The wetting of thin films depends critically on the sign of the spreading coefficient S = γ S G − ( γ S L + γ L G ). We discuss the cases S < 0, S = 0, and S > 0 for transient models with contact line dissipation and find that the use of a dynamic contact angle solves problems for S > 0 that models might otherwise have. For initial data with a non-zero slope and S > 0, we show that there exists a finite time t p at which the contact angle of the thin film goes to zero. Then, a molecular precursor emerges from the thin film and moves outward at a constant velocity.

1.
A.
Oron
,
S. H.
Davis
, and
S. G.
Bankoff
, “
Long-scale evolution of thin liquid films
,”
Rev. Mod. Phys.
69
(
3
),
931
(
1997
).
2.
H.
von Helmholtz
,
Wissenschaftliche Abhandlungen
(
ETH-Bibliothek Zürich
,
1882
), Vol.
1
, pp.
223
.
3.
M.
Doi
, “
Onsager's variational principle in soft matter
,”
J. Phys.: Condens. Matter
23
(
28
),
284118
(
2011
).
4.
G. F.
Teletzke
,
Thin Liquid Films: Molecular Theory and Hydrodynamic Implications
(
University of Minnesota
,
1983
).
5.
T.
Qian
,
X.-P.
Wang
, and
P.
Sheng
, “
A variational approach to moving contact line hydrodynamics
,”
J. Fluid Mech.
564
,
333
360
(
2006
).
6.
U.
Thiele
,
D. V.
Todorova
, and
H.
Lopez
, “
Gradient dynamics description for films of mixtures and suspensions: Dewetting triggered by coupled film height and concentration fluctuations
,”
Phys. Rev. Lett.
111
(
11
),
117801
(
2013
).
7.
C.
Huh
and
L. E.
Scriven
, “
Hydrodynamic model of steady movement of a solid/liquid/fluid contact line
,”
J. Colloid Interface Sci.
35
(
1
),
85
101
(
1971
).
8.
D.
Bonn
,
J.
Eggers
,
J.
Indekeu
,
J.
Meunier
, and
E.
Rolley
, “
Wetting and spreading
,”
Rev. Mod. Phys.
81
(
2
),
739
(
2009
).
9.
O.
Voinov
, “
Inclination angles of the boundary in moving liquid layers
,”
J. Appl. Mech. Tech. Phys.
18
(
2
),
216
222
(
1977
).
10.
R.
Cox
, “
The dynamics of the spreading of liquids on a solid surface. Part 1. viscous flow
,”
J. Fluid Mech.
168
,
169
194
(
1986
).
11.
E.
Lauga
,
M.
Brenner
, and
H.
Stone
, “
Microfluidics: The no-slip boundary condition
,” in
Springer Handbooks
(
Springer
,
2007
), pp.
1219
1240
.
12.
D.
Ausserré
,
A.
Picard
, and
L.
Léger
, “
Existence and role of the precursor film in the spreading of polymer liquids
,”
Phys. Rev. Lett.
57
(
21
),
2671
(
1986
).
13.
M.
Boudoussier
, “
Dry spreading of polymer solutions
,”
J. Phys.
48
(
3
),
445
455
(
1987
).
14.
A.
Cazabat
,
S.
Gerdes
,
M.
Valignat
, and
S.
Villette
, “
Dynamics of wetting: From theory to experiment
,”
Interface Sci.
5
,
129
139
(
1997
).
15.
P.-G.
De Gennes
, “
Wetting: Statics and dynamics
,”
Rev. Mod. Phys.
57
(
3
),
827
(
1985
).
16.
M. N.
Popescu
,
G.
Oshanin
,
S.
Dietrich
, and
A.
Cazabat
, “
Precursor films in wetting phenomena
,”
J. Phys.: Condens. Matter
24
(
24
),
243102
(
2012
).
17.
W.
Ren
and
W.
E
, “
Boundary conditions for the moving contact line problem
,”
Phys. Fluids
19
(
2
),
022101
(
2007
).
18.
J. H.
Snoeijer
and
B.
Andreotti
, “
Moving contact lines: Scales, regimes, and dynamical transitions
,”
Annu. Rev. Fluid Mech.
45
,
269
292
(
2013
).
19.
T. S.
Chan
,
C.
Kamal
,
J. H.
Snoeijer
,
J. E.
Sprittles
, and
J.
Eggers
, “
Cox–Voinov theory with slip
,”
J. Fluid Mech.
900
,
A8
(
2020
).
20.
J.
Eggers
, “
Toward a description of contact line motion at higher capillary numbers
,”
Phys. Fluids
16
(
9
),
3491
3494
(
2004
).
