Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Phil. Trans. R. Soc. A (2008) 366, 1627–1647
doi:10.1098/rsta.2007.2176
Published online 11 January 2008
Meniscus and viscous forces during separation
of hydrophilic and hydrophobic smooth/rough
surfaces with symmetric and asymmetric
contact angles
B Y S HAOBIAO C AI *
AND
B HARAT B HUSHAN
Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM ),
201 West 19th Avenue, The Ohio State University, Columbus,
OH 43210-1142, USA
Adhesive or repulsive forces contributed by both meniscus and viscous forces can be
significant and become one of the main reliability issues when the contacting surfaces are
ultra smooth, and the normal load is small, as is common for micro/nano devices. In this
study, both meniscus and viscous forces during separation for smooth and rough
hydrophilic and hydrophobic surfaces are studied. The effects of separation distance,
initial meniscus height, separation time, contact angle and roughness are presented.
Meniscus force decreases with an increase of separation distance, whereas the viscous
force has an opposite trend. Both forces decrease with an increase of initial meniscus
height. An increase of separation time, initial meniscus height or a decrease of contact
angle leads to an increase of critical meniscus area at which both forces are equivalent.
An increase in contact angle leads to a decrease of attractive meniscus force but an
increase of repulsive meniscus force (attractive or repulsive dependent on hydrophilic or
hydrophobic surface, respectively). Contact angle has a limited effect on the viscous
force. For asymmetric contact angles, the magnitude of the meniscus force and the
critical meniscus area are in between the values for the two angles. An increase in the
number of surface asperities (roughness) leads to an increase of meniscus force; however,
its effect on viscous force is trivial. A slightly attractive force is observed for the
hydrophobic surface during the end stage of separation though the magnitude is small.
The study provides a fundamental understanding of the physics of the separation process
and it can be useful for control of the forces in nanotechnology applications.
Keywords: meniscus force; viscous force; contact angle; hydrophilic; hydrophobic
1. Introduction
Adhesion due to condensation of water from the ambient or the presence of a
thin liquid film on hydrophilic contacting surfaces has been studied extensively
in various biological and technological applications, such as the forces
* Author for correspondence ([email protected]).
One contribution of 7 to a Theme Issue ‘Nanotribology, nanomechanics and applications to
nanotechnology II’.
1627
This journal is q 2008 The Royal Society
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1628
S. Cai and B. Bhushan
developed for attachment by insects, spiders and lizards to various surfaces
(Gorb 2001), stiction of an atomic force microscope (AFM) tip to a sample
during interaction, magnetic storage devices, micro/nano devices and fuel
injectors in automobiles (Bhushan 1996, 1999, 2002, 2005, 2007). The primary
mechanism responsible for the adhesion/stiction is the formation of micro
menisci. The repulsion in the case of hydrophobic contacting surfaces has not
been studied to the best of our knowledge in the past even though this
phenomenon is also common in nature.
Menisci form around the contacting and near-contacting asperities due to
surface energy effects in the presence of a thin liquid film (Adamson 1990;
Israelachvili 1992; Bhushan 1999, 2002, 2003, 2005). Pendular rings are formed
on contacting asperities and liquid bridges are formed on near-contacting
asperities. (A small amount of liquid at the point of contact between the solid
surfaces is often called a pendular ring.) For two hydrophilic surfaces, concaveshaped menisci are formed, and for two hydrophobic surfaces convex-shaped
menisci are formed. For a hydrophilic surface, the lower pressure inside the
meniscus, that is, negative Laplace pressure, results in an intrinsic attractive
force, called the meniscus force, acting on the interfaces. For hydrophobic
surfaces, a repulsive meniscus force will act. When separation of two surfaces is
required, the viscosity of the liquid causes an additional attractive force, ratedependent viscous force during separation. Meniscus and viscous forces govern
the break of a meniscus bridge. The resultant force, adhesive or repulsive, is
highly dependent on the formed meniscus area, contact angles, number of
menisci, separation time, and surface tension and viscosity of the liquid.
For hydrophilic surfaces, many studies on meniscus forces have been carried
out based on identical contact angles with static contact configurations. Fisher
(1926) analysed the mean curvature of axisymmetric menisci and the volume of
trapped liquid and forces due to pendular rings between identical spheres by
using interpolation in the solutions to satisfy the boundary conditions. Woodrow
et al. (1961) solved the Laplace–Young equation under given initial conditions to
calculate the meniscus profile and meniscus forces between identical spheres. An
increase of adhesion force was generally observed when a thin liquid film was
introduced at the contact interface either through adsorption or by deposition.
Orr et al. (1975) solved the Laplace–Young equation for axisymmetric menisci
between a sphere and a flat surface. In their analysis, profiles of pendular rings
had been calculated and expressed in terms of elliptical integrals, and the
corresponding enclosed volumes and capillary forces were reported based on
static contact configurations. Surface roughness, properties of contacting solids,
such as layers, Young’s modulus, hardness and film thickness (RH), have also
been introduced in many studies. Tian & Bhushan (1996) studied the micromeniscus effect based on a multi-asperity contact model for homogeneous solids.
Peng & Bhushan (2001) and Cai & Bhushan (2006) studied the meniscus effects
for rough-layered contact models and examined the dependence of meniscus force
on layer properties.
During separation of two surfaces, both meniscus and viscous forces operate
inside the meniscus. The latter is significant especially when menisci have a
larger size and the separation time is short. Also, asymmetric contact angles
and division of meniscus (which are the real cases) can significantly affect the
magnitudes and behaviour of both meniscus and viscous forces during
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Meniscus and viscous forces
1629
separation. Though many studies have been carried out on meniscus forces and
meniscus profiles, studies on the separation process are rare. Fortes (1982),
Carter (1988), Gao (1997) and Stifter et al. (2000) investigated the meniscus
force–distance relationship which is one of the important relationships during
separation. Though the distance dependence of meniscus forces was well
presented, these studies were confined to a purely static meniscus force
analysis since no viscous force and separation time were considered. Chan &
Horn (1985) calculated the viscous force due to viscous dissipation for the case
of a sphere moving normally to a flat surface with some separation by
considering Reynolds’ lubrication equation. The force equation derived is
suitable for an infinite wetted region. Matthewson (1988) modified the viscous
force equation to be applicable to a finite wetted region, but his assumption of
an infinite break point leads to divergence when integrating over separation.
