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Adhesion of Thermally Reversible Gels to Solid Surfaces Cynthia L. Mowery,† Alfred J. Crosby,† Dongchan Ahn,‡ and Kenneth R. Shull*,† Department of Materials Science and Engineering and Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60208 Received May 30, 1997. In Final Form: September 3, 1997X We have developed methods for quantifying very weak adhesive interactions between two bodies in contact. Our approach is based on the use of a low-modulus material in conjunction with a linear elastic fracture mechanics analysis based on the treatment of Johnson, Kendall, and Roberts (JKR.) The JKR theory can be used to describe the effects of adhesive interactions between a soft, elastomeric lens and a solid, rigid surface, under the assumption that the contact area is small relative to the size of the lens. When the ratio of the contact radius to the height of the lens becomes large, it is necessary to account for finite size corrections to the compliance and displacement of the lens. This situation has been addressed by using results from finite element analyses to modify the JKR equations so that an appropriate expression for G, the energy release rate, can be obtained. Adhesion experiments have been performed on lowmodulus lenses formed by diluting a triblock copolymer, consisting of poly(methyl methacrylate) end blocks and a poly(n-butyl acrylate) midblock, with 2 ethylhexanol. Rheological studies on this swollen copolymer indicate that the material is completely elastic at room temperature and undergoes a rapid, thermally reversible gelation, thus making it an excellent model system. For this low-modulus material, the applied loads are too low to measure directly. Instead, we obtain expressions for G/E, the energy release rate normalized by Young’s modulus. Comparisons to rheological data show that this analysis provides an accurate yet simple method for obtaining this information. Our approach has great potential for quantifying the adhesion of a variety of materials without the need to directly measure the applied load.
Introduction It is well-known that adhesive interactions between contacting materials can significantly affect the shapes of these objects. The contact area between the two materials and the force required to separate them are determined by a balance between the work of adhesion and the stored elastic energy. A useful quantitative description of this energy balance has been given by Johnson, Kendall, and Roberts (JKR).1 These authors considered the general case of two spherical surfaces in contact, with a sphere on a flat surface included as a limiting case where one of the surfaces has an infinite radius of curvature. A variety of experimental work has verified the validity of the JKR treatment for elastomeric materials with relatively low moduli.2-11 An important contribution to this technique was the introduction by Chaudhury of a very simple method for producing spherical elastomers appropriate for JKR adhesion tests.3 In this approach, an un-crosslinked precursor polymer is placed on a low-energy substrate, resulting in the formation of a spherical cap, with dimensions determined by the volume of polymer used and the contact angle the polymer makes with the substrate of interest. This spherical cap is then crosslinked to produce elastomers appropriate for JKR tests. This approach enables one to accurately measure the work of adhesion from a very low sample volume. In almost all * To whom correspondence should be addressed. † Department of Materials Science and Engineering. ‡ Department of Chemical Engineering. X Abstract published in Advance ACS Abstracts, October 15, 1997. (1) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (2) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 1989. (3) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 1013. (4) Barquins, M. Wear 1992, 158, 87. (5) Brown, H. R. Macromolecules 1993, 26, 1666. (6) Creton, C.; Brown, H. R.; Shull, K. R. Macromolecules 1994, 27, 3174. (7) Silberzan, P.; Perutz, S.; Kramer, E. J.; Chaudhury, M. K. Langmuir 1994, 10, 2466. (8) Deruelle, M.; Leger, L.; Tirrell, M. Macromolecules 1995, 28, 7419. (9) Maugis, D. Langmuir 1995, 11, 679. (10) Ahn, D.; Shull, K. R. Macromolecules 1996, 29, 4381. (11) Shull, K. R.; Ahn, D.; Mowery, C. Langmuir 1997, 13, 1799.
