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Dynamics of the Rise around a Fiber: Experimental Evidence of the Existence of Several Time Scales Maria-Jesus Vega, David Seveno,* Ghislain Lemaur, Maria-Helena Ada˜o, and Joe¨l De Coninck Centre for Research in Molecular Modelling, University de Mons-Hainaut, Materia Nova, Av. Copernic, 1, B-7000 Mons, Belgium Received May 20, 2005. In Final Form: July 28, 2005 We study experimentally the dynamics of the spontaneous rise of a meniscus made of polydimethylsiloxane of various viscosities on the outside of a micrometric fiber made of monofilament in polyethyleneterephthalate. Optical methods are used to measure simultaneously the height of the liquid interface and the associated contact angle versus time. Versus the liquid viscosity, we observe asymptotically, for the dynamic contact angle, either a t-1/2 behavior in agreement with the hydrodynamic description or a t-1 behavior predicted by the molecular-kinetic theory, in its linear form, thus confirming the existence of at least two major time scales in the rise phenomenon.
1. Introduction The dynamics of wetting of a liquid on a solid surface has received considerable attention since the pioneering work of Inverarity1 and Blake et al.2 in the late 1960s. Different mechanisms were presented to explain the dependence of the dynamic contact angle on the velocity of the triple line. The energy dissipated by the system is assumed to be consumed either in the inner layers of the liquid (viscosity) according to the hydrodynamic approach3-8 or in the vicinity of the triple line according to the molecular-kinetic theory2. In 1986, for a droplet partially spreading on top of a flat surface, de Gennes9 showed that these different channels of dissipation should coexist during the spreading process. This fundamental statement leads to the elaboration of various combined models,10,11 which among them the model of de Ruijter et al.12 offers a clear understanding of the phenomenon. One prediction of this model is that the early stages of the drop spreading are dominated by the friction at the triple line whereas the viscous dissipation is enhanced at larger times.13,14 Of course, wetting science is not limited to the study of a droplet spreading on top of a flat surface. Many other systems, most of the time more complex, are of both fundamental and industrial importance. The solid can be a pore, a fiber, or a network of these entities, and the liquid can be of finite or effectively infinite size. Hereafter, the fiber geometry is examined. To illustrate the differ(1) Inverarity, G. Br. Polym. J. 1969, 1, 245. (2) Blake T. D.; Haynes J. M. J. Colloid Interface Sci. 1969, 30, 421. (3) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 1, 39. (4) Dussan, E. B. Annu. Rev. Fluid Mech. 1979, 11, 371. (5) Voinov, O. V. Fluid Dyn. 1976, 11, 714. (6) Tanner, L. H. J. Phys. D. 1979, 2, 1473. (7) Hoffmann, R. J. Colloid Interface Sci. 1975, 50, 228. (8) Cox, R. G. J. Fluid. Mech. 1986, 168, 169. (9) de Gennes, P. G. Colloid Polym. Sci. 1986, 264, 463. (10) Petrov, P. G.; Petrov, J. G. Langmuir 1992, 8, 1762. (11) Blake, T. D. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993. (12) de Ruijter, M. J.; De Coninck, J.; Oshanin, G. Langmuir 1999, 15, 2209. (13) de Ruijter, M. J.; Blake T. D.; De Coninck, J. Langmuir 1999, 15, 7836. (14) de Ruijter, M. J.; Charlot, M.; Voue´, M.; De Coninck, J. Langmuir 2000, 16, 2363.
