Langmuir 1997, 13, 1799-1804
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Finite-Size Corrections to the JKR Technique for Measuring Adhesion: Soft Spherical Caps Adhering to Flat, Rigid Surfaces Kenneth R. Shull,* Dongchan Ahn, and Cynthia L. Mowery Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208-3108 Received August 27, 1996. In Final Form: December 13, 1996X Adhesion measurements based on the fracture mechanics analysis of Johnson, Kendall, and Roberts (JKR) provide a very convenient method for measuring the energy of adhesion, G, for elastomeric materials against a variety of substrates. The JKR approach utilizes linear elastic fracture mechanics, and is based on the assumptions that the contact geometry is characterized by a single radius of curvature, and that the relevant dimensions of the adhering bodies are large compared to the dimensions of the contact area. The assumption of large sample size is not necessarily valid for the commonly employed geometry consisting of a soft, spherical cap pressed against a flat, rigid surface. The implications of the resultant finite-size corrections are studied here using two different model systems: a cross-linked poly(n-butyl acrylate) homopolymer and a gel made from an acrylic triblock copolymer diluted with 2-ethylhexanol. The compliance of the spherical caps is found to deviate significantly from the value assumed in a standard JKR analysis. This discrepancy is independent of the contact area, however. Determinations of the fracture energy which are based on the relationship between the load and contact area are, therefore, not affected by this correction to the compliance. The modified compliance does need to be accounted for when the fracture energy is determined from the relationship between the contact area and the relative displacements of the adhering bodies. Use of this relationship is shown to provide a particularly powerful method for determining the modulus and/or adhesion energy for low-modulus solids.
Introduction A commonly used testing geometry for measuring the adhesion between two materials involves bringing a sphere into contact with a flat surface. The fracture mechanics of this test were originally analyzed by Johnson, Kendall, and Roberts (JKR) in 1971.1 By simultaneously measuring the applied load, P, and the radius, a, of the contact patch formed between the two surfaces, the energy release rate, G, can be determined. The energy release rate is the driving force available to separate the two surfaces, in terms of an energy per unit area. Within the JKR analysis, G is related to P and a by the following two equivalent equations:
G) a3 )
(Ka3/R - P)2 6πKa3
(1)
R {P + 3πGR + [6πGRP + (3πGR)2]1/2} (2) K
where K is the effective modulus of the lens and R is the characteristic radius of curvature of the contact region. In general, R and K depend on the elastic properties and radii of curvature of both surfaces. For a soft lens with a Poisson ratio of 0.5 on a flat, rigid substrate, R is the radius of curvature of the lens and K is 16E/9, where E is Young’s modulus for the lens. Adhesion tests based on use of eqs 1 and 2 are referred to as JKR tests, and have become increasingly prevalent in the recent literature.2-6 Much of the earlier work has X Abstract published in Advance ACS Abstracts, February 1, 1997.
(1) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London A 1971, 324, 301. (2) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 1013. (3) Brown, H. R. Macromolecules 1993, 26, 1666. (4) Silberzan, P.; Perutz, S.; Kramer, E. J.; Chaudhury, M. K. Langmuir 1994, 10, 2466. (5) Deruelle, M.; Leger, L.; Tirrell, M. Macromolecules 1995, 28, 7419. (6) Ahn, D.; Shull, K. R. Macromolecules 1996, 29, 4381.
S0743-7463(96)00845-1 CCC: $14.00
Figure 1. Geometry of the JKR adhesion experiment, illustrating the shape of the elastic lens before (a) and after (b) deformation.
involved a rigid sphere in contact with a thick, flat elastic layer. The original experimental work of Johnson, Kendall, and Roberts employed this geometry, as did the classic experiments of Maugis and Barquins.7 More recently, Chaudhury developed a technique based on the use of soft, hemispherical lenses on rigid, flat plates.2 The basic experimental geometry is shown in Figure 1. The size and shape of the lens are specified by any two of the following four parameters: the lens radius of curvature, R, the height of the lens, h, the basal radius, r, and the contact angle, θ, which the lens makes with the surface on which it was prepared. For θ < 90°, these parameters are related to one another by the following expressions: (7) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 1989.
