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Chapter 4 Creep Effects in Nanometer-Scale Contacts to Linear Viscoelastic Materials

Downloaded by UNIV OF MASSACHUSETTS AMHERST on June 2, 2018 | https://pubs.acs.org Publication Date: December 10, 1999 | doi: 10.1021/bk-2000-0741.ch004

W. N. Unertl Laboratory for Surface Science and Technology, University of Maine, Orono, ME 04469 Nanometer-scale contacts to compliant linear viscoelastic materials can be studied experimentally with the scanning force microscope (SFM). Creep significantly modifies the formation and rupture of these contacts compared to contacts to elastic materials. Not only does the maximum contact area depend on the loading history but, unlike elastic materials, it reaches its maximum value well after the maximum load is applied. Effects due to creep are distinctfromthose induced at the periphery of the contact by adhesion. Creep effects dominate adhesion effects in SFM-scale contacts for a wide range of compliant viscoelastic materials. Strategies are presented to optimize experimental parameters for creep studies in SFM-scale contacts to linear viscoelastic materials.

The contributions to this book amply demonstrate the broad interest in using the scanning force microscope (SFM) to make quantitative measurements of materials properties with nanometer-scale resolution. The mechanical properties of interest include the storage and loss moduli, shear strength, and the yield strength. Knowledge of these properties is important for a broad range of applications rangingfromfundamental understanding of tribological processes to the characterization of biological systems. Continuum mechanics models from the field of contact mechanics (1,2) are widely used to analyze nanoscale SFM experiments involving elastic materials. The effects of adhesion are well-understood. Unfortunately, the appropriate analysis has not yet been carried out for viscoelastic materials. In the late 1960's, Ting (3,4) solved the creep problem for cases where adhesion can be neglected, which is generally not true for SFM contacts. Subsequently, Barquins and Maugis (5-7)incorporated viscoelastic response into thefracturemechanics approach to contact mechanics. In this approach, the increase and decrease of contact size is viewed as the opening or closing of a crack at the contact periphery. They ignored the possibility of creep. Johnson's (8) article in this volume provides an excellent overview of the present state of theoretical understanding of the contact mechanics of linear viscoelastic materials.

66

© 2000 American Chemical Society Tsukruk and Wahl; Microstructure and Microtribology of Polymer Surfaces ACS Symposium Series; American Chemical Society: Washington, DC, 1999.


67 In this article, we apply contact mechanics to interpret dynamic SFM loading experiments on compliant linear viscoelastic materials. We start with a discussion of experimental procedures including the advantages of modulation techniques. The interpretation of typical data is then discussed using Ting's analysis. Several general conclusions are reached about data analysis and optimization of experiment design. We then examine modifications due to adhesion and conclude that they have only a small effect on the time response. Based on these results and arguments presented in Johnson's article, we conclude that creep effects should be more important than crack tip effects for SFM-scale contacts to materials with effective moduli below about 100 MPa. The notation used to describe the contacts is shown in Figure 1. P(t) is the time dependent applied load, d\P,t) the deformation, a(P,t) the contact radius, and R\ and Ri the radii of curvature of the two bodies at the point of contact. We consider only flat substrates so that R\=R and R2 = °°. Each elastic material is described by its Young modulus E, Poisson ratio v, and is assumed to be isotopic so that the shear modulus is G = E/2(\ + v). Viscoelastic materials are assumed to be linear with stress relaxation functions E(t) and creep compliance functions J(t). All properties are assumed to be independent of depth. Experimental Aspects of Dynamic Loading This section describes typical SFM studies of the formation and rupture of contacts to a viscoelastic substrate during a single cycle of loading and unloading. Measurements are also possible under static loading conditions but will not be discussed here (9). In analogy with bulk dynamical mechanical analysis (10Jl\ a small mechanical modulation can be applied to the contact and the phase and amplitude of the response measured. In SFM this modulation is either a displacement of the sample or of the base of the cantilever. In either case, the amplitude and phase of the response of the probe end of the cantilever beam are measured. The modulation can be applied either normal(72-74) to the surface or parallel to it (9,75-77). The latter is a shear modulation, shown schematically in Figure 2. Shear modulation has an advantage over normal modulation, in that, for small modulation amplitudes, the contact area remains constant. For elastic materials, the

Figure 1. Geometry of a deformable contact between axially symmetric bodies.

