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Roughness and Adhesion
321
ROUGHNESS AND ADHESION K.N.G.
FULLER
Malaysian Rubber Producers' Res. ASSOC., Hertford, England
ABSTRACT The effect of surface roughness upon the adhesion between rubber and a rigid substrate is analysed.
Except at low roughnesses the theory predicts the relative
adhesion,(the adhesion expressed as a fraction of that for a smooth surface) reasonably well, and accounts for the influence of the rubber modulus and of the pressure bringing.the surfaces together. Although an obvious reference surface for the theory is one consisting of asperities all of the same height, the necessity of using the same, smooth surface as the reference in both theory and experiment is emphasised.
The assumptions of the theory are shown to break down
with slightly roughened surfaces, against which soft rubbers give their maximum adhesion.
This enhancement compared to a smooth surface can nevertheless be at
least partially accounted for, though a precise analysis is not possible.
Provided
the contact region splits into isolated zones during separation, the work to peel the surfaces apart is increased by the elastic energy lost when these 'islands' of contact are finally broken.
INTRODUCTION An optically smooth rubber surface readily adheres to a similarly smooth substrate, whereas metals, even if highly cleaned and polished, show virtually no adhesion unless yielding occurs in the contact region.
The contrast in behaviour
can be ascribed to the effect of surface roughness upon adhesion, and the dependence of this effect upon the moduli of the contacting bodies. This paper describes how the influence of roughness upon adhesion may be analysed theoretically for elastic contacts.
Although much of the basic analysis
is applicable to other contact systems, the main concern here is the adhesion between rubber and a rigid surface, and the phenomena, such as the enhancement of the adhesion by slight roughening, particular to that system.
322
1
0
-1 A
Ig
-2 I
I
-2
-1
1
0.
(roughness average,
R, rm)
1
F i g . 1. Observed r e l a t i v e a d h e s i o n of o p t i c a l l y smooth r u b b e r a s a f u n c t i o n o f t h e roughness of t h e s u b s t r a t e ( r e f . 3 ) . Tp, t h e p e e l i n g e n e r g y f o r t h e rough s u b s t r a t e s , i s n o r m a l i s e d by y t h e p e e l i n g e n e r g y f o r a smooth s u b s t r a t e . The r a t e of p e e l i n g was lmms-l.PyRubber moduli: 0 0.14 MPa; A 1 . 5 MPa. C o n t a c t p r e s s u r e b r i n g i n g s u r f a c e s t o g e t h e r : 0 2 x lo3 Pa; 0 2 . 5 ~l o 4 Pa; A 5 x 1 0 4 Pa. EXPERIMENTAL OBSERVATIONS The e f f e c t o f roughness upon a d h e s i o n h a s been i n v e s t i g a t e d ( r e f . 1) by o b s e r v i n g t h e f o r c e t o s e p a r a t e a smooth s i l i c o n e r u b b e r s p h e r e from a roughened PMMA f l a t .
The r e s u l t s showed a s t e a d y d e c r e a s e o f a d h e s i o n w i t h i n c r e a s e d
roughness;
t h e d e c r e a s e o c c u r r e d a t lower roughness f o r r u b b e r s of h i g h e r modulus.
L a t e r e x p e r i m e n t s ( r e f . 2 ) confirmed t h e s e o b s e r v a t i o n s , b u t a l s o i n d i c a t e d t h a t s o f t e r r u b b e r s a d h e r e more s t r o n g l y t o s l i g h t l y roughened t h a n t o smooth surfaces.
T h i s enhancement i s s e e n ( F i g . 1) i n a d h e s i o n measurements o b t a i n e d by
r o l l i n g a n a t u r a l r u b b e r wheel down a n i n c l i n e d PMMA t r a c k ( r e f . 3 ) .
( I n Figure 4
o f t h a t p a p e r a p o i n t r e f e r r i n g t o t h e l i g h t c o n t a c t l o a d was m i s p l o t t e d . ) Although t h e d a t a f o r t h e s o f t e r r u b b e r s u g g e s t t h a t r a i s i n g t h e c o n t a c t p r e s s u r e h a s l i t t l e e f f e c t upon t h e d e g r e e of enhancement, i t does i n c r e a s e t h e a d h e s i o n a t more s e v e r e r o u g h n e s s e s . ANALYSIS The e n e r g y ,
YP’
t o p e e l a p a r t u n i t a r e a of a smooth p l a n a r c o n t a c t i s , i n t h e
case of r u b b e r , r a t e - d e p e n d e n t and g e n e r a l l y much g r e a t e r t h a n t h e e q u i l i b r i u m
323 surface energy,
yo.
The corresponding energy, ya, given up when surfaces adhere
appears to be almost rate-independent and the same as
yo (ref. 4 ) .
analysis more appropriate to rubber is presented, the case in which are both assumed rate-independent and equal to
y
Before an yp and ya
is considered.
Rate-independent Peeling Energy The adhesion between a rough and a smooth plane is calculated by assuming the roughness to consist of a random series of asperities all of tip radius, p, and with a certain distribution of heights.
The analysis (ref. 5) for the contact of
elastic spheres in the presence of adhesive forces is applied to each asperity contact.
The force-displacement relation for an individual asperity derived from
that analysis is:-6=
(1
{$ ( 5 + 1))
-
35)
5
= (1
6C
where
-
,
+.
p/pc)
The force, p, is normalised by pc =
Y3 E P
yo,
and the displacement
6 (which is the
sum of the deformations of the sphere and the plane at the centre of the contact circle) by 6c, 3
where v1
K =
P Pc 1-v
- -
and v 2 being the Poisson's ratio of the contacting materials, and El and E2 their
Young's modulus. under load
-
displacement
The form of the relation is shown in Figure 2.
