Langmuir 2008, 24, 9401-9409
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Model-Independent Extraction of Adhesion Energy from Indentation Experiments Shilpi Vajpayee,† Chung-Yuen Hui,‡ and Anand Jagota*,† Department of Chemical Engineering, Lehigh UniVersity, Bethlehem, PA 18015, and Department of Theoretical and Applied Mechanics, Cornell UniVersity, Ithaca, NY 14853 ReceiVed March 16, 2008. ReVised Manuscript ReceiVed May 25, 2008 We present a model-independent method for extraction of adhesion energy from indentation experiments in which force, deflection, and contact area are measured. It is based on calculating the difference between external work supplied and stored strain energy, the deficit, which is assigned as work expended in opening (or closing) an interface. The critical step in the method is determination of the elastic indentation response without adhesion, using measured compliance. We show that this model-independent method accurately reproduces results obtained using the Johnson-Kendall-Roberts theory when it is applicable, and is particularly useful when an analytical contact mechanics solution is unavailable.
1. Introduction The energy per unit area required to separate two materials, the adhesion energy, is a fundamental interfacial property that one often needs to measure.1 For macroscopic scale specimens, there are many well-established techniques that allow one to extract the adhesion energy directly from experimental measurements such as force, deflection, and crack length (or interfacial area) in a model-independent manner.1 Extraction of adhesion energy is a special case of the more general problem of measuring fracture toughness, in which the crack is confined to an interface. Not surprisingly, many measurement techniques are similar or identical.1,2 By model-independent, we mean that the extracted value of adhesion energy relies only on directly measured quantities and physical assumptions such as linear elastic constitutive behavior of the body. Recognizing that adhesion energy, G, is the rate of change with area of the difference between supplied external work and stored elastic energy provides one with the expression
G)
P2 dC 2 dA
(1)
under constant load where C is the compliance of the specimen, P is the external load, and A is the decohered area.1,2 Indentation of one surface by another is often a particularly attractive method for adhesion measurement.3 Advantages include simplicity of sample preparation, ease in surface modification, and the ability to make local and spatially resolved measurements.4 The availability of the Johnson-Roberts-Kendall (JKR) theory of adhesive contact5,6 has allowed easy and accurate extraction of adhesion energy from measurements of force and contact area. It can be used to extract rate-independent works of adhesion * Corresponding author. E-mail address:
[email protected]. † Lehigh University. ‡ Cornell University.
(1) Kinloch, A. J. Adhesion and AdhesiVes; Chapman and Hall: London, 1987. (2) Lawn, B. Fracture of Brittle Solids, 2nd ed.; Cambridge University Press: New York, 1993. (3) Shull, K. R. Mater. Sci. Eng. R 2002, 36, 1–45. (4) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7(5), 1013–1025. (5) Johnson, K. L. Contact Mechanics; Cambridge University Press: New York , 1987. (6) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324(1558), 301–313.
or rate-dependent adhesion energy.7 However, the JKR theory applies only for spherical indentation of an elastic half-space. For instance, if either of the contact bodies is finite in extent, or if a sphere is indented into a multilayered elastic medium with nonuniform or anisotropic properties, then the standard JKR theory is no longer applicable, and extension of the theory is required. While this has been done successfully in some cases,3,8 one often needs to rely on numerical solutions on a case-by-case basis. The advantages of the indentation technique for adhesion measurement could be extended to a broader class of materials and specimens if a general, model-independent, method such as equation (1) could be applied. For indentation tests, a similar model-independent procedure can be applied if one can find the underlying force-displacement indentation response that would be obtained absent adhesion, which we call the Hertz curve.3,5,8,9 We begin by following a derivation by Shull,3 which we will rederive in section 2 for more general systems. We then demonstrate how one can extract adhesion energy using measurements of force, deflection, and contact area, under the assumption that the bodies are linearly elastic. Like JKR analysis, it can be applied also when the adhesion energy depends on crack velocity. Where JKR analysis is applicable, we show that the new method yields identical results. Our motivation for this work arises from recent interest in bioinspired and other surface-structured adhesives.10–14 Through indentation and other experiments, it has been shown that these materials can have significantly enhanced adhesion. Indentation experiments provide direct measurements of compliance and (7) Chaudhury, M. K. J. Phys. Chem. B 1999, 103(31), 6562–6566. (8) Sridhar, I.; Johnson, K. L.; Fleck, N. A. J. Phys. D: Appl. Phys. 1997, 30, 1710–1719. (9) Barthel, E.; Perriot, A. J. Phys. D: Appl. Phys. 2007, 40(4), 1059–1067. (10) Majidi, C.; Groff, R. E.; Maeno, Y.; Schubert, B.; Baek, S.; Bush, B.; Maboudian, R.; Gravish, N.; Wilkinson, M.; Autumn, K. Phys. ReV. Lett. 2006, 97(7), 76103. (11) Crosby, A. J.; Hageman, M.; Duncan, A. Langmuir 2005, 21(25), 11738– 11743. (12) Ghatak, A.; Mahadevan, L.; Chung, J. Y.; Chaudhury, M. K.; Shenoy, V. Proc. R. Soc. London, Ser. A 2004, 460(2049), 2725–2735. (13) Glassmaker, N. J.; Jagota, A.; Hui, C. Y.; Noderer, W. L.; Chaudhury, M. K. Proc. Natl. Acad. Sci. U.S.A. 2007, 104(26), 10786. (14) Kim, S.; Sitti, M. App. Phys. Lett. 2006, 89, 261911.
