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Dynamic Contacts on Viscoelastic Films: Work of Adhesion Manish Giri,†,‡ Douglas B. Bousfield,† and W. N. Unertl*,‡ Department of Chemical Engineering and Laboratory for Surface Science and Technology, University of Maine, Orono, Maine 04473 Received November 9, 2000. In Final Form: March 9, 2001 Dynamic mechanical contacts with nanometer to millimeter dimensions are important in scanned probe microscopy, contact mechanics analysis, ultralow load indentation, microelectromechanical systems, compact disks, biological systems, pressure sensitive adhesives, and so forth. The response of these contacts is poorly understood if they involve adhesive viscoelastic materials. We have used indentation to study contacts to styrene-butadiene latex films with a range of glass transition temperatures. Contact times were in the range 0.01-200 s and loads were in the micronewton to millinewton range. Diamond probes with Berkovich and spherical end shapes were used. Load versus displacement data showed substantial adhesion hysteresis between the loading and unloading portions. The hysteresis is at least partially due to creep as indicated by the continued increase in penetration after the start of unloading. We show that an extended Johnson-Kendall-Roberts (JKR) model due to Johnson provides a robust way to extract works of adhesion from data obtained at low loading. The range of the interaction potential between the probe and substrate is also obtained from the fits. Because this model neglects long-range creep effects, it breaks down at high loading rates.
1. Introduction Mechanical contacts between elastic materials with nanometer to micrometer dimensions are well understood, including the increasingly important role of adhesion as the contact size decreases.1-3 For elastic materials, the contact radius a and penetration δ are uniquely determined by the applied load P(t), the elastic properties of the contacting materials, and the thermodynamic work of adhesion W. Fits of a versus P or δ versus P data to the appropriate theoretical models allow W and the effective moduli of the contacting materials to be determined. Contacts involving elastomers are less well understood, but approximate models are available that allow estimates of W and mechanical properties.4-6 In contrast, contacts to viscoelastic materials are poorly understood.7 There is very little quantitative data and, until quite recently, there were no theoretical models including both viscoelastic response and adhesion. In this paper, we present a detailed experimental study of micrometer-sized contacts between a rigid probe and three viscoelastic materials. A preliminary report has been published elsewhere.8 These results are analyzed using two new models. In this paper, we show that, at low loading rates, an approximate model due to Johnson9 provides a robust method to determine W and the effective range of the potential. In a subsequent paper,10 we show that the * To whom correspondence should be addressed. unertl@ maine.edu. † Department of Chemical Engineering. ‡ Laboratory for Surface Science and Technology. (1) Maugis, D. J. J. Colloid Interface Sci. 1992, 150, 243. (2) Hughes, B. D.; White, L. R. Q. J. Mech. Appl. Math. 1979, 32, 445. (3) Barthel, E. J. Colloid Interface Sci. 1998, 200, 7. (4) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 1989. (5) Greenwood, J. A.; Johnson, K. L. Philos. Mag. 1981, A43, 697. (6) Chaudhury, M. K. Mater. Sci. Eng. 1996, R16, 97. (7) Unertl, W. N. J. Adhes. 2000, 74, 195. (8) Giri, M.; Bousfield, D.; Unertl, W. N. Tribol. Lett., in press. (9) Johnson, K. L. In Microstructure and Microtribology of Polymer Surfaces; Tsukruk, V. V., Wahl, K. J., Eds.; American Chemical Society: Washington, DC, 2000; p 24. (10) Giri, M.; Unertl, W. N.; Hui, C.-Y.; Lin, Y. Y. In preparation.
