Lubrication at the Frontier / D. Dowson et al. (Editors) © 1999 Elsevier Science B.V. All rights reserved.
67
Adhesion at single point contacts. J. A. Greenwood University Engineering Department, Tnm~pington St., Cambridge, CB1 6JX, U.K. With the advent of the Surface Force Apparatus (SFA) in which contact is between atomically smooth cleaved mica, and the Atomic Force Microscope (AFM), where an ultra sharp stylus is loaded against a plane, it is for the first time possible to study the behaviour of a single, welldefined, contact area. But because of the difficulties of the experiments, it is of great importance to study the contact mechanics of adhesive contacts in order to provide the experimenter with a theoretical background to help in interpreting the measurements. For many years the only theory available was the one developed by Johnson, Kendall & Roberts (JKR theory) in studying rubber/glass contacts. This allows for the influence of the surface energy of the solids in increasing the area of contact above the Hertzian value, and good agreement has been found with the observed contact areas. But because it is based on surface energy rather than surface forces, it is unable to deal with surfaces approaching each other, and already experiencing attractive forces while still out of contact: or with the 'jumpingon' of contacts, already observed in the rubber/glass experiments, and an important feature of SFA experiments. Three models based on surface forces are described here. One involves full numerical calculations for a particular law of surface force: the other two are analytical models based on greatly simplified laws of force. The Maugis model takes the surface force to be a constant when the gap between the surfaces is positive but less than a critical value h e , and to be zero for larger gaps: when the gap falls to zero the normal elastic laws apply. An alternative model, based directly on Hertz theory, is introduced here, and the results of all the models compared. 1. I N T R O D U C T I O N It is now well u n d e r s t o o d t h a t the roughness of engineering surfaces acts to hide from us the details of w h a t is taking place at the individual microcontacts: in engineering, the real area of contact will tend to be p r o p o r t i o n a l to t h e load w h a t e v e r t h e area/load relation at a microcontact. This is very convenient  except for the physicist t r y i n g to u n d e r s t a n d the m e c h a n i s m of interaction between solids, and so, indirectly, for us: for we too would like to u n d e r s t a n d what are these basic laws which are being 'averaged out' by the roughness? The first a t t e m p t s to eliminate surface roughness were tried in Bowden's laboratory in the 1950's. Bowden recognised t h a t when mica is cleaved, it produces flakes which are molecularly smooth over areas of m a n y mm 2, and Bailey & CourtneyPratt (1955) studied
the contact between two flakes bent into cylinders and loaded together at right angles. The contact area is a circle, j u s t as for a Hertzian contact between two solid cylinders, and its size and the shape of the gap around it can be m e a s u r e d interferometrically. The technique became really powerful when Tabor & Winterton (1969) and Israelachvili & Tabor (1972) learnt how to glue the mica onto glass cylinders to obtain something more nearly resembling a Hertzian contact (Fig 1). From the distances at which the mica jumped into contact, it was for the first time possible to determine directly how the forces between solids vary with their distance apart, down to gaps of a few nanometers. Now the [S]urface [F]orce [A]pparatus is a recognised tool used to study normal and tangential forces either between the mica surfaces themselves, or when the surfaces are coated with monolayers of different materials.
68
a) Molecularly smooth mica sheets about 3tun thick are g/tied to cylindrical formers of radius of curvatatrl lcm. = ~  ~ t m , dm~
mmd~
/ /
/
,I
/
b~ The upper surface h, mapported on a short metal spring. Tim lower m r a m ~ towanis it, first by a ~ w dmmd and tlmn by • p i m m ~ ' w i c transducer ,~t • c:rit~! m l m ~ t k m the t w o mufacu jump ~ .