10.1021/la800817x CCC: $40.75 2008 American Chemical Society Published on Web 07/29/2008
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Figure 1. Experimental set up for the indentation tests. A spherical indenter is lowered into the sample to a certain depth and then pulled out using a motorized stage. The load on the indenter, P, is measured using an in-line load cell, the indenter’s vertical displacement, δ, is measured using a capacitance sensor, and the contact between the indenter and sample is obtained by recording its image through the inverted optical microscope.
pull-off load, but are not straightforward to analyze for adhesion energy.15
2. Energy Release Rate in an Indentation Experiment A typical setup for an indentation test is shown in Figure 1, where a rigid sphere is pushed into contact with a flat elastic layer (typically an elastomer) and then retracted. Figure 2 shows a typical load-deflection trace one might measure in such an experiment. (The traces shown in Figure 2 have been calculated using the JKR model; Figures 8 and 12, discussed later in the paper, show experimental measurements.) For most material systems, if the normal force P (P > 0 for compression) is plotted against the indenter displacement δ, the loading curve (increasing contact area) lies above the unloading curve (decreasing contact area), as shown in Figure 2. Since the air gap between the surface of the indenter and the layer can be viewed as an external crack, this implies that the energy release by crack healing is generally smaller than the energy needed to grow the crack by the same area. It is common to call such behavior hysteretic. We will restrict our attention to cases where hysteresis arises as a result of interfacial processes, rather than because of bulk inelasticity. This restriction implies that for a fixed deformation state, the elastic energy and its release rate from the bulk are uniquely defined. Even with this restriction, it is not possible to fit the experimental data using JKR theory with a single work of adhesion. In some cases, it is possible to obtain a good fit using two different works of adhesion: one for crack healing and the other for crack growth. In the following, we shall assume that the indenter and substrate are elastic bodies. The contact between them is usually circular, with radius a; more generally, it is characterized by an area, A. For example, the indenter can be a spherical lens and the substrate can consist of many elastic layers. Consider first the special case where there is no adhesion. For a spherical indenter on a smooth elastic half-space, the mechanics are described by the classical Hertz solution.5 In general, for any indentation experiment with adhesion, one can imagine a corresponding experiment with adhesion turned off. In the following, we will continue to refer
Figure 2. Typical load-displacement curve for indentation of a halfspace by a sphere as given by the JKR model. The example shown here is for a sphere of radius 4 mm indenting an incompressible elastic halfspace with Young’s modulus of 2 MPa. The works of adhesion for loading and unloading are different. (a) The hatched area represents the strain energy in the system at point O, which has the same contact area as OH, which lies on the Hertz curve. Line OOH is obtained from the known contact compliance at the given contact area. Strain energy, SE ) I + II. (b) The hatched area represents external work done on the system to reach state O, W ) II + III - IV.
to the solution of this adhesion-less case as the Hertz solution (shown by the uppermost thin dotted line in Figure 2). The external load for the adhesion-less problem will be denoted by PH (the Hertz load) and the displacement by δΗ, the Hertz displacement). Note that PH, δH, and the contact area A are not independent variables; any two of these variables can be expressed in terms of the third. We define the incremental compliance C by
C ) dδH ⁄ dPH
(See the red dotted line in Figure 2.) The compliance, C, depends on specimen geometry, elastic moduli, and the contact area. For a given specimen, it is often convenient to express C as a function of the contact radius using chain rule, i.e.,
C ) (dδH ⁄ da)(da ⁄ dPH) ) (dδH ⁄ da)(dPH ⁄ da)-1 (2E*a)-1
(3)
In classical Hertz theory, C ) for a rigid spherical punch, where E* is the plane strain modulus of the half-space. 5
(15) Noderer, W. L.; Shen, L.; Vajpayee, S.; Glassmaker, N. J.; Jagota, A.; Hui, C.-Y. Proc. R. Soc. London, Ser. A 2007, 463, 2631–2654.