It is seen that
controlled conditions separation occurs when p = pc, and under
-
controlled conditions when 6 =
Bc. The latter condition applies in
the present analysis as there are many contact points between the surfaces.
The
total force is evaluated by summing the contribution from all the asperity contacts according to their height distribution. The pronounced effect of the dispersion of the asperity heights on the adhesion was shown in an analysis (ref. 6) which approximated the height distribution by an exponential probability density.
The approximation, however, led to the unrealistic
result that the total force between the surfaces is independent of their separation. Experimental characterisation of roughness (ref. 7) suggests that the asperity heights can often be represented by a Gaussian distribution:n(z) = a(-$ 2n) N where n(z)dz and z
+ dz
exp
(-
6 2 )
*
is the number of asperities per unit area with a height between
2,
above the mean asperity height, u is the standard deviation, and N is
324
0-5
1
Fig. 2. Force-displacement relation of an individual asperity in contact with a plane surface for the case of a rate-independent peel energy which is equal to the adhering energy. The inserts illustrate the meaning of the displacement, 6 , for a rigid asperity against a low modulus plane. The roughness average R
the total number of asperities per unit area.
as
measured with a surface profilometer is approximately numerically equal to
a
(ref. 7). For a rough and a smooth surface coming together the force, Pa, per unit nominal area is obtained by summing the contribution from each of the asperities with a height greater than the planes’ separation, d , as measured from the mean asperity height.
where
h = d/a
,
Normalising all lengths by
A = 6
/a
,
Ac
=
Bc/a
a ,
the adhering force, Pa, is given by:
,
and p/pc = F(A / A c ) is the force-displacement relation for an individual asperity contact.
(Equation 2) differs from equation 11) of reference 1) because the
asperity displacements are here defined with the opposite sense).
The force-
distance relation for separating surfaces is different from that just derived for adhering surfaces.
During separation the rubber at an asperity contact is extended
, per unit out of its original plane by a distance up to BC. Thus the force, P SeP nominal area is given by equation 2) with the upper limit of the integral replaced by the lesser of
A c and (h
-
ho), where ho is the initial normalised distance
325
between the surfaces.
During the early stages of separation, P therefore see and hence the force that brought the surfaces together.
depends upon h
As the separation is increased, P
becomes positive (i.e. tensile force is see required to continue the separation), passes through a maximum (P ) and finally m tends to zero as contact with the highest asperities is broken. The adhesion may be defined by P , the pull-off force, or the energy, T required to separate unit P' nominal area of the contact under displacement controlled conditions. T is simply
P
>O. The relative see adhesion may be defined as P /Np or T /T where Npc is the force and T the m c P S energy required to separate unit nominal area of acontact with a surface of the area under the force-separation curve for the region P
u = 0, i.e. with a surface consisting of asperities all of the same height.
Both
definitions of relative adhesion depend only upon the parameter, a ( = u / 6c) and the contact pressure.
The two functions, shown in figure 3 for the case of zero contact
load, are very similar. a 30.5;
The effect due to contact pressure only exists fo;
moreover its magnitude is limited because additional asperities brought
into contact by a high pressure will be separated from the smooth surface before P becomes positive, the point from which T is reckoned. The 'adhesion see P parameter', a , is seen from equation 1) to be proportional to the roughness and, for a rubber-rigid substrate contact, to (rubber m o d ~ l u s ) ~ / ~ .The theory has been found to fit the experimental measurements for hard rubbers reasonably well;
the
relationship between the rubber modulus and the roughness for a certain level of adhesion appears to be of the right form (ref. 1).
No enhancement of the adhesion
at low roughness is predicted. Rate-dependent peeling energy An analysis more appropriate to rubber contacts must allow the peeling energy, to be rate-dependent. independent.
The adhering energy, ya, is still assumed to be rate-
The elastic contact analysis (ref. 5) is again used.
The introduction of a rate-dependent y
alters the force-displacement relation P for an individual asperity being separated from a surface. The new relation is determined by an approach given in ref. 8. The elastic energy, G, released when unit area of the interface unpeels is (ref. 9):-
where a is the radius of the asperity contact.
The motion of the edge of the
contact can be obtained by solving the differential equation:G = y
(A)
P if 6 is a known function of time.
The dependence of
unpeeling, A, is here approximated by the relation, ye
=
k An
,
y
P
upon the rate of
326
0
Adhesion Parameter o(
Fig. 3. Theoretical results showing the relative adhesion energy, T /Ts, ( - - , rate-independent and , rate-dependent peel energies? and the relative pull-off force, Pm/Npc, ( . . - * * , rate-independent peel energy) as functions of the adhesion parameter, a.
-
where k and n are constants;
this relation fits the observed rate dependence
With the displacement, 6 , and radius of contact, a, known, the
quite well.
force, p, on the asperity is calculated from the equation (ref. 5):p = (3PKa6
-
3
Ka ) / 2 P
Calculated force-displacement curves are shown in figure 4 ; depend upon the rate of separation, separation starts.
(=
6),
they are seen to
and the contact force from which
The loading curve, which is unique because
ya is assumed
constant, is included in the figure. A comparison (ref. 8) between calculated curves and the observed force-displacement relations indicated close agreement be tween them. The total force between the rough and smooth planes is obtained by summing the contribution from each asperity.