© 1997 American Chemical Society
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Shull et al.
tan(θ/2) ) h/r
(3)
R ) (h/2) + (r2/2h)
(4)
Advantages associated with the use of soft hemispherical lenses (as opposed to rigid spherical surfaces on deformable, flat surfaces) include the fact that the surface chemistry of the flat surface can easily be modified. In addition, very smooth lenses with a constant radius of curvature can be readily prepared by placing a viscous precursor solution of the polymer on a low energy surface. One consequence of this geometry, however, is that the deformable solid (in this case the soft, hemispherical lens) can no longer be treated as an elastic half space, i.e., an elastic solid with an essentially infinite thickness in the direction normal to the interface. This feature calls into question one of the basic assumptions used to derive eqs 1 and 2. Previous work involving simultaneous measurement of P, a, and the lens displacement, δ, has indicated that these finite size corrections are potentially important.6,8 Our aim here is to consider how these corrections affect the determination of the energy release rate in common experimental geometries. We begin with a simple derivation of the JKR equations from some basic principles of linear elastic fracture mechanics. Modifications arising from a particular form of the finite size corrections are then introduced. Finally, we compare predictions from the modified JKR theory to experimental results on some model systems. In cases involving very low modulus materials, where the load is too low to be experimentally accessible, we show that the quantity G/K can be obtained from the relationship between the contact area and the lens displacement. Derivation of JKR Equations Consider a rigid, flat cylinder with a radius of a and a cross-sectional area of A (A ) πa2), which is placed into contact with a flat, linearly elastic material. In the presence of adhesive interactions, energy must be supplied in order to separate these two surfaces. This energy is quantified by the energy release rate G, which can be obtained from the relationships between an applied load, the resulting displacement, and the contact area. Suppose a tensile force with a magnitude of P is applied to the cylinder, in the direction normal to the surface on which it rests. If both materials are linearly elastic, and the contact area between the bodies remains constant as the tensile loading is applied, the displacement, δ, at the point of loading will increase linearly with P as shown in Figure 2a. (Positive loads and displacements are defined as compressive here.) The stored elastic energy is equal to Pδ/2 ) P2C/2, where C is the compliance of the system (C ) δ/P). If at this point the contact area decreases, there will be a change in the compliance of the system to C + dC. At constant load, there is an increase in the magnitude of the displacement of the cylinder from point b to point a on Figure 2a. The change in stored elastic energy associated with this change is P2 dC/2. The energy release rate is obtained by dividing this energy difference by the change in contact area, dA, in the limit where these quantities are both very small. In this way, one obtains the following standard expression:9
G)-
P2 dC P2 dC )2 dA 4πa da
Figure 2. Load-displacement relationships for linearly elastic materials, illustrating the origins of eqs 5 (part a) and 6 (part b). The shaded area in each case is the energy required to decrease the contact area between two surfaces in contact by a small amount. Details are described in the text.
The negative sign arises from the fact that energy is required to decrease the contact area, A. One can also speak in terms of a hysteresis loop obtained by loading at constant A, decreasing A at fixed P, and unloading at this new value of A. The area of this hysteresis loop, denoted by the area oab in Figure 2a, is equal to P2 dC/2. Constant load conditions are not required in order for eq 5 to be valid, because it is expressed in terms of differential changes in C and A. Corrections to the area of the hysteresis loop associated with small changes in P or δ become negligible as these changes become vanishingly small. In the development of eq 5, it was assumed that the bodies have a finite contact area for P ) 0, even in the absence of adhesive interactions. We are interested in axisymmetric geometries (such as a sphere on a flat surface) where the surfaces are not necessarily conformal. In general, development of a contact area, A, between nonadhering systems will require the application of a load P′(A), resulting in a corresponding displacement, δ′(A). Adhesive interactions cause the actual displacements, δ, and loads, P, to be less than the respective values of δ′ and P′. The situation is illustrated schematically in Figure 2b, where the solid line represents the relationship between P′ and δ′. This curve represents the loaddisplacement relationship for a completely nonadhesive system. Its shape depends on the elastic constants of the two materials, and on their geometry. The energy release rate for adhesive systems is obtained by the deviations of P and δ from this curve. Use of eq 5 requires that P′-P be substituted for P, and that δ′-δ be substituted for δ:
(5) G)-
(8) Deruelle, M. Ph.D. Thesis, L’Universite Paris VI, 1995. (9) Ward, I. M.; Hadley, D. W. An Introduction to the Mechanical Properties of Solid Polymers; Wiley: New York, 1993.
