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Spreading of Silicone Oils on Glass in Two Geometries Alain Carre´* and Pierre Woehl Corning SAS, Fontainebleau Research Center, 7 bis AVenue de ValVins, 77210, AVon, France ReceiVed July 13, 2005. In Final Form: October 12, 2005 A basic problem in liquid spreading is the hydrodynamic description of the viscous breaking force near the moving contact line. A solution to the problem of divergence at the triple line has been illustrated with two experimental configurations. It consists of observing that the rheological behavior of a Newtonian liquid is modified near the triple line due to high shear rates. Above a critical value of the shear rate, near the triple line and near the solid surface, the liquid becomes shear-thinning, meaning that the apparent viscosity of the liquid decreases as the shear rate increases. As a result, there is no divergence of the viscous energy dissipation and of the braking force as observed for a purely Newtonian behavior. This description of the viscous braking phenomenon in liquid spreading is well supported by spreading experiments of silicone oils on glass substrates in two different wetting configurations. The liquids used are two silicone oils (10 000 and 100 000 cSt). These liquids are Newtonian below a critical value of the shear rate. Above this critical value, the liquid viscosity decreases according to a power law of the shear rate. One series of experiments consider the spreading of silicone oil droplets on treated and untreated glass substrates. The other configuration consists of using the glass substrates as Wilhelmy plates and to determine the advancing dynamic contact angle as a function of the imposed speed of sinking of the plate into oil reservoirs. The two series of experiments satisfy the same dynamic wetting laws. The overall experimental results are compatible with the hypothesis of a Newtonian/non Newtonian transition of the rheological properties of liquids near the wetting front although the main origin of dissipation appears to result from Newtonian viscous braking. The same dynamic law applies for the drop and Wilhelmy plate geometries.
1. Introduction The kinetics of the spreading of liquids on rigid solid surfaces can be described with two basic models. The molecular-kinetic theory proposed by Blake and Haynes1 considers a molecular “hopping” mechanism, derived from the Eyring rate process theory.2 The spreading of Newtonian liquids has also been described in the literature with the hydrodynamic theory.3-5 When a liquid spreads spontaneously on a flat and rigid solid, a dynamic equilibrium is set up in which the nonequilibrated Young force and the corresponding excess capillary energy cause triple line motion. Simultaneously, viscous dissipation in the liquid, chiefly near the wetting front, reduces the spreading speed. This theory has been successfully verified in experiments corresponding to a noninertial, purely viscous regime. In these conditions, a hydrodynamic contribution to dissipation is considered dominant.6 A basic problem in liquid spreading is the hydrodynamic description of the viscous braking force near the moving contact line. With a sharp wedge profile, the viscous energy dissipation becomes infinite, which would mean that a drop is not able to spread on a solid surface. The problem is often currently solved by introducing a “cutoff” length, for example by considering that the drop profile is curved near the solid/liquid/vapor triple line by van der Waals long-range forces.3 We proposed7,8 a new approach to solve the problem of divergence at the solid/liquid/vapor triple line. This approach is based on the hypothesis that the rheological behavior of a * To whom correspondence should be addressed. E-mail: carrea@ corning.com. (1) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (2) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (3) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (4) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85. (5) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (6) Brochard-Wyart, F.; de Gennes, P. G. AdV. Colloid Interface Sci. 1992, 39, 1.
Newtonian liquid may be modified near the triple line due to extremely high shear rates. Above a critical value of the shear rate, near the triple line and near the solid surface, the liquid becomes shear-thinning, so that the apparent viscosity of the liquid decreases as the shear rate increases. As a result, there is no divergence in the calculation of the viscous energy dissipation and of the braking force. In this paper, this description is well supported by spreading experiments of silicone oils on glass substrates in two different wetting configurations. The liquids used are two silicone oils (10 000 and 100 000 cSt). These liquids are Newtonian below a critical value of the shear rate. Above this critical value, the viscosity of the liquid is decreased according to a power law of the shear rate. One series of experiments considers the spreading of silicone oil droplets on treated and untreated glass substrates. The other configuration consists of using the glass substrates as Wilhelmy plates, to determine the advancing dynamic contact angle as a function of the imposed speed of sinking of the plate into oil reservoirs. The two series of experiments satisfy the same dynamic wetting laws. The overall experimental results are compatible with the hypothesis of a Newtonian/nonNewtonian transition of the rheological properties of liquids near the wetting front.