For surfaces brought together to a separation,
di, the number of asperity contacts with a displacement between 6 . and is:
6i + d 6 i
321
P
6 50 2
If
0'
0
9
P
2
4
Displacement, b ( 1O-2mm) -50
Fig. 4 . Force-displacement r e l a t i o n s of an i n d i v i d u a l r i g i d a s p e r i t y ( t i p r a d i u s 0.lmm) a g a i n s t a smooth rubber s u r f a c e f o r a rate-dependent peel energy. The loading curve ( ) and curves f o r s e p a r a t i o n a t 0 . 1 m s - 1 ( ) ) a r e shown; f o r each r a t e t h e curves f o r two d i f f e r e n t and 0.001mm~-~( s t a r t i n g p o i n t s ( A , B ) a r e p l o t t e d . Rubber p r o p e r t i e s : modulus 0.14 MPa; adhering energy 60 mJm-2; p e e l i n g energy = ( 5 . 0 ~104) H 0 . 3 4 , where B (ms-1) i s the r a t e of peeling.
-
---
-..-.
I f the f o r c e on each of these a s p e r i t i e s a f t e r t h e s u r f a c e s have been separated a distance (d
-
di) i s w r i t t e n a s p( 6 i , d
then the t o t a l f o r c e , Psep,
'
-
d . ) f o r a given r a t e of s e p a r a t i o n ,
during s e p a r a t i o n i s given by:-
The peeling energy, T
i s the a r e a under t h e P -d curve i n the region P >O; P' SeP SeP i t depends upon the r a t e of s e p a r a t i o n and, through d i , the load bringing the
surfaces together.
A s i n the previous s e c t i o n t h e r e l a t i v e adhesion may be defined by T /Ts, P where Ts i s t h e p e e l energy f o r a s u r f a c e with a s p e r i t i e s a l l of the same h e i g h t .
Tp/Ts,
which i s evaluated with both T
P
and T s l e f e r r i n g t o the same r a t e of
separation, i s p l o t t e d i n Figure 3 a s a f u n c t i o n of the adhesion parameter, a , f o r the case of s u r f a c e s brought t o g e t h e r under zero c o n t a c t load. a, i s determined using t h e adhering energy (ya
%
yo).
The parameter,
V i r t u a l l y t h e same curve i s
328 o b t a i n e d o v e r a wide range o f s e p a r a t i o n r a t e s and r u b b e r p r o p e r t i e s . i s very s i m i l a r t o those c a l c u l a t e d i n the previous section;
The c u r v e
a g a i n t h e r e i s no
a d h e s i o n enhancement a t low v a l u e s of a.
COMPARISON OF THEORY AND EXPERIMENT The r e l a t i v e a d h e s i o n h a s been o b t a i n e d from t h e t h e o r y by n o n n a l i s i n g T by P
Ts, whereas t h e e x p e r i m e n t a l r e s u l t s have been e x p r e s s e d r e l a t i v e t o
Because Ts i s ( t h e e n e r g y , w) ye. t o s e p a r a t e a n i n d i v i d u a l a s p e r i t y from t h e smooth s u r f a c e ) x ( t h e number of
T
i s n o t f o r a smooth s u r f a c e , i t i s n o t i d e n t i c a l t o
asperities).
If
ye i s r a t e - i n d e p e n d e n t and e q u a l t o
Yay
w = 0.64 A where A i s t h e a r e a o f c o n t a c t f o r a n a s p e r i t y under z e r o l o a d . must be less t h a n much l a r g e r t h a n
y p , s i n c e NA h a s t o be l e s s t h a n u n i t y .
ya, w
may be g r e a t e r t h a n A y
and s o T
In t h i s case T
I f , however,
ye i s
be more t h a n
P' one c a s e t h e e l a s t i c e n e r g y s t o r e d a f t e r t h e a s p e r i t y i s b r o u g h t i n t o c o n t a c t
In
s i g n i f i c a n t l y r e d u c e s t h e e x t e r n a l work r e q u i r e d t o s e p a r a t e t h e a s p e r i t y from t h e smooth s u r f a c e , whereas t h e i n i t i a l e l a s t i c e n e r g y makes a n e g l i g i b l e c o n t r i b u t i o n yP>>ya.
if
Although t h e e x p r e s s i o n o f t h e t h e o r e t i c a l r e l a t i v e a d h e s i o n by T /Ts h a s t h e
P
a d v a n t a g e of removing t h e a s p e r i t y d e n s i t y , N , a s a p a r a m e t e r , i t i s a p p a r e n t t h a t a comparison of t h e o r y and e x p e r i m e n t r e q u i r e s t h a t T be c a l c u l a t e d e x p l i c i t l y . P T h i s n e c e s s i t a t e s a knowledge o f t h e a s p e r i t y t i p r a d i u s , p, and t h e a s p e r i t y S u r f a c e p r o f i l e measurements ( r e f . 2 ) d e n s i t y a p p r o p r i a t e t o e a c h r o u g h n e s s , Ra. -3 s u g g e s t t h a t a v a l u e of p of 10 mm i s n o t u n r e a s o n a b l e f o r a l l t h e r o u g h n e s s e s t o be c o n s i d e r e d h e r e .
Nup
-
The v a l u e of N i s o b t a i n e d from t h e e q u a t i o n ( r e f . 10):-
0.04
u is numerically equal to R a' The r u b b e r p r o p e r t i e s assumed i n t h e c a l c u l a t i o n a r e t h o s e g i v e n i n Fig. 4 ;
along with the assumption t h a t
they were measured f o r t h e r u b b e r (modulus 0.14 MPa), t h e a d h e s i o n r e s u l t s of which a r e g i v e n i n F i g . 1. The t h e o r e t i c a l r e s u l t s , e v a l u a t e d f o r a r a t e of s e p a r a t i o n -1 a r e compared i n F i g u r e 5 w i t h e x p e r i m e n t a l o b s e r v a t i o n s o b t a i n e d a t a
of 0.lmms
,
p e e l i n g r a t e of lnuns-l.
