This document is published at:
Martínez-Calvo, A., Moreno-Boza, D., Sevilla, A.
(2020). The effect of wall slip on the dewetting of
ultrathin films on solid substrates: linear instability
and second-order lubrication theory. Physics of
Fluids, 32(10), 102107.
DOI: https://doi.org/10.1063/5.0028105
“This article may be downloaded for personal use only.
Any other use requires prior permission of the author
and AIP Publishing”.
© 2020 Author(s). Published under license by AIP
Publishing.
Phys. Fluids 32, 102107 (2020); https://doi.org/10.1063/5.0028105 32, 102107
© 2020 Author(s).
The effect of wall slip on the dewetting of
ultrathin films on solid substrates: Linear
instability and second-order lubrication
theory
Cite as: Phys. Fluids 32, 102107 (2020); https://doi.org/10.1063/5.0028105
Submitted: 02 September 2020 . Accepted: 30 September 2020 . Published Online: 15 October 2020
A. Martínez-Calvo , D. Moreno-Boza , and A. Sevilla
Physics of Fluids
ARTICLE
scitation.org/journal/phf
The effect of wall slip on the dewetting
of ultrathin films on solid substrates: Linear
instability and second-order lubrication theory
Cite as: Phys. Fluids 32, 102107 (2020); doi: 10.1063/5.0028105
Submitted: 2 September 2020 • Accepted: 30 September 2020 •
Published Online: 15 October 2020
A. Martínez-Calvo,
a)
D. Moreno-Boza,
b)
and A. Sevilla
c)
AFFILIATIONS
Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid,
Ave. Universidad 30, 28911 Leganés (Madrid), Spain
a)
Electronic mail: amcalvo@ing.uc3m.es
b)
Author to whom correspondence should be addressed: damoreno@pa.uc3m.es
c)
Electronic mail: asevilla@ing.uc3m.es
ABSTRACT
The influence of wall slip on the instability of a non-wetting liquid film placed on a solid substrate is analyzed in the limit of negligible inertia.
In particular, we focus on the stability properties of the film, comparing the performance of the three lubrication models available in the
literature, namely, the weak, intermediate, and strong slip models, with the Stokes equations. Since none of the aforementioned leading-order
lubrication models is shown to be able to predict the growth rate of perturbations for the whole range of slipping lengths, we develop a
parabolic model able to accurately predict the linear dynamics of the film for arbitrary slip lengths.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0028105
.,
s
I. INTRODUCTION
Liquid films play a central role in many engineering applica-
tions, in biological and physiological processes, and in geophysics,
to name a few. Apart from the fascination they hold for theoreticians
for their rich dynamics, their great practical importance is evidenced
by the existence of extensive reviews covering a large number of fun-
damental and applied studies,
1–4
to which the reader is referred to
for a panoramic view of the subject. Many systems, such as plas-
monic devices or biofluids as in the case of a tear film, involve
coatings in the form of ultra-thin liquid films, which can be unsta-
ble if their height is below about 100 nm. Indeed, at these scales,
the long-range van der Waals (vdW) forces can exceed the stabi-
lizing surface tension force if the perturbation wavelength is above
a certain cutoff.
5–7
Many relevant applications involve the use of
thin polymer films such as silicone oils, which are known to expe-
rience a substantial slip when they flow over a solid impermeable
substrate.
8–10
In these cases, the success of continuum mechanics
to account for the observed phenomena requires the use of a slip
length λ
s
, defined as the distance to the wall at which the tangen-
tial velocity extrapolates to zero. The most commonly used wall-slip
model is the linear boundary condition first derived by Navier,
11
and
later on by Maxwell in the case of gases.
12
Slippage is important
in many fields, ranging from microfluidics
13
to fracture and geo-
physical flows, or industrial flows involving polymer melt extruders,
where slip-induced instabilities frequently occur
14
(see Refs. 15 and
16 for a review).
