Dynamics of Wetting Revisited - Langmuir (ACS Publications)

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Dynamics of Wetting Revisited D. Seveno,* A. Vaillant, R. Rioboo, H. Ad~ao, J. Conti, and J. De Coninck Laboratory for physics of surfaces and interfaces, University of Mons, Parc Initialis, Avenue Copernic, MateriaNova, 7000 Mons, Belgium Received March 31, 2009. Revised Manuscript Received September 28, 2009 We present new spreading-drop data obtained over four orders of time and apply our new analysis tool G-Dyna to demonstrate the specific range over which the various models of dynamic wetting would seem to apply for our experimental system. We follow the contact angle and radius dynamics of four liquids on the smooth silica surface of silicon wafers or PET from the first milliseconds to several seconds. Analysis of the images allows us to make several hundred contact angle and droplet radius measurements with great accuracy. The G-Dyna software is then used to fit the data to the relevant theory (hydrodynamic, molecular-kinetic theory, Petrov and De Ruijter combined models, and Shikhmurzaev’s formula). The distributions, correlations, and average values of the free parameters are analyzed and it is shown that for the systems studied even with very good data and a robust fitting procedure, it may be difficult to make reliable claims as to the model which best describes results for a given system. This conclusions also suggests that claims based on smaller data sets and less stringent fitting procedures should be treated with caution.

Introduction Dynamic wetting is key to a wide range of processes both natural and industrial, such as coating, detergents, flotation and oil recovery. It is now generally accepted that dissipation controls the dynamics.1 However, since several channels of dissipation are possible, we have to consider different theoretical models in order to interpret observed behavior. The principle models are based on molecular-kinetic theory (MKT), hydrodynamics (HD) or some combination thereof. Despite considerable efforts by many research groups, it appears that much experimental data can be fitted as well by one model as by another,2,3 and it is therefore difficult to be certain of the underlying mechanism. We have recently developed a systematic statistical tool, G-Dyna (available free of charge at http://crmm.umh.ac.be), for evaluating dynamic contact angle data from spontaneous and forced wetting experiments and comparing them with the current theories. The method has several advantages over conventional fitting procedures. It not only minimizes some error function (the weighted sum of the squares of differences between fitted values and experimental data) using a combination of Simplex and Levenberg-Marquard algorithms, it also explores the robustness of these fits by using the bootstrap technique and systematically varying the initial values of the parameters of the fit. This gives access to a distribution of the fitted parameters and to their possible correlation. Having such a method is important, as we can make the most of hard-won data to improve our understanding of the physical mechanism of wetting. Here, we present new spreading-drop data obtained over a wide range of time scales (milliseconds to seconds) and apply our statistical tool to demonstrate the specific range over which the various models of dynamic wetting would seem to apply for our experimental systems. The paper is organized as follows. Current theories are *Corresponding author. E-mail: [email protected]. (1) De Ruijter, M. J.; De Coninck, J.; Oshanin, G. Langmuir 1999, 15, 2209– 2216. (2) Ranabothu, S. R.; Kaenezis, S.; Dai, L. L. J. Colloid Interface Sci. 2005, 288, 213–221. (3) Schneemilch, M.; Hayes, R. A.; Petrov, J. G.; Ralston, J. Langmuir 1998, 14, 7047–7051.

13034 DOI: 10.1021/la901125a

first briefly reviewed. Then, the experimental setup and materials are presented before results and analysis are given using our fitting procedures. These show that even with very good data and a robust fitting procedure, it may be difficult to make reliable claims as to the model which best describes results for a given system.