The p r e c i s e r e l a t i o n between t h e r a t e a t which t h e p l a n e s
s e p a r a t e i n t h e normal d i r e c t i o n and t h e e x p e r i m e n t a l p e e l i n g r a t e i s u n c e r t a i n ; t h e f o r m e r , however, would be c o n s i d e r a b l y l e s s ' t h a n t h e l a t t e r i f t h e s u r f a c e s meet a t a s m a l l a n g l e a l o n g t h e l i n e of p e e l i n g .
The two t h e o r e t i c a l c u r v e s r e f e r
t o s u r f a c e s b r o u g h t t o g e t h e r under z e r o c o n t a c t l o a d , which a p p r o x i m a t e s t o t h e p r e s s u r e of 2 x lo3 Pa i n t h e e x p e r i m e n t s , and t o s u r f a c e s b r o u g h t t o g e t h e r under a c o n t a c t p r e s s u r e o f 2.5 x
lo4
Pa, which i s t h e same a s t h a t used i n t h e experiments.
329
v
1
contact presswe
X
E
o
\
0
0
0-
experiment theory
\
\
0
1-1
\" \
-1:
I
I
I
b
Fig. 5 . Comparison of calculated relative adhesion T / y results from Fig. 1. for the rubber of modulus 0.14 RPa.'
with the experimental
Considering the approximat ons that have to be used in characterising the surface the agreement between theory and experiment is reasonable, particularly for the case o f zero contact
oad.
The theoretical curves cannot be extended to
lower values of the roughness because the assumption of mechanically isolated asperity contacts (implicit in the use o f the analysis of ref. 5 ) would certainly be violated.
The curves are terminated at a roughness which gives a fractional
area of real contact of about 85% when the surfaces are brought together; above assumption must be breaking down at such a figure.
the
The relative adhesion
for the case of zero contact pressure is just greater than unity at the limit; unless the trends shown by the theory change completely, significant adhesion enhancement at low roughness would be predicted.
The physical basis for the
enhancement is the elastic energy lost when the asperity contacts finally sever. A theoretical curve extending to low values of roughness is shown in reference 11;
the details o f the assumptions made are, however, not given.
ADHESION ENHANCEMENT AT LOW ROUGHNESS Although a precise analysis with the above theory is not possible, some estimate o f the possible degree of enhancement due to the elastic energy l o s s mechanism is possible.
If the contact region becomes split into isolated zones
330 during separation, it can readily be shown (ref. 3) that, provided separation occurs under constant or increasing load, the work done, W area of contact is >, 2
f, where
P to the zones of intimate contact, and
per unit nominal
is the average peel energy applicable f
is the fraction of real contact when the
The peel energy, T is >, W provided that any elastic P’ energy initially stored in the contact is negligible; such would be the case if isolated zones develop.
the peel energy is much greater than the adhering energy or if any contact stresses were able to relax before separation was begun.
A value of
f
>
0.5 would ensure
some enhancement; a figure greater than unity is required to account completely for the observed values of up to 15077.
Thus i t appeaEs that some other mechanism
must also play a role in explaining the observations.
CONCLUSIONS The theory is able to predict the adhesion at high roughnesses reasonably well, and account for the influence of rubber modulus and initial contact pressure. The observed adhesion enhancement produced by slight roughening may be partly explained by the elastic energy lost when asperity contacts are finally broken.
REFERENCES 1 K.N.G. Fuller and D. Tabor, Proc. R. SOC. Lond. A, 345(1975)327-342. 2 G.A.D. Briggs and B.J. Briscoe, J. Phys. D : Appl. Phys., 10(1977)2453-2466. 3 K.N.G. Fuller and A.D. Roberts, J. Phys. D : Appl. Phys., 14(1981)221-239. 4 A.D. Roberts and A.G. Thomas, Wear, 33(1975)45-64. 5.K.L. Johnson, K. Kendall, and A.D. Roberts, Proc. R. SOC. Lond. A, 324(1971) 301-313. 6 K.L. Johnson in A.D. de Pater and J.J. Kalker (Eds.), Proc. IUTAM Symp. on the Mechanics of Contact of Deformable Bodies, Techn. Univ. Twente, August, 1974, Delft UP, Delft, 1975 p.26. 7 J.A. Greenwood and J.B.P. Williamson, Proc. R. SOC. Lond. A, 295(1966)300-319. 8 M. Barquins and D. Maugis, C.R. Acad. Sci. Paris, 289B(1979)249-252. 9 D. Maugis and M. Barquins, J. Phys. D : Appl. Phys., 11(1978)1989-2023. 10 J.F. Archard, Tribology International, 7(1974)213-220. 11 G.A.D. Briggs, in K.W. Allen (Ed.), Adhesion-5, Appl. Science, London,l981, p. 29-47.
331 DISCUSSION Lockwood - How were the roughened surfaces prepared ? Could the roughening process modify the surface chemically, and so a f f e c t the peeling energy ?
Fuller - The rough surface were made by abrading w i t h wet emery or silicon carbide =of various grades. One experiment w i t h roughened (ground) glass gave a relative adhesion that f i t t e d in very well w i t h the PMMA data (see ref.3). This fact suggests t h a t surface roughness is the predomenant factor rather than effects attributable t o surface chemistry.
