Contact Mechanics and Adhesion of Viscoelastic ... - ACS Publications

Chapter 2 Contact Mechanics and Adhesion of Viscoelastic Spheres K. L. Johnson

Downloaded by UNIV OF MINNESOTA on May 23, 2013 | http://pubs.acs.org Publication Date: December 10, 1999 | doi: 10.1021/bk-2000-0741.ch002

College of Engineering and Applied Sciences, University of Cambridge, 1 New Square, Cambridge CB1 1EY, United Kingdom

It has recently become common to use the JKR theory (Johnson, Kendall & Roberts, 1971) to extract the surface and interfacial energies of polymeric materials from adhesion tests with micro-probe instruments such as the Surface Force Apparatus and the Atomic Force Microscope. However the JKR theory strictly applies only to perfectly elastic solids. The paper will review progress in extending the JKR theory to the contact mechanics and adhesion of linear viscoelastic spheres. The observed effects of adhesion hysteresis and ratedependent adhesion are predicted by the extended theory.

Recent years have seen an increased use of microprobe instruments to measure the surface energy and viscoelastic properties of polymeric materials. Such probes commonly comprise an effectively spherical tip pressed against a flat surface or, as in the case of the Surface Force Apparatus, two crossed cylinders of equal radii which are geometrically equivalent to a sphere in contact with a plane. A typical experiment consists of loading and subsequently unloading the contact with a controlled force, while measuring the displacement and/or contact area, until the surfaces separate. If the surfaces adhere during the compressive part of this cycle, a tensile force is required to pull them apart, which is referred to as the * pull-off' force. Microprobe instruments are particularly advantageous for adhesion measurements for two reasons: (i) adhesion forces become increasingly significant with a decrease in the size of the contact; (ii) adhesion is strongly affected by surface roughness which is minimised by using a very small area of contact. In order to extract values of the surface energy or viscoelastic properties of the specimens from such a test a mechanics model of the test is required. Up to the present the so-called JKR theory (/) has been used, e.g. Tirrell (2). The JKR theory, however, is derived for perfectly elastic solids and assumes reversible behaviour in both the adhesive forces and the contacting solids. But the adhesion of viscoelastic solids, notably rubber, is well known not to be reversible. More work is required to separate two surfaces than is returned when they come together, an effect known as

24

© 2000 American Chemical Society

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

25


hysteresis'. This irreversibility may arise from the nature of the adhesive bond, e.g. Israelachvili (3) or from viscoelastic dissipation within the contacting solids (4). This latter mechanism can easily appreciated in a qualitative way. During separation the work done by the external force has to overcome the dissipation in addition to doing work against the surface forces (surface energy). When the surfaces are coming together the surface energy is providing the driving force, which has to provide for the dissipation as well as overcoming the external load. Hence the external force on separation exceeds that on coming together by virtue of the internal viscoelastic dissipation. In common with any non-conservative system, the response is dependent upon the loading path. Several researchers have commented on the need for a mechanics model for the adhesion of viscoelastic solids to assist in the interpretation of microprobe adhesion experiments. This paper is a progress report on work in Cambridge to extend the JKR theory to the adhesion of viscoelastic spheres. The present state of play will be reviewed in a predominantly qualitative way; analytical details will be presented in a separate publication. Elastic Solids: the JKR and Maugis-Dugdale Theories. Continuum mechanics models for the adhesion of perfectly elastic spheres under the action of reversible surface forces are well developed. The essential features of the JKR model are shown in Figure la. The surface traction acting on a contact area of radius a comprises two terms: (i) a Hertz pressure pi(r), caused by the compressive force P \ which flattens the spherical surfaces and (ii) an adhesive tension /? (r)which gives rise to the adhesive force P& The net contact force P can be expressed: F

a

3

p~p

{

- p «,( 4E*a /3R

)-V&rAyE*a

a

(1)

where E* is the combined elastic modulus of the two solids given by 1 E

m

1E

\-v\ +

E

x

2

the relative radius of the two spheres l/R - l/R\ + VR2 and the combined surface energy of the two surfaces Ay - y, + y - y .The surfaces snap apart at a 'pull-off force P given by 2

12

c

P =-{3/2)KRAY C

(2)