ences between a flat surface and a fiber, it is first straightforward to consider the radii of curvature. If gravity is negligible, then when a droplet wets a flat surface its radii of curvatures are equal and of the same sign. These become greater and greater and reach an infinite value in the extreme case of the formation of a flat film. In the fiber geometry, the radii of curvature show different trends. They are of opposite sign, and one of them is determined by the fiber radius. This simple observation outlines the fundamental differences between the two geometries and therefore suggests that the study of the liquid-fiber system should provide distinctive results. This statement is valid both when a droplet of liquid spreads on a horizontal fiber15 (barrel-shaped configuration) and when a fiber is wetted by a bath of liquid.1 In the case of forced wetting, when no film is entrained, Inverarity1 experimentally studied the influence of fiber velocity and liquid viscosity on the advancing contact angle. These data were then reconsidered by Blake11 and compared to an extension of the molecular-kinetic theory that accounts for viscosity effects. Schneemilch et al.16 preferred to adjust their results to different approaches (pure hydrodynamics, pure molecular-kinetic theory, and the combined model of Petrov et al.10) and considered the meaning of the values of the microscopic fitted parameters. Molecular dynamics simulations17 also showed that several channels of dissipation can be taken into account. When a film is entrained, its thickness depends principally on the fiber velocity. Extensive studies have been made of this coating problem (see, for example, the work of Que´re´ et al.18-22) with fibers of micrometric or millimetric order. For spontaneous spreading, when an infinite reservoir of liquid spreads around a monofilament, experimental (15) McHale, G.; Newton, M. I. Colloids Surf., A 2002, 206, 79. (16) Schneemilch, M.; Hayes, R. A.; Petrov, J. G.; Ralston, J. Langmuir 1998, 14, 7047. (17) Seveno, D.; Ogonowski, G.; De Coninck, J. Langmuir 2004, 20, 8385. (18) Que´re´, D.; Di Meglio, J. M.; Brochard-Wyart, F. Europhys. Lett. 1989, 10, 335. (19) Que´re´, D. Europhys. Lett. 1990, 13, 721. (20) Que´re´, D.; Di Meglio, J. M.; Brochard-Wyart, F. Science 1990, 249, 1256. (21) Que´re´, D.; De Ryck, A.; Ou Ramdane, O. Europhys. Lett. 1997, 37, 305. (22) Que´re´, D. Annu. Rev. Fluid. Mech. 2002, 31, 347.
10.1021/la051341z CCC: $30.25 © 2005 American Chemical Society Published on Web 09/01/2005
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Table 1. Kinematic Viscosity, Surface Tension, Capillary Length, and Bond Number of the Silicon Oils
silicon oils
kinematic viscosity (mm2‚s-1)
surface tension (mN‚m-1)
a (mm)
B0
PDMS5 PDMS10 PDMS20 PDMS50 PDMS500
5 10 20 50 500
19.7 20.1 20.6 20.8 21.5
1.479 1.480 1.487 1.486 1.502
0.270 0.270 0.269 0.269 0.266
results23,24 showed that, for at least for some systems, the hydrodynamic approach is suitable. Seveno et al.25 used large-scale molecular dynamics techniques to obtain the dynamic contact angle behavior, close to equilibrium, in agreement with the linear form of the molecular-kinetic model. It is now the aim of this article to reconsider experimentally the spontaneous dynamics of wetting around a fiber to test the possible existence of the different dissipation mechanisms that may appear in the process. The paper is organized as follows. Section 2 is devoted to the experimental setup and materials. The results are presented in section 3, and the interpretations are given in section 4. We end the article with some concluding remarks. 2. Experimental Setup and Materials To measure contact angles and meniscus heights on the outside of a vertical fiber, one can use the classical Wilhelmy method by using a precise balance (KSV 3000 LB for instance). This method is usually chosen to measure advancing and receding contact angles26-30 but is also suitable for following the relaxation of a dynamic contact angle.23,31 Nevertheless, the response of the experiment may prevent study at short times (e.g., within a second). We have therefore developed an optical method based on edge detection with a fit of the associated profile using the Laplace equation. The spontaneous wetting of a fiber is here studied using a PET fiber (poly(ethylene terephthalate) Monofilament, Goodfellow, 0.8 mm diameter) which is put in contact with silicon oils (poly(dimethylsiloxane), PDMS) of different viscosities (Table 1). The motion of the sample is stopped when it touches the liquid. This was carefully checked by analyzing the successive images just before and after the meniscus rise starts. The