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Viscoelastic Braking of a Running Drop Alain Carre´*,† and Martin E. R. Shanahan*,‡ Corning, Fontainebleau Research Center, 7 bis, Avenue de Valvins, 77210 Avon, France, and CNRS, ENSMP, Centre des Mate´ riaux P. M. Fourt, B.P. 87, 91003 Evry Cedex, France Received November 16, 2000. In Final Form: February 7, 2001 Wetting or dewetting on soft elastomeric surfaces is largely controlled by the dissipative properties (viscoelasticity) of the substrate. The component of liquid surface tension acting perpendicularly to the solid causes a deformation, or “wetting ridge”, which must be displaced with the triple line. Previously, triple line motion due to capillary imbalance has been studied, whereas in the present work, we consider liquid motion due to gravity. Liquid drops running down an elastomeric plane move at a speed largely determined by the properties of the polymer. Results obtained agree well with the theory and demonstrate how motion depends on the state of the polymer, the angle of inclination of the plane, and drop volume.
Introduction Wetting equilibrium at a triple line solid (S)/liquid (L)/ fluid (vapor, V) involves a horizontal force balance, as described by the ubiquitous Young equation,1 but the vertical component of liquid surface tension, γ (perpendicular to the solid surface), γ sin θ, where θ is the contact angle, has been largely ignored until fairly recently. It has been shown that γ sin θ induces strain in the solid, leading to a “wetting ridge” of height on the order of γ/G, where G is the shear elastic modulus of the solid,2,3 and that, if the solid is viscoelastic, or lossy, which is likely to be the case with a soft solid for which γ/G is significant, this wetting ridge causes energetic dissipation for a moving triple line.4-6 Thus, wetting or spreading are impeded by viscoelastic losses, which may far outweigh any effects due to liquid viscosity. Till recently, the effects of “viscoelastic braking” have been studied essentially under conditions in which triple line motion is the consequence of a capillary imbalance, θ(t) > θ0 for wetting5 or θ(t) e θ0 for dewetting,6 where θ(t) is contact angle at time t, and θ0 is the equilibrium value. However, there is no reason for the effects of viscoelastic braking to be restricted to uniquely capillarity-controlled motion: external forces may also lead to wetting or dewetting moderated by the lossy properties of the substrate. Indeed, suitable control of the mechanical properties of the substrate may lead to a method of “engineering” wetting or dewetting rates. In this paper, we present results corresponding to gravity-controlled drop motion in which the running rate is moderated by viscoelastic braking. Small drops of liquid were deposited on an inclined elastomeric plane, and their rate of descent was monitored. Theory The key equation to viscoelastic braking has been discussed on several previous occasions (e.g., refs 4-6); † ‡
Corning. CNRS.
(1) Young, T. Philos. Trans. R. Soc. 1805, A 95, 65. (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. (4) Shanahan, M. E. R. J. Phys. D.: Appl. Phys. 1988, 21, 981. (5) Shanahan, M. E. R.; Carre´, A. Langmuir 1995, 11, 1396. (6) Carre´, A.; Shanahan, M. E. R. Langmuir 1995, 11, 3572.
Figure 1. Schematic representation of wetting ridge with liquid front moving toward the left.
suffice it here to quote the basic relation in conjunction with the schematic representation of Figure 1
E˙ ∆ ≈
( )
γ2U U γ2U∆ ≈ 2πG 2πG U0
n
(1)
In this equation, E˙ corresponds to the rate at which work is done (per unit length of triple line) by “lifting” and “stretching” the wetting ridge from the initially flat, deformable solid surface, and ∆ is the fraction of this work dissipated, and therefore not recovered, after passage of the triple line. This is thus the energy expenditure due to the solid deformation (per second and per meter) during triple line motion at speed U. The actual form of ∆ is unknown. An initial suggestion4 of ∆ being relatively constant and equal to ca. 0.1, representative of many elastomers well above their glass transition, was found better supplanted by a power law. The empirical form (∆ ≈ (U/U0)n) used in eq 1, where U0 (.U) and n are constants, the latter often being close to 0.5 and, stemming from work on elastomeric adhesion,7 has been found to give a satisfactory description. The term refers to a microscopic cutoff distance to the elastic strain field.3,4 Let us now consider a relatively small drop of liquid, mass m, deposited on an inclined, elastomeric plane, as depicted in Figure 2, both from the side in panel a and from the top in panel b. The term relatively is operative: we choose a regime in which the drop is sufficiently small so as not to depart significantly from the form of a spherical cap. At the same time, its mass must not be negligible in the overall force balance shown below, otherwise intrinsic wetting hysteresis will prevent drop motion. These two (7) Maugis, D.; Barquins, M. J. Phys. D.: Appl. Phys. 1978, 11, 1989.