The equation derived by taking limit operation based on an infinite break
point is not accurate to estimate viscous force when the break of a meniscus
bridge occurs at a distance comparable to meniscus height. More recently,
Cai & Bhushan (2007a) developed a model to study the meniscus and viscous
forces during separation of two hydrophilic smooth surfaces with symmetric
contact angles. These models were further extended by Cai & Bhushan (2007b)
to investigate asymmetric contact angles by integrating a moving boundary
technique into the previous numerical simulation to capture the liquid–solid
interface differences between the two sides of a meniscus. It was found that
both meniscus and viscous forces are closely dependent on the separation
distance, initial meniscus height, separation time, surface tension and
viscosity. The contact angles significantly affect the break distance of a
meniscus. The magnitude of meniscus force can be largely reduced by choosing
proper asymmetric contact angles. ‘Force scaling’ effects are found to be true
for both meniscus and viscous forces when one larger meniscus is divided into a
large number of identical micro-menisci. Meniscus force is proportional to the
number of divisions whereas viscous force is proportional to the inverse of
the number of divisions (1/N ). During separation, either meniscus or viscous
force could be a dominant one at a given separation time. Viscous force
becomes dominant at a relatively larger meniscus area with larger initial
meniscus height. The results show that viscous force is more likely to become a
dominant force for a liquid with high viscosity at larger contact angles at given
conditions. Though comprehensive studies have been performed on meniscus
and viscous forces for hydrophilic surfaces, the study of the forces for
hydrophobic surfaces has not been carried out to the best of our knowledge.
The effect of hydrophobicity and hydrophilicity of surfaces (which is common
in nature and is seen frequently in applications) on the forces is still needed to
better understand their role and control of the forces.
In the present work, a comprehensive study of both meniscus and viscous
forces is carried out for smooth and rough hydrophilic and hydrophobic surfaces.
The role of these two forces is evaluated during a separation process. The effects
of the parameters, such as separation distance, initial meniscus height,
separation time and contact angles, as well as roughness are presented. The
study provides a fundamental understanding of the forces and it can be useful for
control and use of these forces in nanotechnology applications.
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1630
S. Cai and B. Bhushan
(a)
Fs = Fm + Fv
2
r1
liquid film
( c,
c)
h
1
n
(b)
2
r1
liquid film
1
( c,
c)
h
n
Figure 1. Schematic of the separation of two smooth surfaces: (a) hydrophilic and (b) hydrophobic
surfaces.
2. Approaches and analyses
The separation of two hydrophilic and hydrophobic smooth/rough surfaces with
meniscus bridges having symmetric/asymmetric contact angles is presented. The
two surfaces are assumed to be rigid. Concave and convex arc shapes are
assumed for meniscus curves for hydrophilic and hydrophobic surfaces,
respectively. The meniscus bridge is assumed to be in equilibrium, and the
liquid is incompressible. Thermal effects are considered to be negligible. Body
force and inertia of the liquid are neglected, which has been justified, for
example, by Cameron & McEttles (1981). The pressure is constant on a vertical
cross-section plane, whereas it varies along radial direction through the meniscus
bridge in the process of separation. Based on these assumptions, Reynolds’
lubrication theory is applicable during the separation. Since a separation is
usually done within a very short time, the evaporation of liquid is assumed to
be negligible.
(a ) Forces during separation due to meniscus
In the study we consider separation of two smooth and a rough against smooth
surfaces with a liquid film between as shown in figures 1 and 2a, respectively.
Figure 1a,b shows the configuration of liquid–solid interface for hydrophilic and
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Meniscus and viscous forces
1631
2R
(a)
2R
2R/n
(b)
single meniscus
multiple menisci
h
2R
2R/n
Figure 2. Schematic of (a) rough surface asperity distribution for NZ1!1 with single asperity
diameter 2R, and NZn!n with single asperity diameter 2R/n and (b) a smooth surface with 1 to
NZn!n menisci.
hydrophobic surfaces, respectively, and figure 2a shows the distribution of a
number of identical spherical asperities N on a flat surface with radius R for each
asperity. For the purpose of comparison, the separation of two smooth surfaces
with a number of identical menisci is also presented as shown in figure 2b.
Meniscus and viscous forces are present when separating two surfaces with
liquid-mediated contacts. Meniscus force Fm is contributed by both Laplace
pressure and the surface tension around the circumference. The magnitude of the
meniscus force depends on liquid properties and the size of the meniscus which
relates to the liquid volume and interface geometry. The strength of the viscous
force depends not only on the properties of the liquid and the size of the
meniscus, but also on the separation time and initial gap between the two bodies.
An external force which is larger than the resultant force is needed to make an
initial separation occur. During separation, if the meniscus force is larger than
the viscous force, then the externally applied force to separate two surfaces
depends on the meniscus forces, and vice versa. If the viscous force becomes
larger than the meniscus force, the meniscus will eventually break slowly even
without an increase in the applied force. However, the time taken to separate the
two surfaces would be long.
(i) Meniscus force
Meniscus forming between the two flat surfaces due to surface tension g results
in pressure difference Dp, which is often referred to as capillary or Laplace
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1632
S. Cai and B. Bhushan
Fs = Fm + Fv
R
h
H
R
2
h
d0
r1
D
( c,
c)
M
1
n
Figure 3. Schematic of the separation of a rough surface and a flat surface, and configurations of a
single asperity on a flat surface with a meniscus bridge formed between them.
pressure, and is given by the Laplace equation (Adamson 1990)
1
1
Dp Z g
C
:
r1 r2
ð2:1Þ
The pressure difference Dp can be negative or positive depending on the surface
properties. A hydrophilic surface results in a negative pressure difference,
whereas a hydrophobic surface leads to a positive one. In the equation, g is the
surface tension of liquid; r1 is the meniscus radius as shown in figures 1 and 3; and
r2 is another radius of the meniscus in the orthogonal plane to r1 (not shown in
the figures). 1/r1C1/r2 equals a so-called Kelvin radius r kK1 at equilibrium.
Based on thermodynamic law, r k can be obtained from the Kelvin equation
(Thomson 1870)
rk Z
gV
;
RT logðp=p 0 Þ
ð2:2Þ
where V is molar volume; R is the gas constant; T is the absolute temperature; p
is the ambient pressure; and p0 is the saturated vapour pressure at T. For the
case jr2 j[ jr1 j, r1Zr k and DpZg/r1. The meniscus force can be obtained by
integrating the Laplace pressure over the meniscus area and adding the surface
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Meniscus and viscous forces
1633
tension effect acting on the circumference of the interface (Fortes 1982)
ðð
g
Fm Z
dU C 2pgxn sin q1;2 ;
ð2:3Þ
U rk
where U is the meniscus area; xn is the meniscus radius; and q1,2 in the second
term on the r.h.s. corresponds to the contact angle of the liquid on the surfaces
being pulled. The consideration for this situation is that the two surfaces are
being pulled apart by an external force within a short time, thus the forces on
both sides of the liquid bridge can be different during the pull for asymmetric
contact angles. For initial meniscus radius xn0[r1, one may neglect the surface
tension contribution in equation (2.3) (second term on the right). If jr2 j[ jr1 j
does not hold, r2 may be replaced with the difference between x c, the centre
coordinate of the meniscus curve, and r1. Thus, r2 / xc K r1 (Stifter et al. 2000).