(2)
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Let us return to the adhesion problem and compute the energy release rate G in a load controlled test. In a load controlled test
G)
|
∂PE ∂A P
(4)
where PE is the potential energy of the system, given by
PE ) SE - Pδ
(5)
where SE denotes the strain energy of the system at the elastostatic state (P,δ,A). Because the material is elastic, SE is independent of the deformation history. We compute SE considering a special deformation history. In step 1 of this deformation, the adhesive forces are turned off so that the deformation is Hertzian. The strain energy of the system at the end of the first step is
∫0δ
H
PH(δ′)dδ′
(6)
In Figure 2a, this would be the area under the Hertz curve between the origin and point “OH” (that is, δH in eq 6 is the Hertz displacement at OH). In the second step, the adhesive forces are turned on with the contact area fixed at A, where A is the contact area at the end of the first step (for example, at OH). Because the contact area is fixed, PH drops to P, which is the actual measured load in the test. Likewise, δH drops to δ, the actual displacement. In this step, the geometry of the specimen does not change (since the contact region is fixed) and since the materials are linearly elastic, the compliance of the specimen C is the same as the incremental compliance of a Hertz contact experiment at the same contact area A, i.e.,
C ) dδH ⁄ dPH
(7)
It should be noted that C is the normal compliance of a rigid flat cylindrical punch of area A indenting the substrate. It can be measured experimentally or, for a given specimen, it can be obtained using numerical methods.8,16 Furthermore, because the material is linear and A is fixed, the compliance of the system during step 2 does not change and is related to the displacement and load by
C)
δ - δH P - PH
(8)
Equations 7 and 8 imply that the load and displacement (P,δ) lie on the tangent line to the Hertz curve at (PH,δH) (e.g., line OHO in Figure 2). Since A is fixed in step 2, we have
dδ ) CdP
(9)
H
H
∫0δ
H
C PH(δ′)dδ′ + [P2 - PH2] 2
|
dδH dC ⁄ dA 2 dPH ∂SE ) PH + [P - PH2] - CPH ) ∂A P dA 2 dA dC ⁄ dA 2 [P - PH2] (12) 2
(10)
where we have used eq 9. This change in strain energy is Area∆AOB - Area∆BOHC, where Area∆AOB denotes the area (always positive) of the triangle with vertices A, O, and B. The total SE is the sum of the strain energy change in steps 1 and 2. It is
SE )
the partial derivative of SE with respect to area with P fixed gives
where we have used dδH/dA ) (dδH/dPH)(dPH/dA) ) C(dPH/ dA). The potential energy of the external load is, using eq 8,
The change in strain energy during step 2 is
∫δδ P(δ′)dδ′ ) C∫PP P′dP′ ) C2 [P2 - PH2]
Figure 3. (a) Graphical representation of the energy deficit at point O. Deficit at O ) I - III + IV. (b) The hatched region represents the energy released, i.e., change in deficit when state changes from point O to point O′.
(11)
This is shown in Figure 2a as the sum of two areas I and II, and is obtained by adding the area under the Hertz curve between the origin and point “OH” to Area∆AOB - Area∆BOHC. Taking (16) Shull, K. R.; Ahn, D.; Chen, W. L.; Flanigan, C. M.; Crosby, A. J. Macromol. Chem. Phys. 1998, 199, 489–511.
-Pδ ) P(C[PH - P] - δH)
(13)
Taking the partial derivative of eq 13 with P fixed, we have
-P∂δ ⁄ ∂A|P ) PPHdC ⁄ dA - P2dC ⁄ dA + CPdPH ⁄ dA - PdδH ⁄ dA (14) The energy release rate is obtained by adding eqs 12 and 14, i.e.,
G)
dC ⁄ dA dC ⁄ dA [-P2 + 2PPH - PH2] ) (P - PH)2 2 2 (15)
where we have used CPdPH/dA - PdδH/dA ) 0. The derivation is very similar for a displacement-controlled test where G ) (∂SE/∂A)|δ. The energy release rate is
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(δ - δH)2 dC G)2C2 dA
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(16)
The energy release rate expressions given by eqs 15 and 16 are identical because of eq 8.