liquids were provided by Aldrich Chemical Co. (PDMS10) and ABCR GmbH Germany (PDMS5, PDMS20, PDMS50, and PDMS500). These liquids were chosen because they minimize possible adsorption and evaporation. Table 1 lists the capillary length a of the liquids (a ) xγ/Fg, with γ being the liquid-gas surface tension, F being the liquid density, and g being the acceleration due to gravity) and the Bond number B0 (B0 ) r0/a with r0 being the fiber radius). The Bond number permits one to distinguish the planar from the fiber case. If B0 is large compared to unity, then the fiber reduces to a planar surface. Here Bo is almost constant and equal to 0.27. To obtain a reproducible clean surface for the poly(ethylene terephthalate) fiber (PET fiber), we follow a rigorous cleaning procedure. It consists, first, of pulling off all possible organic contamination by rubbing the fiber with a soap (RBS-35 from Chemical Products) and then sonicating the fiber for 30 min in
Figure 1. Experimental setup. a ultrasound bath of very pure water (Milli-Q water) and finally in another ultrasound bath of very pure 2-propanol (J. T. Baker with a purity of 99.5%) for 1 h. The cleaned fiber is then dried under a flux of nitrogen. The fibers are always cleaned just prior to the experiments. In the experimental setup (Figure 1), the fiber is attached to a support that allows the fiber to move up and down into the liquid reservoir. It is important to note that the size of the reservoir containing the liquid is large enough to avoid any possible effect of the border of the reservoir into the liquid meniscus. In our system, the meniscus profile is captured using a highspeed camera (C-MOS camera, Vossku¨hler HCC-1000) that can reach up to 1840 i/s (images per second) with a resolution of 1024 × 256 pixels and up to 460 i/s with a definition of 1024 × 1024 pixels. The fiber experiment is backlighted with an LED flash, and the camera is connected to a computer that provides the acquisition of the images. Edge Detection. To detect the edges of the meniscus, the first step is to reduce the noise that is always present in images. We thus use a 5 × 5 Gaussian blurring filter32 to reduce the amount of noise. Then, the gradient along two directions can be easily computed. The norm and the direction are easily extracted. The next step usually consists of a threshold to select only significant edges. We follow a different protocol by selecting all pixels that have a maximum gradient in the direction of the gradient (nonmaxima suppression). This is done irrespective of the absolute magnitude of the edges. The image we get is thus full of “random” small edges, but the contour we are interested in is fully connected. Thus, starting from approximations of the position of the fiber, we first find the closest edge and then follow the contour to select only interesting pixels. The exact position of the fiber is paramount for a good estimation of the contact angles, especially for small angles. Thus, a more costly subpixel localization is performed on the basis of the pixel that has already been found. For all of these pixels, we interpolate the denoised image in the direction of the gradient. We then compute the derivative of this 1D signal. A parabola is finally fitted, and the summit denotes the inflection point of the image. This is considered to be the true location of the edge, with much higher accuracy. Laplace Equation. The shape of the free interface in the presence of gravity can be described by the well-known Laplace equation, which relates the surface tension of the liquid to the pressure difference across the liquid/gas interface33
{
dθ y cos θ ) + ds a2 x + r0 dx ) sin θ ds dy ) -cos θ ds
(1)
with the boundary conditions (23) Que´re´, D.; Di Meglio, J. M. Adv. Colloid Interface Sci. 1994, 48, 141. (24) Clanet, C.; Que´re´, D. J. Fluid Mech. 2002, 460, 131. (25) Seveno, D.; De Coninck, J. Langmuir 2004, 20, 737. (26) Lee, Y. L. Langmuir 1999, 15, 1796. (27) Barraza, H. J.; Hwa, M. J.; Blackley, K.; O’Rear, E. A.; Grady, B. P. Langmuir 2001, 17, 5288. (28) Barber, A. H.; Cohen, S. R.; Wagner, H. D. Phys. Rev. Lett. 2004, 92, 186103. (29) Barsberg, S.; Thygesen, L. G. J. Colloid Interface Sci. 2001, 234, 59. (30) Rey, A. D. Langmuir 2000, 16, 845. (31) Sauer, B. B.; Kampert, W. G. J. Colloid Interface Sci. 1998, 199, 28.
{
θ(x ) 0) ) θ0 π θ(y ) 0) ) 2
(2)
θ0 is the equilibrium contact angle at the fiber (x ) 0), and s is the curvilinear coordinate (at s ) 0, x ) 0, and y ) y0) (Figure 2). (32) Bateni, A.; Susnar, S. S.; Amirfazli, A.; Neumann, A. W. Colloids Surf., A 2003, 219, 215. (33) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1969, 30, 323.
