The effect of surface roughness on the adhesion of elastic solids
B. N. J. Persson
a)
IFF, FZ-Ju
¨
lich, 52425 Ju
¨
lich, Germany and International School for Advanced Studies (SISSA),
Via Beirut 2-4, I-34014, Trieste, Italy
E. Tosatti
International School for Advanced Studies (SISSA), Via Beirut 2-4, I-34014, Trieste, Italy, INFM,
Unita’ SISSA, Trieste, Italy, and International Centre for Theoretical Physics, P.O. Box 586, I-34014,
Trieste, Italy
共Received 22 March 2001; accepted 10 July 2001兲
We study the influence of surface roughness on the adhesion of elastic solids. Most real surfaces
have roughness on many different length scales, and this fact is taken into account in our analysis.
We consider in detail the case when the surface roughness can be described as a self-affine fractal,
and show that when the fractal dimension D
f
⬎ 2.5, the adhesion force may vanish, or be at least
strongly reduced. We consider the block-substrate pull-off force as a function of roughness, and find
a partial detachment transition preceding a full detachment one. The theory is in good qualitative
agreement with experimental data. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1398300兴
I. INTRODUCTION
Even a highly polished surface has surface roughness on
many different length scales. When two bodies with nomi-
nally flat surfaces are brought into contact, the area of real
contact will usually only be a small fraction of the nominal
contact area. We can visualize the contact regions as small
areas where asperities from one solid are squeezed against
asperities of the other solid; depending on the conditions the
asperities may deform elastically or plastically.
How large is the area of real contact between a solid
block and the substrate? This fundamental question has ex-
tremely important practical implications. For example, it de-
termines the contact resistivity and the heat transfer between
the solids. It is also of direct importance for sliding friction,
1
e.g., the rubber friction between a tire and a road surface, and
it has a major influence on the adhesive force between two
solid blocks in direct contact. One of us has developed a
theory of contact mechanics,
2
valid for randomly rough 共e.g.,
self-affine fractal兲 surfaces, but neglecting adhesion. Adhe-
sion is particularly important for elastically soft solids, e.g.,
rubber or gelatine, where it may pull the two solids in direct
contact over the whole nominal contact area.
In this paper we discuss adhesion for randomly rough
surfaces. We first calculate the block-substrate pull-off force
under the assumption that there is complete contact in the
nominal contact area. We assume that the substrate surface
has roughness on many different length scales, and consider
in detail the case where the surfaces are self-affine fractal.
We also study pull-off when only partial contact occurs in
the nominal contact area.
The influence of surface roughness on the adhesion be-
tween rubber 共or any other elastic solid兲 and a hard substrate
has been studied in a classic paper by Fuller and Tabor.
3
They found that already a relative small surface roughness
can completely remove the adhesion. In order to understand
the experimental data they developed a very simple model
based on the assumption of surface roughness on a single
length scale. In this model the rough surface is modeled by
asperities all of the same radius of curvature and with heights
following a Gaussian distribution. The overall contact force
was obtained by applying the contact theory of Johnson,
Kendall, and Roberts
4
共JKR兲 to each individual asperity. The
theory predicts that the pull-off force, expressed as a fraction
of the maximum value, depends upon a single parameter,
which may be regarded as representing the statistically aver-
aged competition between the compressive forces exerted by
the higher asperities trying to pry the surfaces apart and the
adhesive forces between the lower asperities trying to hold
the surfaces together. We believe that this picture of adhesion
developed by Tabor and Fuller would be correct if the sur-
faces had roughness on a single length scale as assumed in
their study. However, when roughness occurs on many dif-
ferent length scales, a qualitatively new picture emerges 共see
the following兲, where, e.g., the adhesion force may even van-
ish 共or at least be strongly reduced兲, if the rough surface can
be described as a self-affine fractal with fractal dimension
D
f
⬎ 2.5. We also note that the formalism used by Fuller and
Tabor is only valid at ‘‘high’’ surface roughness, where the
area of real contact 共and the adhesion force兲 is very small.