We now introduce dimensionless variables: P - P / 3^r/?Ay , a « a I (9KF? Ay / 4 £ * ) so that equations (1) and (2) become 2

P-a-(2af and

13

(3)

Pc-1/2 (4) The infinite tension at the edge of the contact in Figure 1(a) is clearly physically unrealistic and it transpires that the above results comprise an asymptotic solution to the general problem of adhesion of elastic spheres, which applies to large spheres of low elastic modulus, i.e. where the elastic deformation is large compared with the range of action of surface forces. Solutions to the general problem have been found numerically,



26

(b)

Figure 1. Adhesive contact of elastic spheres. Surface tractions comprise two terms: Hertz pressure pi and adhesive tension p . (a) JKR model has infinite tension at r = a (b) Maugis model has tension o in annulus acr 5. For a comparison of the JKR and Maugis-Dugdale theories with numerical solutions see Johnson & Greenwood (8). 2

These results remain valid in the presence of frictional shear tractions interface: exactly so if (1 + v )(1 - 2v ) I E »(1 + v )(1 - 2 v ) / E and approximately so if they differ'. x

x

x

2

2

2

Linear Viscoelastic Solids Material Characteristics. To be mathematically tractable analytical continuum mechanics modelling of rate-dependent materials is effectively confined to linear viscoelasticity. We shall use the simplest model which encapsulates the essential features of small strain polymer deformation: the 3-parameter model. It is represented in Figure 2 by two springs and a dashpot. At very low rates of strain the material deforms elastically a with a modulus E„ represented by the two springs in series. At high rates of strain the deformation is again elastic, with a modulus E -E^lk represented by the single spring. At intermediate rates, the strain at time t after a step change in stress is given by the creep compliancefunction:: (0 * (1 / E*)4(t) where 0(f)«{l-(l-*)exp(f/r>}

(6)

where T = (1 -k)r}/ E is the relaxation time of the material. 5, d « a) are assumed. In the purely elastic case the equilibrium relationship between load and contact size is given by equation (1). In the viscoelastic case the surface energy Ay is scaled by the factor /3, given by equation (19) during loading and by equation (18) during unloading. In dimensionless variables equation (3) becomes 0

Q

P-a-^fis?

(20) 1/2

from which

Now

2

2

^ = 3Ja - (afi I if ) - l ^ T da i J (2/3 dP

— / — sb — / J

dP da

da

dt

dl

T \ada\ ol

t

Q

dp da

(21)

\

[a dt J

and daldx is the speed of the edge of the contact, which corresponds to the crack speed V in equation (17), from which 36R(&Y)

da

t

(22) v

v



35

Figure 7. Apparent surface energy of a moving viscoelastic crack (k = 0.1). Exact analysis (Greenwood, unpublished); Approx. (Schapery, equations (18) & (19).


36 Equating (21) and (22) gives

I

i

2/5

Jfl

(23)

v

The simultaneous equations (17),(19) and (23) can be integrated step-by-step to give the variation of j3 with a and hence, by equation (20), the variation of contact size with load. During loading (crack closure) /3 = p i given by equation (19) ; during unloading (crack opening) j8 = Pep given by equation (18). Computations have been carried out using the creep compliance function in equation (6), with k = 0.1, for various values of the loading rate parameter A, where


c

(24) The results are plotted in Figure 8. The elastic (JKR) relationship is shown by the chain line. The dotted lines show the high speed limits (A -*{H(t*)t*\ Ay " fl/*) op

(26)

where H = hld . Using equation (17) to relate t* to the dimensionless crack opening speed v, enables p to be plotted against v in Figure 11 for varying values of the thickness parameter H. It is immediately apparent that reduction in the thickness of the viscoelastic layer attenuates the work of adhesion and, at high peeling speeds, causes a decrease of adhesion with speed. This effect may be understood when it is appreciated that increasing speed brings an increasing volume of material into the high dissipation regime shown by the curve of tan 6 in Figure 2(b). In an unlayered system this process continues until the whole of the dissipation regime is effectively contained within the solids, whereupon the work of adhesion levels off at a maximum value l/k. In a system comprising a thin layer backed by non-dissipative material, the dissipation regime cannot be wholly accommodated within the layer and is progressively excluded at higher speeds. The work of adhesion then decreases with speed. The range of peeling speed for which the adhesion decreases with speed is basically unstable and can lead to unsteady peeling. It is responsible for the vibration and ripping noise which frequently occurs when adhesive tape is pulled rapidly off a flexible sheet. To apply these results to a contact cycle, it is necessary to consider the contact mechanics of the system, as discussed for the Surface Force Apparatus (SFA) by Sridharet al.(i6). 0