For given conditions, one can readily obtain r k using equation (2.2).
For the separation of two smooth flat surfaces with geometry configurations as
shown in figure 1, a meniscus height h can be calculated, hZr k(cos q1Ccos q2),
thus meniscus height is related to meniscus force. For a circular meniscus, the
meniscus force can be calculated using
px 2n gðcos q1 C cos q2 Þ
ð2:4Þ
C 2pgxn sin q1;2 :
h
For the separation of two rough surfaces (figure 2a) with geometry configurations
as shown in figure 3, the meniscus force is the superposition of the separation of a
number of single sphere-on-flat surface. The meniscus force for sphere-on-flat
surface can be calculated using the equation (Orr et al. 1975; Bhushan 2002)
1
1
2
Fm Z px ni g
C
C 2pRg sinðfÞsinðf C qÞ;
ð2:5Þ
ri xci K ri
where the index i represents the i th location of separation.
For the separation of two smooth flat surfaces with N number of identical
menisci, or separation of two rough surfaces with N number of identical spherical
asperities arbitrarily distributed on a flat surface without fully occupying the
total surface area, one can expect a maximum meniscus force
ðFm Þmax Z NFm ;
ð2:6Þ
where Fm is equation (2.4) for flat smooth surfaces and equation (2.5) for rough
surfaces, and this also applies to the case discussed below. For the case of N
number of menisci or the spherical asperities fully occupying the total surface
area, the maximum meniscus and viscous forces can be determined using
equation (2.4) or (2.5) with a proper radius x n or R n. Here, for instance, for a flat
surface with an area 2R!2R, given that the number of asperities is NZn!n, the
radius for each meniscus is x n/n for a flat smooth surface, and R/n for a single
asperity, the maximum meniscus force is
9
!
>
xn
>
ðFm Þmax Z NFm
for smooth surface; >
>
>
=
n
!
ð2:7Þ
>
>
R
>
for rough surface: >
ðFm Þmax Z NFm
>
;
n
Fm Z
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1634
S. Cai and B. Bhushan
(ii) Viscous force
Viscous force occurs due to the viscosity of the liquid when separating two
bodies within a short time. One may ignore viscous force for an infinitely long
separation time t s. However, an infinitely long separation time is not practically
feasible. Thus, characterization of the relevant viscous force is needed in order to
properly estimate the total force needed to separate two surfaces from a liquidmediated contact. In the derivation of the viscous force, we assume that
Reynolds’ lubrication theory applies to the process of separation. The pressure
inside the meniscus bridge consists of horizontal pressure gradients, whereas the
pressure is constant in any vertical plane inside a meniscus bridge, and at the
outside of a meniscus ring rZr b (liquid–air interfacial boundary, which is
exposed in ambient), pðr b ÞZ p b , the ambient pressure.
For separation of two smooth flat surfaces for a liquid with kinematic viscosity
h, the equation for the viscous force has been derived by Cai & Bhushan (2007a)
using Reynolds’ lubrication equation with cylindrical coordinate system for the
separation of two flat surfaces (e.g. Hocking 1973):
v
vh
3 vp
rh
Z 12hr
;
ð2:8Þ
vr
vr
vt
where h is the separation distance. Integrating the equation above and applying
the boundary condition, p(r b)Zpb, the pressure difference at arbitrary radius r
within a meniscus can be obtained as
vh
3h Dp Z 3 r 2 Kx 2n
:
ð2:9Þ
vt
h
From the above equation, one can obtain the maximum pressure difference
occurring at the centre of a meniscus, and the minimum on the outmost
boundary. An average pressure difference is one-half of the summation of the two:
3h
vh
Dpavg ZK 3 x 2n
:
ð2:10Þ
vt
2h
The viscous force thus may be calculated by using the average pressure
difference based on the above equation. Thus, the viscous force can be expressed
as
ðx n
Fv Z
2pDpavg r dr:
ð2:11Þ
0
By integrating the above equation, one can obtain
3phx 4n 1
1
Fv Z
;
ð2:12Þ
K
4t s
h 2s h 20
where h s is the break point and t s is the time to separate two bodies.
For the separation of a sphere and a flat surface, the viscous forces can be
determined using the lubrication equation for separation of a sphere and a flat
surface (Cai & Bhushan 2007a)
v
vp
3
_
rH ðrÞ
Z 12hr D;
ð2:13Þ
vr
vr
where D is separation, H(r) is meniscus height at radius r and H ðrÞZ r 2 =ð2RÞC D.
At the outside boundary rZr b, H ðr b ÞZ r 2b =ð2RÞC D and p(r b)Zp 0. Integrating
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Meniscus and viscous forces
1635
equation (2.13) and applying the boundary conditions gives
1
1
_
Dp Z 3hRD
K
:
ð2:14Þ
H 2 ðrÞ H 2 ðr b Þ
The viscous force can be found by integrating Dp over the meniscus area:
2
D
1 dD
2
:
ð2:15Þ
Fv Z 6phR 1K
H ðr b Þ D dt
H(r b) changes with separation and needs to be solved instantly. For R[r,
conservation of volume leads to
H 2 ðr b Þ Z H 2 ðr0 ÞKD 20 C D 2 ;
ð2:16Þ
where r0 and D0 are initial meniscus radius and gap, respectively. Substituting
equation (2.16) into equation (2.15) and integrating at both sides, the viscous force
can be obtained as
32
2
ðD s
1
D
7 1
6
Fv Z
6phR 2 41K qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5
dD;
ð2:17Þ
t s D0
H 2 ðr0 ÞKD 20 C D 2 D
where t s is separation time and Ds is the distance when separation occurs. The
separation occurs when a meniscus neck radius equals zero; further integrating
equation (2.17) gives
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
D s ½D 0 C H ðr 0 Þ H ðr 0 Þ2 KD 20 C D 2s
6phR
Fv Z
ln
ð2:18Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 :
ts
2
2
2
D 0 H ðr 0 Þ D C H ðr 0 Þ KD 0 C D s
For the separation of two smooth flat surfaces with N number of identical
menisci, or separation of two rough surfaces with N number of identical spherical
asperities arbitrarily distributed on a flat surface without fully occupying the total
surface area, one can expect a maximum viscous force
ðFv Þmax Z NFv ;
ð2:19Þ
where Fv is equation (2.12) for flat smooth surfaces and equation (2.18) for rough
surfaces, and this also applies to the case discussed below. For the case of N
number of menisci or the spherical asperities fully occupying the total surface area,
the maximum viscous force is
9
!
>
xn
>
ðFv Þmax Z NFv
for flat smooth surface; >
>
>
=
n
!