2. Theoretical The equilibrium of a small liquid drop resting on a solid surface is described by the Young equation. In the case where the drop does not completely spread on the solid, a contact angle θe is obtained at equilibrium. This angle satisfies the Young equation 9
γSV ) γSL + γLV cos θe
(1)
(7) Carre´, A.; Woehl, P. Langmuir 2002, 18, 3603. (8) Woehl, P.; Carre´, A. In Contact Angle, Wettability and Adhesion; Mittal, K. L., Ed.; VSP: Utrecht, 2003; Vol. 3, p 427.
10.1021/la0518997 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/19/2005
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dissipation becomes infinite, implying that a drop would be unable to spread on a solid surface due to the infinite braking force, Fb ) Ed/U. This is inherent to the no-slip condition imposed at the triple line for Newtonian liquids. Several hypotheses have been proposed to alleviate the divergence problem of the viscous dissipation in a spreading liquid wedge. One way consists of introducing a cutoff length, xm, below which the liquid wedge profile is curved by the longrange van der Waals forces.3 Under this condition, the energy dissipation due to viscosity is finite and is given by
Ed(J/m‚s) )
Figure 1. Flow profile and the Newtonian/shear-thinning transition in a moving liquid wedge (PDMS 10 000 cSt).
where γSV is the solid surface tension, γSL the interfacial tension between the solid and the liquid, and γLV the liquid surface tension. When a drop spreads on a solid surface, it is under the action of a driving force. As the dynamic contact angle θd is higher than the equilibrium value θe, a force results from the unbalanced Young equation. The excess capillary energy, Ec, per unit time and length of the solid/liquid/vapor triple line, is given by the product of the spreading force and the spreading speed, U, as
Ec(J/m‚s) ) γLV(cos θe - cos θd)‚U
(2)
When we consider now a liquid drop spreading on a solid surface, some time is necessary to reach the equilibrium value of the wetting angle, and this time depends strongly on the liquid viscosity. This means that a braking force exists due to viscosity. This force can be calculated from the hydrodynamic theory of liquid spreading.3 One way to obtain the viscous braking force consists of calculating the energy dissipated by the liquid flow during spreading. The calculation of the viscous dissipation requires knowing the flow profile inside the moving liquid wedge, assuming that the wetting angles are small and that the lubrication approximation is valid. Near the triple line, the wedge profile is approximated by a straight interface,3 leading to a liquid thickness, h, at a distance x from the drop edge given by h ) x tan θd ≈ xθd (Figure 1). For a Newtonian liquid, the flow speed profile is obtained as
V(z) )
3U (2hz - z2) 2h2
(3)
The calculation of the energy dissipation, Ed, in the liquid wedge due to viscosity leads to
Ed(J/m‚s) )
3ηU2 θd
∫0r dxx
(4)
where η is the liquid viscosity and r the drop radius. This calculation raises one problem, namely eq 4 leads to a divergent, infinite result. If we consider a Newtonian behavior in the entire wedge and a sharp liquid wedge, the energy
3ηU2 ln(r/xm) θd
(5)
Another hypothesis to remove the singularity is to allow fluid slip near the moving contact line. In that case, the boundary conditions at the front include a slip length.4,10,11 Assuming that the substrate is never perfectly dry but that evaporation and condensation, or surface diffusion, at the front always produces a thin precursor film permits also to alleviate the singularity at the triple line.12-15 In the case where the liquid is non-Newtonian but shearthinning, the divergence problem does not exist, essentially because the apparent viscosity of a shear-thinning liquid decreases when the shear rate increases, to vanish under infinite shear rate.16,17 The shear-thinning behavior can be described by a power-law between the shear stress, τ(z), and the shear rate, dV(z)/dz as