(P′ - P)2 dC (P′ - P)2 dC )2 dA 4πa da
(6)
where the compliance is given by the following expression:
JKR Technique for Measuring Adhesion
C)
|
δ′ - δ P′ - P A
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(7)
Using this definition of C to substitute for P′ - P in eq 6 gives a second expression for G:
(δ′ - δ)2 dC 4πaC2 da
G)-
(8)
For a rigid, axisymmetric punch on a semiinfinite elastic layer with an effective modulus of K, the compliance is given by the following expression:
Cthick )
2 3Ka
(9)
When this value for C is used in eq 6, we obtain the following expressions for the energy release rate:
G)
G)
(P′ - P)2 3
6πKa
3K(δ - δ′)2 8πa
(12)
(13)
Finite Size Effects
(14)
These assumptions are expected to be reasonable when the thickness of the lens, h - δ, is at least as large as 2a, the diameter of the contact zone. In this regime, the force required to produce the deformation associated with a given contact radius is not significantly affected by the finite thickness of the lens. The decrease in the compliance is simply due to the fact that displacements are truncated by the finite thickness of the lens. Experimental mea(10) Barquins, M. Wear 1992, 158, 87.
a2 2P + - P∆C 3R 3Ka
(16)
δ - (a2/3R) P ) K 2/(3a) - K∆C
The simplest approach for handling the finite size effects for the geometry shown in Figure 1 is to assume that P′ is not affected, and that the lens compliance is decreased by a constant factor, referred to here as ∆C:
2 - ∆C 3Ka
δ ) δ′ + C(P - P′) )
(15)
(11)
These expressions are based on the assumption that the elastic material (either the punch or the flat layer) is a linearly elastic half space. In the following sections, we study the implications associated with the relaxation of this assumption, for the specific case of a soft spherical punch, or lens.
C)
a2 a2 - ∆CP′ ) (1 - Ka∆C) R R
Equation 16 can be rearranged to give the following for P/K:
The corresponding displacement for this geometry is given by
δ′ ) a2/R
δ′ )
(10)
Equation 11 is useful because it allows the quantity G/K (or, equivalently, G/E) to be obtained from measurements of the displacement and contact radius, even if the load itself is too small to be measured. These expressions have been developed from a more rigorous fracture mechanics approach by Maugis and Barquins.7 Expressions for P′ can be obtained for a variety of rigid indenters, as described, for example, by Barquins.10 Equations 1 and 2 for a sphere on a flat are obtained by using the Hertzian expression for P′:
PHertz′ ) Ka3/R
surements of the compliance for real lenses indicate that eq 14 works remarkably well, and that P′ is not affected by the finite thickness of the lens. These experimental results are discussed below. Here, we assume that these assumptions are valid, and we discuss their implications for the interpretation of the JKR experiments. First, because the derivative of the compliance with respect to the contact radius is unchanged by the addition of a constant factor, and because P′ is not affected by the finite size effect, eqs 1 and 2 are still valid. Results based on the relationship between the contact radius and displacement, however, must be modified. For δ′ and δ we obtain
(17)
Finally, substitution of the appropriate expressions for δ′ and C into eq 8 for G gives 3 2 2 G a (1 - aK∆C - (δR)/a ) ) K R26π((2/3) - aK∆C)2
(18)
Equation 12 for P′ is based on the assumption that a/R is small enough that the shape of the spherical indenter in the region of contact can be approximated by a parabola. Maugis has pointed out that, for large values of a/R, it may become necessary to use the form of P′ which appropriately accounts for the spherical shape of the indenter:11
P′sphere )
(
Ka3 3 1 + x2 1 + x 3 ln R 8 x3 1 - x 4x2
)
(19)
with x ) a/R. For the values of a/R commonly encountered, these corrections are not particularly significant. For a/R ) 0.5, for example, P′sphere is only 12% larger than the Hertzian value for P′ as given by eq 12. For the cases described below, the dominant corrections to the JKR theory arise from the finite size and nonlinear stress/ strain relationship for the lens. Corrections for Nonlinear Elasticity Elastomeric materials are somewhat unique in that the stress-strain behavior can be decidedly nonlinear, and yet remain completely elastic. This behavior can be expressed in terms of Frubber, the stored elastic free energy density of a stretched rubber, which for an incompressible material is approximated as follows:12
E Frubber ) ((1 + 1)2 + (2 + 1)2 + (3 + 1)2 - 3) (20) 3 where 1, 2, and 3 are the principal engineering strains. For uniaxial extension or compression, eq 20 reduces to (11) Maugis, D. Langmuir 1995, 11, 679. (12) Flory, P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
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Figure 3. Ratio of strain energies for a rubbery material and for a linearly elastic material with the same modulus, as given by eqs 21 and 22. Uniaxial tension (positive strain) or compression (negative strain) is assumed.