S0743-7463(97)00567-2 CCC: $14.00
cases, adhesion hysteresis is observed, with a higher work of adhesion obtained for a decreasing area of contact than for an increasing area of contact. The thermodynamic work of adhesion is obtained in a limited number of situations, where loading experiments (increasing contact area) and unloading experiments (decreasing contact area) give the same value for the adhesion energy. Because the stored elastic energy for a particular strain state is proportional to the elastic modulus of the material, low-modulus materials are more sensitive to weak adhesive interactions than are materials with a relatively high modulus. In order to obtain an accurate measure of the weak interactions typical of thermodynamic values for the work of adhesion, low-modulus materials are preferred. These materials are typically obtained by lightly cross-linking high-molecular-weight polymers.3,6-8,10 Cross-linked polymers have elastic properties which are adequately described by rubber elasticity theory, with moduli that are inversely proportional to the molecular weight between cross-links. Trapped entanglements between polymer molecules act as effective cross-links and generally place a lower bound on the elastic modulus. A polymer melt with a density, F, of 1 g/cm3 and a molecular weight between entanglements, Me, of 25 000 g/mol has a plateau shear modulus of 105 Pa at room temperature. These values for F and Me are close to those for poly(nbutyl acrylate) (PnBA), a polymer which we have used in previous JKR experiments.10 Light cross-linking of a highmolecular-weight linear polymer with these values of F and Me will give an elastomer with a value for the Young’s modulus, E, near 3 × 105 Pa. The presence of dangling ends which do not contribute to the modulus can result in somewhat smaller values of E. In this previous study, we obtained an acrylic elastomer with E ) 3 × 105 Pa by lightly cross-linking a linear PnBA precursor with a molecular weight of 50 000 g/mol. This value for the modulus is almost certainly close to the lower limit of what can be obtained by these techniques. While measurable changes in the contact area between this relatively low-modulus lens and a PMMA substrate have been attributed to the thermodynamic work of adhesion, the © 1997 American Chemical Society
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effect is not as large as one would like. A lens with an even lower modulus is still preferred. One solution in our quest for a lower modulus lens is to add a low-molecular-weight species which dilutes the entanglement network. These molecules should be added in such a way so that they do not take part in the crosslinking process. Our approach is to accomplish this goal by using physically cross-linked, thermally reversible gels consisting of a triblock copolymer in a selective solvent for the midblock. Network formation at temperatures below the gel point is accomplished by association of the end blocks. Under certain conditions we are able to produce materials which are purely elastic, with no tendency to flow during the adhesion measurement. For the geometry of a soft lens pressed against a rigid substrate, materials with a Young’s modulus as low as 2.5 × 103 Pa are obtained in this manner. Significantly, these values are more than two orders of magnitude less than the values which can be obtained for undiluted elastomers. As a result, the thermodynamic work of adhesion between these gels and a flat surface produces a substantial contact area. In fact, the contact area is often so large that modifications to the JKR theory are required. In this paper, we describe the properties for adhesion of low-modulus, thermally reversible gels to flat, rigid substrates. We begin with a discussion of the linear elastic fracture mechanics approach to axisymmetric bodies in contact, of which the JKR theory is a special case. In addition, we develop an analysis to quantify adhesion without directly measuring the load. This section is followed by a description of the synthesis and characterization of the gels themselves. We then conclude with a description of the adhesion experiments, which are interpreted in terms of the fracture mechanics approach developed earlier in the paper.
The contact between an elastic material and a rigid substrate is determined by the balance of elastic energy stored in the bulk of the elastic material, and the adhesive energy required to separate the two surfaces. We consider axisymmetric systems having a circular area of contact with a radius of a. The energy release rate, G, is obtained from the relationship between this contact area, the applied load, P, and the relative displacement of the two surfaces, δ. In the absence of adhesive interactions (G ) 0), a given contact radius corresponds to a load P′(a) and a displacement δ′(a). Adhesive interactions cause the actual loads and displacements to be less than P′ and δ′. For a linearly elastic material, one obtains the following expression for G:11,12
-(P′ - P)2 dC 4πa da
(1)
C is the compliance at fixed contact area:
C)
δ′ - δ P′ - P
on the experimental relationship between δ and a. Substituting for P′ - P in eq 1, we obtain
G)
-(δ′ - δ)2 dC 4πaC2 da
(2)
The assumption made here is that the load/displacement relationship is completely linear at a fixed contact radius. For the situations in which we are interested, the loads are too low to measure experimentally, and we must rely (12) Ward, I. M.; Hadley, D. W. An Introduction to the Mechanical Properties of Solid Polymers; Wiley: New York, 1993.
(3)
The JKR theory assumes that P′ and δ′ are given by the Hertzian values (PH and δH) for the contact of spheres and that the compliance is equal to C0, the value for a flat punch against a semiinfinite elastic half space:
PH )
Fracture Mechanics
G)
Figure 1. Geometry of the adhesion experiment. Soft spherical cap before deformation (a) and after deformation (b).