21.
S.
Kalliadasis
and
H.-C.
Chang
, “
Apparent dynamic contact angle of an advancing gas–liquid meniscus
,”
Phys. Fluids
6
(
1
),
12
23
(
1994
).
22.
J. A.
Diez
and
L.
Kondic
, “
Computing three-dimensional thin film flows including contact lines
,”
J. Comput. Phys.
183
(
1
),
274
306
(
2002
).
23.
S. B. G.
O'Brien
and
L. W.
Schartz
, “
Theory and modeling of thin film flows
,”
Encycl. Surface Colloid Sci.
1
,
5283
5297
(
2002
).
24.
L.
Zhornitskaya
and
A. L.
Bertozzi
, “
Positivity-preserving numerical schemes for lubrication-type equations
,”
SIAM J. Numer. Anal.
37
(
2
),
523
555
(
1999
).
25.
G.
Grün
and
M.
Rumpf
, “
Nonnegativity preserving convergent schemes for the thin film equation
,”
Numer. Math.
87
,
113
152
(
2000
).
26.
S.
Kalliadasis
,
C.
Bielarz
, and
G.
Homsy
, “
Steady free-surface thin film flows over topography
,”
Phys. Fluids
12
(
8
),
1889
1898
(
2000
).
27.
U.
Thiele
,
L.
Brusch
,
M.
Bestehorn
, and
M.
Bär
, “
Modelling thin-film dewetting on structured substrates and templates: Bifurcation analysis and numerical simulations
,”
Eur. Phys. J. E
11
,
255
271
(
2003
).
28.
D.
Peschka
,
S.
Haefner
,
L.
Marquant
,
K.
Jacobs
,
A.
Münch
, and
B.
Wagner
, “
Signatures of slip in dewetting polymer films
,”
Proc. Natl. Acad. Sci. U. S. A.
116
(
19
),
9275
9284
(
2019
).
29.
R.
Seemann
,
S.
Herminghaus
, and
K.
Jacobs
, “
Dewetting patterns and molecular forces: A reconciliation
,”
Phys. Rev. Lett.
86
(
24
),
5534
(
2001
).
30.
F.
Otto
, “
Lubrication approximation with prescribed nonzero contact angle
,”
Commun. Partial Differential Equations
23
(
11–12
),
2077
2164
(
1998
).
31.
F.
Otto
,
T.
Rump
, and
D.
Slepcev
, “
Coarsening rates for a droplet model: Rigorous upper bounds
,”
SIAM J. Math. Anal.
38
(
2
),
503
529
(
2006
).
32.
S.
Jachalski
,
R.
Huth
,
G.
Kitavtsev
,
D.
Peschka
, and
B.
Wagner
, “
Stationary solutions of liquid two-layer thin-film models
,”
SIAM J. Appl. Math.
73
(
3
),
1183
1202
(
2013
).
33.
L.
Onsager
, “
Reciprocal relations in irreversible processes. I
,”
Phys. Rev.
37
(
4
),
405
(
1931
).
34.
M.-H.
Giga
,
A.
Kirshtein
, and
C.
Liu
, “
Variational modeling and complex fluids
,”
Handbook Mathematical Analysis in Mechanics of Viscous Fluids
(
Springer
,
2017
), pp.
1
41
.
35.
A.
Mielke
, “
A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems
,”
Nonlinearity
24
(
4
),
1329
(
2011
).
36.
L.
Giacomelli
and
F.
Otto
, “
Variatonal formulation for the lubrication approximation of the Hele-Shaw flow
,”
Calculus Var. Partial Differential Equations
13
(
3
),
377
403
(
2001
).
37.
X.
Xu
,
Y.
Di
, and
M.
Doi
, “
Variational method for liquids moving on a substrate
,”
Phys. Fluids
28
(
8
),
087101
(
2016
).
38.
A.
Janečka
and
M.
Pavelka
, “
Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics
,”
Continuum Mech. Thermodyn.
30
(
4
),
917
941
(
2018
).
39.
L.
Schmeller
and
D.
Peschka
, “
Gradient flows for coupling order parameters and mechanics
,”
SIAM J. Appl. Math.
83
(
1
),
225
253
(
2023
).
40.
D.
Peschka
, “
Variational approach to dynamic contact angles for thin films
,”
Phys. Fluids
30
(
8
),
082115
(
2018
).
41.
A.
Münch
,
B.
Wagner
, and
T. P.
Witelski
, “
Lubrication models with small to large slip lengths
,”
J. Eng. Math.
53
,
359
383
(
2005
).