τ(z) ) a
( ) dV(z) dz
n
(6)
where the exponent n is lower than 1 and the coefficient a is constant (for a Newtonian liquid, n ) 1 and a is viscosity). With the boundary conditions of no-slip at the solid surface and no shearing at the free liquid surface, the speed profile is written as17
1 2+ ) ( n z 1 - (1 - ) V(z) ) U ] h 1 [ (1 + n) 1+1/n
(7)
The calculation of the energy dissipation can be made on the entire liquid wedge; the result converges and leads to the expression for Ed for a shear-thinning fluid17
Ed(J/m‚s) ) a‚r
1-n
(2 + n1)U
n+1
(1 - n)θnd
(8)
In previous papers,7,8 we proposed to examine the divergence problem with Newtonian fluids from the evolution of the rheological properties of liquid under high shear rates. In Figure 2 is plotted the viscosity of two silicone oils (viscosity of 10 000 (9) Young, T. Philos. Trans. R. Soc. 1805, A95, 65. (10) Dussan, V. E. B.; Davis, S. H. J. Fluid Mech. 1974, 65, 71. (11) Hocking, L. M. J. Fluid Mech. 1977, 77, 209. (12) Hardy, W. B. Philos. Mag. 1919, 38, 49. (13) Bangham, D.; Saweris, S. Trans. Faraday Soc. 1938, 34, 561. (14) Shanahan, M. E. R. Langmuir 2001, 17, 3997. (15) Shanahan, M. E. R. Langmuir 2001, 17, 8229. (16) Carre´, A.; Eustache, F. Langmuir 2000, 16, 2936. (17) Carre´, A. In Contact Angle, Wettability and Adhesion; Mittal, K. L., Ed.; VSP: Utrecht, 2002; Vol. 2, p 417.
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and 100 000 cSt18) as a function of the shear rate in logarithmic coordinates. Above a critical value of the shear rate, the viscosity of these oils decreases when the shear rate increases. They become shear-thinning above a critical shear rate. The power law dependence is easily deduced as the logarithm of the apparent viscosity, ηa, decreases linearly with the logarithm of the shear rate, γ˘ (ηa ) aγ˘ n-1). This observation leads us to propose a new hypothesis that alleviates the divergence problem; that is, the rheological behavior of the liquid is modified near the contact line due to high shear rates in the moving liquid wedge. Above a critical value of the shear rate, γ˘ c, the liquid is no longer Newtonian but shear-thinning. The transition from the Newtonian to the shear-thinning behavior indicates some type of ordering in the fluid under shear stress. The consequences of the rheological transition in the moving liquid wedge can be simply evaluated. During the spreading of a liquid drop (Figure 1), the distance from the drop edge where the transition occurs can be defined from the expression for the shear rate for z ) 0. Knowing the value of the critical shear rate, γ˘ c, allows us to calculate from eq 3 the distance, xST, in the liquid from the triple line where the transition between the two behaviors occurs
( ) dV(z) dz
3U 3U 3U ) ) γ˘ (0) ) w xST ) z)0 h x‚θd γ˘ cθd
(9)
The different regions and the Newtonian/shear-thinning transition in a moving liquid wedge are shown in Figure 1. In the example of Figure 1, the spreading speed is 1 cm/min, the dynamic contact angle is 1 rad, and the critical shear rate is 1000 s-1 (as for the PDMS of viscosity 10 000 cSt). As a result, the distance xST is 0.5 µm. The Newtonian/shear-thinning transition above the solid surface (z > 0 and 0 < x < xST) can also be determined from
(
)
3U x (h - zST) w zST ) xθd 1 xST h2
γ˘ c )
(10)
Considering the Newtonian/shear-thinning transition and the behavior of the liquid as a function of the shear rate in the three regions shown in Figure 1 allows us to express explicitly the energy dissipation as the sum of three converging terms7 1-n
Ed(J/m‚s) ) )
3
(2n + 1)
n+1
aγ˘ n-1 c 2
2
2
U ηU 3ηlU ‚ + + θ θ θd n (1 - n)(3n + 1 - n ) d d n
[
Figure 2. Rheological behavior of the silicone oils of 10 000 and 100 000 cSt as a function of the shear rate.