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Figure 2. On the left, a schematic meniscus with the coordinates defined by eqs 1 and 2. On the right, an example of a calculation of a contact angle around a fiber. The white solid line is the best fit associated with the liquid-air interface. It gives a contact angle of 24.24 ( 1.00°. This snapshot was taken at 50.4 ms for PDMS5. Contact Angle Measurement. These equations can be solved numerically by using, for instance, a first-order Runge-Kutta algorithm.34 Combining this with a Levenberg-Marquad routine,35 we can fit the best Laplace curve through the experimental edge profile. The contact angle and the meniscus height are then calculated. It is implicitly assumed that the Laplace equation is still valid to describe the shape of nonequilibrium interfaces. This is particularly good if the capillary number (Ca ) ηV/γ, where V is the triple-line velocity and η is the dynamic liquid viscosity) is small, as it is here the case for all of our experiments. Even if this assumption is not valid, the method still gives a good measure of the contact angle because it fits the best curve through the experimental data (just as any other arbitrary fit would do). The algorithm is fast (on the order of a second per profile on a Pentium III computer), and we have checked that the fitted contact angle does not depend on the number of points taken in the profile (up to 1000 points). To validate our method (edge detection and Laplace fit), we have performed two tests. We have first generated theoretical Laplace interface profiles (with a known contact angle using eqs 1 and 2) to reconstruct images, which were then edge detected and fitted with our software G-fiber. Thus, a comparison could be drawn between the theoretical and measured profiles. We found good agreement for contact angle >5° (with a scale of 3 µm/pixel). The error stayed within 1°. All of the following measurements were done with a scale of 2 µm/pixel. Then, as a second test, we measured contact angles by the classical Wilhelmy method and by our optical method. The Wilhelmy experience consisted of plunging a PET fiber into a formamide liquid (Aldrich 99.5+% ACS reagent) at a monitored speed. In the Wilhelmy technique, the measured wetting mass on a stationary fiber is determinated by the difference in mass before and after the fiber is inserted into the liquid. In this case, we have used a Wilhelmy force balance (the KSV 3000 with a precision of 5 × 10-8 N), and the standard force balance equation for a fiber at equilibrium is
cos θ )
FWilhelmy + ∆F Pγ
(3)
where P (m) is the perimeter of the fiber, γ (N‚m-1) is the liquidgas surface tension, FWilhelmy (N) is the force measured by the Wilhelmy balance, and ∆F (N) is the force correction. We found that the Wilhelmy method yielded a contact angle of 58.17 ( 1.13° and the optical method, a contact angle of 57.29 ( 1.00°. Our optical method has thus been validated, and we use it to analyze images of wetting on the outside of the fiber. We considered only perfect monofilaments of constant radius. Deviations from this geometry (i.e., yarns36 or conic cylinders37 are beyond the scope of this work. Moreover, it is very difficult to separate accurately the liquid-air to the fiber-air interfaces (34) Press: W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. In Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1992. (35) More´, J. J.; Garbow, B. S.; Hillstrom, K. E. User Guide for MINPACK-1; Argonne National Laboratory Report ANL-80-74, 1980. (36) Holme, I. Int. J. Adhes. Adhes. 1999, 19, 455. (37) Gu, Y.; Li, D.; Cheng, P. Colloids Surf. 1997, 122, 135.
Figure 3. Meniscus height dynamics vs time for PDMS5 (0), PDMS20 (O), PDMS50 (4) (time scale at the bottom), and PDMS500 (g) (time scale at the top).
Figure 4. Contact angle dynamics vs time for PDMS5 (0), PDMS20 (O), PDMS50 (4) (time scale at the bottom), and PDMS500 (g) (time scale at the top). for very small contact angles,38 so that complete spreading between the silicone oils and the PET fiber cannot be directly measured but should be assessed by the Wilhelmy method.