Tsukruk and Wahl; Microstructure and Microtribology of Polymer Surfaces ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

68


Sample

Figure 2. (a) Schematic diagram of an SFM contact with shear modulation x = X exp(ffi)r) applied to the sample. The response of the contact and tip are x - X exp[/(fi* + /?)] and x = X exp[/(

(8)

0

i f ^ > 0

(9) 2

where S (t) is the solution to the equivalent elastic problem; e.g., S (t) = a (t)/R for the Hertz case. If a(t) is decreasing, but still larger than its value a(0), e

e

( )

P(0 = J ;: ' £(r-T)^[cP (T)>/T f o r r ^ f (

e

(10)

M

where t\(t) is the time at which a reached the value a(t) during the initial loading cycle and cP (t) is the solution to the equivalent elastic problem; e.g., cP (t) = (S/3R)a (t) for the Hertz case with c = HE*. If P(t) is known, equation 10 is used to find t\(t) and thus the value of a(t). Once a(f) is known, the deformation is found using 3

e

e

^ ^ . ( O - ^ ^ ^ A ^ E C r - ^ ^ M ^ r .

(11)

In cases where a decreases to values smaller than a , 0

P{t) = E(t)cP (t) for e

which is solved to find a(t). The deformation is then obtained from


(12)

73


(13)

We now illustrate these solutions for parameters typical of SFM contacts. We use mechanical models whose behavior brackets the range of response expected in real materials but are simple enough that analytical solutions can be obtained. In essentially all realistic cases, analytical solutions will not be possible. Specifically we chose the Maxwell and Voigt/Kelvin models. The Maxwell model is a series combination of an elastic spring with modulus E and dashpot with viscosity 77. The relaxation functions are t/x

E(t) = E* e'

and j(t) = E*(l + t/r)

(14)

where T = rj/E * is the characteristic relaxation time of the model. The Voigt/Kelvin model has the spring and dashpot in parallel with //7

E(t) = E*[H{t) + r8(t)] and /(*) = ( l - e - ) / £ * .

(15)

The standard solid model has intermediate behavior. It consists of a spring in series with a Voigt/Kelvin model with £ (

'

) =

¥ ^ k ( * *

+

£

V - ' " ) and /(,) = J _ _ L ( i - e - * - ) +

(16)

where T, = q/E * and r s rf/(E *, + £ * ). At very short times, the standard model responds like a single spring with E*\ and at long times like springs E*\ and E*z in series. Johnson shows calculations for a standard solid model in his contribution in this book. The range of behavior expected for the linear loading ramp shown in Figure 4a is illustrated in Figure 5 and Figure 6. A linear ramp mimics the load variation in a typical SFM measurement. The total time of contact between the probe and substrate is t and the time of maximum load is yt ntact where 0 < y < 1. The unloading time is tunload = (1-y) t ta t. The slopes of the loading and unloading curves can be varied independently. In order to obtain analytical solution for the Maxwell and Voigt/Kelvin models, we also assume that the probe shape is parabolic or, equivalently, a sphere with R » a. Figure 5 shows the major features of the Ting solution, for the Voigt/Kelvin model (Figure 5a) and the Maxwell model (Figure 5b). The elastic element has £* = 2 MPa and the viscosity of the dashpot is selected to yield a relaxation time of r = 50 s. These values are similar to those of the poly(vinylethylene) sample freshly cast from 2