Figure 5. Measured compliance values for the lens from Figure 4, obtained from the slope of the unloading curve in the high load regime where the contact radius remains constant. The solid line is the prediction of eq 14, using the same values of K and ∆C as were used to fit the data shown in Figure 4.
Experimental Results
subsequent analysis based on this equation. These data were obtained from a cross-linked poly(n-butyl acrylate) lens, with R ) 1.16 mm and K ) 300 kPA (r ) 0.96 mm, h ) 0.51 mm, θ ) 55°). Details of the preparation of this lens, and of the experiment itself, have been published previously.6 Figure 4 shows a comparison of the loaddisplacement relationship for this lens, to the prediction of eq 16, with ∆C ) 4.8 µm/mN (K∆C ) 1440 m-1). Agreement is quite good over the entire range. The adhesion hysteresis exhibited by this system allows us to directly measure the compliance of this lens as well. The contact radius remains unchanged during the initial portion of the unloading curve. The slope of this portion of the curve gives the inverse of the compliance, which can be compared to the prediction of eq 14. This comparison is shown in Figure 5, where we have used the same values for K and ∆C. Taken together, these results indicate that finite size effects give a constant correction to the lens compliance, and that P′ is in accord with the Hertzian prediction (eq 12). As mentioned above, the use of eqs 1 and 2 to obtain energy release rates from the relationship between the load and the contact area is valid. We therefore obtain the encouraging result that previous experiments based on this analysis are unaffected by the finite size corrections. A similar result has, in fact, been obtained by Deruelle et al.8 Finite size effects must be accounted for when using a fracture mechanics approach to obtain G/K in situations where the load is too small to be measured directly. There are many situations where this situation is likely to apply. Examples include materials consisting primarily of water or some other solvent, but with some elasticity arising from the incorporation of a polymeric component. Red blood cells, and other biological entities, are an interesting example of this class of materials. As a model for studying these types of materials, we have developed a system based on a triblock copolymer with poly(methyl methacrylate) (PMMA) end blocks and a poly(n-butyl acrylate) midblock. The copolymer was synthesized by transesterification of an anionically polymerized precursor polymer having a poly(tert-butyl acrylate) midblock, using procedures similar to those described by Varshney et al.13 The respective molecular weights of the midblock and endblocks for the polymer used in our experiments were approximately
The data presented in Figures 4 and 5 are a confirmation of the validity of eq 14 for the compliance, and of the
(13) Varshney, S. K.; Jacobs, C.; Hautekeer, J.-P.; Bayard, P.; Je´roˆme, R.; Fayt, R.; Teyssie´, P. Macromolecules 1991, 24, 4997.
Figure 4. Load-displacement curve for a poly(n-butyl acrylate) lens against a PMMA substrate for loading (circles) and unloading (triangles). The solid line is the prediction of eq 17 with R ) 1.16 mm, K ) 300 kPA, and ∆C ) 4.8 µm/mN.
E Frubber ) (2 + 2 + 2/( + 1) - 2) 3
(21)
where is the applied tensile strain (which assumes a negative value for compressive strains). The free energy for a linearly elastic material is given by the following expression:
Flinear ) E2/2
(22)
At a given level of strain, the approximate form of the strain energy used in a linear elastic fracture mechanics approach (such as JKR theory) overestimates the strain energy in tension and underestimates the strain energy in compression. This behavior is illustrated in Figure 3, where Frubber/Flinear is plotted as a function of the strain. One expects, based on this result, that for tensile loadings typical of JKR experiments G will be overestimated if the average tensile strain is significant. An approximate method for taking into account these nonlinearities is described in the following section.
JKR Technique for Measuring Adhesion
Figure 6. (a) Relationship between contact radius and displacement for an acrylic triblock copolymer lens, swollen with approximately 95 wt % 2-ethylhexanol. Data for loading (circles) and unloading (triangles) are included. The solid and dashed lines represent the predictions of eq 16 for δ′, with K∆C ) 0 (dashed line), and K∆C ) 1100 m-1 (solid line). (b) Values of G/E obtained from the data in part a, using eq 18 with K ) 16E/9. Circles are obtained from loading data, and the triangles and dashed line were obtained from unloading data. Symbols correspond to K∆C ) 1100 m-1, and the dashed line corresponds to K∆C ) 0.
