Langmuir 1995,11, 1396-1402
1396
Viscoelastic Dissipation in Wetting and Adhesion Phenomena M. E. R. Shanahan*lt and A. Carr6*?$ Centre National de la Recherche Scientifique, Ecole Nationale SupCrieure des Mines de Paris, Centre des MatCriawc P.M. Fourt, B.P. 87, 91003 Evry Cedex, France, and Centre de Recherche Corning Europe, 7 bis, Avenue de Valvins, 77210 Avon, France Received December 13, 1994@ When a liquid drop is placed on a smooth, rigid, solid substrate, it spreads until the final thermodynamic equilibrium is attained. The kinetics of spreading of the drop are controlled by conversion of capillary potential energy into viscous dissipation within the liquid. However, if the solid is sufficiently soft, a local deformation, or “wetting ridge”, may form near the wetting front and the motion of the latter may lead to viscoelastic dissipation. We describe cases in which viscoelastic dissipation dominates and thus where spreading speed depends on bulk properties of the solid, rather than on liquid viscosity. This behavior in wetting can be considered to be analogous to dissipation phenomena in the adhesion of elastomers. Therefore, a comparison is made between wetting and adhesion dynamics. It appears clearly that a parallel can be drawn between the wetting of rubber and rubber adhesion to glass when both phenomena involve the same viscoelastic material. Kinetics of formation and breaking of corresponding interfaces is controlled by the damping properties of the soft solid substrate. In this paper, the viscoelastic properties of elastomers are described by two parameters, n and U,; n is the usual speed power factor ( n % 0.5-0.6) and U,a characteristic speed below which a fraction of the elastic strain energy in the wetting ridge is dissipated. 1. Introduction
The phenomena of wetting and adhesion are intrinsically related. Restricting our attention to wetting behavior a t near ambient temperatures, most liquids display surface tensions and interfacial tensions with solids or other (immiscible) liquids, depending mainly on secondary, physical bonds of the van der Waals category (although the importance of acid-base interactions and hydrogen bonding is being increasingly recognized and should not be o~erlookedl-~).Both static meniscus configurations and dynamic behavior are largely controlled by the relative magnitude of these surface properties. (Clearly, body forces, such as those due to gravity, are also relevant, but in the present context, we discuss sufficiently small quantities of liquids for these forces to be neglected.) The adhesion between solids may depend solely on the same types of secondary bonds although other mechanisms involving diffusion and chemical bonding, among others, are often e n c o ~ n t e r e d .That ~ ? ~ van der Waals forces may alone lead to significant adhesion has been shown in experiments in which a steel cylinder adheres after simple contact to the underside of a n elastomeric solid.6 Even in cases in which the final adhesivejunction may well depend on mechanisms other than secondary bonding, usually one phase (generally the adhesive) must go through a n intermediate stage in its liquid state in order to establish good, intimate contact with the second phase by wetting, before solidification occurs. Considering specifically adhesion controlled by secondary interactions, it is well-recognized that the apparent + Centre National de la Recherche Scientifique.
* Centre de Recherche Corning Europe.
Abstract published inAdvunce A C S A b s t r u c t , March 15,1995. (1)Fowkes, F. M. J. Adhes. 1972,4 , 155. (2)vanOss, C. J . ; Chaudhury, M. IC;Good, R. J.Adv.Colloidlnterface Sci. 1987,28, 35. (3)Carr6, A.;Malchere, A. J. Colloid Interface Sci. 1992,149, 379. (4) Wake, W. C. Adhesives; Royal Institute of Chemistry, Lecture Series, No. 4; Royal Institute of Chemistry: London, 1966. (5)Kinloch, A.J . Adhesion and Adhesives: Science and Technology; Chapman and Hall: London, 1987,p 56. (6) Barquins, M.J.Adhes. 1988,26,1. @
0743-7463/95/2411-1396$09.00/0
work of adhesion, W, between two solids, of which a t least one is elastomeric, is often considerably greater than the intrinsic value given by DuprB’s equation’ forW,:
w,= Y1+
Y2 - YlZ
where y1 and y2 are the surface free energies of the materials in contact (to be assimilated to surface tensions, for present purposes) and ylz is their common interfacial free energy (tension). This has been shown using a variety of separation test geometries.*-1° The excess energy, W - W,, corresponds essentially to viscoelastic (andor plastic) dissipation occurring in the elastomer(s) during separation of the two phases. Since elastomeric materials possess a loss modulus, E” (E* = E iE” where E* is the complex Young’s modulus, E’ is the storage modulus and i2 = -11, during the necessary deformation encountered near the fracture front, a certain amount of work is dissipated, even if the material(s) return to their original underformed state after separation. The quantity of work depends both on separation rate and test temperature, but use may be made of the Williams, Landel, and Ferry time-temperature superposition principle1’ and a shift factor, aT, may be used to construct a master curve of apparent energy of adhesion as a function of reduced separation rate. An analogous type of behavior may also be present during the formation of a n adhering interface, but in this case there is a sign change and the excess dissipated energy may be regarded as W, - W*, where W* corresponds to the energy change ofbond formation. Under conditions of interface formation, the dissipative term is of much reduced magnitude and in fact this difference is the basis of the rolling cylinder t e ~ t . ~ A J ~rigid, J ~ smooth
+
(7)DuprB, A. Th6orie MBcanique de la Chaleur; Gauthier-Villars: Paris, 1869;p 369. (8) Gent, A. N.; Schultz, J . J.Adhes. 1972,3 , 281. (9)Andrews, E. H.; Kinioch, A. J. Proc. R . Soc. London 1973,A332, 385. (10)Maugis, D.; Barquins, M. J. Phys. D.: Appl. Phys. 1978,1 1 , 1989
(11)Ferry, J.D. Viscoelastic Properties ofPolymers, 2nd ed.; Wiley: New York, 1970;p 314. (12)Roberts, A. D.; Thomas, A. G. Wear 1976,33, 45.