data can be fit at all loading rates, even when measurements are made near Tg, using a theory developed by Hui and co-workers.11,12 2. Theoretical Background Theoretical models for contacts to viscoelastic materials in the absence of adhesion have been available since the 1960s.13 In the early 1970s, Barquins and Maugis4 and Greenwood and Johnson5 developed models that included adhesion, but viscoelastic effects were confined to the periphery of the contact. Johnson later extended this theory to the case of an entire load-displacement cycle.9 In late 1990s, several experiments demonstrated that viscoelastic effects are not confined to the contact periphery but include the entire contact zone.14-16 Johnson9 and Unertl17 have shown that crack tip and long-range creep phenomena generally occur on much different time scales. Depending on the characteristic relaxation time of the viscoelastic material and the experimental measurement time, both can be important. Recently, Hui and co-workers have put forward a model that accounts for linear viscoelastic response at all length and time scales.11,12 In this section, the theory of Johnson is described. After presenting load-displacement data, we demonstrate that this theory provides a robust method to determine the thermodynamic work of adhesion from data obtained at sufficiently low loading rates. Elastic Fracture Mechanics. Figure 1a shows the notation used to describe a rigid probe in contact with an (11) Hui, C.-K.; Baney, J. M.; Kramer, E. J. Langmuir 1988, 4, 6570. (12) Lin, Y. Y.; Hui, C.-Y.; Baney, J. M. J. Phys. D: Appl. Phys. 1999, 32, 2250. (13) Ting, T. C. T. J. Appl. Mech. 1966, 33, 845; 1968, 35, 248. (14) Falsafi, A.; Deprez, P.; Bates, F. S.; Tirrell, M. J. Rheol. 1997, 41, 1349. (15) Wahl, K. J.; Stepnowski, S. V.; Unertl, W. N. Tribol. Lett. 1998, 5, 103. (16) Basire, C.; Fretigny, C. C. R. Acad. Sci., Ser. IIb 1997, 325, 211. (17) Unertl, W. N. In Microstructure and Microtribology of Polymer Surfaces; Tsukruk, V. V., Wahl, K. J., Eds.; American Chemical Society: Washington, DC, 2000; p 66.
10.1021/la001565b CCC: $20.00 © 2001 American Chemical Society Published on Web 04/21/2001
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by Johnson, Kendall, and Roberts (JKR).18 For this case,
P)
4E*a3 - x8πWE*a3 3R
(1)
and
δ)
a2 R
x8πWa 3E*
(2)
In many cases, however, the potential is not known. This situation is usually treated using the concepts of fracture mechanics.1,4 The periphery of the contact is viewed as the tip of a crack. If a increases, the crack closes; if a decreases, the crack opens. The crack propagates with speed V ) da/dt. We use a sign convention opposite of that in the fracture mechanics literature; for example, we take the speed of a closing crack as positive. The energy flow to the crack tip per unit area of crack extension is given by
KI2 2E*
G)
(3)
where the parameter KI is called the mode one stress intensity factor. KI is related to the potential acting between the surfaces of the crack by19
KI )
Figure 1. (a) Geometry of contact. (b) Dugdale potential.
initially flat, perfectly elastic substrate under timedependent load P(t). P(t) is perpendicular to the substrate surface. Cylindrical coordinates (r,z) are used with positive z into the substrate. The radius of the circular contact is a(t), and the rate at which it changes is V ) da/dt; δ(t) is the deformation along the symmetry axis. The substrate is assumed isotropic and homogeneous with effective modulus E* ≡ E/(1 - ν2) where E is the Young modulus and ν is the Poisson ratio. Equations presented in this section assume the probe is rigid and axisymmetric with parabolic profile f(r) ) r2/2R0, where r is the radial distance from the probe axis parallel to the substrate surface. We will also use analogous expressions for a conical probe; these are given in the Appendix. If the potential between the probe and substrate surface is known, the case of an adhesive elastic contact can be solved using elasticity theory.2,3 However, in many situations, the interaction potential is not known a priori and various approximations are used. Barthel has shown that the behavior of the contact is not very sensitive to the detailed form of the potential as long as the maximum interaction stress σ0 and effective range of the potential h0 are correct. The theoretical models discussed below use the Dugdale approximation: for z < D0, σ f ∞ where D0 is the equilibrium spacing between the probe and substrate surfaces in contact in the absence of load. In the range D0 e z e D0 + h0, the stress is constant (σ ) σ0), and σ ) 0 at z > D0 + h0. Dugdale and Lennard-Jones potentials are compared in Figure 1b. The specific form of the solution in the elastic case is characterized by an elasticity parameter λ ≡ σ0(9R/ 2πWE*)1/3. In this paper, we are interested in the limit of compliant materials (λ > 5) where the solution is given
xπ2∫ σ(x)/xx dx L
0
(4)
where L is the crack length, that is, the distance away from the crack tip over which the potential acts. In the Dugdale model, L is the distance from the crack tip at which the crack surfaces attain separation h0. Because elastic materials have no energy dissipation in the bulk, G ) W at equilibrium. If G > W, the crack advances and a decreases. If G < W, the crack recedes and a increases. In an elastic contact mechanics experiment, KI and W contain equivalent information about the interaction potential. Linear Viscoelastic Fracture Mechanics. Contacts involving viscoelastic materials are more difficult to analyze because not all of the applied energy instantaneously reaches the crack tip. This has the consequence that, in contrast to elastic materials, KI depends on the history of the contact and on the details of the crack opening and closing process.12 Consequently, a and δ are no longer unique functions of the instantaneous load. The analysis of viscoelastic cracks has been carried out in a series of four papers by Schapery.20-23 Figure 2 shows the model used for the crack tip. The line defined by the crack tip T is assumed to lie in a plane. The curvature of the crack in this plane is assumed small compared to the crack length L. Additionally, L , a. The size of failure zone just behind T is characterized by L and height h0. Outside this zone, the material is assumed isotropic, homogeneous, nonaging, and linear viscoelastic with creep compliance function C(t). Roughness of the contacting surfaces must not occur on a length scale comparable to L but is allowed at much longer or much shorter scales. No limitations are placed on the material inside the zone. (18) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (19) Barenblatt, G. I. Adv. Appl. Mech. 1962, 7, 55. (20) Schapery, R. A. Int. J. Fract. 1975, 11, 141. (21) Schapery, R. A. Int. J. Fract. 1975, 11, 369. (22) Schapery, R. A. Int. J. Fract. 1975, 11, 549. (23) Schapery, R. A. Int. J. Fract. 1989, 39, 163.