Ka3 16Ea3 ) R 9R
(4)
a2 R
(5)
2 3 ) 3Ka 8Ea
(6)
δH )
C0 )
where R is the radius of curvature of the sphere, K ) 4E/(3(1 - ν2)), E is Young’s modulus, and ν is Poisson’s ratio, assumed here to be equal to 0.5. Use of PH for P′, δH for δ′, and C0 for C is only expected to be valid when the contact radius is much smaller than the dimensions of the elastic sample. This assumption is not necessarily valid for the geometry of a soft spherical cap between two rigid surfaces, shown schematically in Figure 1. The characteristic dimensions of the lens are the undeformed height, h, the radius of curvature, R, and the basal radius, r, of the lens. For θ < 90°, these quantities are related to one another by the following expressions:
tan(θ/2) ) h/r
(7)
R ) (h/2) + (r2/2h)
(8)
where θ is the contact angle between the lens and the surface on which it is supported. The key quantity which determines the validity of the JKR theory is a/h, the ratio
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Figure 3. Values of C/C0 for the flat punch problem (Figure 2a). The symbols represent the data of Ganghoffer and Gent,13 for ν (Poisson’s ratio) ) 0.48 (O), ν ) 0.49 (4), ν ) 0.495 (+), and ν ) 0.4999 (×). The solid line is the prediction of eq 9, and the dashed line is the prediction of eq 10.
C ) 1 - a/h C0 Figure 2. Experimental geometries for which load-displacement data were obtained: (a) a flat punch of radius a against a thin film of thickness h; (b) a rigid spherical indenter of radius R against a thin film of thickness h.
of the radius of the contact area to the initial height of the lens. We obtain some approximate expressions for C and δ′ from numerical results from finite element predictions. In our case, we are interested in values of a/h between zero and two. Recently, we have compared literature data13 for the compliance of a flat cylindrical punch with a radius of a against a thin, elastic layer and found that the following expression gives a good approximation for the compliance for incompressible materials with ν ) 0.5.14
C ) {1 + 1.33(a/h) + 1.33(a/h)3}-1 C0
(9)
The experimental geometry for which this equation is directly applicable is illustrated in Figure 2a. Our assumption is that corrections for finite size effects for the three geometries illustrated in Figures 1 and 2 have similar dependencies on a/h. Our justification in making this assumption is that the strains in all three cases are largely confined to the region beneath the contact area. Lateral constraints in the compliant material which would result in different compliances for the geometries shown in Figures 1 and 2 are presumed to be unimportant. This assumption is, in fact, supported by direct measurements of the compliance which have been described in a previous publication.11 Figure 3 is a plot of C/C0 as a function of a/h. The symbols represent the data from the finite element modeling of Ganghoffer and Gent,13 and the solid line is the prediction of eq 9. Note that, for a/h < 0.5, C/C0 is a linearly decreasing function of a/h. In this regime, a particularly simple form of the compliance is obtained, and an important equation from the JKR analysis is recovered. For this reason, we begin with the special case of a/h < 0.5 and proceed from there to consider the general case where a/h can be much larger. Case I: a/h < 0.5. In this regime, the finite size correction to the compliance has the following very simple form, shown as the dashed line in Figure 3: (13) Ganghoffer, J.-F.; Gent, A. N. J. Adhesion 1995, 48, 75. (14) Shull, K. R.; Crosby, A. J. Eng. Mater. Technol. 1997, 119, 211.