Fig 1. Early form of the surface force apparatus (Tabor & W'mterton 1969). The S F A is a wonderful device,but it has a drawback: the mechanics of the contact depend on the properties of a) the mica ~ b) the glass substrate and c) t h e glue l a y e r  of u n c e r t a i n m a t e r i a l properties and thickness. The mechanics of adhesion between threelayer solids has been studied by Sridhar, Fleck & Johnson (1997),but the u n c e ~ t y about the properties of the glue layer remains a problem. The alternative way of studying forces between solidshas been betterpublicised,and most engineers know of the A F M and its extension, the [F]riction [F]orce [M]icroscope. Here a n incredibly fine tip mounted on a m i n u t e cantilever is brought near, or into c o n t a c t with, a plane surface, and the deflection of the cantilever measured as the
cantilever is moved normally or tangentially. The difficulties of i n t e r p r e t i n g the m e a s u r e m e n t s are d i s h e a r t e n i n g . A piezoelectric crystal moves the base of the cantilever, and the movement must somehow be calibrated: even the datum from which the separation is, hopeRdly, measured can only be defined by a kink in the force/separation curve: while the forces can only be deduced by measuring, and calibrating, the cantilever deflection. There is nothing comparable with the wavelength of light to set the scale, and there is no direct way of measuring the contact area: the experimenter can try passing a tiny electrical current through the contact and measuring, and interpreting, the c o n d u ~ c e , or can impose tangential vibrations and attempt to deduce, and interpreting, the contact stiffness. But at least the AFM tip is not too far from a Hertzian pamboloid, so that Hertzian contact mechanics, as modified by surface forces, provide a t h e o r e t i c a l background for the i n t e r p r e t a t i o n of the experimental measurements. 2. I. J K R theory. The simplest background was developed by [J]ohnson, [K]endall & [R]oberts (1971) many years ago when Kendall and Roberts in a project on the behaviour of windscreen wipers studied the contact between a rubber sphere and glass, and found the contact areas were considerably larger than predicted by Hertzian theory. They attributed this to the surface energy gained by the formation of a glass/rubber contact, and so available to increase the elastic strain energy. The original analysis used a global minimisation of the total energy (mechanical plus surface) as in Griffith's original analysis of cracks: but tater M a u ~ & Barquins (1978) showed that the standard fracture mechanics ar~,ment gave the same result more quicklyineidentany convincing me for the a r s t time that fraeVare mechanics, with its absurd concept of infinite stresses, was not the complete nonsense it appeared.
69
The Maugis & Barquins argttment is as follows" The H e r t z i a n stresses needed to flatten a sphere of r a d i u s R over a circle r < a are o ( r ) =  2 E ' ~]a 2 _ r 2 where E' is the plane zR strain modulus [ E / (1  v 2) ] of the material. For the contact of two bodies with plane strain moduli E~ a n d E~, E' m u s t be replaced by the contact modulus E* where l/E* =l/E~+l/E~, while R becomes the constant in the equation describing the gap b e t w e e n t h e two b o d i e s a t zero l o a d h ( r ) = r 2 / 2R . This can arise in m a n y ways, such as b e t w e e n two spheres provided t h e i r radii satisfy 1 / R = 1 / Ri + 1 / R2 or b e t w e e n two crossed cylinders each of radius R. Thus, the analysis below is general, b u t it is helpful to t h i n k of the contact as occurring between a rigid sphere and an elastic plane. The Boussinesq 'punch' solution w h e n a cylinder of radius a indents a halfspace by a d i s t a n c e 6 is t h a t t h e s t r e s s e s a r e E*b 1 o(r) =• so if two bodies are ~a 2 _ r 2 brought into contact and t h e n moved a p a r t by a distance b w i t h o u t c h a n g i n g the size o f the contact circle, the stress distribution will be or(r) = E* z
6  2~]a2 r 2 ~]a 2 _ r 2 R
infinite stresses (!?) a t r = a" E*6 1 1 or(r) . . . . . =
42a
4ar
with
[or

or
t} =_! 2 ~ a E*
Ay
Since the Hertz load is
W = ~3E'a3 / R and
the Boussinesq load is W =  2 E * a 6 , the load corresponding to a contact of radius a will be W = ~3 E'a3 / R  2 E ' a _ ! 2~aA7 = E* = ~ E*a3/R
 {8zrE*a3AT} 1/2
This is conveniently w r i t t e n in t e r m s of the fictitious H e r t z i a n load which would give this contact a r e a in the absence of surface energy, W~ = ~3 E*a 3 / R,
w = Wa
{6RAr Wa}1/2
The overall approach of the bodies is the H e r t z a p p r o a c h reduced by the Boussinesq q.itt;': 6 = a 2 / R  ~ 2 z a k 7 / E* It win be clear from these t h a t the usual Hertz equations a p p e a r as the high load limit: b u t a t l o w e r loads t h e r e are i m p o r t a n t differences. According to the J K R theory there will be a finite contact a r e a u n d e r zero load: W
= 0 gives
=
{6 Rar
W ~ o = 6 ~ R A 7 and a ~ = ~ R 3W _ O ~ 3W~
N
4ar
K=~(ZE*Ay)
= 4E'A7
•
},/2
so t h a t
2A7 / E * ;
and there will be a finite pulloff force:
W~
where N is the 'stress i n t e n s i t y factor'. [The more common notation K includes an additional factor 0.399 whose purpose or point I hope someone will explain to me. ] If the energy needed to create unit area of free surface on both bodies is A y, a n d the process is reversible so t h a t the energy gained by b r i n g i n g two surfaces into contact from infinity is also A~, t h e n b o r r o w i n g Irwin's principle from f r a c t u r e mechanics" t h a t the s t r e s s i n t e n s i t y f a c t o r N is g i v e n b y N=~(E*Ay /z),
E*6
or(r) ~
then
] ,
~  3
W~ =3 zRA~,, 2
a n d
~RAy with a ~3  ~ R E A y / E * . 2 J K R found t h a t the complete e x p e r i m e n t a l l o a d / a r e a curves fitted t h e t h e o r y u s i n g a plausible value for the surface energy A y. It is worth noting t h a t the concept of an 'adhesion force' m u s t be t r e a t e d with care" the force of adhesion at p u l l  o f f is 1.5zRAy, b u t it continuosly varies, with the value 6 z2~A7 at zero load. The approximation to the Derjaguin model suggested by Maugis, t h a t the adhesion force should be t a k e n to be a c o n s t a n t , 2 z2~h ~,, has no physical basis.