The present theory, on the other hand, is particularly accurate
for ‘‘small’’ surface roughness, where the area of real contact
equals the nominal contact area.
II. QUALITATIVE DISCUSSION
Assume that a uniform stress
acts within a circular
area 共radius R兲 centered at a point P on the surface of a
semi-infinite elastic body with elastic modulus E, see Fig. 1.
This will give rise to a perpendicular displacement u of P by
a兲
Electronic mail: b.persson@fz-juelich.de
JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 12 22 SEPTEMBER 2001
55970021-9606/2001/115(12)/5597/14/$18.00 © 2001 American Institute of Physics
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a distance which is easy to calculate using continuum me-
chanics: u/R⬇
/E. This result can also be derived from
simple dimensional arguments. First, note that u must be
proportional to
since the displacement field is linearly re-
lated to the stress field. However, the only other quantity in
the problem with the same dimension as the stress
is the
elastic modulus E so u must be proportional to
/E. Since R
is in turn the only quantity with the dimension of length we
get at once u⬃(
/E)R. Thus, if h and represent perpen-
dicular and parallel roughness length scales, respectively,
then if h/⬇
/E, the perpendicular pressure
will be just
large enough to deform the rubber to make contact with the
substrate everywhere.
Let us now consider the role of the rubber–substrate
adhesion interaction. When the rubber deforms and fills out a
surface cavity of the substrate, an elastic energy U
el
⬇Eh
2
will be stored in the rubber. Now, if this elastic energy is
smaller than the gain in adhesion energy U
ad
⬇⫺ ⌬
␥
2
,
where ⫺ ⌬
␥
is the local change of surface free energy upon
contact due to the rubber–substrate interaction 共which usu-
ally is mainly of the van der Waals type兲, then 共even in the
absence of the load F
N
兲 the rubber will deform spontane-
ously to fill out the substrate cavities. The condition U
el
⫽
⫺ U
ad
gives h/⬇(⌬
␥
/E)
1/2
. For example, for very rough
surfaces with h/⬇1, and with parameters typical of rubber
E⫽ 1 MPa and ⌬
␥
⫽ 3 meV/Å
2
, the adhesion interaction
will be able to deform the rubber and completely fill out the
cavities if ⬍ 0.1
m. For very smooth surfaces h/⬃0.01
or smaller, so that the rubber will be able to follow the sur-
face roughness profile up to the length scale ⬃1mm or
longer.
The above-mentioned discussion assumes roughness on
a single length scale . But the surfaces or real solids have
roughness on a wide distribution of length scales. Assume,
for example, a self-affine fractal surface. In this case the
statistical properties of the surface are invariant under the
transformation
x→ x
, z→ z
H
,
where x⫽ (x,y) is the two-dimensional position vector in the
surface plane, and where 0⬍H⬍1. This implies that if h
a
is
the amplitude of the surface roughness on the length scale
a
, then the amplitude h of the surface roughness on the
length scale will be of order
h⬇h
a
共
/
a
兲
H
.
A necessary condition for adhesional-induced complete con-
tact on the length scale is that E
ad
⬎ E
el
, i.e., ⌬
␥
⬎ Eh
2
,
which gives
⌬
␥
⬎ Eh
a
2
冉
a
冊
2H
or
冉
a
冊
2H⫺ 1
⬍
⌬
␥
a
Eh
a
2
. 共1兲
Assume first that H⬎ 1/2. In this case, if
a
⬍ we get
(/
a
)
2H⫺ 1
⬎ 1, and condition 共1兲 gives ⌬
␥
a
/Eh
a
2
⬎ 1.
Thus, adhesion will be important on any length scale
a
⬍ . In particular, if is the long-distance cutoff length
0
in the self-affine fractal distribution, then complete contact
will occur at the interface. More generally, if ⫽ Eh
2
/⌬
␥
⬍
0
, the contact consists of a set of disconnected contact
regions of linear size ; in each such region perfect contact
occurs.