3. Finding Compliance Equations 15 and 16 show that the energy release rate of the external crack is completely determined by the incremental compliance, C, and the Hertz load or displacement. The first step is to find the contact compliance (and contact area) at different measured (P,δ). For material systems that exhibit hysteresis, C can be obtained directly from experiments. Indeed, our comment below eq 8 implies that the compliance C at a given contact area A can be determined by drawing a straight line between two points on the load-versus-deflection curve that have the same contact area (e.g., OOH in Figure 2). Another way to determine the instantaneous compliance is to unload slightly at different points on the load deflection curve in Figure 2. If the material is even slightly hysteretic, the contact line will be pinned for some range of displacements during unloading. This means that the initial slope of the unloading line is 1/C. According to the discussion above (eq 9), this line must be tangent to some point on the Hertz curve. Wahl and co-workers have shown, for example, that during dynamic oscillations, the contact does remain pinned, allowing compliance to be measured.17,18 To summarize: (a) Compliance can be measured experimentally by slight unloading during loading.16–18 If the material is hysteretic, the contact will be pinned for some range of displacement, and hence the initial slope of the force-displacement curve on unloading is the inverse of the compliance. (b) Complete a load-unload indentation cycle. If there is some hysteresis, compliance can be measured by matching selected points on the loading curve with corresponding points on the unloading curve that have the same contact area. The inverse of the slope joining these two points is the compliance. (c) Numerically compute the compliance for a given geometry, or measure it (the compliance) independently with a set of flat punches.
Figure 5. Plot of deficit vs contact area for loading and unloading curves for JKR data sets with and without noise.
Figure 6. Plot of P/a3/2 against a3/2 for loading and unloading curves for exact JKR data sets with and without noise.
In the remainder of this paper, method (b) has been used to obtain contact compliance.
4. Direct Determination of Energy Deficit If the Hertz curve is known, application of eq 15 or 16 to determine adhesion energy requires calculation of dC/dA at every point. We have found that, for simulated JKR data (see section 6), use of numerically evaluated dC/dA yields an accurate estimate of adhesion energy. Practically, since C is the directly measured quantity, this requires numerical differentiation, which can introduce errors. Therefore, we implement a method that calculates the energy deficit directly from experimental data and the Hertz curve. Define the energy deficit as the difference between work done by the external load and the stored elastic energy:
D ) SE Figure 4. Simulated load-displacement curve with Gaussian noise in force, displacement, and contact area. The amplitude of the noise is 1% of the peak value.The thick dotted lines in the figure join the points on loading and unloading curves that have the same contact area. The circles represent the Hertz line found for the simulated noisy “data” using the compliance method. The thin dotted curve is the exact known Hertz curve.
∫0δ P(δ ′ )dδ′
(17)
For example, during loading, the strain energy exceeds the work done by the external force because of the work done by adhesive (17) Ebenstein, D. M.; Wahl, K. J. J. Colloid Interface Sci. 2006, 298(2), 652–662. (18) Wahl, K. J.; Asif, S. A. S.; Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci. 2006, 296(1), 178–188.
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Figure 9. Plot of intercept versus slope of tangents to the Hertz curve for a flat PDMS sample. The negative of the slope of this curve yields the Hertzian displacement at the corresponding point (eq 23). Figure 7. Adhesion energy values calculated for exact JKR data with varying noise amplitude. The underlying Wad values are 0.2 J/m2 (loading) and 0.6 J/m2 (unloading).
forces at the contact edge, and thus the deficit, D, is positive. Our method is based on eqs 4, 5, and 11 in the previous section. If the Hertz curve and compliance at different contact areas are known, then the strain energy (SE) at a point O (Figure 2) in the load-displacement curve can be calculated by numerical evaluation of eq 11, as shown in Figure 2a. The net external work done on the sample is
W)
∫0δ P(δ)dδ ) AreaII + AreaIII - AreaIV
(18)
as shown in Figure 2b. Hence, the deficit
D ) SE - W ) AreaI + AreaII - AreaII - AreaIII + AreaIV (19)
Figure 10. Plots of deficit versus contact area for a flat PDMS sample. The slopes of the plots, the values of adhesion energy, are 30.24 ( 3.08 mJ/m2 (loading) and 239.12 ( 3.70 mJ/m2 (unloading).
The deficit at point O is shown graphically in Figure 3a. We are interested in the change in deficit as the contact area changes.
The change in deficit as the indenter is unloaded from O to O′ is shown graphically in Figure 3b. Note that if there is no change
) AreaI - AreaIII + AreaIV
Figure 8. (a) Experimental force-displacement data for a flat PDMS sample indented with a glass sphere. The dashed lines connect selected points on the loading and unloading curves with the same contact area and are used to obtain compliance and the Hertz curve (black circles). (b) Contact images for two points on loading and unloading curves with the same contact area. The force and deflection corresponding to such pairs of images are used to calculate compliance.