70
value h,, when they fall abruptly to zero: the 'Dugdale model' of fracture mechanics. Then of course
2.2. The Maugis model. Although the original J K R analysis did not need to adduce the fracture mechanics principle, it does equally imply the existence of infinite stresses: a n d a surface e n e r g y approach cannot avoid this. For a more logical theory we need to abandon the use of surface energy and consider the actual surface forces. C e r t a i n l y we m u s t do this if we wish to analyse the b e h a v i o u r before contact takes place, and both the general theory of solids and the Tabor & Wintert~n experiments show t h a t forces act b e t w e e n solids w h e n t h e y m e r e l y a p p r o a c h each other: c o n t a c t is unnecessary. The surface f o r c e s  we refer to t h e m as 'forces' since they are indeed forces acting between individual atoms, but we shall treat t h e m as s t r e s s e s  are a function of the gap, or separation, between the solids, and we have
Ar = ao'h The Maugis stress distribution, like t h a t in the J K R theory, consists of two parts:
Hertz pressures (~(r) =  2E* ~/a2 _ r2 zR over r < a, and adhesive stresses, which are equal to o0 over a < r < c (where h ( c ) = h~ ), and the associated internal stresses such t h a t the action of the complete set of adhesive stresses does not u p s e t the f l a t t e n i n g over r < a produced by the Hertzian s t r e s s e s  a r e q u i r e m e n t u n f o r t u n a t e l y overlooked by Derjaguin et al (1975, 1983) in their analysis. Maugis proves t h a t these i n t e r n a l adhesive stresses m u s t be a(r)=
2 ac ° tza n zl ( ~
a2 a2)r2
o0
A7 =fh_oo(hldh_ It is not at a]I d e a r t h a t the force between u n i t a r e a of plane solids equals the force between u n i t a r e a of curved solids, or even t h a t b e t w e e n inclined plane solids: t a k i n g t h e m to be e q u a l is t h e ' D e r j a g u i n approximation', a n d it is h a r d to see how progress could be made without it. It m a y be noted, w i t h some relief, t h a t B r a d l e y ' s calculation of the force b e t w e e n two rigid s p h e r e s ( B r a d l e y 1932), not u s i n g t h e Derjaguin a p p r o x i m a t i o n , gives the same answer as is found (very much more readily) by using it: b u t the curvature and inclination in the r e l e v a n t a r e a are in this case both rather small. Realistically we expect ¢l(h) to decrease from some m a x i m u m value Clo a t h = 0,
perhaps like (h + Zo)3, but while a numerical analysis m a y incorporate such behaviour, the aim here is to provide a simple analytical model, i d e a l l y w i t h e q u a t i o n s no m o r e complex t h a n the J K R equations. Maugis (1992) has provided j u s t such a theory, by postulating t h a t the surface forces retain the value cro u n t i l the gap reaches a critical
Fig 2 compares t h e J K R a n d Maugis stress distributions, for different values of a (7 0
parameter # = (E.2Ay / R) 1/3 • For # ~ 5 the stresses are very close to the JKR stresses,
II jkr iI ii ~ = 3
¢q
<1 O4
UJ
rr"
2
D u) ........
=5
0 1 0
1
2
radius r*
Fig 2. JKR and Maugis stress distributions for different values of f,.
3
71
course the ' r < c' stresses m e r e l y flatten the surface out to r = c.
r e m a i n bounded by the physical limit (70 .[The quantity plotted
is
(
(7 R~
E*2Ay
= #(7 (7 0
Setting
a n d is b o u n d e d by /~ ]. The l o a d / a r e a a n d load/approach curves are also very close to the JKR curves.