Therefore, the kinetics law of spreading of a liquid drop can be deduced from eqs 2 and 11 as being
ηU ) k‚Ca γ
θd(cos θe - cos θd) ) k
where Ca is the capillary number (Ca ) ηU/γ) and k is a factor that can be considered as constant in our experiments. The weak dependence of k on the nature of the oil (10 000 and 100 000 cSt) is discussed in the next section. In a complete wetting situation at equilibrium when the equilibrium contact angle, θe, is close to 0, eq 13 can be recast as
θd ) k′‚Ca1/3
with the approximation cos θd ≈ 1 - θd2/2. Equation 14 is known as the Hoffmann’s law.19 In the case of immersion of a Wilhelmy plate, the geometry of the liquid wedge is different from the geometry of a spreading drop. We will consider that the main change is that the macroscopic dimension of the liquid wedge, namely the drop radius, is replaced by the height of the meniscus, H. At equilibrium or in quasistatic conditions, H, is given by
H ) x2‚κ-1x1 - sin θd
(15)
]
where κ-1 is the capillary length (κ-1 ) [γ/Fg]1/2, with F being the liquid density and g the gravitation acceleration. κ-1 is 1.5 mm for the liquids considered in this study). We will consider that energy dissipation is mainly localized in the wetting front so that the liquid wedge model and eq 11 stay valid for a plate immersed in a liquid at an imposed speed U. However, the parameter l takes the following form:
with
( )
(14)
2
aγ˘ n-1 31-n(2n + 1)n+1 c ηU2 ‚ + 1 + 3l θd nn(1 - n)(3n + 1 - n2) η (11)
r l ) ln xST
(13)
l ) ln (12)
Equations 11 and 12 describe the energy dissipation only from the rheological properties (η, γ˘ c, a, n) and from the dynamic spreading parameters (U, θd, r). It is important to point out that there is no need to hypothesize the existence of a molecular cutoff or slip length. (18) Arkles, B.; Vogelaar, G. In Metal Organics Including Silicones. A SurVey of Properties and Chemistry; Arkles, B., Ed; Gelest Inc.: Tullytown, PA, 1995; p 397.
( ) H xST
(16)
Thus, our assumption considers simply that the macroscopic typical length for a spreading drop or a solid plate immersed in a liquid reservoir is approximately the same and that the liquid flow profile is identical in these two configurations. For both cases, under this assumption, the exact expression for k can be calculated without a priori parameter and is given by (19) Hoffmann, R. J. Colloid Interface Sci. 1975, 50, 228.
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k ) 3 ln
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( )
rγ˘ c θd aγ˘ cn-1 31-n(2n + 1)n+1 +1+ n ‚ 3 U η n (1 - n)(3n + 1 - n2) (17)
in the case of droplets and by
k ) 3 ln
(
)
γ˘ cx2κ-1 θdx1 - sin θd +1+ 3 U 31-n(2n + 1)n+1
aγ˘ c (18) n (1 - n)(3n + 1 - n ) η n
2
‚
Figure 3. Determination of the advancing dynamic contact angle with the Wilhelmy plate technique.
in the case of Wilhelmy plates. 3. Experimental Section The substrates were clean glass slides and glass slides treated with a fluorinated silane, the perfluorodecyltrichlorosilane (FDS). Clean glass is obtained by burning the glass slides in an oxidizing flame. The liquids were two poly (dimethylsiloxane) (PDMS) oils of viscosities 10 000 and 100 000 cSt. Their molecular weight is respectively 62 700 and 139 000 g/mol. These oils have a surface tension of 21.5 mN/m. In the first series of experiments, liquid drops of 0.2 µL were placed on the substrates and the contact angles were measured at 20 °C with a Rame´-Hart contact angle goniometer equipped with a video camera and a printer. The measurements consisted of measuring both the static and dynamic contact angles, and the drop radius, r, which is typically of the order of 0.5-1 mm, as a function of time after drop deposition, t. The spreading speed, U, defined as dr/dt, was also determined as a function of time after deposition according to the protocol described in ref 7. The second series of experiments consisted of using the glass substrates as Wilhelmy plates, and determining the advancing dynamic contact angle at 20 °C as a function of the imposed speed of sinking (U) of the plate into oil reservoirs (Dynamic Contact Angle Analyzer, DCA-322, Cahn Instruments). In this method, the glass plate whose thickness, e (e ) 0.13 mm), is much smaller than the width, L (L ) 24 mm), is partially immersed vertically in the liquid. A scheme of the device is simply illustrated by Figure 3. The plate hangs from a balance which measures the vertical force, F, acting on the plate. Assuming that the viscous force acting on the plate as it moves in the liquid bath is negligible,20 F is given by F ) W + 2(L + e)γ cos θd - B