(10)
Equivalently, one can write C ) C0 - ∆C, with ∆C ) 3/(8Eh). Substitution into eqs 1 and 3 gives
G)
3(P′ - P)2
(11)
32πEa3
and
{ ( )}
2
G 2(δ′ - δ) a 1) E 3πa h
-2
(12)
Equation 11 is the equation which is commonly used to determine the energy release rate from JKR experiments. It is valid despite the significant correction to the compliance because this correction can be treated as a constant offset to C0. With C0 ) 3/(8Ea), we have C ) C0 - 3/(8Eh). For this reason, dC/da ) dC0/da, and the expression for G in terms of P′ - P reduces to the more familiar JKR expression. Because most JKR experiments with soft lenses have been restricted to the regime where a/h < 0.5, analyses based on eq 11 are completely valid. For example, values for G obtained from eq 11 have been shown to agree quantitatively with more direct measures of energy dissipation, such as the integrated area under the load-displacement curve.10 In situations where the load is too low to be measured accurately, eq 12 can be used to determine G/E, provided that δ′ is known. For a/h < 0.5, P′ ) PH and δ′ is modified from the Hertzian value by an amount determined by the correction to the Hertzian compliance:
δ′ ) δH + PH∆C )
{
a2 2a 1R 3h
}
(13)
Simultaneous measurements of P, δ, and a in this regime give completely self-consistent results, when P′ is assumed to be equal to PH and δ′ is assumed to be given by the previous expression.11 In this previous work on finite size corrections to the JKR technique, a predicted value for the compliance correction, referred to as ∆C, was not given. Instead, a value of K∆C was determined which agreed with the experimental load-displacement relationship. We now have that K∆C should be equal to 2/(3h) or, equivalently, that E∆C should be equal to 3/(8h). Indeed, the value of K∆C reported in this earlier reference (1440
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Figure 4. Normalized values of δ′ as a function of a/h for the rigid sphere against a thin film of thickness h (Figure 2b), as a function of a/h. The symbols represent data we have obtained numerically by finite element analysis, the solid line is the prediction of eq 15, and the dashed line is the prediction of eq 13.
m-1) is quite consistent with the value of 1305 m-1 predicted from our simple analysis (with h ) 0.51 mm). The situation for a/h < 0.5 appears to be quite tractable from an analytic standpoint. Nevertheless, we are often interested in situations where a/h is larger than 0.5, in which case we clearly need to use eq 9 for the compliance rather than the simple linear form valid for lower values of a/h. Case II: a/h > 0.5. In this regime, eq 10 is no longer a suitable approximation for the compliance, and eq 9 must be used instead. Substitution of this expression into eq 3 for G gives
{
2
( )}
G 2(δ′ - δ) a a 1 + 2.67 + 5.33 ) E 3πa h h
()
3
(14)
For a/h < 0.5, this equation gives a value for G which is within 10% of the prediction of eq 11, so its validity is not restricted to large values of a/h. All that remains is to obtain an accurate expression for δ′. For large values of a/h, one can no longer assume that P′ is equal to the Hertzian value. As a result, there is no simple correction to the Hertzian value of δ′. Instead, we assume that δ′ is equal to the value obtained for the related situation of a rigid spherical indenter which is pressed against a thin film of thickness h (Figure 2b). As with the compliance correction, we assume that values of δ′ for given values of R, a, and h are identical for the geometries shown in Figures 1 and 2b. We find from finite element analysis that the quantity δ′/δH decays exponentially from its Hertzian value at a/h ) 0, to some fraction of the Hertzian value at large values of a/h:
δ′ ) 0.4 + 0.6 exp(-18a/h) δH
(15)
The finite element data for δ′/δH are shown in Figure 4, along with the predictions of eqs 13 and 15. Note that, for a/h < 0.5, these two equations give values for δ′ which are within 7% of one another. Equation 15 has been verified by measurements involving a hemispherical elastomer immersed in water, in situations where G is equal to zero.15 Synthesis and Characterization of Thermoreversible Gels The low-modulus elastic solids used in our experiments are thermally reversible gels consisting of triblock co(15) Chen, W.-L.; Shull, K. R. Unpublished results.
Figure 5. Storage and loss moduli at a frequency of 1 Hz as a function of temperature for the triblock copolymer solution with φp ) 0.18.