10.1021/la001600e CCC: $20.00 © 2001 American Chemical Society Published on Web 04/11/2001
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value of I is not very sensitive to variation of n: for n ) 0, I ) 1 and for n ) 1, I ) π/4 ≈ 0.8. In addition to the above viscoelastic dissipation occurring during drop motion at overall speed U, viscous dissipation within the liquid will also be present. This viscous dissipation for translational drop motion, without spreading, has been evaluated previously11 and, assuming small contact angle and the lubrication approximation of pure, shear flow within the liquid, parallel to the solid surface, it is given approximately by
TS˙ ≈
Figure 2. Running drop on an inclined elastomeric plane.
conditions are inevitably antagonistic, but a compromise exists for drops of characteristic linear dimension close to the capillary length, K-1 ) γ/(Fg)1/2, where F is the drop density (reduced by that of the surrounding fluid, here supposed to be vapor, V) and g is gravitational acceleration. Under conditions where the plane is rigid and smooth, and when contact angle hysteresis is negligible, the drop runs down the plane under its own weight, mg. When hysteresis cannot be neglected, for small angles of inclination, R, the drop remains stationary, since the leading edge (lower, in Figure 2) presents a (slightly) higher average value of contact angle than the trailing edge. The slight imbalance of Young’s equation in front and behind compensates the gravitational force tending to cause running (rolling or sliding).8,9 Above a certain critical value of R, the drop moves. In the following, we shall consider the steady-state motion, assuming R is above its critical value. Now, when the inclined plane is soft and lossy (elastomeric), the presence of a wetting ridge round the circumference of the triple line must be taken into account when describing drop descent. We shall consider the energy balance of motion. Assuming the drop to progress at speed U and taking Φ to be the angle between the direction of descent and a given contact radius, r0, in the plane of the solid surface (Figure 2), it can be seen that the local speed of the wetting ridge is U cos Φ. Bearing in mind eq 1 and the fact that the total triple line circumference is 2πr0, we may calculate the energetic dissipation rate, F˙ T, due to motion of the entire triple line
F˙ T )
2
γ (U cos Φ)
∫02π
n+1
2πGU0n
r0
2
dΦ )
2r0γ U
(n+1)
πGU0n
I
(2)
where
I)
∫0
π/2
6πηU2r0L θ
(4)
where η is the liquid viscosity and L ) ln(r0/i) where i is a cutoff distance close to the triple line.12 However, this additional energy sink can probably be reasonably neglected in the following development. In an earlier paper5 concerning dewetting behavior, it was shown that the ratio viscoelastic dissipation (equivalent to F˙ T above)/viscous dissipation (equivalent to TS˙ ) was equal to (p - 1), where Uη ) pU (Uη being the triple line speed in the absence of viscoelastic dissipation). In the present case, Uη is difficult to estimate (the drop motion is far too rapid for measurement), but for dewetting on two elastomers of the same family as those described below, we obtained (p - 1) ≈ 13 and 80 (corresponding respectively to values of Young’s modulus, Y, of 2.1 and 0.65 MPa).13 Assuming similar values for p in the present case, the viscoelastic term will clearly outweigh the viscous term, and thus, the latter may reasonably be neglected to good approximation. (N. B. dissipative properties increase as Y decreases.) Taking eq 2 to give the energy sink of motion, we must relate this to the energy source, which is simply the loss of potential energy as the drop runs down the inclined plane. The “power” supplied, P, is given by
P ) Umg sin R
(5)
Setting F˙ T equal to P, we obtain
[
U ≈ U0
]
πGVFg sin R 2r0γ2I
1/n
(6)
where V is the drop volume and F the liquid density. Only of academic interest here, since running on a rigid surface could not be successfully followed experimentally, we may compare this to the equation obtained from relations 4 and 5:
U≈
θVFg sin R 6πηr0L
(7)
It may be noted that eq 6 leads us to expect U ∼ (sin R)1/n, whereas eq 7 predicts U ∼ sin R, other values being identical. We shall return to this point later. Experimental Section
cos(n+1) Φ dΦ
(3)
Now the value of n is empirical, although its value is often close to 0.5 (this value has been suggested in a simplified, theoretical manner10). Notwithstanding, the (8) Wolfram, E.; Faust, R. In Wetting, Spreading and Adhesion; Padday, J. F., Ed.; Academic Press: London, 1978; p. 213. (9) Carre´, A.; Shanahan, M. E. R. J. Adhes. 1995, 49, 177.