3. Experimental Results The meniscus rise (Figure 3) and contact angle dynamics (Figure 4) are measured for the five different liquids. The experimental average error bars associated with the contact angle and meniscus rise dynamics measurements are respectively equal to 1.75° and 11.12 µm. PDMS5 and PDMS10 show very similar trends, so for clarity, only PDMS5 is plotted in the figures. It must be noted that when the fiber is put into contact with the liquid the meniscus starts to rise spontaneously and instantaneously around the fiber. The relatively low surface tension of the liquids (Table 1) compared to the critical surface tension of the PET fiber (= 42 mN‚m-1) favored the break up of the liquid/air interface without significant additional elastic stress. This was checked very carefully during the experiments. When these data are put on a logarithmic scale (Figure 5), we see that the contact angle dynamics present two different trends during the latest stages of the meniscus rise as the meniscus approaches equilibrium (e.g., small contact angles). On one hand, the liquids with a low (38) Seveno, D. Dynamic Wetting of Fiber. Ph.D. Thesis, Faculte´ Polytechnique de Mons, 2004.
Dynamics of the Rise around a Fiber
Langmuir, Vol. 21, No. 21, 2005 9587 Table 3. Characteristic Time24 liquid
tc (ms)
PDMS5 PDMS10 PDMS20 PDMS50 PDMS500
1.73 1.73 30.00 73.85 722.60
eq 1 reduces to dθ/ds ) y/a2 with the same definition for dx/ds and dy/ds. The shape of the profile is then given by24 Figure 5. Late stage contact angle dynamics vs time for PDMS5 (0), PDMS20 (O), PDMS50 (4), and PDMS500 (g) on a logarithmic scale. The solid lines correspond to the best linear fits. For clarity, the PDMS10 contact angle dynamics is not shown. Table 2. Slopes of the Different Contact Angle Dynamics silicon oils
slope
quality of the fit (R2)
PDMS5 PDMS10 PDMS20 PDMS50 PDMS500
-0.928 ( 0.012 -1.109 ( 0.029 -0.528 ( 0.014 -0.513 ( 0.006 -0.504 ( 0.003
0.986 0.983 0.964 0.982 0.998
viscosity (i.e., PDMS5 and PDMS10) have a slope of around -1.0, and on the other hand, the liquids with higher viscosity (i.e., PDMS20, PDMS50, and PDMS500) have a slope of around -0.5 as shown in Table 2. It should be noted that the slopes of the contact angle dynamics have been assessed for θ e45° so that the errors between the trigonometric functions and their common Taylor expansion are lower than 20%. This was done to allow for the use of asymptotic and scaling arguments that are developed in the Theoretical Considerations section.
yt(θt) ) ax2(1 - sin θt)
(6)
The main results of the following sections are based on asymptotic and scaling arguments. It is therefore necessary to determine if the dynamics have reached this limit. In ref 24, it is shown that for nonviscous liquids on small fibers the static meniscus is reached after a characteristic time tc ≈ xFr03/γ. For viscous liquids, tc ≈ 80ηr0/γ. Table 3 lists these characteristic times for the different liquids. Here, it is assumed that PDMS5 and PDMS10 behave as inviscid liquids and that PDSM20, PDMS50, and PDMS500 are viscous ones. This assumption is verified in the next sections. It is clear that the characteristic times are very much lower than the times corresponding to the late stages of the dynamics. Thus, it is strongly believed that asymptotic and scaling arguments are fully justified. Figures 6 and 7 plot respectively the meniscus rise and the dynamic contact angle as a function of the normalized time t/tc. Evidently, the trend in PDMS5 dynamics is quite different from that of the higher-viscosity liquids.
4. Theoretical Considerations Consider a fiber with radius r0 that is put in contact with a liquid surface. A meniscus forms rapidly around the fiber until equilibrium is reached, as described in Figure 2. (For the following equation, the origin of the coordinates system is considered to be displaced along the x-axis to the axis of symmetry of the fiber.) With gravity, the equilibrium profile y(x) obeys the Laplace equation and can be very well approximated by James’s equation39
( (( ) ) ))
(
x y(x) ) r0 cos(θ0) ln(4) - ln(B0) - E - ln + r0 cos2(θ0)
x r0
1/2
2
-
Figure 6. Meniscus height dynamics vs time for PDMS5 (0), PDMS20 (O), PDMS50 (4), and PDMS500 (g) vs the normalized time t/tc.