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experimental data. To obtain PH ) PH(δH), we note that for any A
mδH ) PH - c
(21)
Differentiating eq 21 by A and using
dδH dPH ) dA dA
(22)
dc ) -δH dm
(23)
m we have
Figure 11. Plot of P/a3/2 against a3/2 for the flat PDMS sample with oxidized surface. From this plot, the values of adhesion energy are 24.1 ( 4.3 mJ/m2 (loading) and 246.2 ( 36.8 mJ/m2 (unloading). For the linear fit to unloading data, the first 10 points after reversal of indenter motion were ignored.
in contact area, which implies that compliance remains unchanged, then there is no change in the deficit. The absolute adhesion energy at a particular state is the rate of change of deficit with contact area. Hence, it can be estimated as the slope of a plot of deficit as a function of contact area. A version of this method was used in a previous work but with the assumption that the Hertz curve is approximately given by the experimental loading curve.15 (This assumption is a good approximation for the energy release rate when the adhesion energy on unloading (or crack opening) is considerably larger than the adhesion energy on loading (or crack closure).) Note that, by construction, we do not need two different works of adhesion to determine the deficit. If there are two different works of adhesion, it helps to determine the deficit and the contact compliance within a single experiment. However, if contact compliance is known, can be measured, say, by slight unloading, by connecting a point on the loading curve with a point on the unloading curve with the same contact, or computed theoretically, then the method applies equally well when there is a single curve with adhesion but without hysteresis.
5. Finding the Hertz Curve For both the proposed methods, a common and necessary first step is the determination of the Hertz curve. It is a well-known result in geometry that any curve in the plane can be represented as the envelope of a family of tangent lines. Since every tangent line on the Hertz curve intersects a point on the loading (unloading) portion of the P-δ curve, the Hertz curve PH ) PH(δH) can be reconstructed as the envelope of the family of tangent lines with slope m ) 1/C(A). We considered three different methods to determine the Hertz curve. Here we present one that we found to be most robust; the other two are described in the Supporting Information. The envelope of the family of tangent lines is described by the equation
m ) m(c)
(20)
where c is the intercept of each tangent line on the P axis. Now, for any point on the unloading curve, m ) 1/C(A) can be determined (see above discussion), and this allows us to determine c. In this way a series of (c, m) pairs can be obtained from
Once a set of (c, m) pairs has been obtained, we obtain a polynomial fit between the two, which yields c(m). We enforce the condition that c ) dc/dm ) 0 at m ) 0, i.e., at δH ) 0 the slope of the curve vanishes and its intercept is zero. Once c(m) is known, eqs 21 and 23 yield a set of points on the Hertz curve.
6. Extracting Adhesion Energy In this section we demonstrate the compliance-based technique for extraction of adhesion energy by showing the following examples of its application: (1) Simulated JKR indentation with noise. We generate simulated “data” based on JKR theory modified by Gaussian noise. In the limit of very low noise, we establish that the compliance-based technique yields the exact value of adhesion energy. We also show how the accuracy of the technique is affected when measured data contain Gaussian noise. (2) Indentation experiments on flat poly(dimethylsiloxane) (PDMS). We compare adhesion energy extracted by the compliance and JKR techniques from experimental data on a flat PDMS specimen. (3) Indentation experiments on a structured surface. We extract adhesion energy for a material system for which it would not be possible to do so in any other way, since a contact mechanics model is not available. 6.1. Simulated JKR Indentation with Noise. As a check of the compliance method, we first apply it to simulated “data” created from the JKR theory but with additional noise, corresponding to moderate indentation of an elastic half-space by a sphere. According to the JKR model for indentation, the loaddisplacement relationship is given by the following equations:5,6
(
P-
)
4E/a3 2 ) 8πWadE/a3 3R 4E/a3 3R δ ) (πa ⁄ 2E/)(po + 2p′o)
w P ) - √8πWadE/a3 + and
where
po ) 2aE/ ⁄ πR p′o ) -(2WadE/ ⁄ πa)1⁄2(24)
where, R is the radius of spherical indenter, Wad is the work of adhesion, and E* is the effective plane strain modulus of elasticity, which is calculated as
1 - υsample 1 - υglass 1 ) + / E Eglass E sample 2
2
(25)
Esample, Vsample, and Eglass, Vglass are the elastic moduli and Poisson’s ratios of the elastic half-space and the glass indenter, respectively. (Since for a typical elastomer Eglass . Esample, E* is effectively the sample plane strain modulus.) Here, force, P, as well as displacement, δ, are related to the contact radius, a.