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intensity factor and crack tip velocity is specific for a material compliance function. Because G is a function of crack speed V and G f W in the limit V f 0,
G(V) ) Wβ(V)
(11)
where β(V) is greater than unity for an opening crack and less than unity for a closing crack. For a closing crack,
βcl = E∞C(τcrack) < 1
Figure 2. Crack tip model.
It may be highly nonlinear, viscoelastic, and discontinuous as in the case of crazing. The stresses σ on the surrounding material are assumed perpendicular to the plane of the crack (Mode I). Schapery derives an expression for V for the case where σ is approximated by a Dugdale model (Figure 1b). The rate of energy flow to the crack tip is
1 G(t) ) C(t)KI2(t) 2
(5)
Equation 5 is similar to the elastic criterion (eq 3) except that the time-dependent C(t) replaces the constant E*, thereby making G a dynamic quantity. In the short time limit, C(t f 0) f C0 ) 1/E/0, where E/0 is the instantaneous effective modulus. In the long time limit, C(t f ∞) f C∞ ) 1/E/∞ where E/∞ is the long time effective modulus. In analytical results, Schapery usually assumes that C(t) can be approximated as a power law over the time interval of interest;
C(t) ) C0 + C1tn
(6)
where C0 and C1 are constants and n is determined from the slope of a double logarithmic plot of C(t) in the time interval of interest. For most polymers, 0 e n e 0.5.21 The time for the crack to propagate distance L is
L τcrack ) g(n)1/n V
(7)
where g(n)1/n is a slowly varying function of n in the range 0.278-0.347. Following Schapery, we use g(n)1/n = 1/3, so that τcrack = L/3V. The physical requirement that the stress at the crack tip be finite leads to a relationship between the stress intensity factor and the crack length L,
( )
d0 π KI 2 L= L= 2 σ0I1 E∞C(τcrack)
(8)
where
d0 )
/ 2 π E∞h0 4 W
(9)
is the crack length in the elastic limit and the constant I1 < 2 if the stress distribution in the cohesive zone is constant. The dependence of crack velocity on stress intensity is given by
V(t) )
( )[
da π KI ) dt 2 σ0I1
2
λnC1
]
C(t)(1 - KI2/KIg2)
1/n
(10)
where KIg is the stress intensity factor in the glassy limit (t f 0) corresponding to the glass modulus, 1/C0. This result clearly illustrates that the relation between stress
(12)
and, for an opening crack,
βop =
1 >1 E∞C(τcrack)
(13)
Note that eqs 8, 12, and 13 use E∞ and not E/∞. Johnson’s Theory of the Viscoelastic Adhesion Cycle. Johnson uses the Schapery results to extend the JKR theory of adhesive contacts to the case of linear viscoelastic materials. He implicitly assumes that the loading is slow enough that material outside the crack failure zone is completely relaxed, that is, τcrack is long compared to the characteristic relaxation time of the viscoelastic material. He determines τcrack by eliminating L from eqs 7 and 8. Substituting τcrack into eqs 12 and 13 yields βop and βcl as functions of crack speed. In the elastic JKR theory, the pressure acting on the contact consists of two parts. One is the Hertz pressure due to the compressive load and is distributed smoothly over the entire contact. The other is due to the adhesive interaction and is most important at the contact periphery. For viscoelastic materials, surface energy W is replaced by its scaled value Wβ(V) so that eq 1, the expression for the net contact force, becomes
j3 P h )a j 3 - x2βa
(14)
where reduced units P h ≡ P/6πRW and a j ≡ a/(9πR2W/ / 1/3 2E∞) have been used. For the case of constant loading or unloading rates (|dP h /dt| ) P h 0/t0), differentiating eq 14 with respect to a j leads to a differential equation
( ) x2βaj AC(τ
β dβ ) 3x2a jβ - 3 da j a j
3
crack)τcrack
(15)
where A ) (P h 0/t0)(36RE/∞W2/πh03)2/3. Equations 7, 8, 12, and 15 can be solved simultaneously by numerical j . The initial integration to find the variation of βcl with a j condition for the loading cycle is given by βcl ) 1 when a ) 0-. Equations 7, 8, 13, and 15 yield the variation of βop with a j . For the unloading portion, the initial condition is j )a j max, the contact radius reached given by βcl ) 1 when a at the end of the loading cycle. Unlike the case of macroscopic contacts, it is usually not possible to measure the contact radius directly in scanning force microscopy or nanoindentation experiments. Therefore, the relationship between contact depth and contact radius is a crucial one if the Johnson extended JKR model is to be compared directly with experimental data. Following Hui, Baney, and Kramer,11 the contact depth at any instant t can be calculated once the contact radius is known:
δ(t) )
a(t)2 - KI(t)C0xπa(t) R
(16)
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latex
styrene-butadiene ratio
degree of cross-linking
diameter (nm)
Tg (°C)
1 2 3
3:2 3:2 4:2
medium high low
130 130 190
5 5 20
Table 1. Latex Properties
where G(t) is given by eq 5 but expressed in the approximate form of eq 11 for the analysis in section 4. 3. Experimental Details The polymers studied were films cast from aqueous suspensions of carboxylated styrene/butadiene copolymer latexes (provided by Omnova Solutions Inc., Specialty Polymers Division, Akron, OH). Three latexes with properties similar to latexes used in coating applications were studied. Their properties are summarized in Table 1. They differed primarily in their glass transition temperatures and degree of cross-linking. The glass transition temperatures, Tg, were 5 °C for latex 1 and latex 2 and 20 °C for latex 3. Latex 1 and latex 2 differed primarily in their degree of cross-linking with latex 2 exhibiting a higher degree. Latex 3 had the lowest degree of cross-linking. Film samples were prepared by depositing the latex suspension on Mylar, drying in air, heating for about 1 h to about 50 °C above Tg, and then washing with deionized water. All samples were tested within 2 days of preparation. The final thickness of these films was 0.1-0.2 mm. These thicknesses are more than 50 times the deepest indentation so that the sample thickness has no influence on the results. The root-mean-square roughness of the films, as measured with contact mode atomic force microscopy (AFM), was typically 10 nm over areas of several square micrometers. AFM images typically show evidence of residual particle structure because the cross-linked parts of the original latex particles remain intact during film formation. This introduces some heterogeneity into the samples and is responsible for most of the intrinsic surface roughness. It is also probably the major cause of variation in the contact properties from point to point on the surfaces. Contacts were made with diamond probes in a nanomechanical testing system (Hysitron PicoIndenter with the sample mounted on a Park Scientific Instruments CP scan base). In the Hysitron instrument, the probe is attached to the center plate of a threeplate capacitor which is mounted above the sample. The outer plates are rigidly fixed, and the center plate is supported by springs whose stiffness was measured to be 159 N/m. The load is changed by applying a voltage between the bottom and center plates. Load sensitivity is about 1 µN. The displacement is determined by measuring the change in capacitance between the top and center plates. The displacement sensitivity is about 1 nm. A detailed description of the performance of this type of indenter is given by Wahl, Asif, and Colton.24 Two probe shapes were used: Berkovich and a 10 µm diameter spherical tip. An ideal Berkovich indenter is a triangular pyramid whose crosssectional area increases as A(h) ) 24h2 with distance h from the tip, but the actual indenter is rounded at the tip. The indenter compliance and actual A(h) were determined using the method of Oliver and Pharr.25 The effective tip radius of the Berkovich indenter was estimated to be R ≈ 200 nm by assuming a spherical end shape and fitting the A(h) data for small values of h. The diameter of the spherical indenter was confirmed to be between 9.5 and 10.5 µm using environmental scanning electron microscope images. The surface roughness of the spherical probe was estimated from 10 µm × 10 µm atomic force microscope images to be less than 30 nm. Contacts were controlled with a predetermined loading/unloading cycle P(t). The displacement (also called deformation or depth of penetration) δ(t) of the rigid probe was measured continuously during the contact cycle. Thermal drift of about 0.05 nm/s limited measurement times to a few hundred seconds. Figure 3 shows results of typical cyclical loading-unloading experiments. Figure 3a shows load and displacement versus time (24) Asif, S. A. S.; Wahl, K. J.; Colton, R. J. Rev. Sci. Instrum. 1999, 70, 2408. (25) Oliver, W. C.; Pharr, G. M. J. Mater. Res. 1992, 7, 1564.