ð2:20Þ
>
>
R
>
>
ðFv Þmax Z NFv
for rough surface:
>
;
n
(b ) Meniscus curvature
It is well known that viscosity starts to drop above a certain shear stress and
the liquid becomes plastic and can only support a constant stress, known as the
limiting shear strength, at higher strain rates (Bhushan 1996). If we assume the
meniscus breaks at the break point, one may consider that point occurs at infinite
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1636
S. Cai and B. Bhushan
distance. However, this may lead to an overestimation of the real viscous force
since a meniscus bridge may break very quickly when it is small and the meniscus
radius is comparable to its height. Thus, the break point should be determined to
give a reasonable estimate of the viscous force. In this paper, the break distance
is assumed to be the distance corresponding to a zero meniscus neck thickness
during separation. The instant meniscus and viscous forces given in equation
(2.3) and (2.17) depend on the solving of the break point and meniscus radius xn
which in turn rely on initial and boundary conditions, and the instant meniscus
curvature needs to be calculated during the process of separation. The meniscus
profile can be found by solving the Laplace–Young equation as done by Orr et al.
(1975), who expressed the meniscus profile in terms of elliptical integrals. For
simplicity, a concave arc-shaped meniscus is assumed to account for meniscus
curve due to hydrophilic surfaces, and a convex arc-shaped curve for
hydrophobic surfaces. For a concave arc-shaped meniscus, we have applied a
simple approach to effectively capture the curvature during separation and avoid
the complexity of handling elliptic integrals (Cai & Bhushan 2007a). For a
hydrophobic surface, one may apply a similar approach with slight modifications
to characterize the meniscus curvature.
Let H and M denote the geometry shapes of the upper boundary and a
meniscus as shown in figure 3. H and M can be chosen as needed
H : y h Z yðxÞ;
ð2:21Þ
and if the meniscus shape is an arc, M may be expressed as
M : ðx K xc Þ2 C ðym K yc Þ2 Z r 2 :
The geometry configurations satisfy a set of boundary conditions
9
dy >
m
>
>
M :
Z tan q1 ;
>
>
dx >
>
ymZ0
>
>
>
>
for hydrophilic surface; yc Z r cosðq1 Þ; >
>
>
>
=
ð2:22Þ
ð2:23Þ
yc Z r cosðpKq1 Þ; >
>
>
>
"
!# >
>
>
>
dy m
dy h
>
>
H wM :
Z tan q2 C arctan
; >
>
>
dx
dx
>
>
;
ym Z y h :
One more condition is needed to fully constrain the problem and uniquely
determine the meniscus curvature instantly. For incompressible fluid, conservation of volume gives
ð2:24Þ
Vi Z V0 :
The volume can be found by integrating the whole area enclosed by H, M and the
two coordinate axes, and the magnitude of V0 can be calculated from initial
conditions. The instant values of xni , ri , xci and yci can be determined with the
boundary conditions and conservation of volume. Correspondingly, the instant
meniscus curvature and meniscus force can be calculated. For separation of two
parallel flat surfaces, one sets yhZh, otherwise, the shape function y(x) for H
should be defined. The case of yhZh corresponds to a constant separation speed.
for hydrophobic surface;
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Meniscus and viscous forces
1637
3. Results and discussion
Separation of two hydrophilic and hydrophobic surfaces relying on symmetric
and asymmetric contact angles with various initial meniscus heights is analysed
with respect to meniscus and viscous forces. The effects for a hydrophilic smooth
surface with symmetric and asymmetric contact angles have been studied by
Cai & Bhushan (2007a,b), and some results will be presented here for
comparison. The two forces are calculated based on the curvatures during
separation. For simulation purposes, we assume that a meniscus breaks at a zero
meniscus neck radius though this is not true practically and it may break at a
certain critical separation distance (Singh et al. 2006), and the forces disappear
at the break point. The force needed to overcome a meniscus force is the force
occurring at the beginning, whereas the force needed to overcome a viscous force
(due to viscosity) equals the force resulting from the break of a meniscus bridge
(which occurs at the break point). In the analyses for the separation of two flat
surfaces, the dimensionless meniscus and viscous forces are defined as follows to
eliminate effects of the contact angles due to surface property, and the liquid
surface tension and viscosity for the purpose of comparison:
1
F m Z
ð3:1Þ
F ;
pxn0 gðcos q1 C cos q2 Þ m
F v Z
4ts h 20
Fv :
3phx 4n0
ð3:2Þ
The effect of a rough surface based on the numerical model is also presented. In
the analyses, liquid bridges formed from water are evaluated. The meniscus
curvatures presented as examples are generated based on an initial h 0Z2 nm for
separating two surfaces. An initial meniscus radius xn0 of 100 nm is used in the
study except for the rough surface case which has a nominal area 100!100 mm2
with a larger initial meniscus height of 100 nm. The forces calculated during
separation are based on various initial meniscus heights from 2 to 6 nm. Contact
angles 08, 908 and 1808 corresponding to 0.0018, 89.9998 for hydrophilic and
90.0018 for hydrophobic surface, and 179.9998 respectively to avoid singularities.
The separation time ts used in the study is 0.1 ms related to the real separation
time of a diesel fuel injector.
(a ) Effects of contact angles for hydrophilic and hydrophobic surfaces
Figure 4 shows examples of instant meniscus curvatures during the separation
of two liquid-mediated surfaces for hydrophilic (figure 4a) and hydrophobic
surfaces (figure 4b). It is shown that for a given set of contact angles and a given
initial meniscus height, asymmetric contact angles lead to a faster break of
meniscus for both hydrophilic and hydrophobic surfaces. Hydrophobic surfaces
with asymmetric contact angles have the shortest break distance as compared to
the corresponding hydrophilic surfaces. This difference will eventually affect the
forces as we can see from the data to be presented later.
Figures 5 and 6 show dimensionless (identified with ) and dimensional meniscus
and viscous forces versus relative separation D (from initial distance h 0 to h 0CD) for
separating two parallel surfaces from various initial meniscus heights h 0Z2–6 nm
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1638
S. Cai and B. Bhushan
(a) 180
(i)
(ii)
150
D (nm)
moving upper
surface
moving upper
surface
120
90
60
fixed lower
surface
30
fixed lower
surface
0
(b) 180
(i)
(ii)
moving upper
surface
150
D (nm)
120
90
moving upper
surface
60
30
0
–120 –80
fixed lower
surface
fixed lower
surface
– 40
0
40
(nm)
80
120
–120
–80
–40
0
40
(nm)
80
120
Figure 4. Meniscus curvatures when separating two parallel flats with initial meniscus height
h 0Z2 nm, gZ72 mN mK1, xn0Z100 nm and contact angles (a(i)) q1Zq2Z608 and (ii) q1Z08,
q2Z608 and (b(i)) q1Zq2Z1208 and (ii) q1Z1808, q2Z1208.
with symmetric and asymmetric contact angles for both hydrophilic and
hydrophobic surfaces. The dimensionless figures (right column of figure 5, and
figure 6a,c) presented here are for the purpose of generalization and for general use
since the effects of liquid and surface properties have been eliminated. One can
obtain the appropriate force magnitude from these figures for various surfaces and
liquids by simply multiplying the surface tension, viscosity and contact angle effects.