2

contact

2

CO

con

C


74

(a) P max 0 ~ •a Lo
The maximum load P x is 2 nN and occurs at yt tact where y - 0.2 in Figure 5. This value of y is typical of SFM force curve measurements. Also shown for reference is the response of an elastic material with the same E* (solid lines). The most striking feature of both models shown in Figure 5 is that the time of maximum contact radius t k does not coincide with the time of maximum load as it does for elastic materials. Instead, maximum radius always occurs after the maximum load. This delay occurs because the substrate continues to respond by creep even during the unloading cycle. Consider first the Voigt/Kelvin model (Figure 5a). The contact radius increases more slowly than in the elastic case and its maximum value is always smaller. As tcontact goes to values much shorter than T, t k shifts closer and closer to the end of the contact (e.g., t k -»t ) and farther and farther away from the time of maxicontact

con

fna

con

pea

pea

pea

contact


75

(a) Voigt/Kelvin Model Downloaded by UNIV OF MASSACHUSETTS AMHERST on June 2, 2018 | https://pubs.acs.org Publication Date: December 10, 1999 | doi: 10.1021/bk-2000-0741.ch004

E* = 2

MPa

contact

100-.

Maxwell Model

(b)

80 E* = 2

MPa

2000 s

contact

Figure 5. Solutions to the Ting model for contact of a rigid probe with a viscoelastic substrate described by (a) the Maxwell model and (b) the Voigt/Kelvin model.



76

contact

Figure 6. Time of maximum contact radius tpeak as a function of %/ tcontactfor the Maxwell and Voigt/Kelvin models. Adhesion is neglected. mum load. Conversely, as t t becomes much longer than T, t k shifts toward the time of maximum load. The expression relating tpeak to the other quantities is COMac

'peak

.

T

— — = j> +

pea

i

l + (y-l)exp(-yr

cow/acr

/T)

(17)

In

Clearly, t depends only on T and is independent of both the maximum load and tip radius. Figure 6 shows equation 17 for the case y = 0.2. The ordinant is the time interval between maximum load and maximum contact area (t k - yhontact) normalized to the total unloading time (t tact - yt ntact)\ it varies between zero and unity. Extremes in behavior are indicated for y = 0.01 and 0.99. Behavior for 0.01 < y < 0.99 is intermediate. For a given T, t k has only a weak dependence on y and the best sensitivity is found for t tact in the range 0.05 T < tcontact < 5T. In contrast to the Voigt/Kelvin model, the contact radius for the Maxwell model (Figure 5b) is always larger than the elastic case and, the longer t u the closer t eak is to the end of the contact. Compared to the Voigt/Kelvin model, the Maxwell model is also much more sensitive to the relative length of the loading cycle y as shown in Figure 6. The radius becomes comparable to the probe radius early in the experiment and parabolic approximation used in the calculation is violated. Ting has given the appropriate form but it cannot be evaluated analytically. However, the qualitative trends should remain the same as shown in Figure 5b. Based on these simple models, several important observations can be made about the response of contacts to creep. These are useful in the design and analysis of experiments to study creep. peak

pea

con

CO

pea

con

contac

P


77 1. The maximum value of the contact radius a always increases as the contact time t ct increases. 2. The contact radius a can be either larger or smaller than elastic limit. It is larger if the viscous component dominates the response. 3. The time t k at which the maximum contact radius occurs during the unloading cycle can either increase or decrease as the total contact time t ntact increases. If viscous behavior dominates, t k increases. 4. Best sensitivity to creep is attained if the loading cycle is as short as possible and the unloading time comparable to the relaxation time. The results in Figure 5 and Figure 6 assume that the load is controlled during the contact formation and rupture. This is not the case for SFM experiments where the displacement A(t) of the sample is usually varied linearly with time (23). Since the cantilever bends in response to the changing load, the distance between the probe and surface plane of the sample and A are related as follows conta

pea

CO


pea

S{P,t) = A{t)-Z(P)

(18)

where b\Pj) is the deformation of the contact under applied load P at time t and Z(P) is the deflection of the cantilever under load P. Under these conditions, the Ting analysis becomes more complicated analytically but the major conclusions reached above should not be significantly modified. Creep in Viscoelastic Contacts with Adhesion. The contact mechanics problem for a viscoelastic material including the effects of adhesion has not been solved. To do this, the elasticity problem must be solved for the viscoelastic continuity equation, equation 6, with the boundary conditions, equation 5, on z = 0 replaced by (27) dw(D) dD(r)