Figure 3. (a) Load and displacement during a typical loadingunloading cycle on latex 3 at a high loading rate. (b) Typical loading-unloading cycle plotted as displacement vs load for latex 1 at a low loading rate. for a high loading rate of 2.5 N/s, and Figure 3b shows displacement versus load for a much lower loading rate of 0.8 mN/s. Initially, the probe is about 2.5 µm out of contact, which is necessary to achieve tensile loads sufficient to overcome the probe-sample adhesive forces during the unloading portion of the cycle. During segment 1, the probe is brought rapidly into contact, which occurs at point A. If the time to achieve contact was less than about 0.1 s, inertial effects cause the indenter motion through air to be nonlinear, as is the case in Figure 3. Once contact is achieved, the stiffness of the contact is large enough that inertial effects are no longer important. In segment 2, the load is increased at a constant rate dP/dt under feedback control to a predetermined maximum value Pmax, which is reached at B. During the compressive portion of unloading (segment 3), the rate was -dP/dt. The unloading rate was not constant during the tensile portion (segment 4), because it is determined entirely by the extension of the indenter springs, which cannot be controlled by the feedback system. The total time of contact tc could varied between 0.04 and 2000 s. All measurements were carried out at room temperature (≈25 °C). Surface contact was identified to within a few nanometers using the small discontinuity in displacement observed in displacement versus load plots of the data after correcting for indenter stiffness [10]. This discontinuity is due to jump-tocontact plus initial elastic deformation of the substrate. Bulk rheological properties were measured using small amplitude oscillatory modulations in a parallel plate rheometer (Bohlin Instruments) over the frequency range 0.01-30 Hz for
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Figure 5. Load-displacement data for latex 1 obtained with a Berkovich probe.
Figure 4. (a) Storage and (b) loss shear moduli of the latexes. temperatures of 5, 25, 35, 60, and 80 °C. Strains were kept in the range 0.0006-0.003 to ensure that deformations were in the linear regime. Samples 1 mm thick with radii of 2 cm were cast in a Teflon mold. Figure 4a,b shows the storage shear moduli E′(ω) and loss shear moduli E′′(ω) obtained from the data using the WLF (William-Landel-Ferry) time-temperature superposition principle.26 The shift factors used to obtain these master curves are shown in the inset. The range of frequencies (assuming ω ∼ 1/tc) corresponding to the response times of the indenter is indicated in Figure 4b. Indentation is sensitive to the rheological response of the sample within this frequency interval.
4. Cyclic Loading-Unloading Measurements Because all the theoretical models assume that the contacts have linear viscoelastic response, it is important to ensure that the strains and strain rates were low enough to avoid any plastic deformation. This was done as follows. First, a cyclic loading-unloading experiment was carried out at a high loading rate of ∼0.024 N/s. Second, the probe was held above the location of this indent for 2 h. After the rest period, the loading-unloading cycle was repeated. The displacements at initial contact, the slopes of the loading curves, the pull-off forces, and areas under the (26) Ferry, J. D. Viscoelastic Properties of Polymers; John Wiley & Sons: New York, 1980.