(4)
where y is the height of the meniscus, E is the Euler constant (E = 0.57721), and x is the abscissa of the considered point. Let us, as before, assume that the liquid during its capillary rise takes an equilibrium meniscus shape or, in other words, that we have local equilibrium. That is to say that this equation holds for any time t on substituting the dynamic contact angle θt for θ0. In particular, at the fiber radius, we have
[(
yt(r0) ) r0 cos(θt) ln
) ]
4 -E B0(1 + sin(θt))
(5)
If the fiber radius becomes large compared to the capillary length (i.e., the rise on a planar wall is considered), then
Figure 7. Contact angle dynamics vs time for PDMS5 (0), PDMS20 (O), PDMS50 (4), and PDMS500 (g) vs the normalized time t/tc.
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Contact Angle Dynamics. The basic idea of this section is to extend to the fiber geometry the model developed by De Ruijter.12 The channels of dissipation associated with flow dissipation and with friction at the triple line are reconsidered and again combined. The relation between the driving force of spreading γ(cos θ0 - cos θt) and the dissipation is provided by the standard mechanical description of dissipative system dynamics40
γ(cos θ0 - cos θt) )
∂(Df + DF) ∂U
(7)
where Df and DF stand for, respectively, the viscous dissipation function and the dissipation function at the triple line. U is the triple-line velocity. In the vicinity of the triple line, the influence of the fiber curvature vanishes because the radius dimension is much greater than the meniscus dimension. Consequently, in this approach, the fiber reduces to a flat wall. de Gennes9 has shown that a simple liquid pulled out from a bath by a plate, having a velocity U, has a parabolic flow field
( )]
[
-1 3 x - x0 + v(x) ) U 2 2 x0
2
(8)
and
[ ] ( )
dθt γ(cos θ0 - cos θt) ∂yt(r0) ) dt xmax ∂θt 6η ζ0 + ln tan θt xmin
∫yy (∫0x (y) η(dv dx) max
0
min
2
)
dx dy
[(
( )
we get for small values of θt and θ0 ) 0°
dθt ≈dt
(10)
(11)
(12)
where ζ0 is the friction associated with the dissipation channel occurring in the vicinity of the triple line. Thus
U)
∂yt(r0) γ(cos θ0 - cos θt) with U ) xmax ∂t 6η ζ0 + ln tan θt xmin
( )
( ) ( ) θt2 2
xmax 6η ln xmin ζ0 + θt
(16)
dθt ≈ -θt3 dt
(17)
θt ≈ t-1/2
(18)
or
xmax is on the order of millimeters, whereas xmin is of molecular size. The logarithm factor is generally approximated to 12.9 The dissipation due to the attachment of the liquid molecules to the substrates at low velocities is
ζ0U2 DF ) 2
γ
Whenever there is no friction at the triple line, we thus obtain
and for the other case with a small viscosity,
dθt ≈ -θt2 dt
(19)
θt ≈ t-1
(20)
or
with
∂yt(r0) U) ∂t
) ]
∂yt(r0) 4 ) -r0 sin(θt) ln -E ∂θt B0(1 + sin(θt)) r0[1 - sin(θt)] (15)
(9)
with x0(y) being the thickness of a liquid layer at a height y, where ymax and ymin delimit the meniscus size. Moreover, x0 ) tan(θt)y. Finally,
xmax 3ηU2 ln Df ) tan θt xmin
(14)
Because
The flow dissipation is then
Df )
-1
(13)
(39) James, D. F. J. Fluid Mech. 1974, 63, 657. (40) Landau, L. D.; Lifschitz, E. M. Me´ canique, 3rd ed.; MIR: Moscow, 1969.
