Langmuir 1994,10, 1647-1649
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Anomalous Spreading of Liquid Drops on an Elastomeric Surface Martin E.R. Shanahan*,+and Main C a d Centre des Matkriaux, Centre National de la Recherche Scientifique, Ecole Nationale Supkrieure des Mines de Paris, BP 87, 91003 Evry Ckdex, France, and Centre de Recherche Corning Europe, 7 bis Avenue de Valvim, 77210 Avon, France Received January 3,1994. In Final Form: March 23,1994' When a liquid droplet is placed on a flat, smooth, and rigid solid, ita spreading to equilibrium can be describedby a dynamic energybalance. Excess free surface (interfacial) energy, resultingfrom the capillary imbalance, is dissipated by viscous motion within the liquid. If the solid is sufficiently soft, a local deformation, or 'wetting ridge", may form near the wetting front, and ita motion may lead to viscoelastic dissipation. We describe the case in which viscoelastic dissipation dominates and thus where spreading speed is controlled by bulk properties of the solid, rather than by liquid viscosity. A parallel is drawn with elastomeric adhesion.
Introduction The kinetics of a spreading drop is controlled by conversion of capillary potential energy into viscous dissipation within the liquid when the solid is rigid.' However, if the solid is sufficiently soft,a 'wetting ridge" near the triple line can also be a dissipative sink as the wetting front moves.2 Following the observation of abnormally slow spreading on elastomeric surfaces, it is shown in this paper how the kinetics is essentially controlled by the viscoelasticrather than the viscous losses in the systems studied (low-viscosity liquids on silicone rubber). Thus, the spreading phenomenon of a liquid becomes comparable to the formation of an adhesive contact involving a viscoelastic solid. It is well-known that the apparent energy of adhesion, W, often greatly exceeds the intrinsic value of Dupr6, W O (=TI + 7 2 - ylz where yl and yz are the surface free energies of phases 1and 2, and 7 1 2 is their interfacial free energy), in, for example, peel The excess, W - Wo, is largely consumed by viscoelastic (or plastic) straining of the polymeric material(s) during separation. When a small axisymmetric sessile drop is first placed on an ideal solid substrate (i.e., flat, smooth, homogeneous, isotropic, and rigid),ita contact angle, 8, subtended between the tangents to the solialiquid and liquidhapor interfaces at the triple line and within the liquid, will generally be greater than ita equilibrium value as predicted by the Young equation.' This capillary imbalance leads to a spreading force of y [cos 80 -cos 8(t)l where y is the liquid surface tension and 80 and 8(t)represent the contact angles, respectively, at equilibrium and at time t . If the triple line advances at a rate U,we may define P as the work done per unit time and per unit length of wetting front:
P = COS eo - COS e(t)i
(1) Under classic conditions of wetting on a rigid solid, this work is dissipated almost entirely by viscous shear within the 1iquid.l Provided the equilibrium contact angle is f i i t e + &ole Nationale Sufirieure des Mines de Paris. Centre de Recherche Coming Europe. e Abstract published in Advance ACS Abstracts, May 1, 1994. (1)de &Mer$, P. G. Rev. Mod. Phys. 1986,57, 827. ( 2 ) Shanahan,M.E. R. J. Phys. D Appl. Phys. 1988,21,981. London 1969, A310,433. (3)Gent, A. N.; Petrich, R. P. h o c . R. SOC. (4)Gent, A. N.; Schdtz, J. J. Adhes. 1972, 3, 281. (5) Andrew, E. H.; Kinloch, A. J. Roc. R. SOC.London 1973, A332, 385. (6) Maugis, D.; BFquins, M. J. Phys. D Appl. Phys. 1978,11,1989. (7)Young, T.Phzlos. Tram. R. SOC.1805, A H , 65.
0743-7463/94/2410-1647$04.50/0
_ _ - - _ _ _ _ - __ __ _ - - - - - - - ---- - -
4
s L-2. Figure 1. Sketch of the wetting ridge near the triple line solid @)/liquid(L)/vapor (V). The dotted line represents the solid without deformation caused by the vertical component of the liquid surface tension, 7 sin 0 0 ) .