adhesion hysteresis curves were compared for the two cycles. These quantities were always identical within the experimental precision. Imaging studies demonstrate that lateral drift is negligible over tens of hours. To eliminate the possibility of vertical drift during the wait, the displacement to reach initial contact was determined at several symmetrically located nearby points. Their average was the same as the initial displacement to contact. Thus, we conclude there is no significant residual plastic deformation under the experimental conditions used in the experiments reported here. For each of the latexes, sets of load-displacement curves were obtained over a wide range of loading rates. The same maximum displacement (1.5 µm ( 5%) was used in each case. Contacts using the Berkovich tip were made deeper than 300 nm to minimize effects of tip rounding. Contacts using the spherical probe were shallower than 2 µm so that the probe profile could be approximated as a paraboloid in the analysis. Indents were also substantially deeper than 10 nm to minimize effects due to surface roughness and possible variations of sample stiffness near the surface. Measurements of the near surface stiffness of latex 1 were made at the Naval Research Laboratory using the force modulation technique described in ref 24. The measured stiffness was found constant for indentations deeper than about 10 nm. Whether the near surface variation is due to surface roughness or true variation in creep compliance is not known. Results for each latex, obtained with a Berkovich indenter, are shown in Figure 5, Figure 6, and Figure 7. The basic features are the same for each set of data and have been discussed in detail in conjunction with Figure 3. The adhesion hysteresis (i.e., the energy dissipated during a loading cycle) is measured by the area enclosed by the cycle. This hysteresis is largest for the shortest contact times and decreases continuously as the contact time increases. For the shortest contact times, the probe continues to penetrate into the latex after the maximum applied load is reached. This is most pronounced for latex 3. The maximum displacement occurs during the unloading portion of the cycle. This is precisely the behavior expected for long-range viscoelastic creep. The inset in Figure 7 plots the portion of the penetration that occurs after maximum load as a function of inverse contact time; ∆h is the difference between the maximum displacement and
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Figure 6. Load-displacement data for latex 2 obtained with a Berkovich probe.
Figure 7. Load-displacement data for latex 3 obtained with a Berkovich probe. The inset shows the additional penetration that occurs after maximum applied load is reached. Also indicated by ωmax is the time corresponding to the maximum in bulk tan δ.
the displacement at maximum load. ∆h reaches its largest value at the shortest tc. Also shown in the inset is ωmax, the frequency at which the bulk tan δ has its maximum value. For latex 3, ωmax ≈ 0.02 s at 24 °C. Creep effects should be largest at ωmax and decrease on either side. This decrease is clearly observed at short 1/tc. Unfortunately, the experiments could not reach the shorter contact times needed to confirm the expected decrease in ∆h for large 1/tc but came closest to ωmax for latex 3. The largest tensile (negative) load achieved is the effective pull-off force, Peff. Unlike the more familiar case of a spherical probe, the displacement is always positive at Peff. This is a general property of conical indenters even for elastic materials. Using the approach of Maugis,27 it is straightforward to show that pull-off occurs when Pel ) -54W2 tan3 R/πE* and hel ) +3W tan R/E* where R is the cone’s enclosed half-angle. The Berkovich indenter is frequently approximated as a cone in the indentation literature, and we use this approximation in the analysis below. The general features of data obtained with the spherical probe are the same as discussed above for the Berkovich (27) Maugis, D. Langmuir 1995, 11, 679.
Figure 8. Typical load-displacement data obtained with a spherical probe.
probe. Figure 8 and Figure 9 show typical data obtained with the spherical probe. In contrast to the Berkovich data, the effective pull-off occurs at displacements near zero as is typically observed for elastic contacts. 5. Discussion In this section, we demonstrate that the data obtained at the lowest loading rates are described well by the extended JKR model. In particular, we determine the appropriate experimental conditions necessary to extract the thermodynamic work of adhesion. The Creep Compliance Function. The analysis in this section requires knowledge of the creep compliance function of the latex C(t). We assume it to be the same as for the bulk and calculate it from the measured shear modulus (Figure 4). To facilitate this calculation and to obtain C(t) in an analytical form more suitable for the
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Figure 9. Load-displacement data for latex 1 obtained with a spherical probe at a loading rate of 0.25 µN/s. (a) The solid lines show fits using the Johnson model for Weff ) 77, 81, and 84 mJ/m2 with h0 ) 0.5 nm. (b) Variance of the fits at fixed h0 for two data sets. (c) Fits for h0 ) 0.45, 0.5, and 0.55 nm with Weff ) 81 mJ/m2. (d) Variance of the fits at fixed Weff. Table 2. Parameters of the Creep Compliance Functions
analysis, we use the following functional form:
{
1 + Mtp E0 C(t) ) 1 E∞ + Nt-q
t < t0 t > t0
(17)
where E0 is the instantaneous modulus (E in the limit ω f ∞), E∞ is the relaxed modulus (limit ω f 0), and p and M are constants. Constants N and q are determined using the requirement that C(t) and its first derivative be continuous at t0. For the latexes studied here, E0 values are 1-3 GPa and E∞ values are in the range 15-30 MPa. E(ω) was calculated directly from C(t) using a sequence of Laplace transformations.28 Constants p and M were determined by fitting this E(ω) to the measured shear moduli assuming a Poisson ratio ν ) 0.5 so that E ) 3G. The resulting values are listed in Table 2. Determination of the Work of Adhesion. The dynamics of contact formation and rupture are controlled by the interplay of the viscoelastic response of the materials and the thermodynamic work of adhesion. We now show how the thermodynamic work of adhesion can (28) Findley, W. N.; Lai, J. S.; Onaran, K. Creep and Relaxation of Nonlinear Viscoelastic Materials; Dover Publications: New York, 1976.