(19)

where p(r) is the radial pressure distribution including contributions from non-contact adhesive forces. The Derjaguin approximation (28) is used in equation 19 to express p(r) in terms of w(D), the free energy of interaction between planar surfaces separated by D. This approximation is valid as long as D and the range of the interaction are much smaller than the radii of the surfaces (29). Additionally, since the actual deformation of the surfaces is assumed to be the same as would occur for flat surfaces under the same conditions, the contact radius a must also be small compared to the radii of curvature. The adhesion energy W and w are related by (27) (20) i.e., the work per unit area required to separate the surfaces from the no-load point to


78 Barquins and Maugis (6) have approached this problem from the view point of fracture mechanics for the special case that viscoelastic losses are localized at the crack tip. They completely ignore effects of creep. Viscoelastic response is assumed to be proportional to the thermodynamic work of adhesion in analogy with the results of peel experiments. Specifically, F - w = w(a v) where T is the strain release rate and (a V) is assumed to be a universal function of crack speed V for a viscoelastic material and a is the WLF shift factor (). For macroscopic glass-polyurethane contacts (a = 50-250 \im), Maugis and Barquins showed experimentally that with n = 0.6 for 10" m/s < V < 10' m/s and a{f)« a . Most SFM experiments are at V much smaller than this range. Greenwood and Johnson (30) showed that these results are consistent with the behavior of a crack acting under the combined effects of surface forces and applied load. Aime* and co-workers (23) have demonstrated that the Maugis-Barquins theory qualitatively describes some SFM experiments on polymers but a quantitative experimental study has not yet been made. Effects of creep on the contact are completely ignored in the fracture mechanics approach of Barquins and Maugis. Yet, the theory of Ting and results like those in Figure 3 suggest creep effects can be significant. Comparison of the relevant time scales for the two processes provides a means to estimate the conditions for which one or the other is expected to dominate and has already been discussed by Johnson (8). Creep effects are due to the bulk deformation whose size is characterized by a, the radius of the contact. Adhesive interactions result in large stresses that are concentrated near the contact edge. These can be characterized by the width d of the Dugdale zone over which the adhesive force with characteristic range h is important or, from the fracture mechanics point-of-view, by the length L of the crack at the contact edge. If the rate at which the contact size changes is v =da/df, two characteristic times can be defined: T

T


T

7

3

n

T

Q

t =a

a

. d L and t =- or V V V d

(21)

If T is a characteristic relaxation time of the substrate, creep will be important when t « T. However, if ?j » T, then adhesion (crack tip) effects at the contact periphery will dominate. Johnson (8) has shown that, in the JKR limit, these two times are very different which allows the two effects to be treated independently. We now ask, "Do creep or crack effects dominate in SFM-scale contacts?" The JKR limit is most appropriate for compliant materials. Following Johnson (8), the width of the Dugdale zone is given by d «{9 n/4)(E */W)h*. Note that d is independent of R. Table I shows values for d and dla u-offi its ratio to the contact radius at pull-off, for a range of values of E* and W. The pull-off radius was calculated using R = 50 nm and h was taken as 0.5 nm. Only when £* becomes as large as 10-100 MPa does the Dugdale zone start to exceed one percent of the contact size. Clearly, creep effects should dominate the viscoelastic response of SFM contacts to most materials with effective moduli smaller than this value. Thus, the conclusions given above for a

m

pu

Q


79 Table I Effective Modulus, E*

d (dla u. ff) pu


0

2

W=20 mJ/m

W= 100mJ/m

1 MPa

0.02 nm (0.02 %)

0.004 nm (0.002 %)

lOMPa

0.20 nm (0.4%)

0.04 nm (0.05 %)

100 MPa

2.0 nm (9%)

0.41 nm (1.1 %)

2

the Ting model should be unaffected by the neglect of adhesion in the model. To confirm this conclusion, we incorporate the effects of adhesion into the Ting model using an empirical approach suggested by Falsafi et al (31) They replaced the load P in equation 8 with an effective load P calculated from JKR theory, i.e., eff