(19)
where W and 2(L + e) are the weight and perimeter of the solid plate and B is the buoyancy force. During the detecting process, the force F varies with the variation of B, which is proportional to the immersion depth of the plate. Thus a linear relationship is obtained as far as the friction force acting along the plate is negligible. During the analysis process, as the plate moves into the liquid reservoir, the three phase contact line also moves. The linear part of the force diagram corresponding to eq 19 allows determining θd at the speed, U, imposed by the moving stage of the microbalance (U varies from 2 to 250 µm/s). The computing software of the DCA calculates F at a zero depth of immersion (B ) 0) from a linear regression. An example of the force measured as a function of the depth of immersion of a clean glass plate in the liquid reservoir is presented in Figure 4. The oil has a viscosity of 10 000 cSt and the speed of immersion is 20 µm/s. The capillary number Ca is about 0.01. The balance is calibrated so that the force F is zero when the solid is not in contact with the liquid. The zero depth of immersion (zdoi) is the position where the solid plate just makes contact with the liquid. The force diagram of Figure 4 has two distinct zones. The first one corresponds to the formation of the meniscus. It is followed by a straight line having a negative slope resulting from buoyancy. The dynamic contact angle is deduced from the linear part of the variation (20) Rame´, E. J. Colloid Interface Sci. 1997, 185, 245.
Figure 4. Example of measurement of the advancing dynamic contact angle of silicone oils on glass. The substrate is clean glass and the silicone oil has a viscosity of 10 000 cSt. The speed of immersion is 20 µm/s. of the force as a function of the immersion depth. In the example of Figure 4, the advancing dynamic contact angle is found to be 50.0°. Duplication of the same measurement (new clean glass slide) led to a value of 50.3°, which shows the good accuracy of the technique (the 95% of confidence level is ( 0.15° for these two experiments). Figure 2 presents the rheological behavior of silicone oils. The oil having a viscosity of 10 000 cSt (9.74 Pa s) becomes shearthinning above a critical shear rate, γ˘ c, of 1000 s-1. Therefore, in spreading experiments, we will consider that the liquid is Newtonian for shear rates lower than 1000 s-1 and is shear-thinning above it with a and n parameters respectively equal to 331.21 and 0.496 (values obtained from log ηa ) (n - 1) log γ˘ + log a, and experimental data of Figure 2). The silicone oil having a viscosity of 100 000 cSt (97.8 Pa.s) has a similar behavior (Figure 2), but the critical shear rate at the transition between the two rheological behaviors is only 193 s-1 and the a and n parameters are 2603.75 and 0.377, respectively. In all the experiments reported, surface tension forces dominate inertia forces (Weber number lower than 9 × 10-6) and viscous forces dominate inertia forces (Reynolds number lower than 4 × 10-5). The capillary number (viscous on surface tension forces) is comprised between 9 × 10-5 and 0.3.
4. Results At equilibrium, the two oils have the same static contact angle, θe, of respectively 0° and 63° on clean and FDS treated glass. The results concerning the spreading of droplets of silicone oils of viscosity 10 000 and 100 000 cSt on clean and FDS treated glass are presented in Figure 5. Equation 13 is evaluated in log/ log coordinates. The linear fit has a slope of 0.949 close to the value of 1 corresponding to the purely viscous dissipation (the R2 factor is the linear correlation coefficient).
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Figure 5. Spreading dynamics of drops of silicone oils on clean and FDS treated glass.
Figure 7. Theoretical spreading dynamic law on clean glass in the Wilhelmy plate configuration (K ) 103).
Figure 6. Verification of the spreading dynamic law on clean and FDS treated glass with the sessile drop and Wilhelmy plate techniques.