92 000 and 23 000 g/mol. At temperatures above approximately 50 °C, this polymer is completely soluble in a variety of alcohols, including methanol, ethanol, nbutanol, and 2-ethylhexanol. At lower temperatures, these alcohols are a good solvent for n-butyl acrylate, but not for PMMA. Upon cooling to room temperature, solutions containing 5-10 wt % polymer in these alcohols form low-modulus gels. The experiments used here utilized gels consisting of about 5 wt % polymer in 2-ethylhexanol, chosen as a solvent because of its relatively low evaporation rate. Hemispherical lenses were produced by placing a small amount of the polymer/solvent mixture on a fluorinated glass microscope slide, and heating this slide to a temperature at which the mixture was able to flow freely. These lenses were then cooled to room temperature. Adhesion of the resultant gels to PMMA surfaces was studied. The PMMA surfaces were obtained by spin casting thin layers of PMMA onto polished silicon wafers. Figure 6a shows the contact radius (against PMMA) as a function of displacement for a triblock gel with a radius of curvature of 0.93 mm (R ) 0.93 mm, r ) 0.73 mm, h ) 0.36 mm, θ ) 52°). The actual data are represented by the symbols, whereas the solid and dashed lines represent the relationship between a and δ for G ) 0, as given by eq 15. Values for K∆C ) 0 (dashed line) and K∆C ) 1100 m-1 (solid line) are included. The horizontal distances between these lines and the actual data points represent δ′ - δ for these two values of K∆C. In order to obtain the measured data points, the lens and substrate were initially
Langmuir, Vol. 13, No. 6, 1997 1803
separated, and were moved toward one another at a rate of 2.5 µm s-1, using a linear stepper motor as described previously.6 The contact radius was monitored by video optical microscopy, giving us a time resolution of 1/30 s. The lens and substrate come into contact at a displacement, δ, of 0, at which point there is a sudden increase in the contact radius due to the adhesive interactions between the lens and substrate. Here the lens begins to spread over the surface of the PMMA. The spreading is limited by the elasticity of the lens, and reaches a fixed value within a tenth of a second. As the displacement is further increased, the contact radius increases as well, as indicated by the circles on Figure 6a. The relationship between a and δ for unloading (decreasing δ) is almost identical to the relationship between these quantities for loading, indicating that there is very little adhesion hysteresis in this system. Detachment of the lens from the substrate occurs at a large negative value of the displacement. Also, the contact radius remains constant at fixed displacement, indicating that the lens is indeed completely elastic, with no discernible tendency to flow over the time scale of the experiment. Part b of Figure 6 shows G/E as a function of the displacement as calculated by eq 18. (We have assumed ν ) 0.5, so that G/E ) (16/9)G/K.) Both E and G are constants for this system, so any dependence of G/E on the displacement can be attributed to inaccurate assumptions made in the theoretical treatment. The importance of the correction due to the finite size of the lens is illustrated by the inclusion of data for K∆C ) 0 and K∆C ) 1100 m-1. The value of 1100 m-1 was empirically chosen to give a value for G/E which is independent of the displacement, at least for the larger displacements. The increasing value of G/E at lower values of δ can be attributed to the effects of nonlinear elasticity. The magnitude of the strain in the deformed region of the lens is maximized for these values of the displacement. To illustrate this effect, we consider the quantity P/EA, obtained from eq 17, with K ) 16E/9 and A ) πa2, and plotted in Figure 7a. This quantity is the average normal strain for the portions of the lens in contact with the substrate. A very rough approximate correction for nonlinearities in the stress-strain behavior can be obtained by substituting -P/EA for in the expressions for Frubber and Flinear (eqs 21 and 22), and multiplying the G/E values obtained from the linear analysis by the quantity Frubber/Flinear.14 This simple correction significantly reduces the G/E values for the lower values of δ, as illustrated in Figure 7b. There remains, however, a clear need for a more rigorous approach which appropriately takes into account the non-uniform nature of the strain field within the lens. Data from both the loading and unloading data are consistent with the assumption that G is equal to the appropriate thermodynamic work of adhesion. For the loading experiment, the lens spreads over a dry PMMA surface. The thermodynamc work of adhesion during loading, Wload, is given by the following expression: Wload ) γlens + γpmma - γlens/pmma
(23)
where γlens and γpmma are the respective surface energies of the lens and of the PMMA substrate, and γlens/pmma is the interfacial energy for the lens/substrate interface. The respective surface energies of PMMA and PNBA at room (14) The negative sign in the relationship between the strain and the normalized load arises from our use of the contact mechanics convention, where compressive loads and displacements are defined in the positive sense, while maintaining the more common convention of defining compressive stesses and strains in the negative sense.