0 1995 American Chemical Society
Viscoelastic Dissipation in Wetting and Adhesion
Langmuir, Vol. 11, No. 4, 1995 1397
cylinder (typically of steel or glass) rolls down a n inclined elastomeric track. In dynamic equilibrium, the rolling speed stays constant since the potential energy lost as the cylinder progresses down the track, instead of leading to a n increase in kinetic energy, is dissipated (mainly) by the separation process a t the trailing edge of the cylinder/ track contact zone. By changing the angle of the track with respect to the horizontal andfor the cylinder weight, the effective adhesion may be assessed as a function of rolling speed and therefore separation rate. As the separation rate approaches zero, the viscoelastic dissipation contribution to apparent adhesion may be expected to disappear, but using the JKR technique,14 there is evidence to suggest that this is not always the case15and some molecular dissipation mechanism may well be supplementing interfacial resistance.16J7 Surface (interfacial) free energies, yo, govern the equilibrium configuration of liquid menisci in contact with solid surfaces. At the triple line solid (S)/liquid (L)/vapor (VI, Young‘s equation18must hold a t equilibrium (at least on a macroscopic scale):
Ysv = YSL + YLV cos 80
(2)
where Bo represents the equilibrium contact angle measured within the liquid phase between the tangents to the LN and S/L interfaces. In the simplest derivation of Young‘s equation, the horizontal force equilibrium a t the triple line (i.e. parallel to the solid surface) is considered, but the vertical balance is often neg1e~ted.l~The component of liquid surface tension perpendicular to the plane of the solid must be equilibrated and this leads necessarily to some distortion of the substrate near the triple line. While this effect will be negligibly small for “hard” solids, since substrate deformation scales with y ~ sin v 6JG where G is the elastic (shear)modulus of the solid, if the material is sufficiently soft, the local “wetting ridge” may lead to interesting behavior including modification to Young‘s equation under some circumstances.20 Equation 2 adequately describes contact angle equilibrium on a macroscopic scale, but the surface (interfacial) tensions ysv,ysL, and ~ L (=y) V are also largely responsible for the spreading behavior of a liquid from a n arbitrary initial state to equilibrium. The spreading process may be considered from the point of view of a dynamic energy balance.21 If the initial contact angle, 6 , is greater than that a t equilibrium, e,, a net spreading force exists a t the triple line, and as the wetting front moves forward, work is expended and dissipated by viscous shear within the liquid. However, if spreading occurs on a soft, viscoelastic (or plastic) substrate, there exists, in addition, another dissipative process retarding wetting front movement.22 As the triple line advances, the wettingridge accompanies it and the straidrelaxation cycle occurring along the spreading path leads to further energy loss. In the following we describe a situation in which the major dissipative losses occurring during spreading are due to (13) Zaghzi, N.; C a d , A,; Shanahan, M. E. R.; Papirer, E.; Schultz,
J. J . Polym. Sci., Polym. Phys. 1987,25,2393.
(14) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. SOC.London
1971,A234, 301. (15)C a d , A.;Roberts, A. D. J . Nut. Rubber Res. 1987,2, 152. (16)Carre, A.; Schultz, J. J . Adhes. 1984,17,135. (17)Shanahan, M. E. R.; Michel, F. Int.J.AdhesionAdhesives 1991, 11, 170. (18)Young, T. Phil. Trans. R.SOC. 1805,A95, 65. (19)Bikerman, J. J . J . Phys. Chem. 1959,63, 1658. (20)Shanahan, M.E.R. J. Phys. D.: Appl. Phys. 1987,20,945. (21)de Gennes, P. G. Rev. Mod. Phys. 1985,57,827. (22) Shanahan, M.E. R. J . Phys. D.: Appl. Phys. 1988,21,981.
