Langmuir 1995,11, 24-26
24
Direct Evidence for Viscosity-IndependentSpreading on a Soft Solid Main Carre* Centre de Recherche Corning Europe, 7 bis Avenue de Valvins, 77210 Avon, France
Martin E. R. Shanahan” Centre National de la Recherche Scientifique, Ecole Nationale Supirieure des Mines de Paris, Centre des Matiriaux P.M. Fourt, B.P. 87, 91003 Evry Cedex, France Received August 22, 1994. In Final Form: October 17, 1994@ The kinetics of a spreading drop is usually 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 solidAiquid/vaportriple line can also be a dissipative sink as the wetting front moves. In this paper we provide evidence to show that when the substrate is soft, this local deformation near the triple line may lead to an energy dissipation that may outweigh effects due to viscosity over a large range of speeds and therefore spreading kinetics becomes viscosity independent.
Introduction Wetting phenomena are of paramount importance in a number of natural and industrial processes such as eye irrigation, the application of insecticides, paints and inks, and tertiary oil recovery. Wetting also plays a fundamental role in the production of adhesive bonds and composite materials, in which sufficient contact and coverage of a solid must be obtained by, for example, a liquid polymer or solder, before solidification occurs. Not only the degree ofwetting but also the speed ofthe wetting process is of great importance. Classically, wetting, or spreading, is considered to be dependent essentially on the capillary properties of the liquid and the solid surface being wetted and the viscosity of the liquid (and perhaps also those related to the environment). Under classic conditions of wetting, in which a small axisymmetric sessile drop is deposited on an ideal solid surface (flat, smooth, rigid, homogeneous, and isotropic) and allowed to spread to equilibrium, the kinetics is controlled essentially by a dynamic energy balance between the rate ofrestitution of capillary potential energy and viscous dissipation occurring due to shear motion within the liquid.’ Nevertheless, ideal substrates are rarely encountered in practice and, although rigidity of the solid may often be assumed to a good degree of approximation, when liquids are put into contact with soft solids, such as elastomers or gels, local deformation will result near the solid (S)/liquid (Lllenvironment (vapor, v> triple line due mainly (but not entirely) to the component of liquid surface tension, y , perpendicular to the (undisturbed) substrate, y sin B.2,3 B represents the conventional contact angle measured in the liquid between the tangents to the LV and (undisturbed) SL interfaces at the triple line. The resulting “wettingridge”, or elastic displacement of the solid has a height, h, of order of magnitude y sin BIG, where G is the shear modulus of the solid. When a sessile drop is deposited on such a soft solid, as spreading toward equilibrium takes place, the aforementioned wetting ridge must accompany the triple line and, as a consequence, the solid surface in the spreading path undergoes a strain cycle. As the triple line ap~~~
Abstract published in Advance ACS Abstracts, November 15, 1994. (1)de Gennes, P.G. Rev. Mod. Phys. l985,57,827. (2)Lester, G.R.J . Colloid Sci. 1961,16, 315. (3) Shanahan, M.E. R.; de Gennes, P. G. In Adhesion 11; Allen, K. W., Ed.; Elsevier Applied Science: London, 1987;p 71. @
proaches, the local solid is raised, and once the wetting front has passed, it descends, but since such soft solids are viscoelastic, or at least dissipative (perhaps plastically in the case of gels), the imposed strain cycle leads to energy 1 0 ~ sIn . ~a recent note,s it was shown how, under certain experimental conditions, the dissipative losses due to viscoelastic effects near the triple line outweigh those related to viscous shear within the liquid and, as a result, an analogy could be drawn between liquid spreading and adhesion phenomena, in particular, rate dependent peeling of elastomers.6 In ref 5, the liquid viscosity was inferred to be unimportant. The purpose of this contribution is to demonstrate in a direct manner that over a large range of viscosities and spreading speeds, this inference is indeed confirmed. When a small axisymmetric sessile drop (i.e., when gravity may be neglected) is placed on a solid surface, its contact angle, 8, will, in general, be greater than the equilibrium value, Bo, and a wetting force will result leading to work $’being done per unit time and per unit length of triple line:
F=
COS e, - COS e(t)i
where U is the triple line speed (U = drldt, where r is contact radius) and 8 is a function of time, t . This work is consumed partially by viscous dissipation, TS, taking the form, for small contact ang1es:l
TS 3ylU%3(t)
(2)
where 7 is liquid viscosity and 1 is a logarithmic factor (approximately constant) involving cutoff distances to the dissipative zone of the drop. In addition, viscoelastic dissipation, BA, occurs a t the wetting ridge?
(3) where G represents, as before, the shear modulus of the solid, E is a cutoff distance near the triple line, below which the behavior may be considered to be no longer linearly elastic (a few nanometers3), and UOand n are constants related to the rate-dependent viscoelastic dissipation of (4) Shanahan, M. E. R. J . Phys. D: Appl. Phys. 1988,21, 981. ( 5 ) Shanahan, M.E. R.; Carr6, A. Langmuir 1994,10,1647.
(6)Maugis, D.; Barquins, M. J . Phys. D: Appl. Phys. 1978,11,1989.