42.
F.
Brochard
, “
Motions of droplets on solid surfaces induced by chemical or thermal gradients
,”
Langmuir
5
(
2
),
432
438
(
1989
).
43.
D.
Peschka
, “
Thin-film free boundary problems for partial wetting
,”
J. Comput. Phys.
295
,
770
778
(
2015
).
44.
D.
Peschka
and
L.
Heltai
, “
Model hierarchies and higher-order discretisation of time-dependent thin-film free boundary problems with dynamic contact angle
,”
J. Comput. Phys.
464
,
111325
(
2022
).
45.
D.
Peschka
, see https://github.com/dpeschka/thinfilm-freeboundary for “
Matlab Code thinfilm_clm_dual.m
” (
2023
).
46.
M.
Chiricotto
and
L.
Giacomelli
, “
Weak solutions to thin-film equations with contact-line friction
,”
Interfaces Free Boundaries
19
(
2
),
243
271
(
2017
).
47.
L.
Giacomelli
,
M. V.
Gnann
,
H.
Knüpfer
, and
F.
Otto
, “
Well-posedness for the Navier-slip thin-film equation in the case of complete wetting
,”
J. Differential Equations
257
(
1
),
15
81
(
2014
).
48.
H.
Knüpfer
, “
Well-posedness for the Navier slip thin-film equation in the case of partial wetting
,”
Commun. Pure Appl. Math.
64
(
9
),
1263
1296
(
2011
).
49.
L.
Giacomelli
,
M. V.
Gnann
, and
D.
Peschka
, “
Droplet motion with contact-line friction: Long-time asymptotics in complete wetting
,” arXiv:2302.03005 (
2023
).
50.
D.
Peschka
, “
Numerics of contact line motion for thin films
,”
IFAC-PapersOnLine
48
(
1
),
390
393
(
2015
).
51.
F.
Bernis
, “
Finite speed of propagation for thin viscous flows when
2 n < 3,”
C. R. Acad. Sci., Ser. I
322
(
12
),
1169
1174
(
1996
).
52.
J.
Hulshof
and
A. E.
Shishkov
et al, “
The thin film equation with 2 n < 3: Finite speed of propagation in terms of the l1-norm
,”
Adv. Differential Equations
3
(
5
),
625
642
(
1998
).
53.
R.
Dal Passo
,
L.
Giacomelli
, and
G.
Grün
, “
A waiting time phenomenon for thin film equations
,”
Annali Sc. Norm. Super. Pisa-Classe Sci.
30
(
2
),
437
463
(
2001
).
54.
J. F.
Blowey
,
J. R.
King
, and
S.
Langdon
, “
Small-and waiting-time behavior of the thin-film equation
,”
SIAM J. Appl. Math.
67
(
6
),
1776
1807
(
2007
).
55.
J.
Fischer
and
D.
Matthes
, “
The waiting time phenomenon in spatially discretized porous medium and thin film equations
,”
SIAM J. Numer. Anal.
59
(
1
),
60
87
(
2021
).
56.
H.
Yin
,
D. N.
Sibley
,
U.
Thiele
,
A. J.
Archer
et al, “
Films, layers, and droplets: The effect of near-wall fluid structure on spreading dynamics
,”
Phys. Rev. E
95
(
2
),
023104
(
2017
).
57.
F.
Brochard-Wyart
,
J. M.
Di Meglio
,
D.
Quére
, and
P. G.
De Gennes
, “
Spreading of nonvolatile liquids in a continuum picture
,”
Langmuir
7
(
2
),
335
338
(
1991
).
58.
For q = ( h , ω h ) and S > 0 the functional F ( q ) is not lower semicontinuous for h H 1 ( Ω ). Let q 0 = ( h , ω h ) such that | ω h | < | Ω | and define h ε ( x ) = h ( x ) + | ε | and q ε = ( h ε , ω h ε ), then ω h ε = Ω and thus F ( q ε ) F ( q 0 ) = S ( | ω h | | Ω | ) < 0. For certain models this property is problematic, as the evolution develops intrinsic instabilities when obtaining energy by enlarging the wet substrate ω at no cost. This is resolved by requireing from q ε = ( h ε , ω ε ) that h ε converges and ω ε converges as a set so that | ω ε | | ω |.
59.
R. G.
Badr
,
L.
Hauer
,
D.
Vollmer
, and
F.
Schmid
, “
Cloaking transition of droplets on lubricated brushes
,”
J. Phys. Chem. B
126
(
36
),
7047
7058
(
2022
).

Supplementary Material

You do not currently have access to this content.