ROUGHNESS AND ADHESION K.N.G.
FULLER
Malaysian Rubber Producers' Res. ASSOC., Hertford, England
ABSTRACT The effect of surface roughness upon the adhesion between rubber and a rigid substrate is analysed.
Except at low roughnesses the theory predicts the relative
adhesion,(the adhesion expressed as a fraction of that for a smooth surface) reasonably well, and accounts for the influence of the rubber modulus and of the pressure bringing.the surfaces together. Although an obvious reference surface for the theory is one consisting of asperities all of the same height, the necessity of using the same, smooth surface as the reference in both theory and experiment is emphasised.
The assumptions of the theory are shown to break down
with slightly roughened surfaces, against which soft rubbers give their maximum adhesion.
This enhancement compared to a smooth surface can nevertheless be at
least partially accounted for, though a precise analysis is not possible.
Provided
the contact region splits into isolated zones during separation, the work to peel the surfaces apart is increased by the elastic energy lost when these 'islands' of contact are finally broken.
INTRODUCTION An optically smooth rubber surface readily adheres to a similarly smooth substrate, whereas metals, even if highly cleaned and polished, show virtually no adhesion unless yielding occurs in the contact region.
The contrast in behaviour
can be ascribed to the effect of surface roughness upon adhesion, and the dependence of this effect upon the moduli of the contacting bodies. This paper describes how the influence of roughness upon adhesion may be analysed theoretically for elastic contacts.
Although much of the basic analysis
is applicable to other contact systems, the main concern here is the adhesion between rubber and a rigid surface, and the phenomena, such as the enhancement of the adhesion by slight roughening, particular to that system.
322
1
0
-1 A
Ig
-2 I
I
-2
-1
1
0.
(roughness average,
R, rm)
1
F i g . 1. Observed r e l a t i v e a d h e s i o n of o p t i c a l l y smooth r u b b e r a s a f u n c t i o n o f t h e roughness of t h e s u b s t r a t e ( r e f . 3 ) . Tp, t h e p e e l i n g e n e r g y f o r t h e rough s u b s t r a t e s , i s n o r m a l i s e d by y t h e p e e l i n g e n e r g y f o r a smooth s u b s t r a t e . The r a t e of p e e l i n g was lmms-l.PyRubber moduli: 0 0.14 MPa; A 1 . 5 MPa. C o n t a c t p r e s s u r e b r i n g i n g s u r f a c e s t o g e t h e r : 0 2 x lo3 Pa; 0 2 . 5 ~l o 4 Pa; A 5 x 1 0 4 Pa. EXPERIMENTAL OBSERVATIONS The e f f e c t o f roughness upon a d h e s i o n h a s been i n v e s t i g a t e d ( r e f . 1) by o b s e r v i n g t h e f o r c e t o s e p a r a t e a smooth s i l i c o n e r u b b e r s p h e r e from a roughened PMMA f l a t .
The r e s u l t s showed a s t e a d y d e c r e a s e o f a d h e s i o n w i t h i n c r e a s e d
roughness;
t h e d e c r e a s e o c c u r r e d a t lower roughness f o r r u b b e r s of h i g h e r modulus.
L a t e r e x p e r i m e n t s ( r e f . 2 ) confirmed t h e s e o b s e r v a t i o n s , b u t a l s o i n d i c a t e d t h a t s o f t e r r u b b e r s a d h e r e more s t r o n g l y t o s l i g h t l y roughened t h a n t o smooth surfaces.
T h i s enhancement i s s e e n ( F i g . 1) i n a d h e s i o n measurements o b t a i n e d by
r o l l i n g a n a t u r a l r u b b e r wheel down a n i n c l i n e d PMMA t r a c k ( r e f . 3 ) .
( I n Figure 4
o f t h a t p a p e r a p o i n t r e f e r r i n g t o t h e l i g h t c o n t a c t l o a d was m i s p l o t t e d . ) Although t h e d a t a f o r t h e s o f t e r r u b b e r s u g g e s t t h a t r a i s i n g t h e c o n t a c t p r e s s u r e h a s l i t t l e e f f e c t upon t h e d e g r e e of enhancement, i t does i n c r e a s e t h e a d h e s i o n a t more s e v e r e r o u g h n e s s e s . ANALYSIS The e n e r g y ,
YP’
t o p e e l a p a r t u n i t a r e a of a smooth p l a n a r c o n t a c t i s , i n t h e
case of r u b b e r , r a t e - d e p e n d e n t and g e n e r a l l y much g r e a t e r t h a n t h e e q u i l i b r i u m
323 surface energy,
yo.
The corresponding energy, ya, given up when surfaces adhere
appears to be almost rate-independent and the same as
yo (ref. 4 ) .
analysis more appropriate to rubber is presented, the case in which are both assumed rate-independent and equal to
y
Before an yp and ya
is considered.
Rate-independent Peeling Energy The adhesion between a rough and a smooth plane is calculated by assuming the roughness to consist of a random series of asperities all of tip radius, p, and with a certain distribution of heights.
The analysis (ref. 5) for the contact of
elastic spheres in the presence of adhesive forces is applied to each asperity contact.
The force-displacement relation for an individual asperity derived from
that analysis is:-6=
(1
{$ ( 5 + 1))
-
35)
5
= (1
6C
where
-
,
+.
p/pc)
The force, p, is normalised by pc =
Y3 E P
yo,
and the displacement
6 (which is the
sum of the deformations of the sphere and the plane at the centre of the contact circle) by 6c, 3
where v1
K =
P Pc 1-v
- -
and v 2 being the Poisson's ratio of the contacting materials, and El and E2 their
Young's modulus. under load
-
displacement
The form of the relation is shown in Figure 2.
