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Langmuir 1996,11,679-682
Extension of the Johnson-Kendall-Roberts Theory of the Elastic Contact of Spheres to Large Contact Radii D.Maugis Laboratoire des Matdriaux et des Structures du a n i e Civil, Unitd Mixte CNRS-LCPC, UMR 113,2 a116e Kepler, 77420 Champs, France Received August 1, 1994. I n Final Form: October 17, 1994@ As with the Hertz theory, the Johnson-Kendall-Roberts (JKR)theory of the elastic contact of spheres with adhesion uses the parabolic approximation for the profile of the sphere which is only valid for small contact radii. Recent experiments on adhesion of small particles on soft elastic substrates have revealed that the contact radius under zero load can be rather large and does not vary as the particle radius to the 2/3 power. The JKFt theory is extended by using the exact expression for the profile of the sphere and the results are in good agreement with published experiments.
I. Introduction As with the Hertz theory, the JKR theory of the elastic contact of spheres with adhesion1 uses the parabolic approximation for the profile of the sphere, which is only valid for small contact radii. Recent experiments on adhesion of small particles on elastic substrate^^-^ have revealed that the contact radius under zero load can be rather large and does not vary a s the particle radius to the 2/3 power, as required by the JKR theory, but rather to the 3/4 power. 11. Theory As in refs 5-7, we will make use of the Sneddon equations*for the problem of the frictionless axisymmetric punch indenting a n elastic half-space. For a punch whose profile is given by fie) (a is the radius of contact, e = rla, and f i 0 ) = 0), Sneddon has shown that the depth of penetration (The depth of penetration is positive if the tip of the punch is below the level of the half-space.) 6 of the tip in the elastic half-space (Young modulus E, Poisson ratio Y ) and the loadP that must be applied to achieve this penetration are given by
P = -J’x(t) naE 1-v2
dt O
where
=~ ( 1 sin-’ )
e
-
L’x’(t) sin-’
Furthermore, the distribution of pressure c@,O) under the punch and the displacement u,(e,O) of the surface of
(Compared with classical fracture mechanics, the factor ’12 is introduced in G because the punch is undeformable (If the punch has a Young modulus E’and a Poisson ratio v’, the quantity (1 - v2)lEmust be replaced by (1- v2)/E (1 - V’~)/E’.), so that energy is released by the half-space only; we are faced with half a Griffith crack). Note that eq 7 is a n expression for G that is only valid in the linear elastic fracture mechanics approximation (see ref 7 for the JKR-DMT transition when this expression is no longer valid). Inserting for the profile of the spherical punch (radius R ) the parabolic approximation
Abstract published in Advance ACS Abstracts, December 15,
fie)= ga2j e2
( 1 )Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R.SOC.London Ser. A 1971,324, 301. ( 2 )Rimai, D. S.;DeMejo, L. P.; Bowen, R. C. J . Appl. Phys. 1989,66,
(3) DeMejo, L. P.; Rimai, D. S.; Bowen, R. C. J . Adhes. Sci. Technol. 1991,5,959. (4)Rimai, D. S.; DeMejo, L. P.; Vreeland, W. B.; Bowen, R. C. Langmuir, in press. (5) Barquins, M.; Maugis, D. J . Mkc. ThBor. Appl. 1982,1,331. (6) Maugis, D.; Barquins, M. J . Phys. D 1984,16, 1843. (7)Maugis, D. J . Colloid Interface Sci. 1992,150,243. (8) Sneddon, I. N. Int. J . Eng. Sci. 1965,3,47.
0743-7463/95/2411-0679$09.00/0
dt (5)
and a n energy release rate
1994.
nrn, a0 14.
e
As discussed in refs 5-7, if ~ ( 1f) 0 these stresses and displacements are those offracture mechanics with a stress intensity factor
+
(3)
@
the indented plane are given by
one has
Equations 2 and 7 become
0 1995 American Chemical Society
Maugis
680 Langmuir, Vol. 11, No. 2, 1995
d = - +a2 3R
-i5) a2
G=-(63K
2~
(9)
3aK = ( ( a 3 N R )- P)2
8na
6na3K
(10)
with
E Writting the equilibrium as G = w , with w the Dupr6 energy of adhesion, the JKR results are immediately recovered. In particular, the radius of contact under zero load, ao, is given by
-2
-1
1
0
3
2
4
5
Figure 1. Reduced values of the cube of the radius of contact versus normalized load P h w R for various values of the parameter m = (RW~CW)”~.