Theoretical Models As demonstrated by de Gennes,4 the out-of-balance interfacial tension forces F = γ(cos θ0 - cos θd) (where γ is the liquid/air interfacial tension, θd and θ0 are the dynamic and equilibrium contact angles respectively) can be compensated by three channels of dissipation: viscous dissipation, dissipation in the precursor film associated with the complete wetting case (θ0 = 0), and dissipation in the close vicinity of the solid near the contact line. According to this view, when partial wetting is considered (Figure 1), the viscous dissipation described by the hydrodynamic approach (HD)5,6 and the dissipation near the contact line described by the molecular-kinetic theory (MKT)7 are the dominant channels. When the HD approach is considered, Voinov5 established a relation between θd and microscopic contact angles, θm (for θd e 3π/4), measured at a microscopic height hm. Voinov did not consider any phenomena occurring below this height, especially any dissipation link to a friction process between the liquid and the solid. Cox6 then assumed that the liquid slips on the solid in a region of length Ls (the slip length (m)) near the wetting line and that θm does not depend on the velocity of the contact line and can be set equal to the apparent equilibrium contact angle θ0 giving eq 1 for Ca , 1 where Ca is the capillary number (Ca= (ηV )/γ). Blake and Haynes7 were the first to effectively take into account a dissipation process occurring in the close vicinity of the contact line. The contact line motion is modeled by the displacements of length λ (the jump length (m)) of the molecules from one adsorption site to an other at a jump frequency K0 (Hz) (eq 2). Both theories predict evolution of the dynamic contact angle as a (4) (5) (6) (7)

De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. Voinov, O. V. Fluid Dyn. 1976, 11, 714–721. Cox, R. G. J. Fluid Mech. 1986, 16, 169–194. Blake, T. D.; Haynes, J. J. Colloid Interface Sci. 1969, 30, 421.

Published on Web 10/21/2009

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Both dissipations occur at the same time but in different regions. It should be mentioned that when the linear approximation of the MKT is valid, i.e. when dealing with contact angles close to equilibrium, Petrov’s combined model can also be simplified (eq 5). 

θd

Figure 1. A liquid drop partially wetting a solid surface. The hydrodynamics parameters are shown on the left of the droplet. The molecular-kinetic parameters are shown on the right of the droplet.

3

   Vζ ηV L þ9 ln ¼ arccos cos θ0 γ γ Ls 3

ð5Þ

To the best of our knowledge, this model has never been tested. De Ruijter et al. used a purely mechanical scheme and the linear version of the MKT model (eq 3) yielding eq 6. V ¼

γðcos θ0 - cos θd Þ   ζ þ 6ηΦðθd Þ ln La

ð6Þ

function of the contact-line velocity, V. "  # -1 γ 3 L 3 ðθ - θ0 Þ ln V ¼ 9η d Ls

ð1Þ

with V the contact-line velocity (m/s), γ the liquid-gas surface tension (N/m), η the liquid viscosity (Pa 3 s), L a characteristic length scale of the droplet (m). ! γλ2 ðcos θ0 - cos θd Þ ð2Þ V ¼ 2K0 λ sinh 2kB T with kB the Boltzmann’s constant and T the temperature (K). The density of absorption site is set to λ-2. When dealing with contact angles close to θ0, the MKT can be simplified, if the argument of the sinh function is small (eq 3). γ V ¼ ðcos θ0 - cos θd Þ ζ

ð3Þ

with ζ the friction at the contact-line given by ζ=kBT/K0λ3 (Pa 3 s). These approaches lead to different predictions for small dynamic contact angles and for a vanishing θ0: HD leads to Rd ∼ t1/10 and θd ∼ t-3/10, whereas the MKT leads to Rd ∼ t1/7 and θd ∼ t-3/7 where Rd is the dynamic drop radius and t the time. Brochard et De Gennes8 have pointed out that for small angles the dynamics is more likely to be controlled by viscous dissipation, whereas for large angles, contact-line friction would be the governing channel of dissipation.9 To overcome the inconsistency between these two models, Petrov and Petrov10 and de Ruijter et al.1 argued that both channels of dissipation should coexist and therefore combined these two approaches, though in different ways. Petrov considered that the microscopic contact angle appearing in the HD approach is the dynamic contact angle described by the MKT model (eq 4). 2

θd 3

2

33 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 ffi 2k T V V B þ ¼ arccos3 4cos θ0 - 2 ln4 þ155 2K0 λ 2K0 λ λγ   ηV L ln ð4Þ þ9 γ Ls

(8) Brochard-Wyart, F.; Gennes, P. G. D. Adv. Colloid Interface Sci. 1992, 39 1–11. (9) Gentner, F.; Ogonowski, G.; De Coninck, J. Langmuir 2003, 19, 3996–4003. (10) Petrov, P. G.; Petrov, J. Langmuir 1992, 8, 1762–1767.