Consider now instead H⬍ 1/2. In this case, if
a
⬍ we
get (/
a
)
2H⫺ 1
⬍ 1, and condition 共1兲 no longer guarantees
that ⌬
␥
a
/Eh
a
2
⬎ 1. In fact, it is easy to show that at short
enough length scale
a
, ⌬
␥
a
/Eh
a
2
⬍ 1. Thus, without a
short-distance cutoff, adhesion and the area of real contact
will vanish. Hence, in spite of the fact that the contact at first
may seems to be perfect on large scales 共since ⌬
␥
⬎ Eh
2
兲,
there is, in fact, no contact at all since ⌬
␥
a
⬍ Eh
a
2
holds at
short enough length scale
a
. In reality, a finite short-
distance cutoff will always occur, but this case requires a
more detailed study 共see Sec. III兲. Also, in the above-
mentioned analysis we have neglected that the area of real
contact depends on h 共i.e., it is of order
2
only when h/
Ⰶ 1兲. A more accurate analysis follows.
III. INTERFACIAL ELASTIC AND ADHESION
ENERGIES FOR ROUGH SURFACES
Assume that a flat rubber surface is in contact with the
rough surface of a hard solid. Assume that because of the
rubber–substrate adhesion interaction, the rubber deforms
elastically and makes contact with the substrate everywhere,
see Fig. 2.
Let us calculate the difference in free energy between the
rubber block in contact with the substrate and the noncontact
case. Let z⫽ h(x) denote the height of the rough surface
above a flat reference plane 共chosen so that
具
h
典
⫽ 0). Assume
first that the rubber is in direct contact with the substrate over
the whole nominal contact area. The surface adhesion energy
is assumed proportional to the contact area so that
U
ad
⫽⫺⌬
␥
冕
d
2
x
关
1⫹
共
ⵜh
共
x
兲兲
2
兴
1/2
FIG. 1. A uniform stress
, acting within a circular area 共radius R兲 on the
surface of a semi-infinite elastic medium, gives rise to a displacement u.
FIG. 2. The adhesion interaction pulls the rubber into complete contact with
the rough substrate surface.
5598 J. Chem. Phys., Vol. 115, No. 12, 22 September 2001 B. N. J. Persson and E. Tosatti
Downloaded 21 Dec 2006 to 134.94.122.39. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
⬇⫺ ⌬
␥
冋
A
0
⫹
1
2
冕
d
2
x
共
ⵜh
兲
2
册
, 共2兲
where we have assumed
兩
ⵜh
兩
Ⰶ 1. Now, using
h
共
x
兲
⫽
冕
d
2
qh
共
q
兲
e
iq"x
we get
冕
d
2
x
共
ⵜh
兲
2
⫽
冕
d
2
x
冕
d
2
qd
2
q
⬘
共
⫺ q"q
⬘
兲
⫻
具
h
共
q
兲
h
共
q
⬘
兲
典
e
i(q⫹ q
⬘
)"x
⫽
共
2
兲
2
冕
d
2
qq
2
具
h
共
q
兲
h
共
⫺ q
兲
典
⫽ A
0
冕
d
2
qq
2
C
共
q
兲
, 共3兲
where the surface roughness power spectrum is
C
共
q
兲
⫽
1
共
2
兲
2
冕
d
2
x
具
h
共
x
兲
h
共
0
兲
典
e
⫺ iq"x
, 共4兲
where
具
¯
典
stands for ensemble average. Thus, using Eqs.