acr(r) drdhdr
completes the analysis: it leads to E* 2 z Ay  k(1 + k)~~z(ca) (c + 2 a ) with
cro _ 2 kE* ~/c2 _ a2 zR The b e h a v i o u r is conveniently found from these equations, j u s t as in the Maugis' theory, by s e t t i n g c =ma a n d t r e a t i n g m as a parameter: t h e n kz z____~zf t3 (m 1) 1/2 (m + 2) l+k12 ( m + l ) 3/2
2.3. The DoubleHertz model. H e r e we show t h a t a n a n a l y s i s m o r e familiar to tribologists leads to very s i m i l a r r e s u l t s ( G r e e n w o o d & J o h n s o n 1998). The a s s u m p t i o n t h a t (7(h) = (70 h a s no physical basis; so we a b a n d o n it for a n e q u a l l y u n p h y s i c a l law: one t h a t will a r i s e in t h e course of the analysis. We s t a r t by n o t i n g t h a t two H e r t z i a n pressure distributions

where
E.ZA~,
/~ = ao
/~ is a form of the T a b o r p a r a m e t e r (Tabor 1977) which d e t e r m i n e s the b e h a v i o u r of the system, a n d in p a r t i c u l a r w h e t h e r the range of action of the surface forces is large or small c o m p a r e d to t h e gap b e t w e e n t h e solids. C l e a r l y t h e w o r k done in s e p a r a t i n g the surfaces to infinity can only be r e l e v a n t if the surfaces are, effectively, s e p a r a t e d to infinity.
~ ( r ) =  2 E* ~/a 2 _ r 2 and zJ~ o ( r ) = _ ___2E*~]c z _ r z
zR
will each f l a t t e n t h e c o n t a c t i n g bodies, b u t over circles r < a a n d r < c respectively. Hence or(r) = 2E* gr~ { 4(c2
~Jo c~(h)dh=
A~'
 r2 )  ~](a2  r2 ) }
[
where ( ) indicates t h a t the t e r m should be set to zero if n e g a t i v e , will not c h a n g e t h e c u r v a t u r e over r < a, a n d so can p l a y the same role as t h e Maugis adhesive stresses. A scale factor k will not disturb this, so we m a y choose k to m a k e the g r e a t e s t stress
.
~R [ k~](c=
r2)(l+k)~(a
2
r 2)
.
.
.
.
.
.
.
.
.
.
.
.
b
.
.
.
II
.
:
..............
il . . . . . . .
:lu=2
:
II
.. . . . . . . . . . . . . .
.. . . .

O
...........
Mau.g]s"
b
0.5
] 0.5
T h e r e will be no gap for r < a, while for
a
.
1.5
o ( a )  2kE* ~/c z _ a 2 equal to G 0. Adding ~R the (unscaled) H e r t z i a n stresses over r < a, the complete stress distribution will be cr(r)=2E*
.
.
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
.
.
2
.
.
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.
.
.
.
.
.
.
.
.
3
radius r* Fig. 3. Stress d i s t r i b u t i o n s at zero load for JKR, Maugis and DoubleHertz theories.
72
Fig. 3 compares the stress distributions at zero load for the JKR, M a u g i s and DoubleHertz theories, for a n i n t e r m e d i a t e value t t  2: for l a r g e r values of ft it becomes increasingly h a r d e r to see the differences. The different assumptions about the stress at the contact edge have a negligible effect on the contact radius, or on the stress distributions away from the edge. [For ft  2 the stresses at r  0 for t h e t h r e e m o d e l s a t zero l o a d a r e r e s p e c t i v e l y 0.5131 c~0, 0.5150 (~o and 0.5158 (~o ]. Fig. 4 compares the puUoff forces from the three models, and brings out the role of ft very clearly: the J K R value Tn~ ffi 1.5 ~ A $ is the a s y m p t o t e for high ft, and is a good approximation for ft ~ 5: at the other end, the value calculated by Bradley for a rigid sphere, Tm~ = 2 z ~ A y is the a s y m p t o t e as ft * 0, and is a good e s t i m a t e for ft < 0.1. But of course it m u s t be recalled t h a t the value of the
:
:
:
:
:
::!
:
! !:i;
0.95
N
....
:
:
:
:
I~adle7:.
i: : ii!i!i :: ::::
:
::!
: " :..:
:
:
:
:
'
::..;..::.:
:
:
i iiiii
decreases when h increases in a physically realistic way, it is not actually a physical law at all. Fig. 5 shows however t h a t it is very n e a r l y one: even w h e n the contact is "all f r i n g e "  a = 0  the difference from the law for c ~ a is minor, and in all cases it does a p p r o ~ m a t e to realistic physical behaviour.
:::
" ;iiiii ..... ii iiiiiii\~dHertz ...........
0.9 ..:.i. i i. ii.Maugis.~i
surface e n e r g y AT' is r a r e l y k n o w n to an accuracy of 25%, so it is h a r d l y possible to d i s t i n g u i s h b e t w e e n t h e s e two l i m i t s e x p e r i m e n t a l l y . Note t h a t while it is no surprise t h a t the Bradley value is independent of the elastic properties, since these were i g n o r e d in t h e c a l c u l a t i o n , it is m o s t remarkable t h a t the JKR value should be! But w h a t surface force/separation law does the Double Hertz t h e o r y use? This is its weakness: the Maugis t h e o r y m a y use a n unrealistic law, b u t at l e a s t the force is a function of the separation, even if only in the rudimentary form o ( h ) = ~o for h < h~. In the p r e s e n t model (~ = f ( r ) : so a l t h o u g h it
i i i : ::::
0.9 O8 o0.7 .~ 0.6
X
~0.85
!