2

2

2

p (t) = p{t)+3nRW+^6nRWP(t) + 9n R W . Assuming a modulus obtained from rheometric measurements, they solved equation 8 numerically with W as a fitting parameter. For contacts between poly(ethylene)-poly(ethylene-propylene) spheres (R « 1 mm), they obtained values of W in good agreement with previous contact angle measurements (32). This result supports the assumption made in using equation 8 that Wis the thermodynamic work of adhesion. We have extended this approach, to explore the entire range of behavior expected in dynamic contacts to linear viscoelastic materials, including contacts with decreasing area. We focus on the simple Maxwell and Voigt/Kelvin models because they represent extremes of response and also allow analytical solutions to equations 813. For the same reason, we assume the adhesion to be described by a DMT model. Thus, the load P is replaced by P(t)—> P(t)+2nWR. Figure 4b shows the linear loading and unloading assumed in the calculations. Jump-to-contact occurs at t = 0 and its magnitude is determined relative to the pull-off load (-3KRW) by the parameter £ In the results presented here, £ = 0.2. The other parameters are defined as in Figure 4a. For the Maxwell model, adding adhesion has little effect except for the expected increase in the contact area. The time at which the maximum contact area occurs is not changed by inclusion of adhesion so that the results shown in Figure 6 also apply for DMT adhesion. Because the short time response of the Maxwell model is that of an elastic spring, the contact radius increases discontinuously at jump-tocontact. In the case of the Voigt/Kelvin model, not only does the contact radius increase, it shifts slightly toward shorter times as illustrated in Figure 7. This shift is at most about ten percent. Thus, the results shown in Figure 6 are only weakly affected by adhesion. eff


80 80

70-

Voigt Model with DMT Adhesion £* = 2MPa, T=50S W (mJ/ml


S a

Probe Radius

O c o

U

100

150

Contact Time (s)

Figure 7. Effect of adhesion of the contact response of the Voigt/Kelvin model. Conclusions Recent SFM measurements using shear modulation techniques have increased interest in developing quantitative methods to measure viscoelastic response of nanometerscale contacts. We used the theory developed by Ting (3) to characterize the response of such contacts during loading and unloading. Although this theory neglects the effects of adhesion, it can be solved analytically for several simple models. The major prediction of this theory is that creep can cause the contact area to reach its maximum value well after the maximum load has been applied to the contact. Results for Maxwell and Voigt/Kelvin models are used to determine how to optimize experiments to study creep. Specifically, we find that, if the loading cycle is much shorter than the unloading cycle, the delayed maximum occurs in the middle of the unloading cycle if the length of the unloading cycle is about twice characteristic relaxation time of the material. For typical SFM instruments, it should be possible to study creep processes with characteristic relaxation times in the range from roughly a millisecond up to a few hundred seconds. Using an empirical approach suggested by Falsafi et al. we also demonstrate that adhesion can be expected to have only a small effect on this conclusion. Stated another way, this result shows clearly that quantitative characterization of compliant viscoelastic materials requires that force vs. distance curves be carried out over as wide a range of contact times as possible. This observation should be particularly relevant for many biological materials. Additionally, it suggests that studies