Since the dimensionless and dimensional figures show the same trends, we will
mainly discuss the dimensional one for brevity.
For the case of hydrophilic surfaces, the meniscus forces are attractive. The
results show that the asymmetric contact angles (one of them kept fixed, here,
q2Z608) lead to a larger meniscus force, and the smaller the other contact angle,
the larger the meniscus force (figure 5). It is noted that the asymmetric contact
angles play a major role in the quick break of a meniscus. It is observed that for
q1Z08 and q2Z608 there is a smaller break distance as compared to the case
q1Zq2Z608. For a given set of asymmetric contact angles, the effect of initial
meniscus height h 0 on the break distance D is insignificant as shown in figure 6b.
For the case of hydrophobic surfaces, repulsive meniscus forces are observed in
general as shown in figure 5. A slight attractive force is observed at the later
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1639
Meniscus and viscous forces
Fm*
30 60 90 120 150 180
(i)
2 = 60°
meniscus height:
(2 –6 nm)
0
5
= 180°,
10
2
30 60 90 120 150 180
0
–0.5
meniscus height:
(2–6 nm)
1 = 2 = 120°
–2.0
1.0
2.0
0.5
1.5
0
–0.5
–1.0
0
20
40
60
meniscus height:
(2 – 6 nm)
1
0.030
0.015
1.5
1 = 2 = 60°
0
meniscus height:
1.0
(2–6 nm)
– 0.015
0.5
– 0.030
–1.5
(ii)
30
35
0.015
–0.015
0.5
–0.030
15
0
30
45
60
0
–0.5
meniscus height:
(2–6 nm)
=
1 180°, 2 = 120°
–1.5
25
0.030
1 = 0°, 2 = 60°
meniscus height:
1.0
(2–6 nm)
–1.0
= 120°
15 20
(nm)
Fm (×103 nN)
1 = 0°,
(ii)
–1.0
Fm*
(b) 60
50
40
30
20
10
0
–10
–20
–30
– 40
–50
– 60
Fm (×103 nN)
2.0
1.0
(a) 60 (i)
50
0.5
=
=
60°
1
2
40
0
meniscus height:
30
–0.5
(2 – 6 nm)
20
–1.0
10
0
0
–10
–20
meniscus height:
–30
(2 – 6 nm)
– 40
1 = 2 = 120°
–50
– 60
–2.0
0
5
10
15 20
(nm)
25
30
35
Figure 5. (a(i),b(i)) Dimensionless and (a(ii),b(ii)) dimensional meniscus forces versus separation for various initial meniscus heights (h 0)Z2–6 nm for a separation time t sZ0.1 ms with
gZ72 mN mK1, xn0Z100 nm, and contact angles (a) q1Zq2Z608 and q1Zq2Z1208 and (b) q1Z08,
q2Z608 and q1Z1808, q2Z1208. The insets show zoomed-in views.
stage of separation; however, the magnitude is small. The attractive effect
disappears if one of the contact angles equals 1808 (zoomed figures in figure 5a).
This is believed to be the effect of the second term on the r.h.s. of equation (2.4),
the force due to surface tension of the liquid on the circumference of the solid–
liquid interface. The results show that the asymmetric contact angles (one of
them kept fixed, here, q2 equals 1208) lead to a larger value of the absolute
meniscus force in magnitude, and the larger the other contact angle, the larger
the meniscus force (figure 5), which is different from the hydrophilic situation.
Again, it is observed that the asymmetric contact angles play a major role in the
quick break of a meniscus. q1Z1808 has a much smaller break distance as
compared to the other case. Also, the effects of a given set of asymmetric
contact angles on the break distance are more significant than an initial
meniscus height h 0, which has the same trend as for hydrophilic surfaces
(figure 6d ). An intersection is observed in figure 6c for q1Z1808 and q2Z1208.
This is due to the multiplication factors varying with various initial meniscus
heights. These observations may be useful for the design of travel distance of
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1640
S. Cai and B. Bhushan
(i) meniscus height: (2 – 6 nm)
separation speed:
1.78, 2.02, 2.20, 2.38, 2.52 m s–1
Fv*
1.0
(expanded scales)
0.5
(b) 10.0
Fv (× 10 2 nN)
(a) 1.5
0
(ii) meniscus height: (2 – 6 nm)
separation speed:
0.86, 0.98, 1.06, 1.13, 1.19 m s–1
Fv*
1.0
(expanded scales)
0.5
7.5
(ii)
separation speed:
0.86, 0.98, 1.06, 1.13, 1.19 m s–1
meniscus height
5.0
2 nm
2.5
3 nm
4 nm
5 nm
6 nm
Fv*
0.5
(d) 10.0
Fv (× 10 2 nN)
(i) meniscus height: (2 – 6 nm)
separation speed:
1.22, 1.38, 1.48, 1.56, 1.63 m s–1
(expanded scales)
0
7.5
(i)
separation speed:
1.22, 1.38, 1.48, 1.56, 1.63 m s–1
meniscus height
5.0
2 nm
2.5
3 nm
4 nm
5 nm
6 nm
0
Fv*
(expanded scales)
0.5
5
10 15 20 25 30 35
(nm)
10.0
Fv (× 10 2 nN)
(ii) meniscus height: (2 – 6 nm)
separation speed:
0.37, 0.42, 0.45, 0.48, 0.50 m s–1
1.0
0
3 nm
4 nm
5 nm
6 nm
2.5
0
1.0
1.5
2 nm
5.0
10.0
0
(c) 1.5
meniscus height
0
Fv (× 10 2 nN)
1.5
7.5
(i)
separation speed:
1.78, 2.02, 2.20, 2.38, 2.52 m s–1
7.5
(ii)
separation speed:
0.37, 0.42, 0.45, 0.48, 0.50 m s–1
meniscus height
5.0
2.5
0
2 nm
3 nm
4 nm
5 nm
6 nm
50
100 150
(nm)
200
250
Figure 6. For various initial meniscus heights (h 0)Z2–6 nm for a separation time t sZ0.1 ms with
hZ0.89 cSt, xn0Z100 nm, and contact angles (a(i)(ii)) dimensionless viscous force versus
separation for q1Zq2Z608 and q1Z0, q2Z608, (b(i)(ii)) dimensional viscous force versus separation
for q1Zq2Z608 and q1Z0, q2Z608, (c(i)(ii)) dimensionless viscous force versus separation for
q1Zq2Z1208 and q1Z1808, q2Z1208 and (d(i)(ii)) dimensional viscous force versus separation for
q1Zq2Z1208 and q1Z1808, q2Z1208.