.
:!!!!.
. .
...... " "i'i" ..... !i" " i" i'~i~;
!!.!!i.
GreenwOOd : : :
Len ini?:._~,,p~~i
.~ o.5
i ....... :...
.0.4
0.8
i iil ..... !!!!
iiii i
:
!!!!!i.!
4
0.3
"
0.2
0.75 .
.
.
.
.
.
i
10 1
.
.
.
.
.
.
I
100
,
.
.
.
.
.
.
0.1
.
.......
i ...............
101 0
L
0
0.2
0.4 gap
Fig. 4 Pulloff forces from the different models. The asymptotes are the JKR value TH~ ffi 1.5 ~RAy for high fe and Bradley's rigid sphere value, Tmax = 2 yrRA~/ as ft ~ 0. The transition between the two is similar on Maugis, DoubleHertz and LennardJones models.
0.6 h / h
0.8
1
C
F i g . 5. D e p e n d e n c e o f s u r f a c e f o r c e o n separation in the DoubleHertz model. Although the model postulates a dependence on radial position, this is almost equivalent to a dependence on separation. Even when the contact is"all fringe"  a = 0  the difference from the law for ¢ ~ a is minor.
73
2.4. A realistic force/separation law. When two a t o m s are moved a p a r t , a tensile force develops between them. When they are moved them together, a compression force attempts to keep them apart. The same holds between two planes of atoms, or between two solids in general: t h e r e is no physical reality to our division into contact, in which elastic contact stresses exist, and noncontact, in which there are tensile surface forces. For a van der Waals' solid, the law of force is often taken to be
{(
3 < 9}
w h e r e Z is t h e d i s t a n c e b e t w e e n t h e outermost planes of atoms of the two solids: we shall examine how to relate it to the separation h between the solids below. z = zo represents the equilibrium spacing. The maximum surface force is when z = zo31/6 and is 2 A / 3 , ~ " we set this equal to Go. Compressive stresses correspond to z < zo" tensile forces to z > zo. But does z > zo imply a gap between the solids? In other words, when have we separated the solids? S t a t e d thus, this is a philosophical question a n d should simply be ignored: though we note t h a t in an ordinary tensile test we should certainly not r e g a r d the tensile specimen as broken when the force/extension curve is still rising. The meaningful question is, how can we best match this model to the ones used in the analytical theories? In the J K R model, the surfaces are in contact over the whole of the stressed region, even though the stresses there reach (tensile) infinity. In both finite stress models, the stress attains its maximum value inside the contact region: it then either remains at a0 or decreases in the 'cracked' region outside the contact. To m a t c h the models, we s h o u l d t a k e t h e gap to be h = z  z03~/6, and write the force law as o0
°=
f _._ ~(h)= 2
e
h+e
w h e r e we have w r i t t e n
e
h+ e
e  z0 3~/6 =1.2Zo;
then tensile contact stresses will exist as the atomic planes are moved a p a r t from z = z0 to z=l.2z0 [h=0.167e toh=0.] Then a b e t t e r model of contact between a sphere and a plane is to regard the solids as elastic continua up to the final plane of atoms, and as force fields above this plane; these force fields being such t h a t when the final planes of atoms are a distance Zo apart, no force will act  j u s t as if the prospective surface were merely an internal plane within the solid. The numerical problem is to find a contact stress distribution or(r) which deforms the final planes of atoms, according to the ordinary elastic laws, to leave a gap h(r) such t h a t or(r) ffi f ( h ( r ) ) , where f (h) is the assumed force law. This proves to be straightforward, using the obvious iterative process, for/~ < l, and to be d i s t i n c t l y h a r d w o r k for /~ > l , b u t solutions have been obtained up to tt = 5 (Greenwood 1997). Typical stress distributions and shapes are shown in Fig 6. For the higher values of tt, there is a welldefined 'contact area', in which the gap varies only from h   O . 0 5 e up to h = 0, while outside this region the gap quickly increases to large values while the force quickly drops to small values  a picture not very different from the analytical models, and which encourages a belief t h a t it would be possible to identify a 'contact area' experimentally. But for small values of tt the picture is quite different: nothing in the shape or the stress distribution points to a contact area or a contact edge. Our definition (" contact is w h e n h < 0") still holds but it is not clear t h a t we should trust it. Over what area of the surface in Fig 6d should we expect friction forces to act opposing motion? Is it possible to define a frictional shear stress when the friction force m u s t be divided by an apparently arbitrary area ? The safer q u a n t i t i e s to examine, and perhaps the more interesting ones, are the load/approach curves. Fig 7 shows the results of the different theories for tt = 1. There is an extensive region in which all four solutions agree well, the curves having similar shapes
74
i
!