a: = 6nwR2fK and varies as RU3. If instead, the exact profile is used
f l e ) = R - m Let us introduce the dimensionless parameters one has
A=
a ( x w R2fKl‘I3
p- = - P
(19)
nwR
with u = e2/t2. Hence 2 at R + a t x(t) = i d - In G)
With these notations, eqs 15 and 16 become
Equations 2 and 7 become
p 4a
G =a
In ‘ R -2 a )
R+a2 d- 1 n G )
Equations 9 and 10 are recovered with the power expansion
R+aln--2tanhR-a The equilibrium relations 6(a)and P(a) are obtained by making G = w (If w = 0 one obtains the extension of the Hertz theory for large contacts.): (15)
ln>R -E R-a 2
4%)
(16)
and the equilibrium radius of contact under zero load is given by
= %m2 + A 2 )In m S A - 3m2A-A&
4
(23)
to be compared with the JKR approximations A =A2- (2/3)&
(24)
P=A3 -A&
(25)
Figures 1 and 2 display the curves a3(P)andP(G)in reduced coordinates for various values of the parameter m. The adhesion force a t fured load (the point with a vertical tangent in Figure 1 or with an horizontal tangent in Figure 2) is no longer independent of the Young modulus. One can see that the JKR approximation is valid for m > 10. The height h of the solid meniscus, i.e., the height of the edge of the contact above ( h > 0) or below (h < 0) the level of the half-space, is given by
h=H-6 where H is the height of the spherical cap (H = R - (R2 Using the same normalization as for the depth of penetration, one has
Elastic Contact of Spheres to Large Contact Radii
Langmuir, Vol. 11, No. 2, 1995 681 1.5 l ” ”
5
0 .
0
Figure 2. Load displacement curves, in reduced coordinates, for various values of the parameter m. 2.5
I
2
3
5
4
6
7
8
I0
9
Figure 4. Normalized penetrationunder zero load as a function of the parameter m.
@
1.5
I
.s
0 0
I
2
3
4
5
6
7
8
9
I0
Figure 3. Normalized radius of contact under zero load as a function of the parameter m. It may be-more instructive to have 61R and hlR instead of A and h. They are given by
6IR = N m 2
R
hlR = &lm2 Case of Zero Applied Load. The radius of contact under zero load is given, in reduced coordinates, by the implicit equation
3m -(m2 8
-
m +A, + A:) In -A o ( Y+ m -Ao
-
(in)
Figure 5. Radius of contact under zero load a0 as a function of the sphere radius R , in log-log plot, for various values of
wlK.
a)
= 0 (26)
-
and is displayed in Figure 3 as a function of m. When m 0 one has Adm 1; i.e., a0 R. The corresponding depth of penetration A of the tip into the half-space is displayed in Figure 4. These figures show how much the JKR approximation is inexact for m 10 but are not very explicit from an experimental point of view. Figure 5 displays the radius of contact a0 as a function of the sphere radius R, in log-log plot, for various values of wlK. One observes a continuous increase of the slope from V3, the JKR value, to 1 as R decreases. Figures 6 and 7 display the variations of 6dR and hdR with m, respectively. When m 0, both values tend toward whereas in the JKR approximation one has
-
0
I
2
3
4
S
6
7
8
9
10
Figure 6. Variation of the relative penetration 6dR under zero load with the parameter m. increase from V3 to 1 as R decreases. The difference between a contact under zero load in the JKR approximation and for m 0 is schematically shown in Figure 8.
-
6, = a33R = 2h0 In a plot In ho - In R or In 60 - In R the slope should
111. Comparison with Experimental Results Rimai et al. have observed in a scanning electron microscope the contact under zero load of small spherical
682 Langmuir, Vol. 11,No. 2, 1995
Maugis
1
-4
-5
-5.5
0
I
2
3
4
5
5
7
E
9
18
Figure 7. Variations of the relative height of the meniscus hdR under zero load with the parameter m.
-6
-6.5 -6.5
-6
-5.5
-5
-4.5 Lag R ( m )
-4
Figure 9. Radius of contact under zero load. Comparison of glass experimental results of Rimai et al. with the theory: (0) , ~ glass beads on beads on soft polyurethane s ~ b s t r a t e(*) polyurethane s ~ b s t r a t e ,and ~ (+I polystyrene spheres on polyurethane.
-
Figure 8. Schematic diagram of a contact under zero load in the JKR approximation (a, top) and for m 0 (b, bottom). particles on elastic substrates: polystyrene (radii 1-6pm) on polyurethane2 and glass (radii 0.5-100 pm) on polyret thane.^^^ Their data points are shown in the log-log plot of Figure 9. They clearly show a continuous bending from the slope 213(dotted line) corresponding to the JKR theory to the slope 1as R decreases, which explains the slope 3/4 obtained when trying to draw a straight line through these data points. The curves giving the best fit correspond to wlK = 1.7 x lo+, 2.7 x and 9 x m, as given in the figure. For the data points, ofthe upper curve the values: of m appear to vary from 1.1to 3.3. The heights of menisci plotted in Figure 5 of ref 4 are plotted in Figure 10 and compared to both the JKR and the extended JKR theory. Due to the scatter of the data and to the larger particle radii, the departure from the JKR simple theory is not so clearly visible.
IV. Conclusion For spherical particles of small radii on a soft elastic solid,the ratio of the radius of contact over the ball radius
Figure 10. Height of the meniscus as a function of particle radius. Data from ref 4. The dotted line is the JKR approximation and the limit ho = R/2. can be so large that the parabolic approximation for the sphere profile used in the JKR theory can no longer be valid. The use of the exact expression for the sphere profile allows the theory to be extended to larger radii of contact, in agreement with experimental results. LA9406158