polymers diluted with a solvent that preferentially dissolves the middle block. The end blocks are poly(methyl methacrylate) (PMMA), each with a molecular weight of 23 000 g/mol. The midblock is poly(n-butyl acrylate) (PnBA), with a molecular weight of 92 000 g/mol. This polymer was synthesized by conversion of a precursor polymer which had a poly(tert-butyl acrylate) midblock. The anionic polymerization of the precursor polymer and the subsequent conversion of the midblock to poly(n-butyl acrylate) were carried out in a fashion very similar to that described by Varshney et al.16 Briefly, a difunctional initiator is used to initiate the polymerization of the tertbutyl acrylate monomer in THF. After a small amount of the PTBA homopolymer is removed for characterization, methyl methacrylate monomer is added to the living polymer solution to form the triblock copolymer. The molecular weight of the PTBA midblock was determined from size exclusion chromatography of the midblock, and the relative number-average block lengths were determined by NMR. The polydispersity index for the midblock, and for the overall triblock copolymer, was approximately 1.11, as determined by size exclusion chromatography in toluene. Thermoreversible gels were prepared by making solutions of the triblock copolymers in warm 2-ethylhexanol. Gels can also be produced by making solutions in lower alcohols such as ethanol or n-butanol, but the higher vapor pressure of these solvents leads to rapid drying of the gels, thereby complicating the adhesion measurements described in the following section. Rapid solvent evaporation was not a problem with 2-ethylhexanol. For this reason, our experiments were confined to gels prepared with this solvent. Gelation proceeded rapidly as the solutions were cooled to room temperature. Rheological characterization of the gels made from different polymer volume fractions, φp, was performed, using a Bolin VOR rheometer with a Couette (concentric cylinder) geometry, in order to verify that the gels satisfied the following requirements: (1) At room temperature, the polymer solutions form gels with a completely elastic response, with a frequencyindependent value of G′ which far exceeds G′′. These aspects of the gels are illustrated in Figure 5, where G′ and G′′ at a frequency of 1 Hz are plotted as a function of temperature for a gel with φp ) 0.18. The gel point for this polymer, defined as the temperature at which G′ and G′′ are equal to one another, is equal to 65 °C for this value of φp. In Figure 6, G′ and G′′ are plotted as a function (16) Varshney, S. K.; Jacobs, C.; Hautekeer, J.-P.; Bayard, P.; Je´roˆme, R.; Fayt, R.; Teyssie´, P. Macromolecules 1991, 24, 4997.
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Figure 6. Storage and loss moduli at 25 ° C as a function of frequency for a triblock copolymer solution with φp ) 0.036.
Figure 8. Schematic diagram of the apparatus used for the adhesion measurements.
Figure 7. Time dependence of the storage and loss moduli at a frequency of 1 Hz at 25 °C for a triblock copolymer solution with φp ) 0.036.
of frequency at 25 °C, for a gel with φp ) 0.036. The response of this material is also purely elastic, even though the copolymer volume fraction is quite low. (2) At an elevated temperature, the polymer solutions behave as viscous liquids with values for G′ and G′′ which are typical of homopolymer solutions. This point is illustrated by the data shown in Figure 5. At 70 °C, this polymer has a complex viscosity of approximately 10 Pa‚s at 1 Hz and a value of G′′ at this frequency which exceeds G′ by nearly an order of magnitude. (3) When a solution at elevated temperature is cooled, gelation occurs very rapidly, giving an elastic gel with a modulus that does not change with time. In fact, we have been unable to detect any time dependence at all of the rheological properties of the gels at room temperature. A steady state value of the modulus is obtained as soon as the temperature in the rheometer stabilizes. The data in Figure 7 show that, for a gel at room temperature with φP ) 0.036, no discernible change in the modulus is observed for 30 min, a time which exceeds the time required to conduct the adhesion measurements described in the following section. This rapid, reversible gelation is perhaps the most useful characteristic of our particular system. Adhesion Measurements The experimental geometry used for the adhesion measurements is shown schematically in Figure 8. Hemispherical lenses of the gel-forming material are produced on glass microscope slides by a procedure similar to that introduced by Chaudhury.3 First, glass slides are fluorinated by dipping them in a 10% solution of 1H,1H,2H,2H-perfluorodecyltrichlorosilane (PCR, Inc.) in
hexane. A small portion of a gel is placed onto the glass slide and heated to a temperature at which it is able to flow, thereby forming a spherical cap with a shape defined by the contact angle which the gel-forming solution makes with the substrate. For our experiments, θ is typically close to 55°. The gel is then mounted upside down under an optical microscope so the contact area between the gel and an opposing silicon substrate can be viewed in reflected light. A thin film of polystyrene or poly(methyl methacrylate) is typically spun cast onto the silicon wafer, in order to provide a reproducible surface to which the gel will adhere. The distance between the lens and the substrate is then decreased, and the relationship between the contact radius (a) and displacement (δ) is recorded. Our sign convention for δ is that it increases as the lens is brought into contact with the substrate. A value of zero for δ corresponds to the situation where the lens and substrate initially come into contact. Figure 9a shows the relationship between a and δ for a gel with R ) 1.93 mm, h ) 0.784 mm, and φp approximately equal to 0.12 in contact with a PMMA substrate. Note the large increase in the contact radius as the gel first comes into contact with the substrate at δ ) 0. The increase in the contact area is driven by the thermodynamic work of adhesion and is retarded by the energy penalty associated with the elastic deformation of the gel. The equilibrium value of a at a given value of δ is determined by the balance between these energetic effects, as quantified by the analysis presented above. Because the elastic moduli of the gels used in our experiments are quite low, the elastic energy associated with a given deformation of the lens is also low, and large contact areas are obtained from the relatively weak adhesive interactions which are present in these systems. One consequence of the large deformations induced by the adhesive forces is that we must apply an energy balance based upon eqs 14 and 15. The value of a/h at δ ) 0 is equal to 0.56, which exceeds the value 0.5, beyond which eqs 12 and 13 are no longer valid. By inspection of Figure 9a, it is easy to see that very similar results are obtained during loading (increasing δ) and unloading (decreasing δ), indicating that the adhesion hysteresis in this system is very low. There are no significant dissipative processes operating in this system, and the only energy involved in separating two surfaces is the thermodynamic energy penalty associated with
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Figure 10. Comparison of elastic moduli obtained from oscillatory shear rheometry (O) and from the adhesion measurements (4), assuming G ) 70 mJ/m2 and G ) 68mJ/m2 for contact with PMMA and PS, respectively.