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1641
Meniscus and viscous forces
(b) 2
2
2 nm
3 nm
4 nm
5 nm
6 nm
Fm (× 103 nN)
1
2
0
2
–1
= 120°
= 60°
1
Fv (× 103 nN)
(a)
–1
–2
2
2
1
90° <
0
0° <
–1
2
2
= 105
2 = 120
2 = 135
2 = 150
2 = 165
2 = 180
= 60°
2
= 120°
(d)
3
2
< 180°
< 90°
–2
2
Fv (× 103 nN)
=0
= 15
2 = 30
2 = 45
2 = 60
2 = 75
2 = 90
2
3
Fm (× 103 nN)
2
–2
(c)
–3
0
1
0
0° <
–1
2
< 90°
90° <
2
< 180°
–2
0
30
60
90 120
(deg.)
1
150
180
–3
0
30
60
90 120
(deg.)
1
150
180
Figure 7. (a) Meniscus and (b ) viscous forces for various initial meniscus heights (h 0)Z2–6 nm for
a separation time t sZ0.1 ms with gZ72 mN mK1, hZ0.89 cSt, xn0Z100 nm, at fixed contact angle
q2Z608 or q2Z1208 and various contact angles q1; (c ) meniscus and (d ) viscous forces at
fixed initial meniscus heights (h 0)Z2 nm for a separation time tsZ0.1 ms with gZ72 mN mK1,
hZ0.89 cSt, xn0Z100 nm, at various contact angles q1 and q2.
two surfaces to achieve optimal size of a device. For both hydrophilic and
hydrophobic surfaces, the effect of contact angle on viscous force is insignificant
as shown in figure 6b,d.
Figure 7 summarizes the effects of contact angles on both forces for both
hydrophilic and hydrophobic cases. The l.h.s.s of figure 7a,b show the effects on
meniscus and viscous forces for fixed q2 equals 608 and various q1 from 0 to 908.
The r.h.s.s of figure 7a,b show the effects to these forces for fixed q2 equals 1208
and various q1 from 90 to 1808 for a set of various h 0Z2–6 nm. It is observed that
the contact angles have a large effect on the absolute magnitudes of meniscus
forces for hydrophilic surfaces: the larger the q1, the smaller the meniscus forces.
And a larger q1 can also help decrease the effects of initial h 0 on the magnitude of
meniscus forces. For hydrophobic surfaces (r.h.s.s of figure 7a,b), the trends are
opposite. Though asymmetric contact angles largely affect meniscus forces, the
effects on viscous forces are trivial (figure 6b). As compared to figure 7a,b,
figure 7c,d shows the effects on both forces for various q1 and q2 from 0 to 1808 for
a fixed h 0Z2 nm. The results show that the increase of both or any one of the
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1642
S. Cai and B. Bhushan
(a) 103
(b)
critical meniscus area (µm2)
(i)
force (µN)
102
10
1
critical point
10 –1
meniscus height:
(2 – 6 nm)
10 –2
10 –3
103
(ii)
critical meniscus area (µm2)
force (µN)
10
1
critical point
10 –1
10 –3
meniscus height:
(2 – 6 nm)
0
1
10 –1
meniscus height:
(2–6 nm)
10 –2
10 –3
10 – 4
(ii)
10
102
10 –2
(i)
10
0.25
meniscus
0.50
area (µm2)
0.75
1
10–1
meniscus height:
(2–6 nm)
10–2
10 –3
10 – 4
0
0.25
0.50
ts (µs)
0.75
1.00
Figure 8. (a) Maximum meniscus and viscous forces versus meniscus area with separation time
tsZ0.1 ms, and (b) effect of separation time ts on critical meniscus area (at which meniscus force and
viscous force are comparable during separation of two parallel flats) for (a(i),b(i)) q1Zq2Z608
and (a(ii),b(ii)) q1Z08, q2Z608 at various initial meniscus heights h 0Z2–6 nm with gZ72 mN mK1
and h=0.89 cSt. Solid lines indicate Fm and dotted lines indicate Fv.
two contact angles leads to a noticeable decrease of the absolute magnitudes of
meniscus forces for the hydrophilic case as shown on the l.h.s. of figure 7c but an
increase of the absolute magnitudes of meniscus forces for the hydrophobic case
as shown on the r.h.s. of figure 7c. Again, the effects on viscous forces are small
(figure 7d ). From the analysis, an increase in contact angle leads to a decrease of
attractive meniscus force but an increase of repulsive meniscus force (attractive
or repulsive dependent on hydrophilic or hydrophobic surface, respectively). The
contact angle has limited effect on the viscous force. For asymmetric contact
angles, the magnitude of the meniscus force is in between the values for the
two angles.
The contact angle affects the critical meniscus area as well, as shown in
figure 8a,b. It is observed that a decrease of contact angle leads to an increase of
critical meniscus area. For the given contact angle sets, the asymmetric contact
angle pair leads the critical meniscus area to move to a larger value. This is
expected since the decrease of one of the contact angles results in a larger
meniscus force, and thus a larger meniscus area is needed for the viscous force to
match the meniscus force. For asymmetric contact angles, the critical meniscus
area is in between the values for the two angles.
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Meniscus and viscous forces
1643
(b ) Effects of separation distance, separation time and initial meniscus height
for hydrophilic and hydrophobic surfaces
For both hydrophilic and hydrophobic surfaces, the effects of separation to
meniscus and viscous forces can be observed in figures 5 and 6. Figure 5
shows dimensionless and dimensional meniscus forces versus relative separation D
for separating two parallel surfaces from various initial meniscus heights h 0Z2–6 nm.
The dimensionless one is for the purpose of generalization. Figure 6 shows
dimensionless and dimensional viscous forces versus relative separation D for
separating two parallel surfaces in the same conditions. Meniscus force decreases with
an increase of separation distance, whereas the viscous force has an opposite trend.
Both forces decrease with an increase of initial meniscus height. Attractive and
repulsive meniscus forces are observed for hydrophilic and hydrophobic surfaces,
respectively. In either case, both types of meniscus and viscous forces change rapidly
at the beginning. This trend is the same for both the dimensionless and dimensional
results. The larger rate of change in the force at the beginning of separation is due to
the larger change rate in volume at the beginning and becomes gradual thereafter
(figure 4), and a larger volume change rate leads to a relatively larger decrease of
pressure difference (separation at a constant speed).