The liquid used was o-tricresyl phosphate (99% o-TCP, Acros Organics), which has low volatility. Droplets of a few microliters that were exposed to ambient atmosphere did not show weight loss for several weeks. Surface tension, density, and viscosity of (10) Greenwood, J. A.; Johnson, K. L. Philos Magn. 1981, 43, 697. (11) Shanahan, M. E. R. J. Phys. D.: Appl. Phys. 1990, 23, 321. (12) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (13) Carre´, A., Shanahan, M. E. R. J. Colloid Interface Sci. 1997, 191, 141.
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Carre´ and Shanahan
the liquid are respectively γ ) 40.5 mN m-1, F ) 1.20 × 103 kg m-3, and η ) 0.19 Pa s, at a temperature of 20 °C. The polymer used was a two-component transparent silicone elastomer (RTV 615, General Electric Co.). By varying the ratio of the two components, resin and cross-linking agent, constituting the final rubber, it was possible to control the Young’s modulus, Y. The cross-linking density was modified by varying the curing agent content from 2 to 20 parts (by weight) for 100 parts of resin. The values of elastic moduli, Y, were assessed for the elastomers using hardness measurements (in all cases but for the softest rubber, where hardness could not be readily measured; estimation is this case was made using a simple tensile test). The selected values for Y were 2.1, 0.65, and 0.1 MPa (the relationship between Young’s modulus, Y, and shear modulus, G, for an elastomer is simply Y ) 3G). The equilibrium contact angle, θ0, of TCP on the silicone rubber was found to be on the order of 45 ( 3°, with a small apparent dependence on Y, which may possibly be linked to a slight degree of swelling of the elastomer by the liquid, being more accentuated in the case of lower Y. The variability being nevertheless small, it will be neglected in the following. The running drop experiments were made by using a tilting plane. The angles of inclination were chosen between 30° and 90°. Drops of TCP having volumes of 3, 5, and 8 µL were formed with a microsyringe (Gastight # 1705). The running speed was calculated by measuring the time necessary for a drop to cover a distance of 4 cm on the tilted rubber track. For each set of experimental conditions (Young’s modulus, Y, angle of inclination, R, and drop volume, V), the running speed was determined from five measurements.
Figure 3. Running speed, U, of drops of TCP of different volume vs the product V2/3 sin R (V ) volume, R ) plane angle). Both axes are logarithmic. Young’s modulus of elastomer Y ) 2.1 MPa.
Results and Discussion Equation 6 gives us a basic description of the running speed of drops on an inclined elastomeric plane if viscous dissipation may be neglected. Unfortunately, some parameters are not readily, directly accessible, viz. U0, n, , and, of less importance due to its lack of sensitivity on the value of n, I. However, we may write drop volume as V ) r03f(θ), where f(θ) ) π(2 + cos θ)(1 - cos θ)/[3 sin θ(1 + cos θ)], assuming the drop cap shape to be spherical. Then, on taking logarithms, eq 6 becomes
log U ≈ log U0 +
[
Figure 4. As for Figure 3, but Y ) 0.65 MPa.
]
πGFgf1/3(θ) 1 log + n 2γ2I 1 log[V2/3 sin R] (8) n
The interest of this form of eq 6 is that, for a given pair elastomer and liquid, the first two terms on the righthand side are essentially constant, being dependent on the intrinsic properties of the pair of materials under consideration. Thus, we may treat volume, V, and angle of inclination, R, as the chief variables. Since only one liquid has been studied here, we may group the two terms in question and treat them as a constant, KY, for each of the elastomers of differing degree of cross-linking
log U ≈ KY +
1 log[V2/3 sin R] n
(9)
where Y refers to the value of Young’s modulus in MPa (2.1, 0.65, and 0.1). Figures 3-5 present the results of log U vs log[V2/3 sin R] respectively for Y ) 2.1, 0.65, and 0.1 MPa. Various values of drop volume and angle of inclination are included (All experimental data are gathered in Tables 1-8). It can be seen that a highly satisfactory linear relationship is obtained in each case with gradients, respectively, of 3.4, 4.0, and 3.7, as obtained by regression analysis. These figures correspond to values of n of 0.29, 0.25, and 0.27.