op

Discussion and Conclusions To extract solid surface properties, surface energy in particular, from a contact experiment an appropriate contact mechanics model of the test is required. Under perfectly elastic reversible conditions the JKR theory of adhesion of elastic spherical surfaces has proved effective, but there are features of the adhesion of rate-dependent (viscoelastic) solids which are not modelled by the JKR theory. The most notable are: (i) the apparent surface energy (work of adhesion) is rate dependent and (ii) the work of adhesion in separating the surfaces is greater than is released when they come together (adhesion hysteresis). This paper reports progress in extending the JKR theory to linear viscoelastic solids. It successfully predicts both rate-dependent apparent surface energy and adhesion hysteresis. For the purpose of exposing the effects of viscoelasticity, computations have been restricted to a simple 3-parameter model with a single relaxation time T and a ratio of instantaneous to relaxed modulus E I = l/k (see Figure 2). It is recognised that this may be a poor model for a real polymer, but the theory is not restricted to the creep compliance function of equation (6). A simple alternative would be the power law: Q

(29) In the case of polyurethane rubber a 3-parameter model gave a reasonable fit to the Maugis & Barquins' experimental curve of G I Ay (Figure 5) by choosing l/k = 2 E3 op

andT= 1.3E-3 s. A fundamental restriction in the theory developed above is that the length d of the zone at the edge of contact where the adhesive forces act (Dugdale zone) should be small



39

Figure 10. Interface crack between viscoelastic layers, mounted on elastic substrates.

100

101

102

103

104

Non-dimensional peeling speed v = VT/do Figure 11. Effect of layer thickness h on the apparent surface energy of an opening interface crack.


40 compared with the radius of the contact area. As in fracture mechanics, this enables relaxation effects at the edge of contact, which govern adhesion, to be separated from effects in the bulk. Taking typical values: El-2 GN/m , Ay = 0.05 J/m , h = 0.5 nm, equation (27) gives d = 8 nm. Apparatus on the scale of the SFA generally has values of a > 10 pirn, so that the condition of d « a is clearly satisfied but, with nanoprobe instruments like the AFM, a may be measured in nanometres. An analysis of the situation where d is not necessarily small compared with a, when edge and bulk relaxation effects cannot be regarded as independent, is under investigation. The contact mechanics of viscoelastic solids in the absence of adhesion is well established, but the case of a layered system calls for further work. In a loading/unloading cycle the maximum contact area is generally reached after the load begins to decrease, as shown in Figure 4. This feature has been demonstrated experimentally by Wahl & Unertl (17) using polyvinylethylene in a Scanning Force Microscope. The key to analysing the adhesion of viscoelastic solids lies in Schapery's linear viscoelastic analysis of mode I fracture of an interface (12 & 13). He shows that, for an opening crack, the work of adhesion (apparent surface energy) is an increasing function of the opening (peeling) speed whereas, for a closing (healing) crack, the work of adhesion decreases with closing speed (equations (18) & (19) and Figure 7). These equations are not exact, but provide a good approximation provided that again d « a. Of course the Dugdale (uniform) variation of adhesive stress is not realistic, but an analysis by Greenwood & Johnson (14) using an inverse cube force-separation law only had the effect of changing the coefficient in equation (17) from id 12 to 1/2. Figure 7 is plotted for a material in which Ilk = 10; increasing this ratio extends the plot at the high speed end, but does not change the low speed end. A loading/unloading adhesion experiment has been modelled by applying Schapery's fracture mechanics to the motion of the edge of the circular contact region between two adhering spheres. A growing contact during loading corresponds to a closing crack, while a contracting contact during unloading corresponds to an opening crack. In most experimental arrangements the load is applied through a flexible element whose stiffness is small compared with the (Hertz) stiffness of the contact itself. Thus control of the displacement of the remote end of this element leads to control of load rather than displacement of the contact. Therefore the simple ramp cycle, shown in Figure 3 is assumed, in which the load is increased and decreased at a uniform speed. The calculated variation of contact radius a with load is shown in Figure 8. Hysteresis loops are clearly revealed, which increase in width with an increase in the rate parameter A. As shown in Figure 9, the magnitude of the pull-off force (P | also increases monotonically with A.. The parameter A, defined in equation (24), can have a very wide range of values. It incorporates the loading rate (PJt ) the viscoelastic properties of the material (k and 7), the surface energy (Ay) and the radius of the tip of the probe (R). For example, in the tests on polyurethane rubber shown in Figure 5, A varies between 10 and 10 as the peeling speed varies from 10" to 10 jAin/s. Other things being equal, the change in tip radius from the SFA (R = 1 cm) to the AFM (R = 50 nm) reduces A by four orders of magnitude. It is not easy, therefore, to make useful general statements about the values of A; each material and each experimental arrangement must be considered ab initio. In experiments where a thin film of polymer is applied to an elastic substrate, the thickness of the film strongly influences the apparent surface energy, as expressed in equation (26) and shown in Figure 11. In applying this result, it must also be 2