81

of pull-off behavior that have relied on rather short contact times will need to be reevaluated. We also show that the time dependent response of SFM-scale contacts to linear viscoelastic materials should be dominated by creep response in the bulk, rather than adhesion effects at the contact periphery even for materials with effective moduli as large as 10-100 MPa. The analytical calculations presented here are obviously too simple to be used for quantitative analysis of experimental data. In many of the results, the assumption that the probe radius is much less than the contact radius is not satisfied. Ting describes how to do this correctly, but the calculations cannot be carried out analytically. However, we expect that the changes will be qualitative and that the major conclusions reached here will remain correct. In any case, a more rigorous analysis that correctly includes adhesion needs to be carried out. Several attempts are under way (33-35). Acknowledgements The author gratefully acknowledges financial support from the Department of Energy, the Office of Naval Research, the Paper Surface Science Program of the University of Maine, and the Maine Science and Technology Foundation. Discussions with K.L. Johnson and K.J. Wahl have also been invaluable. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Johnson, K.L. Contact Mechanics;Cambridge University Press: Cambridge, 1987. Savkoor, A.B. in Microscopic Aspects of Adhesion and Lubrication; J.M. Georges, J.M., Ed.; Elsevier Sci. Publ. Co.: Amsterdam, 1981, p.279. Ting, T.N.C. J. Appl. Mech. 1966, 33, 845. Ting, T.N.C. J. Appl. Mech. 1968, 353, 248. Maugis, D.; Barquins, M. J. Phys. D 1978, 11,1989. Barquins, M.; Maugis, D. J. Adhesion 1981, 13,53. Barquins, M . J. Adhesion 1982, 14, 63. Johnson, K.L. this proceedings. Wahl, K.L.; Stepnowski, S.V.; Unertl, W.N. Tribology Lett. 1998,5, 103. Findley, W.N.; Lai, J.S.; Onarian, K. Creep and Relaxation of Nonlinear ViscoelasticMaterials;Dover Publications: New York, 1976. McCrumm, N.G.; Read, B.E.; Williams, G. Anelastic and Dielectric Effects in Polymeric Solids; Dover Publications: New York, 1967. Marganov, S.N.; Whangbo, M.H. Surface Analysis with STM and AFM; VCH Publishers: New York, 1996. O'Shea, S.J.; Welland, M.E.; Pethica, J.B. Chem. Phys. Lett. 1994,223, 336. See for example; Tanaka, K.; Taura, A.; Ge, S.R.; Takahara, A.; Kajiyama, T. Macromolecules 1996, 29, 3040. Yamanaka, Y.; Tomita, E. Jpn. J. Appl. Phys. 1995, 34, 2879. Carpick, R.W.; Ogletree, D.F.; Salmeron, M . Appl. Phys. Lett. 1997, 70, 1548.


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17. 18. 19.


20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

Lantz, M.A.; O'Shea, S.J.; Welland, M.E. Phys. Rev. B 1997, 55, 10776. Hertz, H. /. Reine Angew. Math. 1882, 92, 156. Burnham, N.A.; Gremaud, G.; Kulik, A.J.; Gallo, P.J.;Oulevey, F. J. Vac. Sci. Technol. B 1996, 14, 1308. Johnson, K.L.; Kendall, K.; Roberts, A.D. Proc. Roy. Soc. London A 1971, 324, 301. Derjaguin, B.V.; Muller, V.M.; Toporov, Y. J. Colloid Interface Sci. 1978, 53, 314. Fretigny, C.; Basire, C.; Granier, V. J. Appl. Phys. 1997, 52, 43. Aimé, J.P.; Elkaakour, Z.; Odin, C.; Bouhacina, T.; Michel, D.; Curé1y, J.; Dautant, A. J. Appl. Phys. 1994, 76, 754. Lantz, M.A.; O'Shea, S.J.; Hoola, A.C.F.; Welland, M.E. Appl. Phys. Lett. 1997, 70, 970. Maugis, D. Langmuir 1995, 11, 679. Lee, E.H.; Radok, J.R.M. J. Appl. Mech. 1960, 27, 438. Barthel, E. J. Colloid Interface Sci. 1998, 200, 7. Derjaguin, B.V. Kolloid-Z. 1934, 69, 155. Hunter, R.J. Foundations of Colloid Science; Oxford Univ. Press: Oxford, 1995; Vol. I, pp. 191. Greenwood, J.A.; Johnson, K.L. Phil. Mag. A 1981, 43, 697. Falsafi, A.; Deprez, P.; Bates, F.S.; Tirrell, M . J.Rheol.1997, 41, 1349. Zhao, W.; Rafailovich, M.H.; Sokolov, J.; Fetters, L.J.; Plano, R.; Sanyal, M.K.; Sinha, S.K.; and Sauer,B.B. Phys. Rev. Lett. 1993, 70, 1453. K.L. Johnson, private communication. P. Kleban, private communication. A. Falsafi, private communication.


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