Figure 5. As for Figure 3, but Y ) 0.1 MPa. Table 1. Running Speeds of TCP for Various Angles of Inclination, r. Y ) 2.1 MPa, V ) 3 µL R (deg) log U (m/s)
45 -4.432
53 -4.418
60 -4.055
90 -3.832
Table 2. Running Speeds of TCP for Various Angles of Inclination, r, at Y ) 2.1 MPa, V ) 5 µL R (deg) log U (m/s)
30 -4.376
45 -3.699
60 -3.448
90 -3.232
The values seem perhaps a little low, but in these experiments, the elastomers were not preswollen by the liquid beforehand. It has previously been observed that the lack of preswelling can indeed lead to values of n lower than may be expected14 (although, a priori, no particular values of n should be predicted, as discussed above). (14) Shanahan, M. E. R., Carre´, A. J. Adhes. 1996, 57, 179.
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Table 3. Running Speeds of TCP for Various Angles of Inclination, r, at Y ) 2.1 MPa, V ) 8 µL R (deg) log U (m/s)
30 -3.792
36 -3.637
45 -3.354
60 -3.114
90 -2.968
Table 4. Running Speeds of TCP for Various Angles of Inclination, r, at Y ) 0.65 MPa, V ) 3 µL R (deg) log U (m/s)
69 -4.378
76 -4.313
90 -4.133
Table 5. Running Speeds of TCP for Various Angles of Inclination, r, Y ) 0.65 MPa, V ) 5 µL R (deg) log U (m/s)
45 -4.272
53 -3.908
60 -3.797
90 -3.520
Table 6. Running Speeds of TCP for Various Angles of Inclination, r, at Y ) 0.65 MPa, V ) 8 µL R (deg) 30 33 36 45 60 90 log U (m/s) -4.311 -4.184 -3.846 -3.624 -3.363 -3.121 Table 7. Running Speed of TCP as a Function of the Angle of Inclination, r, at Y ) 0.1 MPa, V ) 5 µL R (deg) log U (m/s)
39 -4.250
on U0 and . Nevertheless, it can be seen that for a given value of V2/3 sin R, the running speed increases with Young’s modulus and therefore with decreasing dissipative properties as expected.
90 4.284
Conclusion
Table 8. Running Speeds of TCP for Various Angles of Inclination, r, Y ) 0.1 MPa, V ) 8 µL R (deg) log U (m/s)
Figure 6. Superposition of Figures 3-5 demonstrating shift to higher speeds with increasing value of Y.
45 -4.079
60 -3.804
90 -3.554
In the theoretical section, we pointed out that a scaling relation of the form U ∼ sin R, leading to a gradient of unity on a double logarithmic plot, would be expected if viscous dissipation were to be the primary energy sink. Considering individual sets of results (i.e. for Y constant) in Figures 3-5, it can be seen that the gradients are all far from unity, allowing us to conclude that viscosity is (relatively) unimportant. Figure 6 combines the results of Figures 3-5, and it can be seen that the linear relationships are acceptably parallel, as expected from the above. However, the shift between sets of results for different values of Y cannot really be analyzed quantitatively, given the dependence
Following previous studies on the effects of viscoelastic braking in capillary-controlled wetting or dewetting of elastomeric surfaces, we have considered wetting and dewetting behavior (simultaneously) due to external forces (gravity) by studying drops of tricresyl phosphate running down inclined elastomeric planes. Having developed the basic equation giving running speed (eq 6), we have effected experiments on three polymers with different degrees of cross-linking and used drop volume and plane inclination angle as principal parameters. Results give good agreement with predicted behavior and demonstrate clearly how running speed depends essentially on the mechanical properties of the elastomer: the less the elastomer is cross-linked, the higher its dissipative character and the lower its running speed. The possibility of “engineering” wetting and draining surfaces to a specific need is clear. LA001600E