As already discussed, very nice experimental evidence of the first regime has been obtained by Que´re´ et al.23 and Sauer et al.31 Equating the contact angle behaviors in the two regimes, we get the following estimate for the crossover time T* between the friction regime and the hydrodynamic regime:
T* ≈
2ζ02r0 xmax 3γη ln xmin
( )
(21)
The same procedure can be followed in the case of the rise on a planar wall with the condition that eq 15 is replaced by
∂yt(θt) -cos θ ) a ∂θ x2(1 - sin θ)
(22)
For small contact angles, eq 15 leads to ∂yt(θt)/∂θ ≈ - r0, and eq 22, to ∂yt(θt)/∂θ ≈ - a/x2. Thus, the scaling laws remain the same for either a small fiber or a planar wall.23,24 The crossover time T* also keeps the same form as long as r0 is replaced by a/x2. It should be emphasized that the planar case is not synonymous with the spreading
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Table 4. Fitted Parameters from Equation 14 Associated with the Data of Figure 8 silicon oils
ln(xmax/xmin)
PDMS5 PDMS10 PDMS20 PDMS50 PDMS500
0.065 ( 0.076 0.004 ( 0.015 5.160 ( 0.270 4.630 ( 0.340 7.490 ( 0.660
ζ0 (Pa‚s)
crossover time quality of T*(s) the fit (R2)
0.490 ( 0.042 0.380 ( 0.054 0.900 ( 0.060 0.100 ( 0.019 1.380 ( 0.124 0.104 ( 0.027 9.190 ( 0.730 0.277 ( 0.069
0.994 0.988 0.989 0.995 0.996
of droplets12 or stripes41 on a flat surface because here the liquid is of infinite size. The different experimental contact angle dynamics were fitted by the integration of eq 14 with the initial contact angle θ(t0,exp) with t0,exp, the time corresponding to the first measured angle. It output two parameters, ζ0 and ln(xmax/xmin), characterizing respectively the friction dissipation at the triple line and the viscous dissipation. Using these parameters, the relaxation of the contact angles was extrapolated from the time corresponding to the first measured angle, t0,exp to zero time (θ ) 90°). It was then possible to compare the theoretical value of 90° to the value of the calculated contact angle and therefore to assess any possible offset. For PDMS5, PDMS10, PDMS20, and PDMS50, the calculated contact angle at time zero is equal to the theoretical one, indicating that the dynamics are properly scaled in time. For PDMS500, a slight offset of 0.19 s was found and taken into account during the fitting procedure. Note that, with or without this offset, the values of the fitted parameters are the same within the error bars. For each liquid, the parameters are given in Table 4 as well as the estimated crossover time between the friction and the viscous regime. In the inviscid domain, the viscous parameter is clearly nonphysical and can be set to zero, indicating that the regime is dominated by the frictional dissipation. In the viscous domain, this parameter is now physically relevant, as well as the frictional one, so we can estimate the crossover time. Comparing T* with Figures 3 and 4, we can see that for PDMS20, PDMS50, and especially PDMS500 the frictional regime is very short compared to the viscous one so that the viscous dissipation is dominant here. Thus, it is in complete agreement with the observations that PDMS5 and PDMS10 showed t-1 behavior corresponding to a molecular-kinetic description in its linear form whereas PDMS20, PDMS50, and PDMS500 showed t-1/2 behavior corresponding to the viscous regime. Furthermore, these experimental results confirm the molecular dynamics results obtained in ref 25. It should be noted that in Figures 5 and 8, for PDMS5, the fitting procedures are done for t > 0.02 s. Below the threshold value, inertia may have a predominant role. This assumption is verified in the following section. Meniscus Dynamics. Without any fitting procedure, the meniscus dynamics of PDMS500, PDMS50, and PDMS20 follow the quasi-steady state described by eq 5 fairly well where we have used the measured value for θt. As an example, Figure 9 shows the comparison for the PDMS50 meniscus height dynamics for the measured data and the estimated values from eq 5 In contrast, PDMS5 meniscus height dynamics (Figure 10) shows a serious discrepancy between the experimental data and the quasi-steady-state dynamics predictions, leading to the conclusion that PDMS5, and of course PDMS10 dynamics, cannot be predicted by this approach. Clanet et al.24 showed that in the so-called inviscid domain inertia has to be taken into account so that at the (41) McHale, G.; Newton, M. I.; Rowan, S. M.; Banerjee, M. J. Phys. D 1995, 28, 1925.
Figure 8. Contact angle dynamics vs time for PDMS5 (0), PDMS20 (O), PDMS50 (4), and PDMS500 (g). The solid line corresponds to the best fit associated with the integration of eq 14.