It is seen that
controlled conditions separation occurs when p = pc, and under
-
controlled conditions when 6 =
Bc. The latter condition applies in
the present analysis as there are many contact points between the surfaces.
The
total force is evaluated by summing the contribution from all the asperity contacts according to their height distribution. The pronounced effect of the dispersion of the asperity heights on the adhesion was shown in an analysis (ref. 6) which approximated the height distribution by an exponential probability density.
The approximation, however, led to the unrealistic
result that the total force between the surfaces is independent of their separation. Experimental characterisation of roughness (ref. 7) suggests that the asperity heights can often be represented by a Gaussian distribution:n(z) = a(-$ 2n) N where n(z)dz and z
+ dz
exp
(-
6 2 )
*
is the number of asperities per unit area with a height between
2,
above the mean asperity height, u is the standard deviation, and N is
324
0-5
1
Fig. 2. Force-displacement relation of an individual asperity in contact with a plane surface for the case of a rate-independent peel energy which is equal to the adhering energy. The inserts illustrate the meaning of the displacement, 6 , for a rigid asperity against a low modulus plane. The roughness average R
the total number of asperities per unit area.
as
measured with a surface profilometer is approximately numerically equal to
a
(ref. 7). For a rough and a smooth surface coming together the force, Pa, per unit nominal area is obtained by summing the contribution from each of the asperities with a height greater than the planes’ separation, d , as measured from the mean asperity height.
where
h = d/a
,
Normalising all lengths by
A = 6
/a
,
Ac
=
Bc/a
a ,
the adhering force, Pa, is given by:
,
and p/pc = F(A / A c ) is the force-displacement relation for an individual asperity contact.
(Equation 2) differs from equation 11) of reference 1) because the
asperity displacements are here defined with the opposite sense).
The force-
distance relation for separating surfaces is different from that just derived for adhering surfaces.
During separation the rubber at an asperity contact is extended
, per unit out of its original plane by a distance up to BC. Thus the force, P SeP nominal area is given by equation 2) with the upper limit of the integral replaced by the lesser of
A c and (h
-
ho), where ho is the initial normalised distance
325
between the surfaces.
During the early stages of separation, P therefore see and hence the force that brought the surfaces together.
depends upon h
As the separation is increased, P
becomes positive (i.e. tensile force is see required to continue the separation), passes through a maximum (P ) and finally m tends to zero as contact with the highest asperities is broken. The adhesion may be defined by P , the pull-off force, or the energy, T required to separate unit P' nominal area of the contact under displacement controlled conditions. T is simply
P
>O. The relative see adhesion may be defined as P /Np or T /T where Npc is the force and T the m c P S energy required to separate unit nominal area of acontact with a surface of the area under the force-separation curve for the region P
u = 0, i.e. with a surface consisting of asperities all of the same height.
Both
definitions of relative adhesion depend only upon the parameter, a ( = u / 6c) and the contact pressure.
The two functions, shown in figure 3 for the case of zero contact
load, are very similar. a 30.5;
The effect due to contact pressure only exists fo;
moreover its magnitude is limited because additional asperities brought
into contact by a high pressure will be separated from the smooth surface before P becomes positive, the point from which T is reckoned. The 'adhesion see P parameter', a , is seen from equation 1) to be proportional to the roughness and, for a rubber-rigid substrate contact, to (rubber m o d ~ l u s ) ~ / ~ .The theory has been found to fit the experimental measurements for hard rubbers reasonably well;
the
relationship between the rubber modulus and the roughness for a certain level of adhesion appears to be of the right form (ref. 1).
No enhancement of the adhesion
at low roughness is predicted. Rate-dependent peeling energy An analysis more appropriate to rubber contacts must allow the peeling energy, to be rate-dependent. independent.
The adhering energy, ya, is still assumed to be rate-
The elastic contact analysis (ref. 5) is again used.
The introduction of a rate-dependent y
alters the force-displacement relation P for an individual asperity being separated from a surface. The new relation is determined by an approach given in ref. 8. The elastic energy, G, released when unit area of the interface unpeels is (ref. 9):-
where a is the radius of the asperity contact.
The motion of the edge of the
contact can be obtained by solving the differential equation:G = y
(A)
P if 6 is a known function of time.
The dependence of
unpeeling, A, is here approximated by the relation, ye
=
k An
,
y
P
upon the rate of
326
0
Adhesion Parameter o(
Fig. 3. Theoretical results showing the relative adhesion energy, T /Ts, ( - - , rate-independent and , rate-dependent peel energies? and the relative pull-off force, Pm/Npc, ( . . - * * , rate-independent peel energy) as functions of the adhesion parameter, a.
-
where k and n are constants;
this relation fits the observed rate dependence
With the displacement, 6 , and radius of contact, a, known, the
quite well.
force, p, on the asperity is calculated from the equation (ref. 5):p = (3PKa6
-
3
Ka ) / 2 P
Calculated force-displacement curves are shown in figure 4 ; depend upon the rate of separation, separation starts.
(=
6),
they are seen to
and the contact force from which
The loading curve, which is unique because
ya is assumed
constant, is included in the figure. A comparison (ref. 8) between calculated curves and the observed force-displacement relations indicated close agreement be tween them. The total force between the rough and smooth planes is obtained by summing the contribution from each asperity.