共2兲 and 共3兲:
U
ad
⬇⫺ A
0
⌬
␥
冋
1⫹
1
2
冕
d
2
qq
2
C
共
q
兲
册
. 共5兲
Next, let us calculate the elastic energy stored in the
deformation field in the vicinity of the interface. Let u
z
(x)be
the normal displacement field of the surface of the elastic
solid. We get
U
el
⬇⫺
1
2
冕
d
2
x
具
u
z
共
x
兲
z
共
x
兲
典
⫽⫺
共
2
兲
2
2
冕
d
2
q
具
u
z
共
q
兲
z
共
⫺ q
兲
典
. 共6兲
Next, we know that
5
u
z
共
q
兲
⫽ M
zz
共
q
兲
z
共
q
兲
, 共7兲
where
M
zz
共
q
兲
⫽⫺
2
共
1⫺
2
兲
Eq
, 共8兲
E being the elastic modulus and
the Poisson ratio. If we
assume that complete contact occurs between the solids, then
u
z
⫽ h(x) and from Eqs. 共4兲 and 共6兲–共8兲,
U
el
⬇⫺
共
2
兲
2
2
冕
d
2
q
具
u
z
共
q
兲
u
z
共
⫺ q
兲
典
关
M
zz
共
⫺ q
兲
兴
⫺ 1
⫽
A
0
E
4
共
1⫺
2
兲
冕
d
2
qqC
共
q
兲
. 共9兲
The change in the free energy when the rubber block
moves in contact with the substrate is given by the sum of
Eqs. 共5兲 and 共9兲:
U
el
⫹ U
ad
⫽⫺⌬
␥
eff
A
0
,
where
⌬
␥
eff
⫽ ⌬
␥
冋
1⫹
冕
q
0
q
1
dq q
3
C
共
q
兲
⫺
E
2
共
1⫺
2
兲
⌬
␥
冕
q
0
q
1
dq q
2
C
共
q
兲
册
. 共10兲
The above-given theory is valid for surfaces with arbi-
trary random roughness, but will now be applied to self-
affine fractal surfaces. It has been found that many ‘‘natural’’
surfaces, e.g., surfaces of many materials generated by frac-
ture, can be approximately described as self-affine surfaces
over a rather wide roughness size region. A self-affine fractal
surface has the property that if we make a scale change that
is appropriately different along the two directions, parallel
and perpendicular, then the surface does not change its
morphology.
6
Recent studies have shown that even asphalt
road tracks 共of interest for rubber friction兲 are 共approxi-
mately兲 self-affine fractal, with an upper cutoff length
0
⫽ 2
/q
0
of order of a few millimeters.
7
For a self affine
fractal surface:
6,8
C(q)⫽ 0 for q⬍ q
0
, while for q⬎ q
0
:
C
共
q
兲
⫽
H
2
冉
h
0
q
0
冊
2
冉
q
q
0
冊
⫺ 2(H⫹ 1)
, 共11兲
where H⫽3⫺D
f
共where the fractal dimension 2⬍ D
f
⬍ 3兲,
and where q
0
is the lower cutoff wave vector, and h
0
is
determined by the rms roughness amplitude,
具
h
2
典
⫽ h
0
2
/2. We
note that C(q) can be measured directly, using many differ-
ent methods, e.g., using stylus instruments or optical
instruments.
9
Substituting Eq. 共11兲 in Eq. 共10兲 gives
⌬
␥
eff
⌬
␥
⫽ 1⫹
1
2
共
q
0
h
0
兲
2
g
共
H
兲
⫺
Eh
0
2
q
0
4
共
1⫺
2
兲
⌬
␥
f
共
H
兲
, 共12兲
where
f
共
H
兲
⫽
H
1⫺ 2H
冋
冉
q
1
q
0
冊
1⫺ 2H
⫺ 1
册
,
g
共
H
兲
⫽
H
2
共
1⫺ H
兲
冋
冉
q
1
q
0
冊
2(1⫺ H)
⫺ 1
册
.
If we introduce the length
␦
⫽ 4(1⫺
2
)⌬
␥
/E, then Eq. 共12兲
takes the form
⌬
␥
eff
⌬
␥
⫽ 1⫹
共
q
0
h
0
兲
2
冉
1
2
g
共
H
兲
⫺
1
q
0
␦
f
共
H
兲
冊
. 共13兲
In Fig. 3 we show f(H) and g(H) as a function of H. Note
that the present theory is valid only if (q
0
h
0
)
2
g(H)/2⬍ 1,
otherwise the expansion of the square-root function in Eq.