.8
................
!
:._.~+,
~
i
I
. . . . . . . . . . . . . . .
i
!
=!2
.... i ....... i.......
o.+
~.,,~./~IF
o
D
0.6
t3
......
e0.4
, ~._2!55 0.2
"~
~ 0.5 1
........ i. ..........
i
0.5
1
1.5 radius r*
2
2.5
3
0
2
(a) Pressure
;
6 . . . . . . ~ =0i2 : .........
..... '~i=2 .... i....... i ....... i ....... i~' 8
6
8
:
! .............
,,...
/
5
o
O
N
N
.......
6
. .......
.......
C2. e"
4 radius r*
(c) Pressure
10
N
i ..........
o~.2.55].. : i / :
.......
4
.......
....... ,~., , . ,
]
f
./
....
.....
~4
........
//
N
J
~=1 o. 3
Xi=274 :~_::.
.....................
:.I: I /
:
j
. . . . . . . . . .
. . . . .
,~
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
0
0.5
1
1.5
radius r*
(b) Shape
2
l
•
: 2
0
]"
e
......
¢~
:
/
: .~
2.5
0
3
:
"
:
. . . . . . . . .
: . . . . . . . . . . .
.
.
4
6
!
0
2
radius r* (d) Shape
Fig 6. Typical s t r e s s d i s t r i b u t i o n s (a,c) and s h a p e s (b,d) from the full n u m e r i c a l solution. W h e n a 'neck' exists, as in Fig 6b, a n d t h e 'separation' is a l m o s t c o n s t a n t w i t h i n t h e neck, t h e a r e a over w h i c h friction forces would act opposing m o t i o n seems d e a r . B u t over w h i c h a r e a in Fig 6d should we expect friction forces to act ?
8
75 1
iv,
0.9
.
o.8
. . . . . . . . .
0.7
.
p=l .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
:
.
. . . . . . . .
+ + + Greenwood i
. . . . . . . .
I+.z~,.,. ::.' .' .'' "i:, ,.]~Her!z.. +,
.
•. . . . . . . . .
....
................. .
I~ I: ~
v
. . . . . . . .
.
<1
~ 0.6
.
!
.
.
+" .
.
..... .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.5 I
"~ 0.4 0
0.3
.:./.
...... . . ....... . . . . . ! ........... .....
0.1
•
02
: ....... 1
,
......
I .............. " ""i' Maugis • .¢,,4:.4= 0
1
2
considering a small p e r t u r b a t i o n t h a t P 2 will be an u n s t a b l e e q u i l i b r i u m point: P 1 a n d P3 stable ones. As the tip is b r o u g h t towards the surface (point R far to the right), there will only be a single equilibrium point P 1, a n d as R moves left, equilibrium will move along the curve until point J1 is reached. For a n y f u r t h e r approach the only equilibrium point is n e a r E 1 and there will be an u n s t a b l e j u m p into contact, with a large change in the cantilever deflection (recall t h a t in the AFM this is all t h a t can be detected). Unloading will be an equilibrium process until point J2 is reached: b u t t h e n the tip will "jumpoff'to point E2.
separation  a
Fig 7 load/approach curves for/z ffi 1. and values, always in the order JKR"
Maugis"
DoubleHertz" Numerical
B u t w h e n the 'approach' is n e g a t i v e a n d the bodies are well s e p a r a t e d , t h e n u m e r i c a l solutions show the long r a n g e a t t r a c t i o n which the others omit. The best a t t e m p t to reproduce this is the B r a d l e y rigid sphere model, for a t large s e p a r a t i o n s the elastic deformation is a secondary feature.