Figure 9. Results for a gel with φp ∼ 0.12, R ) 1.93 mm, and h ) 0.784 mm adhering to a PMMA substrate: (a) relationship between contact radius and displacement (The dashed and solid lines represent the predictions of eqs 13 and 15, respectively, for δ′.) (b) values of G/E (energy release rate normalized by Young’s modulus) obtained from the data from part a, using eqs 14 and 15.
creating a new surface area. In other words, the critical energy release rate is equal to the thermodynamic work of adhesion, W, associated with the elimination of a gel/ substrate interface in favor of a gel surface and a substrate surface:
W ) γ(gel) + γ(substrate) - γ(gel/substrate) (16) In this case, G ) W both on loading and on unloading, and this value of G should be independent of the actual values of a or δ. The value of G (normalized by Young’s modulus), obtained from the data in Figure 9a by the application of eqs 14 and 15, is shown in Figure 9b. The slight variation in G/E for different values of δ can be attributed to the effect of approximations which are inherent in our treatment. Specifically, the gels are not linearly elastic, but they obey a more complicated constitutive equation. One expects that the linear approximation that we have used will underestimate the available strain energy for compressive loads and overestimate the available strain energy for tensile loads.11 At very high tensile loads, this linearly elastic assumption begins to break down. The linearized approach we have adopted gives what we believe to be the best trade-off between simplicity and accuracy. In fact, the values of G/E obtained are in very good agreement with values which can be independently determined for our gels. In order to illustrate this point, we have measured the concentration dependence of the elastic modulus for a series of lenses by oscillatory shear rheometry. Values for Young’s moduli, E (E ) 3G′), are plotted as the circles in Figure 10. We have also measured the relationship between the contact radius and the displacement for a series of different gels and have used eqs 14 and 15 to obtain plots of G/E vs δ. These plots are shown in Figure 11. Values for E were then obtained by assuming that G is equal to the thermodynamic work of adhesion W, given by eq 16. Because the polymer volume fractions in the gels are relatively low and because the solvent has the lowest surface energy of the different
Figure 11. Values for G/E obtained from eqs 14 and 15, for gels with different compositions: (a) φp ) 0.06, (b) φp ) 0.062; (c) φp ) 0.10.
components in the gel, we know that γ(gel) must be very close to the surface energy of the solvent, which is 30 mJ/m2 for 2-ethylhexanol.17 For the PMMA substrate, we have γ(substrate) ) 42 mJ/m2.18 For the substrate/gel interfacial energy we expect a value typical of liquid/liquid interfaces, i.e., something less than 5 mJ/m2. The specific value chosen for γ(gel/substrate) will not significantly affect our results. A value of 70 mJ/m2 for W (and, hence, for G) is expected to be within 10% of the actual value. Similarly, for contact between a gel lens and a PS surface, (17) Solvents Guide; Marsden, C., Mann, S., Eds.; Interscience: New York, 1963. (18) Polymer Handbook, 3rd ed.; Brandrup, J., Immergut, E. H., Eds.; Wiley: New York, 1989.