and the liquid viscosity fairly low, spreadingto equilibrium is usually rapid. Although the viscous dissipation term scales as TVl8(t),where 7 is the liquid viscosity, even for small values of 80 equilibrium is quickly attained on rigid solids. However, considering now spreading on a *soft" solid, an elastomer, the vertical component of the liquid surface tension, y sin O(t) (Figure 11, leads to local deformation of the substrate and a wetting ridge with height of order ?[sin 8(t)]/E,where E is the substrate Young's modulus, re~ults.*1~As the triple line advances, so does ita accompanying wettingridge and work is effected in 'lifting up" the solid. Clearly, after the wetting front has passed a given zone of the solid, the surface is released, and thus for a perfectly elastic substrate, no net work is done. However, for a viscoelasticsolid,the straintrelaxation cycle leads to a certain fraction of the strain energy being dissipated. In the case considered, this viscoelastic dissipation largely exceedsthe viscous term evoked above, and the latter may thus reasonably be ignored as an energy sink. The work done (per unit time and per unit length of triple line) has been calculated previously? and when Poisson's ratio for the solid is taken to be 112 (elastomer), this may be expressed as E
y2U/(2?rGc)
(2)
where G is the solid shear modulus and c is a cutoff distance near the triple line, below which the behavior of the solid may be considered to be no longer linearly elastic (of the (8)Lester, G. R. J. Colloid Sci. 1961, 16, 315. (9)Shanahan,M. E. R.; de Gennes, P. G . In Adhesion 11; Allen, K. W., Ed.;Elsevier Applied Science: London, 1987; p 71.
0 1994 American Chemical Society
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1648 Langmuir, Vol. 10, No. 6,1994
order of a few nanometersg). Taking the fraction of this work, A, to be rate dependent, as has been shown in adhesion experiments? we write A = (U/Uo)" (3) where UOand n are constants and may thus define a viscoelastic dissipation term, EA. Assuming the viscous term to be small compared to &A, we may equate P and E A to a first approximation. Using eqs 1-3, we obtain
u = u ~ ( ( ~ ~ G ~ /eo~-) COS [ c ~o (s t ) i ) l / ~
90
k1'
10
4
-0-0 -0
PFA
-0
(4)
Preliminary evidence of the viscoelastic braking phenomenon has been reported,1° but further results allow us now to consider the validity of the multiplicative factor of cos Bo - cos O(t) in the above equation which can be expressed in the form log[cos Bo - cos B ( t ) ] = n log U log (y/G) + constant (5) if it is assumed that the cutoff distance E remains approximately constant (this is reasonable for the case reported below where two relatively unreactive liquids are studied on the same elastomeric substrate).
5
10
I
I
1
0
15
20 t (min)
+
Experimental Results The contact angle B of small sessile drops of (a) formamide and (b) N-methylpyrrolidone, NMP, has been measured as a function of time after deposition, t, on flat, smooth, horizontal surfaces of three solids a t 20 "C (drop volume 2 pL, viscosities (a) 7 = 3.2 CPand (b) I ] = 2.0 cP, and (b) y = 41.2 surface tensions (a) y = 58.2 "am-' "om-'). In the case of the (relatively) rigid solids (Teflon PFA, Du Pont de Nemours, and silica glass, Quartz et Silice), equilibrium contact angles are attained after only ca. 15 s as shown in Figure 2. Viscous dissipation, which scales with V P / B ( t ) , where U is the spreading front speed, is the main energy sink in these cases, and although the intrinsic contact angle on the glass is small compared to that on Teflon PFA, the kinetics of spreading are very similar, tending to confirm our hypothesis of viscous effects being relatively minor. The elastomer (a two-component silicone rubber, RTV 630, General Electric Co.) possesses an equilibrium contact angle, Bo, between those of the rigid solids, and yet spreading to equilibrium is much slower (Figure 2). This anomalous behavior can be attributed to local deformation of the soft solid-the wetting ridgecaused near the wetting front by the vertical component of the liquid surface tension, y. Motion of the wetting ridge involves hysteretic straining of the solid surface, and this type of dissipation largely outweighs effects due to viscosity. The hypothesis of viscoelastic braking of the spreading of these low-viscosity liquids on silicone rubber is supported in Figure 3 in which the difference between the cosines of the equilibrium contact angle, Bo, and the instantaneous contact angle, e(t),is plotted as a function of liquid spreading speed, U,for drops of formamide and N-methylpyrrolidone (NMP) on the silicone elastomer, both scales being logarithmic. The spreading rate was calculated directly from observations of the drop contact radius, r, as a function of time, t. In order to avoid any potential artifact of slow spreading caused by a degree of swelling of the elastomer by the spreading liquid, the substrate was immersed in the relevant liquid for several days prior to the wetting experiment until equilibrium swelling was noted (by gravimetry). The solid was then carefully dried before depositing drops of the same liquid (10)Carre, A.; Shanahan, M.E.R. C. R . Acad. Sci. ZZ 1993,317,1153.