/", /
".. ',
".. t ',
3. J U M P I N G O N A N D O F F . W h a t will h a p p e n w h e n a contact of this sort is m o u n t e d on t h e end of a cantilever, as in the AFM? The s i m p l e s t w a y to r e p r e s e n t the s i t u a t i o n is u s i n g a 'loadline d i a g r a m ' as in electrical circuitry (Fig 8). The b e h a v i o u r of the contact is plotted, as a curve T = F ( u ) where T is t h e t e n s i l e load  W a n d u t h e r e v e r s e d a p p r o a c h  a ( a =  h ( 0 ) : on t h e same d i a g r a m the cantilever response T ffi Kx is plotted w i t h t h e correct stiffness K b u t d r a w n b a c k w a r d s from a p o i n t R w h e r e the combined gap is (x + u). At a n y i n t e r s e c t i o n P1, P2, or P3 t h e forces on c a n t i l e v e r a n d contact will be the same: OQ will correctly r e p r e s e n t the contact "approach" a n d QR the cantilever d i s p l a c e m e n t . It is e a s y to see by
g=t:, ( x ~ l
Fig 8. Loadline diagram for d e t e r m i n i n g jumpingon andoff. The behaviour of the contact is plotted, as a curve T = F(U) where T is the tensile load  W and u the minimum gap h (0): on the same diagram the cantilever response T ffi Kx is plotted with the correct stiffness K but drawn backwards from a point R where the combined gap is (x + u ). At any intersection P1, P2, or P 3 the forces on cantilever and contact will be the same: OQ will correctly represent the contact "approach" and QR the cantilever displacement. P 2 will be an unstable equilibrium point: P 1 and P 3 stable ones.
76
Note t h a t the force at the j u m p will be less t h a n the m a x i m u m value: only with a very sot~ spring will it be close to the maximum. [The a c t u a l m e c h a n i s m of the j u m p s is w o r t h some c o n s i d e r a t i o n : for only t h e i n t e r s e c t i o n s of t h e c u r v e s a r e p o s s i b l e equilibrium points. It seems likely t h a t the inertia of the contact is small, so t h a t its state will always be on the static curve, b u t t h a t the cantilever will have a finite inertia, so its state need not lie on the static curve. In a jump, therefore, the state will r u n round the curve for t h e contact, o v e r s h o o t p o i n t E l , a n d oscillate a b o u t this point until the s u r p l u s energy has been d a m p e d out. This raises the question of w h a t h a p p e n s once the contact curve has a vertical tangent, as happens in the numerical calculations for tt a 0 . 9 , and for all cases in the analytical models.]
curves of load/contact r a d i u s seem to r u n parallel to the J K R curves r a t h e r t h a n to r u n in to them. However, Fig 9 also gives the results of the full calculations for the LennardJones force law, and these also r u n parallel to, and close to, the J K R curve. It seems t h a t the correct i n t e r p r e t a t i o n is t h a t t h e different models are a l r e a d y in a g r e e m e n t for tt  l: the problem is t h a t n e i t h e r the i n n e r radius a nor the outer radius c properly corresponds to the 'contact radius' I n d e e d , in A F M e x p e r i m e n t s by L a n t z et al (1998), it was found t h a t good a g r e e m e n t b e t w e e n t h e Maugis radii and the e x p e r i m e n t a l r a d i i a s deduced from the tangential stiffness could be obtained by t a k i n g the theoretical radius to be b = a + 0.4 x (c  a)  which is also the rule for matching the Maugis and JKR theories.
4. D I S C U S S I O N .
Tabor's parameter
it,
defined
here
as
1/3
3
#
=
cr0 E.2A 7
, was i n t r o d u c e d as t h e 2.5
ratio of the 'neck h e i g h t '  e s s e n t i a l l y the a A7  t o the range Boussinesq lift 6 = J 2 z E*
*o
2
.'.... /..~..,
_ #.~
. . . . . . . . .
~.~".....~..~
: ..........
.
. ...........
i
of action of the surface forces. The m i n i m u m value of lift 6ram is the value at pulloff, when
a  a~  [9~sR2Ay/E*~/3
#ving
6n~n = [ 3 z 2 R A y Z / E * 2 ] 1/3" so the surface
~ 1.5
1 0.5
e n e r g y a p p r o a c h will be valid near p u l l  o f f
when 6mm = [3 z2RA)'2 / E*2] 1/3 > 3h~, where the 3 on the RHS is r a t h e r arbitrary. Setting h~ = A7 / ~1o this gives, approximately,
cro R~ E .2A7
= ~ > 1
At larger loads, a a n d 6 will be l a r g e r , a n d we m i g h t exoect t h e s u r f a c e e n e r g y approach should be valid for smaller values of /~. This does not seem to be the case: for both the Maugis and the DoubleHertz models, the
0 1
0.5
0 0.5 load W / 2~:RA7
1
Fig 9 Variation of contact radii with load f o r p ffi 1. The radius in the LennardJones model, defined by the maximum in the force/separation curve, is always close to the JKR radius. Except near pulloff, these lie between the inner and outer radii (a, c) of the Maugis and DoubleHertz models.
77
5. CONCLUSION. The JKR t h e o r y gives a r e m a r k a b l y good description of singlepoint contact over a wide range of conditions: surface energy is indeed the determining quantity, and the details of the variation of the surface force with the gap are of secondary importance once contact has occurred. But it cannot take into account the very real interactions before contact occurs: and for some m a t e r i a l combinations, and p a r t i c u l a r l y for small contacts, it is very difficult even to define 'contact'.