We report the influence of wall slip on vdW-unstable liquid
films, contemplating their linear stability. Since slippage is often
found in the flow of polymeric films, the effects of inertia will be
neglected herein, rendering the flow effectively Stokesian. More than
40 years ago, de Gennes
8
conjectured that the no-slip boundary con-
dition at a rigid solid in contact with a semi-dilute or concentrated
high-molecular-weight polymer solution may not apply. He argued
that for any shear rate, under the ideal conditions of a perfectly
smooth non-adsorbing wall, the monomers do not create strong
binds with the solid, and thus, viscous forces in the liquid domi-
nate over the friction with the substrate, leading to slip lengths as
large as λ
s
∼100 μm. Even larger slip lengths of λ
s
∼1 cm can be
achieved by covering the substrate with a lubricant. Under semi-ideal
conditions, it was shown a decade later both theoretically
9
and exper-
imentally
10,17
that the slip length in polymer melt flows depends on
Phys. Fluids 32, 102107 (2020); doi: 10.1063/5.0028105 32, 102107-1
Published under license by AIP Publishing
Physics of Fluids
ARTICLE
scitation.org/journal/phf
the interaction between the monomers and the solid and on the
shear stress acting on the liquid. In particular, in this weakly grafted
regime, usually known as the mushroom regime, where bounded
polymer chains do not overlap each other, it was shown by the lat-
ter authors that the slippage is strongly reduced (λ
s
∼10 nm) due
to the local binding of chains. However, as the shear rate increases,
the slip velocity undergoes a sharp transition due to the coil-stretch
transition of the polymer blobs, whose elongation reduces the fric-
tion, inducing an increase in the slip length. Non-ideal conditions
occur when the chains are bound to many locations along the sub-
strate, as in the case of a strong brush, and where the slippage is
almost completely suppressed.
18
Therefore, for thick films such that
λ
s
/h
o
≪1, where h
o
is the initial height of the film, a pressure gradi-
ent induces the standard semi-parabolic velocity profile. In contrast,
for ultra-thin films where λ
s
/h
o
≫1, a plug-flow velocity profile is
expected.
To date, most of the theoretical and numerical studies of the
dynamics of non-wetting ultra-thin liquid films rely on leading-
order lubrication models. However, it has been recently shown that
the leading-order no-slip lubrication model fails to predict the near-
rupture dynamics of ultra-thin films.
19
In the case of slipping films,
there are three lubrication models available in the literature, namely,
the weak,
20
intermediate,
21
and strong
22
slip models. In Ref. 22,
the dispersion relation of a slipping vdW-unstable liquid film was
deduced from the complete Navier–Stokes equations. However, it
was not used to quantify the error of the two lubrication models
available at that time. Here, we show that none of the leading-order
models can cover the whole range of slip lengths, from the no-slip
limit, λ
s
/h
o
≪1, to the opposite limit of a free film, λ
s
/h
o
≫1. Indeed,
the dispersion relations derived from the leading-order models fail at
predicting the linear growth rates of small disturbances for arbitrary
values of λ
s
/h
o
. Specifically, we will show that the strong-slip model
(SSM) provides a reasonably accurate approximation for λ
s
/h
o
≳10
and any value of h
o
, whereas the validity of the weak-slip model
(WSM) is restricted to λ
s
/h
o
≲0.1, provided that h
o
/a ≳4, where a
is the molecular length defined below. Finally, the intermediate-slip
model (ISM) is only valid within a narrow range of λ
s
/h
o
. It should
be emphasized that these three models are still actively employed
to describe the linear and nonlinear dynamics of the liquid film, for
instance in a recent numerical study,
23
where the intermediate lubri-
cation model is used, to explain the instability in slipping dewetting
rims
24,25
or in numerical and experimental dip-coating studies with
porous substrates, where the strong-slip model is used.
26
In view of these limitations and given the present rele-
vance of thin-film flows, the need to develop accurate lubrication
approximations naturally arises. Thus, one of the main contribu-
tions of the present work is to develop a second-order lubrication
model able to accurately predict the linear stability properties of
the slipping film for arbitrary values of λ
s
/h
o
. Our development is
inspired by previous studies dealing with second-order lubrication
theory in similar contexts such as falling films
27–30
and axisymmetric
liquid threads.
31–33
This paper is organized as follows. In Sec. II, we present an exact
continuum formulation of the flow in terms of the Stokes equations
subject to a Navier slip condition at the substrate. In Sec. III, we
present the three leading-order lubrication models developed in pre-
vious studies, followed by the development of a novel second-order
lubrication approximation in Sec. IV. A detailed derivation of the
parabolic lubrication model is presented in the Appendix.