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Figure 12. (a) Force-displacement plot for fibrillar sample with fibril height of 60 µm and interfibrillar spacing of 38 µm. The lines with constant compliance have been found by matching the points with the same contact area on loading and unloading curves. The inverse of the slope of these lines gives the contact compliance. (b) Example of contact images for points on loading and unloading curves with the same contact area. The force and deflection corresponding to such pairs of images are used to calculate compliance.
Figure 13. Plot of deficit versus contact area for a fibrillar sample with fibril length of 60 µm and interfibrillar spacing of 38 µm. Table 1. Values of Work of Adhesion and Separation for the Fibrillar Samples with Fibril Height of 60 µm and Three Different Interfibrillar Spacingsa interfibrillar spacing (µm) 87 62 38
work of adhesion (mJ m2)
work of separation (mJ/m2)
27 ( 3 24 ( 4 19 ( 1
417 ( 8 346 ( 6 257 ( 3
a The adhesion energy values during loading and unloading increase with the interfibrillar spacing, as discussed in previous work.15
Therefore, given a series of different contact radii during the indentation test, corresponding values of force and displacement can be found. The following values of these parameters are used in the example below: R ) 0.004 m Wad for loading ) 0.2 J/m2 Wad for unloading ) 0.6 J/m2 Esample ) 2 MPa νsample ) 1/2 Random Gaussian noise is introduced in the three “measurements” generated from the exact JKR model: force, displacement, and contact area. By assuming that the noisy data are from an
underlying exact JKR model, we can compare the compliance method for extraction of adhesion energy to that obtained by plotting P/a3/2 against a3/2 for both loading and unloading data. According to eq 24, this plot, which we call the JKR plot, should be a straight line. Together, the slope and “y”-intercept of this line will give the modulus of elasticity and the work of adhesion. Figure 4 shows simulated load-displacement data when the root-mean-square (rms) amplitude of the noise is 1% of the peak value of the corresponding quantity. Specifically, rms noise amplitude in force ) 4.23 × 10-4 N rms noise amplitude in displacement ) 3.4204 × 10-7 m rms noise amplitude in contact radius ) 4.5 × 10-6 m The compliance is determined following method (b) in section 3. The Hertz curve is then obtained using the technique discussed in section 5. Figure 5 shows the plot of deficit versus contact area both for the ideal JKR case without noise as well as for data in which the Gaussian noise is introduced in the contact radius, force, and displacement data. The values for estimated adhesion energy with 1% noise as obtained using the compliance method are 0.27 ( 0.01 J/m2 and 0.67 ( 0.02 J/m2 for loading and unloading, respectively (compared to the underlying values of 0.2 and 0.6 J/m2). Figure 6 shows the plot of P/a3/2 against a3/2 for data both with and without noise. The initial points with very low contact radii are not used for the linear fit to obtain adhesion energy, as the errors are very large at low contact area. We first find the range of initial points at which the difference between P/a3/2 values in data with noise and without noise is more than 15% of the maximum P/a3/2 in loading data. The length of this range is the number of initial points ignored for the linear fit. The estimated values of adhesion energy for data with 1% noise are 0.23 ( 0.02 J/m2 (loading) and 0.65 ( 0.02 J/m2 (unloading). Figure 7 shows how the estimated adhesion energy from the compliance method as well as JKR plots depend on rms noise amplitude. In the compliance method, values of adhesion energy are obtained by a linear fit on the deficit versus contact area curve. The covariance matrix of the coefficients found from a linear fit is taken as the confidence bounds for the calculated adhesion energy values. Similarly, in the JKR plot, adhesion energy is obtained from the linear fit on the P/a3/2 versus a3/2 curve. Hence, the confidence bounds for the calculated adhesion energy values are found from this plot. As mentioned previously, the linear fit is not done at very low contact areas to avoid huge
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errors. For this analysis, 10 initial points are ignored in the linear fit. For low noise amplitude, the compliance method yields values for the adhesion energy close to the expected value. As the noise amplitude increases, the scatter in the estimated adhesion energy is greater in the compliance method compared to the JKR plot method. Also, confidence intervals obtained from the linear fits are considerably smaller than the true error or uncertainty in estimated parameters. Note that the estimate of adhesion energy from the JKR plot does not need displacement data. Also, in this case, systematic error in contact area measurement does not bias the estimated adhesion energy, as shown below: Assume, Pm ) RP and am ) βa, where Pm and am are the measured force and contact radius, and P and a are the exact force and contact radius without any error, and departure of parameters R and β from values of unity represents systematic error. From eq 24,
