Langmuir 1994,10,4361-4366
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Adhesion Induced Deformations of a Highly Compliant Elastomeric Substrate in Contact with Rigid Particles D. S. Rimai,* L. P. DeMejo, and W. B. Vreeland OfficeImaging Research and Technology Development, Eastman Kodak Company, Rochester, New York 14653-6402
R. C. Bowen Analytical Technology Division, Eastman Kodak Company, Rochester, New York 14650 Received April 18, 1994. In Final Form: August 1, 1994@ Glass particles having radii between approximately4.0and 100pm were deposited onto a highly compliant polyurethane substrate (Young's modulus approximately 4.5 x lo4 Pa). The resulting adhesion induced contact radii and heights of the menisci were measured using scanning electron microscopy. It was found that the contact radius varied as the particle radius to the 314 power. This is inconsistent with the 213 power law dependence predicted by adhesion theories which assume elastic deformations, such as that proposed in the JKR model (Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R . SOC.London,A 1971,324, 301)or the 112 power law dependence predicted by plastic deformation theories such as that advanced by Maugis and Pollock (MaugisD.; Pollock, H. M. Acta Metall. 1984,32,1323).These results are consistent with previous contact radius measurements made using smaller particles on a more rigid elastomeric substrate (Rimai, D. S.; DeMejo, L. P.; Bowen, R. C. J . Appl. Phys. 1989,66,3574.DeMejo, L.P.; Rimai, D. S.;Bowen, R. C. J . Adhesion Sci. Technol. 1991,5,959).The thermodynamicwork of adhesion between the glass particles and polyurethane substrate was estimated to be 0.044J/m2 and the experimentally determined heights of the menisci are consistent with the predictions of the JKR model.
I. Introduction The adhesion of particles to surfaces is a problem of vast significance both scientifically and technologically. Specifically, the ability to adhere and remove particles in a controlled manner is exceedingly important in areas as diverse as xerography, pharmacology, graphic arts, paint, and semiconductors. Understanding particle adhesion has been an objective of scientific interest for over 60 years. The adhesion of particles to various substrates involves both interactions between the materials and the mechanical response of these materials to the adhesion induced stresses. Indeed, the adhesion generated stresses can be sufficiently large so as to exceed the elastic limit of at least one of the contacting As discussed by Krupp,l the occurrence of plastic flow can have a strong effect on particle adhesion because of the elimination of the socalled "elastic rebound" occurring when a particle is removed from a surface when only elastic deformation has occurred. Adhesion induced deformations were first postulated, independently, by Bradley5s6and Derajaguin.' Derjaguin calculated the contact radius between a particle and a surface by assuming that the particle acted as a Hertzian indentor with the applied load, P",arising from van der Waals interactions. Accordingly,he found that the contact radius, a , was related to the particle radius, R , by
where Y and E are the Poisson ratio and Young's modulus of the substrate and Pois given by
hii, p" = -R 87dz2,2
Here, zo is the separation distance between the particle and the substrate (typically 4 A for van der Waals bonded crystals) and hii, is the Hamaker coefficient. In the course of measuring the contact radius between homogeneous combinations of gelatin and polyurethane spheres, Johnson, Kendall, and Roberts* (hereafter referred to as JKR)realized that the contact radius predicted by the Derjaguin model, which assumed that the stresses were solely compressive,was smaller than that determined by their measurements. By assuming that tensile, as well as compressive, stresses contributed to the size of the contact and that all stresses were within the contact zone, they found that
where W A is the thermodynamic work of adhesion and is related to the surface energies, y l and yz, and the interfacial energy, y12, by
* To whom correspondence should be addressed. Abstract published inAdvunceACSAbstracts, October 1,1994. (1)Krupp, H.Adv. Colloid Interface Sci. 1967,I , 111. (2)Maugis, D.; Pollock, H. M. Acta Metall. 1984,32,1323. (3)Rimai, D.S.; Moore, R. S.; Bowen, R. C.; Smith, V. K.; Woodgate, P. E. J. Muter. Res. 1993,8, 662. (4)Rimai, D.S.;DeMejo, L. P.; Bowen, R. C. J.Appl. Phys. 1990,68, 6234. ( 5 ) Bradley, R. S. Philos. Mug. 1932,13,853. (6)Bradley, R. S. Trans. Furuduy SOC.1936,32,1088. (7) Dejaguin, B. V. Kolloid 2.1934,69, 155. @
P represents any externally applied load in eq 5 and
(8)Johnson, K. L.; Kendall, IC;Roberts, A. D. Proc. R . SOC.London, A 1971,324,301.
0743-746319412410-4361$04.50/00 1994 American Chemical Society
Rimai et al.
4362 Langmuir, Vol. 10, No. 11, 1994 where
k.=-
1 - vi2 nEi
(6)
and Eiand vi are the Young's modulus and Poisson ratio of each of the two materials, respectively. In the absence of any externally applied load, eq 3 reduces to
(7) Upon application of a negative load, the contact radius decreases. However, the requirement that the contact radius be real in eq 3 implies that particle-substrate separation occurs when the separation force, P,, equals
Moreover, the contact radius at separation, as,does not vanish. Rather, the JKR model predicts that the particle separates from the substrate when a equals approximately 63% of the contact radius under no externally applied load. Recent studiesghave found experimental verification of this prediction for the case of glass particles on an elastomeric substrate. A n alternative theory of particle adhesion which still assumed elastic response of the materials and allowed for tensile contributions to the contact radius was proposed by Derjaguin and co-workers,1° hereafter referred to as the DMT model. Their approach was to calculate the contact radius from a microscopic point of view assuming that the shape of the contact was Hertzian. According to their theory, the contact radius decreases continuously with an increasing magnitude of a removal force and separation occurs when the contact radius vanishes. Moreover, as shown by Tabor," the DMT and JKR models both predict that the contact radius varies as RU3and However, for a given value of W A , the DMT theory predicts a separation force and contact radius under no externally applied load of approximately 413 and 112, respectively, of that predicted by the JKR model. Due to the weak dependence of the contact radius on the work of adhesion, the latter result implies that, for a given contact radius, the work of adhesion determined by the DMT model would be approximately an order of magnitude greater than that calculated using the JKR theory. Therefore, it is essential that the model which correctly describes the system be determined. Following a heated debate in the literature,12J3Muller et aZ.l4J5showed that the discrepancy arose from the differing assumptions regarding the shape of the contact zone and the range of the interaction potentials. Moreover, they showed that both theories had ranges of validity and that the transition between these ranges is defined by a dimensionless parameter, p, such that
where
1/E* =
Sci. Technol. (10)Dejaguin, B. V.;Muller,V.M.;Toporov,Y.P.J . ColloidInterface Sci. 1975, 53, 314. (11)Tabor, D. J . Colloid Interface Sci. 1977, 58, 2. (12) Dejaguin,B.V.;Muller,V.M.;Toporov,Y.P. J . Colloidhterface Sci. 1978, 67, 378. (13) Tabor, D. J . Colloid Interface Sci. 1978, 67, 380. (14) Muller, V. M.; Yushchenko, V. S.; Dejaguin, B. V. J . Colloid Interface Sci. 1980, 77, 91. (15) Muller,V.M.;Yushchenko,V.S.; Dejaguin,B.V.ColbidsSurf. 1983, 7,251.
1 - v2
(10)
For p >> 1,corresponding to large particles, large values of W A , and low elastic moduli, the particle-substrate adhesion is described by the JKR model. Conversely, if p