II. THE STOKES EQUATIONS
Figure 1 shows a schematic of the flow configuration under
study, together with the definition of the main governing variables.
The flow equations are made dimensionless using the unperturbed
liquid film height h
o
as the length scale, the vdW-induced velocity
A(6πμh
2
o
)as the velocity scale, their ratio 6πμh
3
o
A as the time scale,
and the characteristic disjoining pressure A(6πh
3
o
)as the pressure
scale, where μ is the liquid viscosity and A is the Hamaker constant.
34
Neglecting the inertia of the liquid, the flow is governed by the Stokes
equations,
∇ ⋅u =0, and 0 =−∇ϕ + ∇ ⋅T, (1)
where u = (u, v) is the two-dimensional fluid velocity field in Carte-
sian coordinates (x, y), T = −pI + ∇u + (∇u)
T
is the stress tensor
of an incompressible Newtonian fluid, p is the pressure field, and
ϕ = h
−3
is the dimensionless vdW potential. The accompanying
boundary conditions include the Navier slip
11,12
and no penetration
conditions,
u =ℓ
s
∂u
∂y
, v =0, (2)
at the substrate wall y = 0, where the non-dimensional parameter ℓ
s
= λ
s
/h
o
compares the slip length, λ
s
, with the initial height of the film,
h
o
. The stress balance and the kinematic boundary condition,
T ⋅n + Ca
−1
(∇ ⋅n)n =0, n ⋅(∂
t
x
s
−u)=0, (3)
are imposed at the interface y = h(x, t), where Ca = A(6πσh
2
o
)
=(ah
o
)
2
is the Capillary number, a = [A/(6πσ)]
1/2
is the molec-
ular length scale,
1
σ is the liquid–air surface tension coefficient,
FIG. 1. Schematic of the flow configuration with an accompanying system of reference. Note that the complete streamwise domain −π/k < x < π/k is not fully represented in
the sketch.
Phys. Fluids 32, 102107 (2020); doi: 10.1063/5.0028105 32, 102107-2
Published under license by AIP Publishing
Physics of Fluids
ARTICLE
scitation.org/journal/phf
x
s
= (x, h(x, t)) is the parameterization of the free surface, and ∇ ⋅n
= C = −∂
2
x
h[1 + (∂
x
h)
2
]
−3/2
is the mean curvature of the inter-
face, with associated unit normal vector n. The flow is thus governed
by two dimensionless parameters, namely, Ca (or equivalently h
o
/a)
and ℓ
s
.
III. LEADING-ORDER LUBRICATION MODELS
Let us now present the three leading-order lubrication models
developed in previous studies. These simplified models took advan-
tage of the existence of three possible dominant balances in the slen-
der flow of liquid films, depending on the slip length. Specifically,
these are the weak, intermediate, and strong slip limits, derived in
Refs. 20, 21, and 22, respectively.
The weak-slip model holds when p = ϕ = O(ε
−1
) and ℓ
s
= O(1),
where ε ≪1 measures the slenderness of the film, thus being the
effect of a slip small correction to the flow driven by vdW forces.
This scaling yields the classical lubrication balance leading to a semi-
parabolic profile of an axial velocity, accommodating the slip condi-
tion at the substrate and the stress-free condition at the interface. In
this case, the evolution of the thin film is described by the following
weak-slip equation for h(x, t):
20
∂
t
h −∂
x
h
2
(h + 3ℓ
s
)
3
∂
x
Ca
−1
C + h
−3
=0, (4)
where the standard no-slip lubrication model is recovered in the reg-
ular limit ℓ
s
→0,
4,35
which has been used extensively in thin-film
problems.
2,4,36,37
The assumption of a parabolic axial velocity profile fails when
ℓ
s
≳1 due to a change in the dominant balance. In particular, the
strong-slip limit ℓ
s
≫1 has associated characteristic scales p = ϕ
= O(ε) and ℓ
s
= O(ε
−2
), thus the flow being driven by the slip veloc-
ity at the substrate, which, at a leading order, provide the strong-slip
equations describing the coupled evolution of h(x, t) together with
the plug-flow velocity u(x, t),
∂
t
h + ∂
x
(hu)=0, (5a)
4
h
∂
x
(h∂
x
u)−∂
x
Ca
−1
C + h
−3
=
u
ℓ
s
h
, (5b)
which, in the limit ℓ
s
→∞, reduce to the free-film equations derived
in Ref. 38 and have been used in a myriad of relevant configura-
tions.