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where a is a cutoff length below which the viscous dissipation in the core of the drop is considered to be negligible in the scheme of the Seaver-Berg approximation.11 Φ(θd) is a geometrical term which relates the base radius of the drop, Rd to its volume, Vol (spherical cap approximation): 

Rd

3

 3Vol Φðθd Þ ¼ π

ð7Þ

sin3 θd 2 -3cos θd þ cos3 θd

ð8Þ

with Φðθd Þ ¼

Within this model, dissipations can occur over different time scales; the dissipation in the vicinity of the triple line (MKT) usually first predominates, then the dissipation due to the friction in the inner layers of the liquid (viscosity, η) controls the spreading of the droplet as the contact angle falls. This model is specific to droplet geometry where the spherical cap approximation is valid. Nevertheless, it can be adapted to others geometries such as meniscus rise around a fiber12-14 or imbibition by porous substrates.15,16 The idea that the dynamic contact angle depends on the contact-line velocity rather than the overall flow field of the liquid has been critisized by Shikhmurzaev in his interface formation theory. Shikhmurzaev17 considered that the surface tensions deviate from their equilibrium values in the vicinity of the contact-line as the liquid particles travel from one interface to another. As Ca goes to zero and in leading order in Ca, this model also provides a simple relation, hereafter referred to as Shikhmurzaev’s formula (eq 9) between the contact-line velocity and the dynamic contact angle: " #1=2 γ ½1þðcos θ0 -σ sg Þð1-FsG Þðcos θ0 - cos θd Þ2 ð9Þ V ¼ ηSc 4ðcos θ0 þ BÞðcos θd þ BÞ

(11) Seaver, A. S.; Berg, J. C. J. Appl. Polym. Sci. 1994, 52, 431. (12) Seveno, D.; De Coninck, J. Langmuir 2004, 20, 737–742. (13) Seveno, D.; Ogonowski, G.; De Coninck, J. Langmuir 2004, 20, 8385–8390. (14) Vega, M. V.; Gouttiere, C.; Seveno, D.; Blake, T. D.; Voue, M.; De Coninck, J. Langmuir 2007, 23, 10628–10634. (15) Martic, G.; De Coninck, J.; Blake, T. D. J. Colloid Interface Sci. 2003, 263, 213–216. (16) Martic, G.; Gentner, F.; Seveno, D.; De Coninck, J.; Blake, T. D. J. Colloid Interface Sci. 2004, 270, 171–179. (17) Shikhmurzaev, Y. D. Capillary flows with forming interfaces; Chapman and Hall CRC: London, 2008.

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Seveno et al. Table 1. Free Parameters of the Models Considered

Table 2. Main Physical Properties of the Liquids Used in the Experiments

free parameters model HD MKT linear MKT De Ruijter Petrov linear Petrov Shikhmurzaev’s formula

θ0 ln (L/Ls) K0 λ θ0 ζ θ0 ζ ln (L/a) θ0 θ0 ln (L/Ls) K0 λ ζ θ0 ln (L/Ls) Sc FsG σsg θ0

squalane PDMSOH1 PDMSOH2 DBP

η (Pa 3 s)

γ (N 3 m-1)

F (kg 3 m-3)