Initial meniscus height h 0 affects both meniscus and viscous forces for either
hydrophilic or hydrophobic surface, which can be observed from figures 5 to 8. It
is shown that a lower meniscus height leads to a larger meniscus force (attractive
or repulsive) and viscous force. The increase of h 0 leads to a decrease of the
magnitudes of these forces. The dimensionless and dimensional results have the
same trend. This is because at a fixed meniscus area, a higher h 0 results in a
larger Kelvin radius and lower absolute pressure difference Dp, and thus lower
meniscus force (attractive or repulsive). It is no surprise to see this trend for
viscous force since it is the inverse of the square of the meniscus height. An
increase of initial meniscus height leads to the critical meniscus area to move to a
larger value since viscous force decreases much faster than meniscus force with
an increase of initial meniscus height. It is observed that h 0 plays a significant
role in the theoretical break point. Smaller h 0 leads to a quick break of the
meniscus. This is because the meniscus bridge with smaller h 0 has a smaller
liquid volume for a given initial meniscus area. Both meniscus and viscous forces
disappear at the break point. As compared to hydrophilic surfaces, the effect of
initial meniscus height h 0 to the break of meniscus becomes less significant for
hydrophobic surfaces (figure 6b,d ). The initial meniscus height affects the critical
meniscus area as well. An increase of initial meniscus height leads to an increase
of critical meniscus area. This is because viscous force increases faster with the
decrease of initial meniscus height than meniscus force.
Figure 8a shows the behaviours of each of the forces as a function of meniscus
area for various initial meniscus heights for contact angles q1Zq2Z608 and q1Z08,
q2Z608, and figure 8b shows the critical meniscus area as a function of separation
time t s for various initial meniscus heights for contact angles q1Zq2Z608 and
q1Z08, q2Z608. Since viscous force is a function of the inverse of separation time
t s, an increase of separation time, initial meniscus height leads to an increase of
critical meniscus area. For a given initial meniscus height and a separation time t s,
one can readily determine the dominating force from the figure during the
separation process, based on meniscus size information.
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1644
(a)
S. Cai and B. Bhushan
15
(i)
0.5
5
0
= 0°,
1 = 60°,
1 = 120°,
1 = 180°,
1
–5
–10
–15
(b)
15
1
2
2
2
2
= 60°
= 60°
= 120°
= 120°
= 0°,
1 = 60°,
1 = 120°,
0.4
(Fv) max (mN)
(Fm ) max (mN)
10
(ii)
1 = 180°,
0.3
2
= 60°
2
2
= 60°
= 120°
2
= 120°
(expanded scales)
0.2
0.1
0
(i)
15
(ii)
(Fv) max (mN)
(Fm ) max (mN)
10
5
0
–5
10
5
–10
–15
0.2
0.4
0.6
N (× 10 4)
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
N (× 10 4)
Figure 9. Maximum meniscus and viscous forces (a(i)(ii)) versus number of asperities N for a rough
surface and (b(i)(ii)) versus number of menisci N for a smooth surface for a surface area 2R!2RZ
100!100 mm2 at initial meniscus heights h 0Z100 nm for a separation time t sZ0.1 ms with
gZ72 mN mK1, hZ0.89 cSt, with various contact angles q1Zq2Z608, q1Z08, q2Z608, q1Zq2Z
1208 and q1Z1808, q2Z1208. The meanings of the symbols in (b ) are the same as those in (a ).
(c ) Roughness effects on meniscus and viscous forces
In the study of roughness effects, the number of asperities N used here ranges
from 1 to 104, and we assume that these surface asperities are identical and have
spherical shapes, and these asperities fully occupy the nominal flat area.
Figure 9a shows the effects of the number of asperities on meniscus and viscous
forces at different contact angles for both hydrophilic and hydrophobic rough
surfaces. It is observed that the increase in the number of asperities leads to an
increase of meniscus force (an increase of attractive meniscus force for
hydrophilic surfaces and an increase of repulsive meniscus force for hydrophobic
surfaces) for a given fixed nominal flat surface area (here 100!100 mm2). As
compared to meniscus force, the effect of the number of asperities on viscous
force is trivial for both hydrophilic and hydrophobic surfaces. A noticeable
decrease of viscous force is observed for q1Z1808 and q2Z1208. This is believed to
be due to the quick break of meniscus under the given condition.
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1645
Meniscus and viscous forces
Table 1. Summary of effect due to initial meniscus height and contact angle on meniscus and viscous
forces. (Meniscus height dominates the effect on viscous force as compared to contact angles.)
q1Zq2
Fm
q1sq2
q1 [ q2 [
q1 Y q2 Y
(a) meniscus forces for hydrophilic and hydrophobic surfaces
hydrophobic surface (attractive)
Y
[
n.a.
n.a.
fixed h 0
h0 [
Y
n.a.
n.a.
n.a.
h0 Y
n.a.
[
n.a.
n.a.
Y
Y
n.a.
[
n.a.
[
hydrophobic surface (repulsive)
[
Y
fixed h 0
n.a.
Y
h0 [
h0 Y
[
n.a.
n.a.
n.a.
n.a.
[
n.a.
[
Y
Y
n.a.
q1 Y q2 [
q1 [ q2 [
q1 Y q2 Y
(b) viscous forces for hydrophilic and hydrophobic surfaces
not much
not much not much
not much
fixed h 0
h0 [
Y
Y
Y
Y
h0 Y
[
[
[
[
not much
Y
[
not much
Y
[
[
Y
q1Zq2
Fv
[
q1 [ q2 Y
n.a.
n.a.
n.a.
q1 Y q2 [
q1sq2
Y
q1 [ q2 Y
For the purpose of comparison, meniscus and viscous forces for the separation
of two smooth hydrophilic and hydrophobic surfaces with number of N identical
menisci are also calculated as shown in figure 9b. The initial separation of two
surfaces is the same as for the rough surface case. It is observed that for the study
cases the attractive meniscus force slightly increases with the increase in N for
hydrophilic smooth surfaces, whereas it slightly decreases with an increase in N
for hydrophobic smooth surfaces. The rate change of force is gradual. Part of the
reason is it may be due to the insignificant change of total meniscus area with N
for smooth surfaces. A quick decrease of viscous force is observed with the
increase of N for either hydrophilic or hydrophobic surface. This is because each
meniscus has a smaller meniscus area at larger N, and the meniscus can be
broken very quickly. As compared to the rough surface case, both forces are
much larger for smooth surfaces at smaller number of N. This indicates that the
introduction of a small number of asperities can help to reduce both forces
significantly and thus reduce stiction.
4. Conclusions
Both meniscus and viscous forces during the separation of hydrophilic and
hydrophobic smooth/rough surfaces with symmetric and asymmetric contact
angles are calculated. The effects of the separation distance, initial meniscus
height, separation time, contact angles and roughness are presented.