For surfaces brought together to a separation,
di, the number of asperity contacts with a displacement between 6 . and is:
6i + d 6 i
321
P
6 50 2
If
0'
0
9
P
2
4
Displacement, b ( 1O-2mm) -50
Fig. 4 . Force-displacement r e l a t i o n s of an i n d i v i d u a l r i g i d a s p e r i t y ( t i p r a d i u s 0.lmm) a g a i n s t a smooth rubber s u r f a c e f o r a rate-dependent peel energy. The loading curve ( ) and curves f o r s e p a r a t i o n a t 0 . 1 m s - 1 ( ) ) a r e shown; f o r each r a t e t h e curves f o r two d i f f e r e n t and 0.001mm~-~( s t a r t i n g p o i n t s ( A , B ) a r e p l o t t e d . Rubber p r o p e r t i e s : modulus 0.14 MPa; adhering energy 60 mJm-2; p e e l i n g energy = ( 5 . 0 ~104) H 0 . 3 4 , where B (ms-1) i s the r a t e of peeling.
-
---
-..-.
I f the f o r c e on each of these a s p e r i t i e s a f t e r t h e s u r f a c e s have been separated a distance (d
-
di) i s w r i t t e n a s p( 6 i , d
then the t o t a l f o r c e , Psep,
'
-
d . ) f o r a given r a t e of s e p a r a t i o n ,
during s e p a r a t i o n i s given by:-
The peeling energy, T
i s the a r e a under t h e P -d curve i n the region P >O; P' SeP SeP i t depends upon the r a t e of s e p a r a t i o n and, through d i , the load bringing the
surfaces together.
A s i n the previous s e c t i o n t h e r e l a t i v e adhesion may be defined by T /Ts, P where Ts i s t h e p e e l energy f o r a s u r f a c e with a s p e r i t i e s a l l of the same h e i g h t .
Tp/Ts,
which i s evaluated with both T
P
and T s l e f e r r i n g t o the same r a t e of
separation, i s p l o t t e d i n Figure 3 a s a f u n c t i o n of the adhesion parameter, a , f o r the case of s u r f a c e s brought t o g e t h e r under zero c o n t a c t load. a, i s determined using t h e adhering energy (ya
%
yo).
The parameter,
V i r t u a l l y t h e same curve i s
328 o b t a i n e d o v e r a wide range o f s e p a r a t i o n r a t e s and r u b b e r p r o p e r t i e s . i s very s i m i l a r t o those c a l c u l a t e d i n the previous section;
The c u r v e
a g a i n t h e r e i s no
a d h e s i o n enhancement a t low v a l u e s of a.
COMPARISON OF THEORY AND EXPERIMENT The r e l a t i v e a d h e s i o n h a s been o b t a i n e d from t h e t h e o r y by n o n n a l i s i n g T by P
Ts, whereas t h e e x p e r i m e n t a l r e s u l t s have been e x p r e s s e d r e l a t i v e t o
Because Ts i s ( t h e e n e r g y , w) ye. t o s e p a r a t e a n i n d i v i d u a l a s p e r i t y from t h e smooth s u r f a c e ) x ( t h e number of
T
i s n o t f o r a smooth s u r f a c e , i t i s n o t i d e n t i c a l t o
asperities).
If
ye i s r a t e - i n d e p e n d e n t and e q u a l t o
Yay
w = 0.64 A where A i s t h e a r e a o f c o n t a c t f o r a n a s p e r i t y under z e r o l o a d . must be less t h a n much l a r g e r t h a n
y p , s i n c e NA h a s t o be l e s s t h a n u n i t y .
ya, w
may be g r e a t e r t h a n A y
and s o T
In t h i s case T
I f , however,
ye i s
be more t h a n
P' one c a s e t h e e l a s t i c e n e r g y s t o r e d a f t e r t h e a s p e r i t y i s b r o u g h t i n t o c o n t a c t
In
s i g n i f i c a n t l y r e d u c e s t h e e x t e r n a l work r e q u i r e d t o s e p a r a t e t h e a s p e r i t y from t h e smooth s u r f a c e , whereas t h e i n i t i a l e l a s t i c e n e r g y makes a n e g l i g i b l e c o n t r i b u t i o n yP>>ya.
if
Although t h e e x p r e s s i o n o f t h e t h e o r e t i c a l r e l a t i v e a d h e s i o n by T /Ts h a s t h e
P
a d v a n t a g e of removing t h e a s p e r i t y d e n s i t y , N , a s a p a r a m e t e r , i t i s a p p a r e n t t h a t a comparison of t h e o r y and e x p e r i m e n t r e q u i r e s t h a t T be c a l c u l a t e d e x p l i c i t l y . P T h i s n e c e s s i t a t e s a knowledge o f t h e a s p e r i t y t i p r a d i u s , p, and t h e a s p e r i t y S u r f a c e p r o f i l e measurements ( r e f . 2 ) d e n s i t y a p p r o p r i a t e t o e a c h r o u g h n e s s , Ra. -3 s u g g e s t t h a t a v a l u e of p of 10 mm i s n o t u n r e a s o n a b l e f o r a l l t h e r o u g h n e s s e s t o be c o n s i d e r e d h e r e .
Nup
-
The v a l u e of N i s o b t a i n e d from t h e e q u a t i o n ( r e f . 10):-
0.04
u is numerically equal to R a' The r u b b e r p r o p e r t i e s assumed i n t h e c a l c u l a t i o n a r e t h o s e g i v e n i n Fig. 4 ;
along with the assumption t h a t
they were measured f o r t h e r u b b e r (modulus 0.14 MPa), t h e a d h e s i o n r e s u l t s of which a r e g i v e n i n F i g . 1. The t h e o r e t i c a l r e s u l t s , e v a l u a t e d f o r a r a t e of s e p a r a t i o n -1 a r e compared i n F i g u r e 5 w i t h e x p e r i m e n t a l o b s e r v a t i o n s o b t a i n e d a t a
of 0.lmms
,
p e e l i n g r a t e of lnuns-l.