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experiment, by an amount which is consistent with the expected ratio between Wload and Wunload. Assuming that these values correspond to the actual values of G for the loading and unloading experiments, we obtain 9000 Pa for Young’s modulus, E, of the lens. This is quite a low modulus in comparison to undiluted elastomeric materials. An advantage of the modified JKR techniques employed here is that a wide range of moduli can be accurately measured from a very small sample volume. Experiments based on this type of analysis can be applied in a variety of applications, including, for example, the evolution of mechanical and adhesive properties for coatings applied as emulsions, and the nature of wetting and curing properties for adhesive formulations.
Figure 7. (a) Normalized load (equivalent to the average strain in the regions of the lens in contact with the PMMA surface) for the loading and unloading data from Figure 6, according to eq 17. (b) Data from Figure 6b, corrected to take into account the effects of nonlinear elasticity. Values of G/E were multiplied by Frubber()/Flinear(), with ) -P/EA.
temperature are 42 and 34 mJ/m2,15 and the surface energy of 2-ethylhexanol at room temperature is 30 mJ/m2.16 Because the lens consists primarily of 2-ethylhexanol, which also has the lowest surface energy of the different components in the lens, we can reasonably equate the lens surface energy with the surface energy of 2-ethylhexanol. Similarly, the lens/PMMA interfacial tension will be quite low, because 2-ethylhexanol is only a moderately poor solvent for PMMA. One expects a value for γlens/pmma which is close to 1 mJ/m2,17 giving Wload ) 71 mJ/m2. Because 2-ethylhexanol completely wets the PMMA substrate, it is energetically favorable for a thin layer of the solvent to remain on the PMMA when the lens and substrate are separated. During the unloading experiment (decreasing a), fracture can be assumed to propagate through a thin layer of 2-ethylhexanol, giving Wunload (the thermodynamic work of adhesion during unloading) ) 2γsolvent ) 60 mJ/m2. The measured values of G/E obtained from the unloading experiment are, in fact, slightly less than the values obtained from the loading (15) Polymer Handbook, 3rd ed.; Brandrup, J., Immergut, E. H., Eds.; Wiley: New York, 1989. (16) Solvents Guide; Marsden, C., Mann, S., Eds.; Interscience: New York, 1963. (17) Helfand, E.; Sapse, A. M. J. Polym. Sci., Polym. Symp. 1976, 54, 289.
Summary and Conclusions Simultaneous measurements of load, displacement, and contact area have been carried out on model, hemispherical lenses on flat, rigid substrates. Experiments based on a poly(n-butyl acrylate) lens have been described, in addition to experiments based on lenses produced from a gelforming acrylic triblock copolymer, diluted with 2-ethylhexanol. Results based on both of these systems are the subject of ongoing work. Our primary conclusions from the present paper concern general approaches for obtaining energy release rates and/or elastic moduli from hemispherical lenses pressed against flat, rigid substrates. These conclusions are as follows: 1. Due to finite-size effects, the compliance typically assumed in a standard “JKR” analysis of the adhesive interactions is larger than the actual value by a constant factor. The Hertzian load is not significantly affected by these finite-size effects, so that the standard JKR analysis where the energy release rate, G, is obtained from the relationship between the applied load and contact area, is still valid. 2. For lenses with a very low modulus, it may become impractical to directly measure the resultant low values of the load. The ratio of the energy release rate to the modulus of the lens can be obtained from the relationship between the lens displacement and the contact area between the lens and substrate, provided that the correction factor to the compliance, associated with the finite size of the lens, is known. We have used this correction factor as a fitting parameter in our experiments, adjusting it to satisfy the requirement that the thermodynamic work of adhesion be independent of the contact area. 3. For large strains, the effects of non-linear elasticity must be taken into account. For a rubber under tensile loading, a linearized theory of elasticity overestimates the available strain energy. The net effect is that the energy release rate is overestimated for negative (tensile) loads and underestimated for positive (compressive) loads. An approximate method for taking these non-linearities into account has been introduced. Acknowledgment. This work was supported by the National Science Foundation under Grant DMR-9457923. Acknowledgment is also made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. LA960845H