K. L. Johnson, K. Kendall & A. D. Roberts (1971) Proc. Roy. Soc. London A324 p301313. Surface energy and the contact of elastic solids. D. Maugis & M. Barquins (1978) J Phys. D: Appl. Phys. 11 p1989 2023. Fracture mechanics and the adherence of viscoelastic bodies. R. S. Bradley (1932) Phil. Mag. 13 p853862. The cohesive force between solid surfaces and the surface energy of solids. D. Maugis (1992) J. Colloid and Interface Science 150 p243269 Adhesion of spheres: the J I ~  D M T transition using a Dugdale model. B.V.Derjaguin, V.M.Muller and Yu.P.Toporov (1975) J. Colloid Interface Sci 53 p314326 Effect of contact deformations on the adhesion of particles V. M. Muller, B.V. Derjaguin and Yu. P. Toporov (1983) Colloids and Surfaces 7 p251259. On two methods of calculation of the force of sticking of an elastic sphere to a rigid plane.
REFERENCES.
A I Bailey & J S CourtneyPratt (1955) Proc. Roy. Soc A227 p 500The area of real contact and shear strength of monomolecular layers of a boundary lubricant.
J A Greenwood & K L Johnson (1998). To appear in J Phys D: Appl. Phys. (see Cambridge University Engineering Department report CUED/CMech/TR75, March 1998). An alternative to the Maugis theory of adhesion between elastic spheres. D Tabor (1977) J Colloid and Interface Science 58 p213 Surface forces and surface interactions.
D Tabor & R H S Winterton (1969) Proc. Roy. Soc A312 p 435The direct measurement of van der Waals ' forces
J A Greenwood (1997) Proc. Roy. Soc. A 453 pp 12771297 Adhesion of Elastic Spheres.
J N Israelachvili & D Tabor (1972) Proc. Roy. Soc A S l p 19The measurement of van der Waals ' dispersion forces in the range 1.5 to 130nm.
M A Lantz, J O'Shea, M Welland & K L Johnson (1995) J. Vac. Sci. Technol. B13 p 19451953. An atomic force microscope study of contact area and friction on NbSe 2.
I Sridhar, K. L. Johnson and N.A. Fleck (1997) J Phys. D: Appl. Phys. 30p17101719. Adhesion mechanics of the surface force apparatus.
78
A p p e n d i x : S u m m a r y of e q u a t i o n s .
Approach
di = a 2 / R  ,d2~aA7 / E" All 'horizontal' distances are scaled according to *[ R2Ay
afa
l/3
,
rr
E*
_[
[ R 2 A y ],,3 "g"
or
~"

a "2 
42~a"
~rertical' distances are scaled according to Maugis e q u a t i o n s .
The Tabor p a r a m e t e r is defined as 1/3 /.t
ffi
The surface energy relation becomes ,2 a* = a l~fl(m)+ 4 /z2f2(m)
R
t70
'~2by
where m
c / a,
=
fl (m) ffi ~/m 2  1 + (m 2  2) * sec 1 m,
JKR equations.
f 2 ( m )  ~ m 2 . 1 * sec cx(r)
=
cr(r) I
% =
E*
[J2=aAy
1
[~ E
2 ~/a2
4aZ_r 2
R
r2 ] 
and
~.~ ,={~.~~ }"~
W= = ~3E'a3 / R:
cx( r)
R~ar = ~
,2
0 gives
s
W,,o =
ao 3 = ~2t t a ~ = ~ R
~r
3
[W=o
=
6zRAy
]
and *
~
3~ Pulloff force:
and
1
m]
1.
1

4a2_r 2
~4m2  1 +a~ 2
2A1,,/E*]
W
OW OW*

sec
4a
2 
r 2
~
DoubleHertz equations.
Zero load =
2Cro arctan
=
2
+m
a'~
w. . w: _ {~ . w: } ~ W
2 a ,3  /~ a .2 [ ~ m 2 _ i l 3~ , ,2 a* 6 a 2/~ ~m 2
W = W=  { 6~RA ? . W=} '/2
W'~ = Wa / { 2 zJ~Ay } = [email protected]'a~ /
mm+l.
~l'nen W * ==
d,[_g__ [ lld ~ a"_~;  ~~ "/a'~  r'~ ]
1
and
=O=,,W~ =3~RAy
6
,
a
•
a
3 ,2
• / 2
qm
+ 1) / [ ( m  1
+ m + 1)
ma] +1
1
2
[Wa=3/41
W~f3:rr~Ay [W'f3/4]with 2 3 ao = ~tR2Ar 8 /E" a " 3 = ~ t .
(m
~ l~a
8
2
2 _ 3 ( m
or(r) . where
cro_az [ '~c2  r 2 (l + k ) ~ l a 2  r 2 ] k k = ~
/~
2 ,dca _ a ,2