2

0


Q

c

Q

y

4

1

3


41 recognised that the compliance of the contact is governed by the elastic modulus of the substrate. So the JKR equation (1) may be written: P - (4E* 13R) e

fap A Ey op

(30)

Y

where E* is an 'effective modulus' of the layered system and fi is given by equation (26). In general E* is not constant, but is a function of the ratio of the contact radius a to the film thickness h but if this ratio is always much greater than unity, the film makes a negligible contribution to the compliance of the contact and E* becomes the (constant) value for the substrate. These considerations in relation to the Surface Force Apparatus have been discussed by Sridhar et al.(76). e

op


e

Acknowledgments The author wishes to acknowledge the help of his colleagues: Dr.J.A.Greenwood for close collaboration particularly over the section on viscoelastic fracture mechanics and DrJ.Woodhouse for assistance with the computations. Literature cited (1) Johnson,K.L.; Kendall,K. & RobertsA.D.Proc.Roy.Soc.Lond. 1971, A324. 301-313. (2) Mangipudi,V.; Tirrell,M, & Pocius,A.V. Langmuir 1995, 11, 19-23. (3) Israelachvili,J In Fundamentaals of friction; Ed.I.L.Singer & H.M.Pollock, NATO ASI Ser.E, Vol.220, Klewer, Dordtrecht, The Netherlands, pp.351. (4) Maugis,D. & Barquins,M. J.Phys.D: Appl.Phys.1978, 11, 1989-2023. (5) Muller,V.M.; Yuschenko,V.S. & Derjaguin,B.V. J.Coll.InterfaceSci.1980, 77, 91-101. (6) Greenwood,J.A. Proc.Roy.Soc.Lond. 1997,453, 1277-1297. (7) Maugis,D. J.Coll.InterfaceSci.1992, 150,243-269. (8) Johnson,K.L. & Greenwood,J.A. J.Coll.InterfaceSci.1997, 192, 326-333. (9) Ting,T.C.T. ASME, J.Appl.Mech. 1966, 33, 845-854. (10) Graham, G.A.C. lnt.J.Eng.Sci. 1967, 5, 495-514. (11) Johnson,K.L. Contact Mechanics; C.U.P. Cambridge, 1985. (12) Schapery,R.A. Int.J.Fracture, 1975, 11, 141-159,369-388. (13) Schapery,R.A. Int.J.Fracture, 1989, 39, 163-189. (14) Greenwood,J.A. & Johnson,K.L. Phil.Mag. 1981, A 43, 697-711. (15) Huntley,J.M. Proc.Roy.Soc.Lond. 1990, A 430, 525-539. (16) Sridhar,I; Johnson,K.L. & Fleck,N.A. J..Phys. D: Appl.Phys. 1997, 30, 1710-1719. (17) Wahl,K.J. & Unertl,W.N. In Tribology Issues and Opportunities in MEMS, Ed.B.Bushan, Klewer, Dordtrecht, The Netherlands, 1998.


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