latex
E0 (MPa)
E∞ (MPa)
p
M (s-p/MPa)
t0 (s)
1 2 3
1600 1650 2600
24 25 16
0.45 0.45 0.5
0.04 0.035 0.003
0.01 0.01 0.1
be determined. In the limit of very long contact times (tc f ∞), the contact would behave elastically with modulus equal to the relaxed modulus E∞. In this case, the loaddisplacement data could be analyzed using the standard JKR theory.18 For a spherical probe, the work of adhesion could then be determined directly from the pull-off force because W ) -2Pel/3πR. Johnson has suggested that Pel might be determined by plotting the effective pull-off force as a function of loading rate (Peff vs dP/dt) and extrapolating to the limit dP/dt f 0. However, for the latexes studied here, the smallest dP/dt obtainable with our apparatus is too large for such an extrapolation to be made reliably. To overcome this difficulty, we use Johnson’s extended JKR theory to analyze the entire hysteresis cycle. In this model, only the creep compliance function C(t) and two parameters, an effective work of adhesion Weff and the effective range of the interaction potential h0 (see Figure 1), are required to calculate an entire load-displacement cycle. We assume that the bulk C(t) is valid near the surface and determine Weff and h0 by fitting the entire set of loaddisplacement data for each latex, including results for
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Giri et al.
Figure 10. Variation of Weff with loading rate for latex 1.
both probe shapes. The load-displacement cycle was divided into three sections: loading (AB in Figure 3), unloading under compression (BCD), and unloading under tension (DE). In each region, the applied load P(t) was fit to a simple polynomial. This polynomial was nearly linear in the first two sections, where feedback control was possible, but contained significant higher order contributions for unloading under tension, because of the lack of feedback control. Using this P(t), the differential equation, eq 15, was solved numerically for various Weff and h0. Typical results for latex 1 are shown in Figure 9 for a spherical probe. For this data set, the best fits were obtained for h0 ) 0.5 nm and Weff ) 80 mJ/m2. Figure 9a compares fits for Weff ) 77, 81, and 84 mJ/m2 at constant h0 ) 0.5 nm. Visual comparison easily distinguishes the best fit to within (5 mJ/m2. Parts b and d of Figure 9 plot the variance of the fits as a function of Weff and h0, respectively. The solid dots are for the data shown in Figure 9a, and the open circles are for another load-displacement cycle obtained under nearly identical conditions but at a different point on the sample. Figure 9c shows the sensitivity of the fits to h0 for Weff ) 80 mJ/m2; there is no difference in location of the minimum variance from point to point on the sample. We estimate the uncertainties in h0 and Weff by their values at which the variance has increased 10% above its minimum. By this criterion, Weff can be determined for a particular data set to within 1 mJ/m2 and h0 can be determined to within 0.05 nm. The Johnson model is valid only if the loading rate is slow enough to avoid significant bulk creep. The range of dP/dt that satisfies this criterion can be determined from the requirement that the same Weff must fit the entire load-displacement curve independent of dP/dt. Figure 10 shows Weff versus dP/dt for latex 1. Weff values extracted from data obtained at low loading rates are all the same. However, once dP/dt increases above a few µN/s for latex 1, bulk creep effects become important and Weff values increase. Thus, we empirically determine the range of dP/dt for which Weff is constant and conclude that this value is W, the thermodynamic work of adhesion of the contact. The data also has the expected geometrical scaling. Figure 11 compares examples of the best fits to data obtained with the Berkovich indenter for each latex. In the calculations, the indenter was approximated as a cone with an enclosed half-angle of 70.40°. Fits are as good as those obtained for the spherical probe. Furthermore, as shown in Table 3, very similar values of the work of adhesion are obtained for each probe shape. The uncertainties quoted are the standard deviations of Weff values determined for a range of low loading rates.