Figure 9. PDMS50 (4) meniscus height dynamics vs time and quasi-steady-state meniscus height (given by the measured contact angle using eq 5) vs time (solid line).
Figure 10. PDMS5 (0) meniscus height dynamics vs time and quasi-steady-state meniscus height (given by the measured contact angle using eq 5) vs time (solid line).
early stage (i.e., when the contact angle is large) of the spreading the height is now given by eq 23
( )
γr0t2 yt ≈ R F
1/4
(23)
where R is an adjustable parameter. Taking the logarithms of the height, as shown in Figure 11, we can confirm that in the inviscid domain and in the early stage of the rise the meniscus height dynamics follow eq 23. At larger times, the contact angles become small enough that eq 20 is valid. Provided that the dissipation at the
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Vega et al.
Figure 11. PDMS5 (0) meniscus height dynamics vs the inviscid contribution given by eq 23. The solid line corresponds to the best linear fit with a slope of 0.240 ( 0.006 (R2 ) 0.9989).
triple line is still dominant, the molecular-kinetic approach can fully describe the dynamics of spreading as shown in the previous section. Therefore, the triple line velocity can be written in the form11
V)
dyt γ ) (cos θ0 - cos θt) dt ζ0
(24)
For small contact angles and for θ0 ) 0°, it yields the scaling law
dyt ≈ θt2 dt
(25)
Using eq 20, it leads to
dyt ≈ t-2 dt
(26)
yt ≈ yequilibrium - βt-1
(27)
then
with yequilibrium being the liquid height at equilibrium and β being a numerical factor. Thus, at large times or equivalently at small contact angles, the liquid reaches the equilibrium height with a t-1 scaling law. Figure 12 shows very good agreement between the numerical fit of eq 27 and the PDMS5 meniscus height dynamics. In particular, the equilibrium height is estimated to be equal to 822.44 ( 3.02 µm. For the planar case, the theoretical equilibrium height, calculated using eq 6 with θ ) 0°, is equal to 2091.62 µm. The difference in the equilibrium heights emphasized the role of the radius of curvature (fiber radius) that inhibits the spreading. In addition, the crossover time between the inertia and friction regimes occurs around 0.02 s, as assumed previously. Consequently, in the inviscid domain, the meniscus height dynamics is described by two regimes: first, the meniscus progresses as a function of
Figure 12. PDMS5 (0) meniscus height dynamics. The solid line corresponds to the best fit of equation yt ≈ yequilibrium βt-n with yequilibrium ) 822.44 ( 3.02, β ) 6301.65 ( 882.39, and n ) 1.05 ( 0.04 (R2 ) 0.993).
t(1/2) (inertial regime), and then, close to equilibrium, the rise slows down following a t-1 regime (frictional regime). 5. Concluding Remarks We have developed an optical method to measure the dynamics of the rise of a liquid meniscus around a fiber. The validation of our technique has been established using different tests. We have then applied our procedure to measure the dynamics of the rise of silicon oils with various viscosities on a PET fiber. The results confirm that there are indeed different regimes: for high-viscosity liquids, the hydrodynamic regime is predominant, whereas for low-viscosity ones the molecular-kinetic theory regime in its linear form is preceded by a short inertial regime. We have estimated the associated parameters for our liquids in contact with the PET fiber. The equilibrium contact angles are zero in all cases, but the dynamic contact angle θt behaves as t-1/2 for high-viscosity liquids and as t-1 for low-viscosity liquids. Our measurement are in good agreement with the theoretical prediction that the dynamics of the rise of a liquid meniscus should present first a linear molecular-kinetic regime followed by a viscous hydrodynamic regime with a crossover time given by eq 21. There is no doubt that a direct experimental confirmation of the validity of this equation would be very interesting. Acknowledgment. We are indebted to T. D. Blake for very stimulating discussions on the subject. It is also a pleasure to acknowledge, D. Que´re´, A. Clarke, and the referees for valuable comments and suggestions. This research has been partially supported by Structural European Funds and by the Re´gion Wallonne. Note Added after ASAP Publication. This article was released ASAP on September 1, 2005. An abstract was added, and the correct version was released on September 12, 2005. LA051341Z