The p r e c i s e r e l a t i o n between t h e r a t e a t which t h e p l a n e s
s e p a r a t e i n t h e normal d i r e c t i o n and t h e e x p e r i m e n t a l p e e l i n g r a t e i s u n c e r t a i n ; t h e f o r m e r , however, would be c o n s i d e r a b l y l e s s ' t h a n t h e l a t t e r i f t h e s u r f a c e s meet a t a s m a l l a n g l e a l o n g t h e l i n e of p e e l i n g .
The two t h e o r e t i c a l c u r v e s r e f e r
t o s u r f a c e s b r o u g h t t o g e t h e r under z e r o c o n t a c t l o a d , which a p p r o x i m a t e s t o t h e p r e s s u r e of 2 x lo3 Pa i n t h e e x p e r i m e n t s , and t o s u r f a c e s b r o u g h t t o g e t h e r under a c o n t a c t p r e s s u r e o f 2.5 x
lo4
Pa, which i s t h e same a s t h a t used i n t h e experiments.
329
v
1
contact presswe
X
E
o
\
0
0
0-
experiment theory
\
\
0
1-1
\" \
-1:
I
I
I
b
Fig. 5 . Comparison of calculated relative adhesion T / y results from Fig. 1. for the rubber of modulus 0.14 RPa.'
with the experimental
Considering the approximat ons that have to be used in characterising the surface the agreement between theory and experiment is reasonable, particularly for the case o f zero contact
oad.
The theoretical curves cannot be extended to
lower values of the roughness because the assumption of mechanically isolated asperity contacts (implicit in the use o f the analysis of ref. 5 ) would certainly be violated.
The curves are terminated at a roughness which gives a fractional
area of real contact of about 85% when the surfaces are brought together; above assumption must be breaking down at such a figure.
the
The relative adhesion
for the case of zero contact pressure is just greater than unity at the limit; unless the trends shown by the theory change completely, significant adhesion enhancement at low roughness would be predicted.
The physical basis for the
enhancement is the elastic energy lost when the asperity contacts finally sever. A theoretical curve extending to low values of roughness is shown in reference 11;
the details o f the assumptions made are, however, not given.
ADHESION ENHANCEMENT AT LOW ROUGHNESS Although a precise analysis with the above theory is not possible, some estimate o f the possible degree of enhancement due to the elastic energy l o s s mechanism is possible.
If the contact region becomes split into isolated zones
330 during separation, it can readily be shown (ref. 3) that, provided separation occurs under constant or increasing load, the work done, W area of contact is >, 2
f, where
P to the zones of intimate contact, and
per unit nominal
is the average peel energy applicable f
is the fraction of real contact when the
The peel energy, T is >, W provided that any elastic P’ energy initially stored in the contact is negligible; such would be the case if isolated zones develop.
the peel energy is much greater than the adhering energy or if any contact stresses were able to relax before separation was begun.
A value of
f
>
0.5 would ensure
some enhancement; a figure greater than unity is required to account completely for the observed values of up to 15077.
Thus i t appeaEs that some other mechanism
must also play a role in explaining the observations.
CONCLUSIONS The theory is able to predict the adhesion at high roughnesses reasonably well, and account for the influence of rubber modulus and initial contact pressure. The observed adhesion enhancement produced by slight roughening may be partly explained by the elastic energy lost when asperity contacts are finally broken.
REFERENCES 1 K.N.G. Fuller and D. Tabor, Proc. R. SOC. Lond. A, 345(1975)327-342. 2 G.A.D. Briggs and B.J. Briscoe, J. Phys. D : Appl. Phys., 10(1977)2453-2466. 3 K.N.G. Fuller and A.D. Roberts, J. Phys. D : Appl. Phys., 14(1981)221-239. 4 A.D. Roberts and A.G. Thomas, Wear, 33(1975)45-64. 5.K.L. Johnson, K. Kendall, and A.D. Roberts, Proc. R. SOC. Lond. A, 324(1971) 301-313. 6 K.L. Johnson in A.D. de Pater and J.J. Kalker (Eds.), Proc. IUTAM Symp. on the Mechanics of Contact of Deformable Bodies, Techn. Univ. Twente, August, 1974, Delft UP, Delft, 1975 p.26. 7 J.A. Greenwood and J.B.P. Williamson, Proc. R. SOC. Lond. A, 295(1966)300-319. 8 M. Barquins and D. Maugis, C.R. Acad. Sci. Paris, 289B(1979)249-252. 9 D. Maugis and M. Barquins, J. Phys. D : Appl. Phys., 11(1978)1989-2023. 10 J.F. Archard, Tribology International, 7(1974)213-220. 11 G.A.D. Briggs, in K.W. Allen (Ed.), Adhesion-5, Appl. Science, London,l981, p. 29-47.
331 DISCUSSION Lockwood - How were the roughened surfaces prepared ? Could the roughening process modify the surface chemically, and so a f f e c t the peeling energy ?
Fuller - The rough surface were made by abrading w i t h wet emery or silicon carbide =of various grades. One experiment w i t h roughened (ground) glass gave a relative adhesion that f i t t e d in very well w i t h the PMMA data (see ref.3). This fact suggests t h a t surface roughness is the predomenant factor rather than effects attributable t o surface chemistry.