l".-oELASTOMER
60
A -I
0
I
5
10
20 t(min)
15
Figure 2. Evolution of the contact angle of formamide (a) and N-methylpyrrolidone(b) on silicon elaetomer, Teflon PFA,and silica glass. I log [cose,
- cose(t)l
0
/Po* /
- 1.0 - 1.5 -
@
Formamide
o NMP
- 2.0 -
0.54 1
Figure 3. Supporting evidence of the hypothesis of viscoelastic braking for formamide and N-methylpyrrolidone (NMP)spreading on silicon rubber. The value 0.54 is the mean gradient (n) and 0.19 the vertical shift (log[~1G2/(7&1)] with 1 formamide and 2 NMP).
in order to conduct the spreading experiment. In addition, measurements were effected in an atmosphere saturated with the vapor of the contact liquid to eliminate any evaporation effects. The mean gradient for both liquids of the graph, n = 0.54, corresponds to the (inverse) exponent of eq 4, or the prefactor of log U in eq 5. From eq 5, the vertical shift of 0.19 log unit between the parallel or y1Gd(y2G1) = lines corresponds to log[y~Gd(yzG~)l, 1.55, where 1 and 2 represent, respectively, formamide and NMP. The values of yllG1 and y2/G2 obtained from direct measurements of the surface tension and modulus (after the swelling period) were found to be, respectively, 4.20 X 10-8 and 2.59 X 10-8 m, leading to a ratio of 1.62. This good agreement between direct and indirect experimental evaluations of the vertical shift corroborates eq 5.
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Discussion and Conclusion This behavior in wetting can be considered to be analogous to dissipation phenomena already well-established in the adhesion of elastomers. The Young-DuprB equation relates the thermodynamic work of adhesion, WO, to the surface tension and equilibrium contact angle [WO= y(1 + cos eo)]. Using $$’ to represent the “instantaneous” work of adhesion corresponding to angle e@),eq 5 may be rewritten as log[(W o- m / y l = n log U + constant (6) It has long been recognized that, during the separation process of an elastomer from a substrate, the apparent energy of adhesion, W, often far exceeds the thermo~ that this excess, W - WO, dynamic value, W O ,and corresponds essentially to viscoelastic dissipation. Indeed the following relationship has been proposeds (using the present nomenclature):
wo
w - = WOC$(aTu) (7) where C$ is a dimensionless function of the crack speed, u, and temperature and UT is the Williams,Landel, and Ferry shift factor.” It was shown that C$(QTU) varies as (uTu)”, where the constant n = 0.6. By changing the sign of the left-hand member of eq 7, the process of formation, rather than separation, is described and a direct analogy between (11)Ferry, J. D.Viscoelastic Properties of Polymers, 2nd ed.;Wiley: New York, 1970; p 314.
relations 6 and 7 is then evident. The dissipativeproperties of the elastomer used in the present wetting experiments have also been determined using the rolling cylinder adhesion test1”14 and the value of n found to be 0.54, which is equal to the mean value of n obtained for liquid spreading. Despite the clear analogy, it is perhaps worth emphasizing that the prefactor of W Oon the right-hand side of eq 7 becomes effectively y for the case of liquid spreading. In conclusion,the present study showsthat the spreading of a liquid on an elastomer may be largely dependent on viscoelasticdissipation in the wetting ridge of the substrate near the triple line. Under “classic”conditions,spreading behavior is assumed to be dependent entirely on liquid properties, but these may be unalterable in a given situation. It now appears possible to control the spreading kinetics of a liquid on a solid by judiciously modifying the mechanical properties of the substrate itself, for example, by changing the degree of cross-linking in the case of a polymer. This may be of direct benefit in a variety of practical and medical applications. Apart from practical implications in spreading processes on soft solids, this interpretation allows a parallel to be drawn between the phenomena of wetting and adhesion and leads to a certain unification of the concepts involved. _
_
_
_
_
~
~
~
(12)Roberta, A. D.;Thomas, A. G. Wear 1975,33,45. (13)Zaghzi, N.; CarrB, A.; Shanahan, M.E.R.; Papirer, E.;Schultz, J. J. Polym. Sci., Polym. Phys. 1987,25, 2393. (14)Barquins, M.J. Adhes. 1988, 26, 1.