共2兲 is invalid.
Let us emphasize that the present theory is strictly valid
only for purely elastic solids; many real solids 共e.g., most
polymers兲
10
behave in a viscoelastic manner, and in these
cases ⌬
␥
may be much larger than in the adiabatic limit, and
the theory presented in this paper is no longer valid. Vis-
coelastic effects may be particularly important for rough sur-
faces, where, during pull off, the roughness introduces fluc-
tuating forces with a wide distribution of frequencies. The
same effect operates during sliding as described in a recent
work on rubber friction.
11
5599J. Chem. Phys., Vol. 115, No. 12, 22 September 2001 Effect of surface roughness on the adhesion of elastic solids
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Consider first an elastically very soft solid, e.g., jelly. In
this case, using E⬇10
4
Pa and ⌬
␥
⬇3 meV/Å
2
, we get
␦
⬇10
m, and since typically q
0
⫽ 2
/
0
⬃(10
m)
⫺ 1
and
g(H)Ⰷ f(H), we expect ⌬
␥
eff
⬎⌬
␥
. Thus, for an 共elastically兲
very soft solid the adhesion force may increase upon rough-
ening the substrate surface. This effect has been observed
experimentally for rubber in contact with a hard, rough
substrate,
12,13
and the present theory explains under exactly
what conditions that will occur 共see the following兲.
Note that if the condition g(H)/2⬎ f (H)/(q
0
␦
) is satis-
fied, the adhesion force 共for small enough h
0
兲 will increase
with increasing amplitude h
0
of the surface roughness. We
may define a critical elasticity E
c
such that if E⬍ E
c
, ⌬
␥
eff
increases with increasing h
0
, while it decreases if E⬎ E
c
.
E
c
is determined by the condition g(H)/2⫽ f(H)/(q
0
␦
),
which gives
E
c
⫽ 2
共
1⫺
2
兲
⌬
␥
q
0
g
共
H
兲
/f
共
H
兲
.
This expression for E
c
depends on the nature of the surface
roughness via the cutoff wave vector q
0
and the fractal ex-
ponent H⫽ 3⫺ D
f
. These quantities can be obtained from
measurements of the surface roughness power spectra C(q).
Such measurements have not been performed for any of the
systems for which the dependence of the adhesion on the
roughness amplitude h
0
has been studied. However,
measurements
9
of C(q) for similar surfaces as those used in
the adhesion experiments have shown that typically H⬇0.8
and
0
⫽ 2
/q
0
⬇100
m. For H⬇0.8, Fig. 3 gives
g(H)/f(H)⬃100 and with the measured 共for rubber in con-
tact with most hard solids兲⌬
␥
⬇3 meV/Å
2
we get E
c
⬇1 MPa. This is in very good agreement with experimental
observations. Thus, Briggs and Briscoe
12
observed a strong
roughness-induced increase in the pull-off force for rubber
with the elastic modulus E⫽ 0.06 MPa, but a negligible in-
crease when E⫽ 0.5 MPa. Similarly, Fuller and Roberts
13
ob-
served an increase in the pull-off force for rubbers with E
⫽ 0.4, 0.14, and 0.07 MPa, but a continuous decrease for
rubbers with E⫽ 1.5 and 3.2 MPa. It would be extremely
interesting to perform a detailed test of the theory for sur-
faces for which the surface roughness power spectra C(q)
has been measured.
According to Eq. 共13兲, the roughness-induced contribu-
tion to ⌬
␥
eff
scales as ⬃h
0
2
. This scaling is exact for the
contribution from elastic deformations 共as long as complete
contact occurs兲, but is only valid for small enough h
0
for the
adhesion contribution. For large h
0
the expansion in Eq. 共2兲
is invalid, and one obtains instead
U
ad
⬇⫺ ⌬
␥
冕
d
2
x
兩
ⵜh
共
x
兲
兩
,
which varies linearly with h
0
. Thus, for large enough h
0
the
共negative兲 contribution to ⌬
␥
eff
from the elastic deforma-
tions will always dominate, and this explains why the pull-
off force always decreases for large enough h
0
, even when
the elastic modulus of the rubber is very small.
