49.
A.
Riad
, H.
Nazaripoor
, A.
Mohammadtabar
, and M.
Sadrzadeh
, “Electrohydrodynamic patterning of non-Newtonian thin films
” in 72nd Annual Meeting of the APS Division of Fluid Dynamics
[American Physical Society (APS), 2010], pp. Q34.004; available at https://ui.adsabs.harvard.edu/abs/2019APS..DFDQ34004R/abstract.50.
G.
Karapetsas
, N. T.
Chamakos
, and A. G.
Papathanasiou
, “Thermocapillary droplet actuation: Effect of solid structure and wettability
,” Langmuir 33
, 10838
(2017
).51.
G. L.
Wilkes
, “An overview of the basic rheological behavior of polymer fluids with an emphasis on polymer melts
,” J. Chem. Educ.
58
, 880
(1981
).52.
V. S.
Mitlin
, “Dewetting of solid surface: Analogy with spinodal decomposition
,” J. Colloid Interface Sci.
156
, 491
(1993
).53.
R.
Khanna
and A.
Sharma
, “Pattern formation in spontaneous dewetting of thin apolar films
,” J. Colloid Interface Sci.
195
, 42
(1997
).54.
W. H.
Vandevender
and K. H.
Haskell
, “The SLATEC mathematical subroutine library
,” ACM SIGNUM Newsl.
17
, 16
(1982
).55.
L. R.
Petzold
, “Description of DASSL: A differential/algebraic system solver
,” https://www.osti.gov/biblio/5882821-description-dassl-differential-algebraic-system-solver (1982
).56.
K. E.
Brenan
and L. R.
Petzold
, “The numerical solution of higher iindex differential/algebraic equations by implicit methods
,” SIAM J. Numer. Anal.
26
, 976
(1989
).57.
H.
Nazaripoor
, C. R.
Koch
, M.
Sadrzadeh
, and S.
Bhattacharjee
, “Thermo-electrohydrodynamic patterning in nanofilms
,” Langmuir
32
, 5776
(2016
).58.
E.
Schäffer
, S.
Harkema
, R.
Blossey
, and U.
Steiner
, “Temperature-gradient–induced instability in polymer films
,” Europhys. Lett.
60
, 255
(2002
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