The spreading dynamic of silicone oils on clean and FDS treated glass studied by the sessile drop and Wilhelmy plate techniques are presented in Figure 6. All of the data (both sessile drops and Wilhelmy plates) can be fitted with the same linear regression, showing that the spreading dynamic law, given by Equation 13 in log/log coordinates, applies as well for both different geometries. The linear fit has a slope of 0.931 which is close to the expected value of 1 for the case of a purely viscous dissipation. The slopes of Figures 5 and 6, slightly lower than unity, may indicate that a small fraction of the energy dissipated can be actually attributed to the shear-thinning behavior of fluids under high shear rates. In the case of a purely shear-thinning behavior the slope takes the value of the power factor n.16,17 The dynamic spreading law for a shear-thinning liquid is obtained by equating eqs 2 and 9. From eqs 13 and 18, the exact numerical resolution for k in the case of the Wilhelmy plate (clean glass, θe ) 0°) has been obtained and is reported in Figure 7 for all silicone oils used in the Experimental Section (1000 cSt to 100 000 cSt). The values for the slopes for each PDMS oils can be found in Figure 8. The dependence of the slope upon the type of oil is minor (within a few percents) and the behavior of all oils can be considered following the same law as a function of log Ca. A linear regression for the complete dataset gives a slope of 0.921, which is close to the 0.931 obtained experimentally (within a one percent match between the two values). When the experimental data (drop spreading and Wilhelmy plate) are limited to clean glass (θe ) 0), eq 14 is satisfactorily verified in Figure 9, where θd is plotted as a function of Ca in
Figure 8. Slope in log-log scale (PDMS silicone oils 1000 cSt to 100 000 cSt).
Figure 9. Verification of the Hoffmann’s law (θe ) 0) on clean glass with the sessile drop and Wilhelmy plate techniques.
log/log coordinates. The slope of the linear fit is equal to 0.348 which is very close to the expected 1/3 exponent. It can be deduced that the moving liquid wedge model established for a spreading drop can also be used in the case of a solid plate immersed in a liquid reservoir at an imposed speed. However, to determine the advancing dynamic contact angle with the Wilhelmy plate technique, the viscous drag force along the plate must be negligible relative to the viscous braking force
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Table 1. Rheological Properties of Silicone Oils PDMS 1000, 2000, 5000, 10 000, 60 000 and 100 000 cSt PDMS, cSt
η, Pa s
n
a
γ˘ c, s-1
1000 2000 5000 10 000 60 000 100 000
9.971 1.944 4.865 9.74 58.56 97.7
0.757 0.660 0.599 0.496 0.400 0.377
7.642 30.832 75.648 331.207 1692.778 2603.754
18408 6456 1349 1000 275 193
located in the liquid meniscus, which limits the usage of the method to low capillary numbers.
5. Conclusions This study proposes a simple description of the wetting dynamics of silicone oils on glass substrates for a drop spreading or for a plate immersed in the liquid at an imposed speed. Although often considered as model of Newtonian fluids, high molecular weight silicone oils do not show a Newtonian behavior at high shear rates. A Newtonian/shear-thinning transition is observed at high shear rates. This transition can be localized in a liquid wedge moving on a solid substrate and suppresses the divergence problem at the triple line resulting from the hydrodynamic analysis of the spreading of Newtonian liquids.
The hydrodynamic theory applies to different wetting configurations such as droplet spreading and a liquid plate forced into a liquid bath with imposed velocity. The main contribution to the total dissipation is viscous dissipation in the Newtonian region (verification of the wetting dynamic for partial wetting and Hoffmann’s law for complete wetting). These results indicate that the same kinetic law of spreading can be used to describe the dynamic of moving of the liquid/ solid contact line, as well as for a spreading drop and for an immersed solid plate. However, in the case of the plate, the viscous force along the plate must be negligible relative to the viscous braking force located in the meniscus, which was the case in the experimental conditions considered in this paper. This work shows that the Wilhelmy plate technique can be successfully used to study the wetting dynamic in advancing mode, with equivalent performance than the sessile droplet method. This method offers numerous advantages in the determination of the dynamic of liquid spreading at imposed spreading speeds, among which are the good resolution of the method in the determination of the contact angle, the possibility to investigate spreading velocities arbitrarily, the very good practicality of implementation as compared to the spreading of a sessile droplet. LA0518997