12,13
In fact,
we can derive an expression for ⌬
␥
eff
which is approxi-
mately valid also for large h
0
, as follows: Let us write Eq.
共2兲 as 共see Appendix B for the derivation of the exact result兲
U
ad
⫽⫺⌬
␥
A
0
具
关
1⫹
共
ⵜh
共
x
兲兲
2
兴
1/2
典
⬇⫺ ⌬
␥
A
0
关
1⫹
具
共
ⵜh
共
x
兲兲
2
典
兴
1/2
,
where
具
共
ⵜh
共
x
兲兲
2
典
⫽
1
A
0
冕
d
2
x
共
ⵜh
共
x
兲兲
2
⫽ 2
冕
q
0
q
1
dq q
3
C
共
q
兲
.
Thus, for a self-affine fractal surface Eq. 共13兲 is replaced
with
⌬
␥
eff
⌬
␥
⬇
关
1⫹
共
q
0
h
0
兲
2
g
共
H
兲
兴
1/2
⫺
共
q
0
h
0
兲
2
1
q
0
␦
f
共
H
兲
. 共14a兲
If we denote
⫽ h
0
q
0
g
1/2
then Eq. 共14a兲 becomes
⌬
␥
eff
⌬
␥
⫽
共
1⫹
2
兲
1/2
⫺
E
2E
c
2
. 共14b兲
This function is shown in Fig. 4 for E
c
/E⫽ 1 and 2 共dashed
lines兲. The solid lines in Fig. 4 are obtained using the exact
result derived in Appendix B 关see Eq. 共B2兲兴. If we assume
that the pull-off force is proportional to ⌬
␥
eff
关as expected
for a rubber ball, see Eq. 共21兲兴, we obtain the h
0
dependence
of the pull-off force shown in Fig. 4, which is in good quali-
tative agreement with experiment.
13
If it would be possible to prepare surfaces with different
roughness amplitude h
0
but constant q
0
共and H兲, then it is
easy to prove from Eq. 共14b兲 that the maximum of ⌬
␥
eff
as a
function of h
0
is
FIG. 3. The functions f(H) and g(H) are defined in the text.
5600 J. Chem. Phys., Vol. 115, No. 12, 22 September 2001 B. N. J. Persson and E. Tosatti
Downloaded 21 Dec 2006 to 134.94.122.39. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
共
⌬
␥
eff
兲
max
⫽
⌬
␥
2
冉
E
E
c
⫹
E
c
E
冊
.
The maximum occurs for h
0
⫽ h
c
:
q
0
h
c
⫽ g
⫺ 1/2
冋
冉
E
c
E
冊
2
⫺ 1
册
1/2
.
Thus if, e.g., E
c
/E⬇10, the maximal pull-off force should
be ⬃5 times larger than for perfectly smooth surfaces. This
type of enhancement of ⌬
␥
eff
has been deduced from rolling
friction experiments
13
using very soft rubbers 共with E
⬇0.07 MPa兲, but the interpretation of the data is complicated
by the fact that the rubber is not perfectly elastic, but rather
exhibit 共rate-dependent兲 viscoelastic properties.
For most ‘‘normal’’ solids, ⌬
␥
⬇Ea, where a is an
atomic distance 共of order ⬃1Å兲 and E the elastic modulus.
Thus,
␦
⬃a⬃1 Å and typically 1/q
0
␦
⬃10
5
so that the 共re-
pulsive兲 energy stored in the elastic deformation field in the
solids at the interface, and proportional to f(H), largely
overcomes the increase in adhesion energy derived from the
roughness induced increase in the contact area, described by
the term (q
0
h
0
)
2
g(H)/2.
Let us note the following very important fact. Many sol-
ids respond in an elastic manner when exposed to rapid de-
formations, but flow plastically on long enough time scales.