1
X
0
Figure 1. Schematic representation of the wetting ridge. The terms 5 and u correspond,respectively,to vertical displacement of the solid surface caused by y sin Bo and stretching due t o y cos Bo. The cutoff distance for linear elastic behavior, E , has been estimated to be typically on the order of a few nanometers for an elastomer.26
viscoelastic processes, and as a consequence, we obtain a direct parallel between dynamic adhesion and wetting behavior. 2. Theory 2.1. Description of the Wetting Ridge. Although the derivation of Young‘s equation in which forces are equilibrated parallel to the solid surface is, perhaps, a little unsatisfying, more rigorous demonstrations of its validity e ~ i s tand ~ ~the, expression ~ ~ may be regarded as correct, for most purposes. Nevertheless, the component of liquid surface tension perpendicular to the solid surface must leadto areaction in the substrate. This was pointed out several years and it has indeed been shown that, in principle a t least, local deformation of the solid must occur, leading to a “wetting ridge”.25,26As pointed out above, in order of magnitude, this wetting ridge has a height of y sin BJG and, assuming the solid to be incompressible as in the case of a n elastomer (Poisson’s ratio, Y = l/2, takes the formz2(See Figure 1):
(3) where 5 is the vertical displacement of the solid, d is a constant (essentially a datum depth, which may be eliminated by treating the problem as three-dimensional rather than two-dimen~ional,~~ but which is of no consequence in the following), x is the distance measured from the triple line parallel to the undisturbed substrate surface, and E is a cutoff length, below which the solid no longer behaves in a linearly elastic manner (typically on the order of a few nanometers for a n elastomer26).28 Clearly, for a typically “hard” solid, such as a metal or a ceramic, the disturbance to the solid due to the presence of a drop of liquid is totally negligible, but when a “soft” solid, such as a n elastomer, is being considered, this (23) Collins, R. E.; Cooke, C.E. Trans. Faraday SOC.1969,55,1602. (24)Johnson, R. E. J . Phys. Chem. 1969,63, 1655. (25)Lester, G. R. J . Colloid Sci. 1961,16,315. (26)Shanahan, M. E. R.; de Gennes, P. G., Adhesion 11; Allen, K. W., Ed.; Elsevier Applied Science: London, 1987;p 71. (27)Hills, D.A.;Sackfield, A. J.Phys. D.: Appl. Phys. 1989,22,371. (28)It should be noted that 8. and 8 retain their meaning as representing the contact angle subtended within the liquid phase between tangents to the W and undeformed S L interfaces. The angle between the actual W and S/L interfaces will be greater, but as described elsewhereF0 Young‘s equation remains valid under most conditions provided that the conventional definition of Bo is retained.
Shanahan and C a r d
1398 Langmuir, Vol. 11, No. 4, 1995 correction may become significant. Indeed, for a soft solid, the horizontal component of liquid surface tension, y cos e,, may also lead to a degree of stretching of the 0, either on the vapor side of the triple line for 8, < 90" or on the liquid side for 6, 90":
However, in the case under consideration, when effects due to the wetting ridge formed on the soft substrate may not be neglected, a viscoelastic dissipation term must be added to that due to viscosity. Using eqs 6-8, the dynamic energy balance is now given by
(4) where d' is a constant. Let us now consider a triple line moving a t speed U , corresponding to a drop spreading with contact angle 6(t) decreasing toward 60. The wetting ridge will move with the triple line and work, E , is effected per unit time and per unit length of the triple line:22
where both differentials are evaluated just ahead of the triple line, i.e. a t x = 6. Were the solid purely elastic, this work would be restituted immediately after passage of the triple line as the solid relaxed. No net work would be done. However, most soft solids, and in particular elastomers, are viscoelastic and a fraction, A, ofthe energy expended in eq 5 will therefore be dissipated. Using eq 3-5 and allowing for the fact that 8, must be replaced by 6(t) in eqs 3 and 4, it may be shown that the quantity of energy dissipated is given by
(This expression is in fact a simplified form of eq (8)in ref 22. Terms of lesser importance involving contact angle have been omitted.) Equation 6 must be allowed for in a n adequate description of spreading on a soft solid. 2.2. Dynamics of Spreading. The behavior during spreading of a small axisymmetric drop of liquid on a n ideal solid (Le. flat, smooth, homogeneous, isotropic, and rigid) has been previously described.21 Briefly, a dynamic energy balance is invoked in which the excess free energy, due to the fact that the drop has not attained capillary equilibrium, is consumed by dissipation within the liquid caused by viscous shear. The rate a t which capillary free energy is given up, per unit length of triple line, is given by
P = YUCOS eo - COS e(t)i
(7)
eq 7 being obtained simply by considering the spreading force a t the triple line due to Young's equation of equilibrium not being satisfied for contact angle 6%) and multiplying this by spreading speed. Viscous dissipation has been calculated approximately,21 albeit for small values of 6(t), and is given by
when 17 is liquid viscosity and 1 represents the logarithm of the ratio of two lengths, the numerator corresponds approximately to drop contact radius, and the denominator is a small distance from the contact line necessary to avoid a singularity in the flow field adopted. It is worth noting that viscous dissipation is greater for small contact angles. For spreading on rigid solids, approximate spreading behavior may be estimated by equating relations 7 and 8.
If the contact angle is large, the viscosity term is small, and anyway, in the experimental work described below, it has been satisfactorily demonstrated that TS