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
1646
S. Cai and B. Bhushan
The results show that meniscus and viscous forces change at a rapid rate at the
early stages of separation. The meniscus force decreases with an increase of
separation distance, whereas the viscous force has an opposite trend. Both forces
decrease with an increase of initial meniscus height. Also, larger initial meniscus
height has a longer meniscus break distance. An increase of separation time,
initial meniscus height or the decrease of contact angle leads to an increase of
critical meniscus area at which both forces are equivalent. An increase in contact
angle leads to a decrease of attractive meniscus force but an increase of repulsive
meniscus force (attractive or repulsive dependent on hydrophilic or hydrophobic
surface, respectively). For asymmetric contact angles, the magnitude of the
meniscus force and the critical meniscus area are in between the values for the
two angles. Though the contact angle significantly affects meniscus force, it only
has a limited effect on the viscous force. A slightly attractive force is observed for
the hydrophobic surface during the end stage of separation though the magnitude
is small.
The combination effects of initial meniscus height and contact angles on both
forces are summarized in table 1. At a fixed initial meniscus height, an increase of
contact angle leads to a decrease of attractive meniscus force for a hydrophilic
surface. An increase of both initial meniscus height and contact angle leads to a
decrease of attractive meniscus force and vice versa. For a hydrophobic surface,
an increase of contact angle leads to an increase of repulsive meniscus force.
An increase of initial meniscus height and a decrease of contact angle lead to a
decrease of repulsive meniscus force and vice versa. As compared to contact
angle, the initial meniscus height dominates the effect on viscous force. An
increase of initial meniscus height leads to a decrease of viscous force and
vice versa.
For a rough surface, an increase in the number of surface asperities
(roughness) leads to an increase of meniscus force; however, its effect on viscous
force is trivial. As compared to a smooth surface, the introduction of a small
number of asperities can help to reduce both forces.
This study provides a comprehensive analysis of meniscus and viscous forces
during separation of two hydrophilic or hydrophobic surfaces from liquid
menisci. It helps in better understanding the physics of both hydrophilic and
hydrophobic phenomena. Control of forces may be achieved by proper
manipulation of the surface properties. It is also useful for solving real
technological problems by understanding the behaviour of these forces.
References
Adamson, A. W. 1990 Physical chemistry of surfaces, 5th edn. New York, NY: Wiley.
Bhushan, B. 1996 Tribology and mechanics of magnetic storage devices, 2nd edn. New York, NY:
Springer.
Bhushan, B. 1999 Principles and applications of tribology. New York, NY: Wiley.
Bhushan, B. 2002 Introduction to tribology. New York, NY: Wiley.
Bhushan, B. 2003 Adhesion and stiction: mechanisms, measurement techniques, and methods for
reduction. J. Vac. Sci. Technol. B 21, 2262–2296. (doi:10.1116/1.1627336)
Bhushan, B. 2005 Nanotribology and nanomechanics: an introduction. Heidelberg, Germany:
Springer.
Bhushan, B. 2007 Springer handbook of nanotechnology, 2nd edn. Heidelberg, Germany: Springer.
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 16, 2017
Meniscus and viscous forces
1647
Cai, S. & Bhushan, B. 2006 Three-dimensional dry/wet contact analysis of multilayered
elastic/plastic solids with rough surfaces. ASME J. Tribol. 128, 18–31. (doi:10.1115/1.2114947)
Cai, S. & Bhushan, B. 2007a Meniscus and viscous forces during normal separation of liquidmediated contacts. Nanotechnology 18, 465704. (doi:10.1088/0957-4484/18/46/465704)
Cai, S. & Bhushan, B. 2007b Effects of symmetric and asymmetric contact angles and division of
menisci during separation. Philos. Mag. 87, 5505–5522. (doi:10.1080/14786430701652869)
Cameron, A. & McEttles, C. M. 1981 Basic lubrication theory. New York, NY: Wiley.
Carter, W. C. 1988 The forces and behavior of fluids constrained solids. Acta Metall. 36, 2283–2292.
(doi:10.1016/0001-6160(88)90328-8)
Chan, D. Y. C. & Horn, R. G. 1985 The drainage of thin liquid films between solid surfaces.
J. Chem. Phys. 83, 5311–5324. (doi:10.1063/1.449693)
Fisher, R. A. 1926 On the capillary forces in an ideal soil; correction of formulae given by W. B.
Haines. J. Agric. Sci. 16, 492–505.
Fortes, M. A. 1982 Axisymmetric liquid bridges between parallel plates. J. Colloid Interface Sci.
88, 338–352. (doi:10.1016/0021-9797(82)90263-6)
Gao, C. 1997 Theory of menisci and its applications. Appl. Phys. Lett. 71, 1801–1803. (doi:10.1063/
1.119403)
Gorb, S. 2001 Attachment devices of insect cuticle. Dordrecht, The Netherlands; Boston, MA;
London, UK: Kluwer Academic Publishers.
Hocking, L. M. 1973 The effect of slip on the motion of a sphere close to a wall and of two adjacent
spheres. J. Eng. Math. 7, 207–218. (doi:10.1007/BF01535282)
Israelachvili, J. N. 1992 Intermolecular and surface forces, 2nd edn. San Diego, CA: Academic
Press.
Matthewson, M. J. 1988 Adhesion of spheres by thin liquid films. Philos. Mag. A 57, 207–216.
(doi:10.1080/01418618808204510)
Orr, F. M., Scriven, L. E. & Rivas, A. P. 1975 Pendular rings between solids: meniscus properties
and capillary force. J. Fluid Mech. 67, 723–742. (doi:10.1017/S0022112075000572)
Peng, W. & Bhushan, B. 2001 A numerical three-dimensional model for the contact of layered
elastic/plastic solids with rough surfaces by a variational principle. ASME J. Tribol. 123,
330–342. (doi:10.1115/1.1308004)
Singh, S., Houston, J., Swol, F. V. & Brinker, C. J. 2006 Drying transition of confined water.
Nature 442, 526. (doi:10.1038/442526a)
Stifter, T., Marti, O. & Bhushan, B. 2000 Theoretical investigation of the distance dependence of
capillary and van der Waals forces in scanning force microscopy. Phys. Rev. B 62, 667–673.
(doi:10.1103/PhysRevB.62.13667)
Thomson, W. 1870 On the equilibrium of vapour at a curved surface of liquid. Proc. R. Soc. Ed. 7,
63–68.
Tian, X. & Bhushan, B. 1996 The micro-meniscus effect of thin liquid film on the static friction of
rough surface contact. J. Phys. D: Appl. Phys. 29, 163–178. (doi:10.1088/0022-3727/29/1/026)
Woodrow, J., Chilton, H. & Hawes, I. 1961 Forces between slurry particles due to surface tension.
J. Nuclear Ener. B: Reactor Technol. 2, 229–237.
Phil. Trans. R. Soc. A (2008)