4E/a3 3R Pm am 3 4E/ am 3 + w ) - 8πWadE/ R β 3R β 4E/Ram3/2 R 3/2 / wPm ⁄ am ) - 3⁄2 √8πWadE + β 3Rβ3 3/2 3⁄2 In plot of Pm ⁄ am vs am : P ) - √8πWadE/a3 +
( )
( )
( )
4E/R 3Rβ3 × Slope w E/ ) 3 4R 3Rβ R and Intercept ) - 3⁄2 √8πWadE/ β (Intercept)2 β3 wWad ) 8πE/ R2 2 (Intercept) 1 ) 6πR × Slope R 1 ) Wadm R wWadm ) RWad Slope )
( )
sphere was cleaned in Piranha solution (30:70 volume ratio of hydrogen peroxide and 98% sulfuric acid) for 30 min and then dried with nitrogen gas before the test. After cleaning, the glass indenter was indented and retracted from the flat PDMS sample several times to condition its surface in order to reduce time effects in the actual test. The test was performed by indenting the glass sphere into the specimen to a certain depth and then retracting until it lost contact with the specimen. The indentation and retraction was at a constant rate of 0.1 µm/s. The force on the indenter was measured using an in-line load cell (Transducer Techniques), displacement was measured by a capacitive sensor (MTI instruments), and contact images were obtained through an inverted microscope. The contact area corresponding to the force-displacement data was measured via image analysis using edge detection in Matlab. For this flat unstructured sample, Wad can also be estimated by the JKR model, by applying the correction for the finite value of specimen thickness.3 Since the specimen thickness (h) is 650 µm and the contact radius (a) reaches values as high as 0.3 mm in the test, the Hertzian load, PH should be multiplied by a correction factor, f, given by Shull3 as
( ha )
f)1+β
(26)
()
()
()
where Wadm is the adhesion energy obtained from the measured experimental data. This shows that the measured adhesion energy is affected only by R, that is, it is unaffected by systematic error in contact radius. Therefore, adhesion energy extracted from the JKR plot is affected by systematic and random errors in force data and only by random errors in contact area data, while the compliance method is affected by errors in force, contact radius, as well as displacement. Additionally, if there is any dissipation or hysteresis (other than at the interface we are attempting to interrogate) in the loading system between the two points where displacement is measured, it will contribute to the overall deficit, presumably lowering the adhesion energy on loading (crack closure), and increasing it during unloading (crack opening). 6.2. Experimental Data for a Control Sample. We compare the compliance method for extraction of adhesion energy to the JKR-plot method by applying it to experimental data obtained on a PDMS (Sylgard 184, Dow Corning, Midland, MI) specimen. A specimen 650 µm in thickness was fabricated by cross-linking liquid PDMS (10:1 mass ratio of elastomer base to curing agent) at 80 °C for 2 h, as described previously.13 Its top surface is oxidized in an oxygen plasma at 7.2 W, 200 mTorr pressure for 30 s, to introduce some hysteresis. The indentation test was performed on a custom-built apparatus (Figure 1) described previously13 using a glass sphere 4 mm in diameter. The glass
(27)
where β is equal to 0.15 for frictionless contact. Hence, assuming frictionless contact fmax ) 1.015. Since the maximum change introduced in the value of Hertzian load, PH, is only 1.5% of its value without using this correction, the contact area was assumed to be small enough so that JKR model was used without any correction. That is, the values of adhesion energy for loading and unloading found using the compliance method are compared with the values obtained from the JKR plot (P/a3/2 versus a3/2). The experimental force-displacement data along with the Hertz curve calculated using the compliance method are shown in Figure 8. Figure 9 shows the plot of intercept (c) against the slope (m) of tangents to the Hertz curve. The values of deficit obtained from this Hertz curve are plotted against contact area in Figure 10. The values of adhesion energy obtained from this plot are 30.24 ( 3.08 mJ/m2 (loading) and 239.12 ( 3.70 mJ/m2 (unloading). Adhesion energy values from the plot of P/a3/2 versus a3/2 (Figure 11) are 24.1 ( 4.3 mJ/m2 (loading) and 246.2 ( 36.8 mJ/m2 (unloading). The values of adhesion energy found from the compliance method are in good agreement with the values estimated from the JKR plot. Recall that the true uncertainty is likely to be considerably larger than the value given by confidence intervals. We note that, in cases of low hysteresis, such as the example shown in Figure 8, our method of obtaining compliance is the main source of error. Another technique, such as partial unloading, may yield better results. 6.3. Adhesion Energy Measurement on a Structured Interface. Having shown that the compliance method can be used to extract the absolute adhesion energy in an indentation test, we now demonstrate its application to a material with a fibrillar surface for which an exact analytical model is difficult to obtain. These samples are made using PDMS (Sylgard 184, Dow corning). Their structure consists of pillars with 14 µm square cross-section arranged in a hexagonal pattern on a 650 µm thick base. These pillars are topped with a 4 µm thick film to provide more stability to the pillars and to increase the surface area during contact. The fabrication process of these samples has been discussed previously.13 The indentation test was performed in the same manner as for flat PDMS samples as explained in section 6.2. Figure 12 shows typical experimental force-displacement data for a fibrillar sample along with the Hertz curve obtained using the compliance method. Also, the straight lines joining