Thermally Reversible Gels
W is approximated as 68mJ/m2. By using these values for G, we obtain values for E which are indicated by the triangles in Figure 10. The excellent agreement between these values and the values determined by direct measurement is further confirmation that an analysis based on eqs 14 and 15 is accurate. We conclude here by mentioning several advantages of the experimental methodology which we have utilized. One very useful feature of these experiments is that the only quantities which need to be measured are the displacement and the contact radius. While the displacement is controlled in our device with a piezoelectric inchworm motor, a much simpler device driven manually by a micrometer screw would work just as well. One can readily design a portable device which can easily be used with any optical microscope. We have found, for example, that the contact radius between two polymer surfaces immersed in water is very difficult to visualize by standard, reflected light optical microscopy. Other microscopes, based on fluorescence imaging or the use of Nomarski contrast, might be much more suitable for these types of experiments. A simple micrometer-driven assembly can be easily used in a variety of specialized microscopes, without requiring that they be modified in any way. Similar issues arise if one is interested in using these techniques to measure adhesive interactions in biological systems. In this case, one could use our techniques to quantify the adhesion between a soft gel and a layer of living cells. The availability of a simple, transportable device for measuring adhesive interactions is essential here, since the experiments need to be done in carefully controlled, sterile environments. Finally, the small sample volumes necessary in our experiments can also be a significant advantage when using highly specialized materials that might be available only in small quantities. The volumes of the gels used in measurements of G/E are in the microliter range, which is about 3 orders of magnitude less than the volumes required for the standard rheological experiments. Clearly, there is enormous potential for these types of experiments to expand our understanding of both the elastic and adhesive properties of a variety of important materials. Summary and Conclusions We have developed a linear elastic fracture mechanics approach to analyze the adhesive interactions between two axisymmetric bodies in contact, while only measuring the resulting contact area and displacement. Our analysis has been applied to adhesion tests where a low-modulus, hemispherical lens is pressed against a rigid substrate. Modifications to the standard JKR theory are necessary when the contact area becomes large relative to the thickness of the elastic layer. The spherical cap which is formed from a swollen, triblock copolymer with poly(methyl methacrylate) end blocks and a poly(n-butyl acrylate) midblock serves as an excellent model system for these experiments. The synthesis and rheological characterization of these unique, thermally reversible gels
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have been described in detail. These materials have been used to demonstrate the accuracy and simplicity of our methods for measuring the energy release rate without directly measuring the applied load. Our primary conclusions are as follows: (1) It is necessary to account for finite size effects when the JKR assumption of a small contact area relative to the dimensions of the lens is no longer valid. Two different approaches have been outlined for taking these finite size effects into account: a simplified theory valid for a/h < 0.5 and a slightly more complicated version which is approximately valid for all commonly encountered values of a/h. (2) The modified equations to the JKR theory provide an accurate measure of G/E, the energy release rate normalized by Young’s modulus, as shown by the excellent agreement with moduli measured rheologically. This comparison could be made because the gels used in our studies showed reversible adhesion to the substrates used, thereby allowing G to be approximated as W, the thermodynamic work of adhesion. (3) The fracture mechanics analysis based on measurements of contact area and displacement opens the door to a wide variety of adhesion tests. For experiments where it is either inconvenient to measure the applied load or necessary to use a small sample size of material, this approach proves to be very useful. It extends the possibility for probing adhesive interactions in other fields such as biomaterials, where environmental conditions must be rigorously controlled, and direct measurement of the applied load may be difficult. (4) The a vs δ curves from the JKR adhesion tests indicate that this polymer system is very sensitive to weak adhesive interactions, thus making it an excellent model system to probe subtle interfacial effects. We are able to probe these weak adhesive interactions between two bodies in contact by using a low-modulus material in which solvent molecules dilute the entanglement networks of the triblock copolymer. (5) The gel-forming acrylic copolymer swollen with 2 ethylhexanol serves as an ideal model system for adhesion experiments. Rheological studies show that the swollen triblock copolymer exhibits a rapid, thermally reversible gelation, in which the gels have a completely elastic response at room temperature and do not exhibit any tendency to flow over the time period needed to complete adhesion experiments. Acknowledgment. The authors would like to thank W. Burghardt and W.-L. Chen for many helpful discussions. Acknowledgment is made to the National Science Foundation for support of this project under Grant DMR9457923 and to the donors of the Petroleum Research Fund for partial support of this work. Access to the ABAQUS finite element package is made available through a HKS academic site license. LA9705672