This is clearly the case for non-cross-linked glassy polymers,
but it is also to some extent the case for rubbers with cross
links. The latter materials behave as relative hard solids
when exposed to high-frequency perturbations, while they
deform as soft solids when exposed to low-frequency pertur-
bations. Thus, when such a solid is squeezed rapidly against
a substrate with roughness on many different length scales, a
large amount of elastic energy may initially be stored in the
local 共asperity induced兲 deformation field at the interface.
However, if the system is left alone 共in the compressed state兲
for some time, the local stress distribution at the interface
will decrease 共or relax, because of thermal excitation over
the barriers兲, while the area of real contact simultaneously
increases. This will result in an increasing adhesion bond
between the solids, and a decrease in the elastic deformation
energy stored in the solids: both effects will tend to increase
of the pull-off force. 共Note: The elastic energy stored at the
interface during the compression phase is almost entirely
given back during slow pull-off.兲 Since we use a frequency
independent elastic modulus, such time-dependent effects
are, of course, not taken into account in the analysis pre-
sented previously.
The interfacial free energy is a sum of the adhesive part
U
ad
, which is proportional to the area of real contact, and the
elastic energy U
el
stored in the strain field at the interface. As
long as ⌬U⫽ U
ad
⫹ U
el
⬍ 0, a finite pull-off force will be nec-
essary in order to separate the bodies. When the amplitude of
the surface roughness increases, ⌬U will in general increase
and when it reaches zero, the pull-off force vanishes. Sup-
pose now that an elastic slab has been formed between two
solids from a liquid ‘‘glue layer,’’ which has transformed to
the solid state after some hardening time. For example, many
glues consist of polymers which originally are liquid, and
slowly harden, e.g., via the formation of cross bridges. In this
case, if the original liquid wets the solid surfaces, it may
penetrate into all surface irregularities and make intimate
contact with the solid walls, and only thereafter harden to the
solid state. Ideally, this will result in a solid elastic slab in
perfect contact with the solid walls, and without any interfa-
cial elastic energy stored in the system, i.e., with U
el
⫽ 0. 共In
practice, shrinkage stresses may develop in the glue layer,
which will lower the strength of the adhesive joint.兲 Thus the
last term in the expression for ⌬
␥
eff
vanishes, and ⌬
␥
eff
will
increase with increasing surface roughness in proportion to
the surface area. This will result in an increase in the pull-off
force, but finally the bond breaking may occur inside the
glue film itself,
14
rather than at the interface between the glue
film and the solid walls 共see Fig. 5兲; from here on no
strengthening of the adhesive bond will result from further
roughening of the confining solid walls.
Thus, the fundamental advantage of using liquidlike
glues 共which harden after some solidification time兲, com-
pared to pressure-sensitive adhesives which consist of thin
solid elastic (E⬇10
4
–10
5
Pa) films, and which develop tack
only when squeezed between the solid surfaces, is that in the
former case no elastic deformation energy is stored at the
interface 共which would be given back during the removal
process and hence reduce the strength of the adhesive bond兲,
while this may be the case for the latter type of adhesive,
unless the interfacial stress distribution is able to relax to-
ward the stress-free state 共which requires the absence of
cross links, or such a low concentration of cross links that
‘‘thick’’ liquidlike polymer layers occur at the interfaces兲.
If we define
FIG. 4. The effective change in surface energy as a function of the dimen-
sionless parameter h
0
q
0
g
1/2
for E
c
/E⫽ 1 and 2. The solid lines are obtained
using the exact result given by Eq. 共B2兲, while the dashed lines are obtained
using the approximation 共14b兲.
FIG. 5. When the interaction between the ‘‘glue’’ film and the substrate is
‘‘strong,’’ the separation may involve internal rupture of the glue film rather
than detachment at the interface.
5601J. Chem. Phys., Vol. 115, No. 12, 22 September 2001 Effect of surface roughness on the adhesion of elastic solids
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