Figure 11. Typical load-displacement data obtained with a Berkovich probe showing best fits to Johnson theory. Table 3. Work of Adhesion Results Berkovich probe
spherical probe
latex
W (mJ/m2)
h0 (nm)
W (mJ/m2)
h0 (nm)
1 2 3
84 ( 3 64 ( 2 75 ( 1
0.5 ( 0.05 0.5 ( 0.05 0.5 ( 0.05
80 ( 3 68 ( 3 73 ( 2
0.5 ( 0.05 0.5 ( 0.05 0.5 ( 0.05
The values of W obtained for the three latexes are comparable to previous determinations of W (70-85 mJ/ m2) measured for similar styrene-butadiene latexes.29,30 These studies were carried out on individual particles by a very different method. AFM was used to measure the contact angles after the particles had spread on various substrates including calcite, glass, and polystyrene. Fits to the data using the extended JKR model are excellent. Clear minima in the variance make it possible (29) Unertl, W. N. Langmuir 1998, 14, 2201. (30) Woodland, D. D.; Unertl, W. N. To be published.
Dynamic Contacts on Viscoelastic Films
Langmuir, Vol. 17, No. 10, 2001 2981
to determine W to within about (1 mJ/m2 and h0 to within about (0.05 nm. However, the optimum values of W determined from different data sets vary by more than (1 mJ/m2. For example, the minimum variance for the two sets of data shown in Figure 9b differs by about 2 mJ/m2. The most likely cause of this variation is small lateral variations in the latex surface energy. Such variations could also explain the anisotropic spreading of individual latex particles on various substrates.29,30 The effective range of the interaction potential was h0 ) 0.5 ( 0.05 nm for all of the latexes regardless of probe shape. This distance is comparable to spacings in molecular or inert gas solids and hydrogen bond lengths. It clearly indicates that the interaction between the diamond probe and the latexes is short range. The fits are poorest on the loading portion of the cycle (see Figure 8, Figure 9, and Figure 11). Sometimes the curvature is slightly higher than the data, and sometimes slightly lower. This variation is likely due to local variations in the compliance resulting from the residual cross-linked structure of the latex particles. Thus, more systematic studies of these variations could be used to characterize local variations in mechanical properties at submicrometer length scales.
mostly by point-to-point variations in surface properties. Thus, the technique can be used to study small heterogeneities in surface energy at the submicrometer scale. At higher loading rates, bulk creep effects become increasingly important, the assumptions of the Johnson model are no longer valid, and the more complex HBKL model must be used. However, the HBKL analysis is simplified by the knowledge of the work of adhesion and range parameter obtained by the extended JKR model. In a subsequent paper, we apply the HBKL model to our data and discuss the resulting stress intensity factors.
6. Conclusions
P h )a j 2 - x2βa j3
Cyclic loading/unloading experiments were carried on three styrene-butadiene copolymer latexes with a nanoindenter. Loading conditions were chosen to avoid plastic deformation. Hysteresis occurs because of unrecoverable work done during each cycle. Specifically, a higher force is needed to reach a certain displacement during loading than during unloading. The amount of energy dissipated in each cycle increases with increasing loading rate for the range of loading rates that could be achieved with the nanoindenter. These data are used to verify an extended JKR model proposed recently by Johnson.9 This model combines classical JKR theory with a fracture mechanics model of crack initiation and growth in linear viscoelastic materials proposed by Schapery. The major assumption of the model is that viscoelastic effects are limited to the periphery of the contact. Longer-range creep effects are ignored. This limits applicability of the model to low loading rates. The extended JKR model provides excellent fits to the data obtained at low loading rates. The fitting is robust because distinct minima are found for both fitting parameters, the work of adhesion and the effective range of the potential acting between the probe and substrate. Works of adhesion are determined to within a few mJ/ m2 and are consistent with previous determinations on similar materials. This uncertainty appears to be caused
Acknowledgment. We thank the sponsors of the University of Maine Paper Surface Science group and Dr. Nick Triantafillopolous and Dr. Gary Jailanella at GenCorp Specialty Polymers Division for donating the samples as well as for invaluable discussions. We are grateful to Dr. S. A. Syed Asif for making measurements of the near surface stiffness and for discussions of the results. Appendix: Equations for a Conical Indenter For a conical probe with half angle R, the equations corresponding to eqs 14, 15, and 16 are easily calculated using the method of Maugis.27 The analogue of eq 14 is
(A.1)
where
P h ) PπE/∞/512W2 tan3 R and
a j ) aπE/∞/32W tan2 R The analogue of eq 15 is
dβ ) da j
x8βaj - 3βaj - AC(τ)τx2βaj 3
(A.2)
where
A)
(
)
3P h 0 8W tan R t0 πh E/ 2 0 ∞
2
The analogue of eq 16 is
δ(t) ) LA001565B
2a(t) - KI(t)C0xπa(t) tan R
(A.3)