0743-746319512411-0024$09.0010 0 1995 American Chemical Society
Letters 90
Langmuir, Vol. 11, No.1, 1995 25
4 v
l 1
- 1250 CP
* - 150 CP -2cP
03
Q'-----
.- *-
0.1
1 10"
0
5
10 tfmin)
15
20
the solid.6 Considering the energy balance between eq 1 and eqs 2 and 3 together, we obtain
3qlU
'+)"('
ea - COS e(t) x yO(t) 2xGc U,,
150 CP 2 CP
, 104
10"
10''
10-1
U(mm.s-')
Figure 1. Evolution with time, t , of contact angle, 8, of NMP (qa = 2 cP, ya = 41.2 d a m - ' ) , NMP containing 17.5w t % of polyimide (PI) (qb = 150 cP, yb = 41.5 "am-'), and NMP containing 28 wt % of PI (ae= 1250 cP, ye= 42.8am-'). The dotted lines are extrapolated to equilibnum values of contact angle, 0 0 , for the three liquids.
COS
1250 CP
(4)
which may be taken as the basic equation governing spreading. Typically, n > 8 0 in this range, the contribution of the latter may be neglected. Using the relation cos 6 x 1 - e2/2,a much simplified version of eq 4 may be recast as 6v1U x ye3(t). Taking the transition (ii) from viscous to viscoelastically controlled spreading to occur at similar values of U,we see that, to a first approximation, q = 03(t)and thus the ratio of differences ofvalues of e@),even in the viscoelasticregime (iii),should
Figure 2. Variation of [cos 80 - cos 19(t)las a function of the spreading speed U,both on logarithmic scales, corresponding essentially to regime (iii),or viscoelastically controlled spreading, for the three systems described in Figure 1. be roughly in agreement with the ratio of differences of values ofqy3. (vcy3- ?7bv3)/(vb'" - q,1'3) x 1.35.Using the same subscripts for contact angles, it is found that (e&) - &(t))/(eb(t)- ea@)>is approximately equal to 1 over the accessible time range. Thus, despite viscosities ranging over nearly three decades, the approximate argument involving the third power of v satisfactorily accounts for relative contact angle evolution of the three liquids. Equation 4 represents an energy balance in which, in regime (iii) which is of the most interest in the present context, capillary free energy is transformed into viscoelastic dissipation. This is rate, rather than state, dependent. After sufficient spreading, when regimes (i) and (ii)have been left, eq 4 may be written for regime (iii) as
Since the second member on the right-hand side is essentially constant for the solid and liquids considered, we should expect a plot of log [cos 60 - cos 8(t)l vs log U to give straight lines of gradient n which are directly superposable, irrespective of liquid viscosity. Figure 2 gives the results corresponding to Figure 1 plotted in this manner. As can be seen, the relationship between log [cos 60 - cos 8(t)l and log U is satisfactorily linear over a range covering approximately three decades of speed (80 was measured after 2 h when the systems had apparently reached equilibrium). Not only is the linearity convincing, but in addition the results obtained for the three viscosities of 2, 150,and 1250 CPlie on the same line, within the limits of experimental scatter, thus showing that the spreading behavior in regime (iii) is effectivelyviscosity independent and suggesting strongly that the motion is essentially governed by the viscoelastic properties of the systems which are virtually identical. The values of the gradient n are respectively 0.24,0.24, and 0.22 for the viscosities 2,150,and 1250 CPgiving an average value of 0.23f 0.01. These values would seem to be rather low, but it should be pointed out that the elastomer was not preswollen for the experimentsreported here, although the experiments were effected in atmospheres saturated with the vapor of the relevant liquid to eliminate potential evaporation effects. Experiments of a similar nature, but less extensive, using the preswollen polymer suggest a value ofn in the range of 0.5-0.6,more in keeping with values typically found in adhesion experiment^.^,^ Clearly, swelling of the substrate by the
Letters
26 Langmuir, Vol. 11, No. 1, 1995 liquid does influence the behavior and modifies the value of n, but the mechanism is not at present understood.
Discussion and Conclusion Our results convincingly show viscosity-independent wetting kinetics over a wide range of viscosities and spreading speeds. Despite the acceptable linearity of the relationship exhibited by Figure 2, there would seem to be a slight degree of concavity away from the abscissa at the highest values of spreading speed, U. Considering eq 4, it may be seen that on the logarithmic scales used in Figure 2, in the viscosity-controlled regime (i), the gradient should be approximately unity (the dependence of the right-hand side member on 8 will in fact reduce this slightly, but not significantlly in the ranges considered). It may well be that this slight concavity is related to the fact that, for higher spreading speeds, the results reported are entering into the transition regime (ii). However, the effect is small and although speeds here correspond to spreading times of the order of 10 s (smaller times were not readily
accessible to experimental measurements), we can reasonablytake it that the results presented constitute regime (iii)essentially. At this stage, without further data from regime (i), it would be difficult to define the range of spreading speeds corresponding to the transition regime (ii). In conclusion, the results presented here corroborate the predictions of ref 4, although the earlier paper failed to take into account the rate dependence ofthe viscoelastic dissipation of the elastomeric substrate occurring as a result of the moving wetting ridge. It would appear that three regimes exist during spreading on a soft substrate: (i) that related essentially to viscous dissipation within the liquid at relatively high speeds and occurring at the start of spreading, (ii) a transition regime, and finally, (iii)that governed mainly by viscoelastic properties of the substrate, independently of liquid viscosity, and covering most of the (normally) experimentally accessible measurement time scale (typicallyaRer 10s of contact between the liquid and the substrate). LA940660F