Model-Independent Extraction of Adhesion Energy
selected points with same contact area on loading and unloading curves are shown. Figure 13 shows a typical plot of deficit versus contact area for the fibrillar samples with fibril height of 60 µm. The values of adhesion energy during loading and unloading for these fibrillar samples are listed in Table 1. As established previously,15 the adhesion energy on crack opening is considerably larger than that on crack closure and increases with increasing interfibrillar spacing. When hysteresis is large, the Hertz and experimental loading curves lie close to each other. Then, the estimate for unloading adhesion energy is far more accurate than that for loading adhesion energy. Possibly a different technique for compliance determination for the loading curve will yield more accurate results for loading adhesion energy.
7. Discussion We have developed a model-independent method to measure the effective adhesion energy by indentation. The method requires one to measure force, displacement, and contact area simultaneously. With a little hysteresis, compliance can be determined as a function of contact area using experimental data. We have successfully demonstrated the performance of the modelindependent compliance method for calculating the absolute adhesion energy for specimens with different geometries. If the conditions of linear elasticity are fulfilled, this method should work well with any kind of specimen. In this paper, to obtain adhesion energy for different samples, contact compliance is found by matching the contact area at points on loading and unloading curves. Then, the Hertz curve has been obtained by using the compliance method (section 5), and finally adhesion energy has been calculated from the slope of energy deficit as a function of contact area. Although two other methods for obtaining Hertz curve are presented in the Supporting Information, they were found to be more sensitive to noise in experimental data. Also, calculation of adhesion energy using eq 16 is found to be affected by noise more in comparison to the method of plotting deficit against contact area. The compliance method is attractive because of its applicability to arbitrary specimens. However, it is less accurate than the JKR plot in cases where the exact analytical model applies. When adhesion hysteresis is small, the method used in this paper of matching points on the loading/
Langmuir, Vol. 24, No. 17, 2008 9409
unloading curves becomes increasingly error-prone. To get accurate results in such cases, precise displacement measurements are very important. Also, if the compliance is calculated by matching contact areas on loading and unloading curves, careful contact area measurements are equally important for correct compliance calculation. Alternatively, for cases of low adhesion hysteresis, compliance could be determined through independent experiments or by computational analysis. In the examples shown here, the interface appears to have two different but constant adhesion energies for loading and unloading, i.e., for crack closure and crack opening respectively. In practice, the adhesion energy often depends on the crack velocity.7 If there is a strong effect of crack velocity on the adhesion energy and the crack velocity changes appreciably during the experiment, then the plot of deficit versus contact area will be nonlinear. However, our analysis still applies, since it calculates the energy release rate, which depends only on the bulk material being linearly elastic. In such a situation, the instantaneous adhesion energy will be the local slope of the curve (i.e., dD/dA). Note that, since our analysis is based on quasi-static crack growth, it is not valid in dynamic crack growth conditions. The calculation of energy release rate does not assume a particular shape of the contact, since the analysis is based on area. When the contact is circular, as in indentation of a homogeneous material by a sphere, we can interpret the measured energy release rate as the effective work of adhesion. In other cases, for example, in the fibrillar sample shown in Figure 12, the contact may not be circular. The extracted effective work of adhesion should then be interpreted as some average of its local value. Acknowledgment. This work was supported by a grant from the Department of Energy (DE-FG02-07ER46463). The authors would also like to acknowledge the reviewers’ helpful suggestions. Supporting Information Available: Supporting Information contains two other methods to determine the Hertz curve, given loading and unloading data, and a geometrical result for energy release rate applicable to standard JKR theory. This material is available free of charge via the Internet at http://pubs.acs.org. LA800817X