39–44
Finally, the intermediate-slip limit can be deduced from either
(4) or (5). Indeed, it was shown in Refs. 21 and 24 that it can be
obtained from (4) upon letting t →ℓ
−1
s
t and ℓ
s
→∞, or from (5)
taking u →ℓ
s
u and ℓ
s
→0, yielding the intermediate-slip equation,
∂
t
h −ℓ
s
∂
x
h
2
∂
x
Ca
−1
C + h
−3
=0, (6)
with a certain range of validity around ℓ
s
∼1 to be compared against
the other models below.
IV. SECOND-ORDER LUBRICATION THEORY
A large number of higher-order lubrication models have been
derived in the past to describe a wide variety of free-surface flows. In
particular, regular expansions in powers of the slenderness parame-
ter and weighted-residual approximations have been used to derive
second-order models for cylindrical liquid threads
31,33
and falling
liquid films,
27–30
respectively. However, to the authors’ knowledge,
there is no second-order lubrication model available in the literature
to describe the dynamics of ultra-thin liquid films on horizontal sub-
strates, neither in the standard no-slip case nor to account for wall
slip. To fill this gap, here, we present a second-order parabolic model,
which has O(ε
2
) accuracy.
The model, derived in detail in the Appendix, consists of three
coupled equations for the evolution of the film height, h(x, t), and
the leading- and second-order longitudinal velocities, u
0
(x, t) and
u
2
(x, t), respectively, and reads
∂
t
h + hu
0
1 +
h
2ℓ
s
+
h
2
u
2
3
′
=0, (7a)
−
Ca
−1
C + h
−3
′
+ 3u
′′
0
+ 2u
2
+
(h
2
u
0
)
′
hℓ
s
′
−
2(h
2
u
′
0
)
′
h
′
h
+
h
2
2
(u
′′
0
+ 2u
2
)
′
−2(h
2
u
2
)
′
′
=0, (7b)
u
′′
0
2
+ 3u
2
′′
−
3u
0
ℓ
s
h
3
+
3u
′′
0
h
2
+
12 h
′
u
′
0
h
3
−
6u
2
h
2
+
3 h
′2
u
0
ℓ
s
h
3
+
9u
′′
0
2ℓ
s
h
+
12 h
′
u
′
0
ℓ
s
h
2
−
3 h
′2
u
′′
0
h
2
+
6 h
′2
u
2
h
2
+
12 h
′
u
′
2
h
=0, (7c)
where primes indicate derivatives with respect to x. The accuracy of
the parabolic model (7) to account for the linear dynamics of the film
will be assessed below.
V. LINEAR STABILITY ANALYSIS
In this section, we revisit the stability of slipping ultra-thin liq-
uid films destabilized by the long-range vdW forces. Our aim is to
perform a systematic comparison of the dispersion relations pre-
dicted by using the different lubrication models presented in Secs. III
and IV with those obtained from the Stokes equations (1)–(3).
A. Stokes equations
To obtain the dispersion relation ω = ω(k) relating the lon-
gitudinal wavenumber k with the temporal growth rate ω, the
flow is decomposed into normal modes of the form (u, v, p, h)
= (0, 0, 1, 1)+ δ(
ˆ
u,
ˆ
v,
ˆ
p,
ˆ
h)exp(ikx + ωt), where hats denote the
eigenfunctions and δ ≪1 is the disturbance amplitude. Introducing
the normal-mode decomposition into (1)–(3) yields the following
closed-form solution:
ω =
3 −Ca
−1
k
2
2k
2kℓ
s
cosh(2k)+ sinh(2k)−2k(ℓ
s
+ 1)
1 + (4ℓ
s
+ 2)k
2
+ 2ℓ
s
k sinh(2k)+ cosh(2k)
, (8)
describing the temporal instability modes of the film for 0 <k <k
c
=
√
3Ca and any value of Ca >0. Equation (8) is a particular case
of the dispersion relation derived in Ref. 22 in the limit of negligible
Phys. Fluids 32, 102107 (2020); doi: 10.1063/5.0028105 32, 102107-3
Published under license by AIP Publishing