0.0314 0.0272 0.0926 0.0167

0.0311 0.0205 0.0201 0.0339

810 960 970 1040

with B ¼

ð1þFsG u12 ðθd , kμ ÞÞ -σ sg 1 -FsG

ð10Þ

and u12 ðθd , kμ Þ ¼

ðsin θd - θd cos θd ÞKðθ2 Þ - kμ ðsin θ2 - θ2 cos θ2 ÞKðθd Þ ðsin θd cos θd - θd ÞKðθ2 Þþkμ ðsin θ2 cos θ2 - θ2 ÞKðθd Þ ð11Þ

where θ2 = π - θd, K(θ) = θ2 - sin2 θ, kμ is the gas-to-liquid viscosity ratio, FsG is the dimensionless equilibrium surface density of a liquid-gas interface, σsg the dimensionless surface tension of the liquid-gas interface and Sc a scaling factor depending on the material properties of the fluid and the interfaces. Except for the linear version of the Petrov’s model, these theoretical developments have been widely tested against experimental data (HD,18-21 MKT, linear MKT,22-24 De Ruijter,25,26 Petrov,3,27-29 and Shikhmurzaev’s formula17,30). The effectiveness of a given model is usually assessed by fitting the experimental data to the relevant equation with its free parameters, as given in Table 1, where θ0 is the only measurable parameter. To evaluate this effectiveness, the first criteria is to calculate the error between the experimental data and the best fit. The error is given by

error ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uN uP u ½ðθnfit - θn Þ=σθn 2 tn ¼1 N

ð12Þ

where N is the number of experimental points, θfit n is the value of the contact angle predicted by the theory, θn is the measured contact angle, and σθn is the experimental error associated with the (18) Hoffman, R. L. J. Colloid Interface Sci. 1974, 50, 228. (19) Fermigier, M.; Jenffer, P. J. Colloid Interface Sci. 1991, 146, 226. (20) Cazabat, A. M.; Gerdes, S.; Valignat, M. P.; Villette, S. Interface Sci. 1997, 5, 129–139. (21) Biance, A. L.; Clanet, C.; Quere, D. Phys. Rev. E 2004, 69, 016301–1. (22) Blake, T. D. Dynamic Contact Angles and Wetting Kinetics. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993. (23) Petrov, J. G.; Ralston, J.; Hayes, R. A. Langmuir 1999, 15, 3365–3373. (24) Nguyen, A.; Alexandrova, L.; Grigorov, L.; Jameson, G. Miner. Eng. 2006, 19, 651–658. (25) De Ruijter, M. J.; De Coninck, J.; Blake, T. D.; Clarke, A.; Rankin, A. Langmuir 1997, 13, 7293–7298. (26) De Ruijter, M. J.; Charlot, M.; Voue, M.; De Coninck, J. Langmuir 2000, 16, 2363–2368. (27) Petrov, J. G.; Ralston, J.; Schneemilch, M.; Hayes, R. A. J. Phys. Chem. B 2003, 107, 1634–1645. (28) Karakashev, S.; Phan, C.; Nguyen, A. J. Colloid Interface Sci. 2005, 291, 489–496. (29) Phan, C.; Nguyen, A.; Evans, G. J. Colloid Interface Sci. 2006, 296, 669– 676. (30) Blake, T. D.; Bracke, M.; Shikhmurzaev, Y. D. Phys. Fluids 1999, 11, 1995– 2007.

13036 DOI: 10.1021/la901125a

Figure 2. Snapshots of the droplet relaxation. Images taken with the high speed camera (left column), with the CCD camera (right column).

measurement of the contact angle, i.e., either the standard deviation or the error due to the magnification. The function proposed is classical;14,26 it represents the average distance between the experimental data and the predicted ones weighting by the experimental error bars. More complex equations taking into account sophisticated weighting functions or manipulations of the data could be used but it would require some arbitrary choices and may add complexity in the understanding of the error function. If the error is large, the model is not well adapted to the data. If it is small, the fitted parameters must then be assessed to endure that the procedure provides values that are physically reasonable.31 It is important to keep in mind that a successful fit does not guarantee that the model is physically adequate, it just indicates that the mathematical function modeling the physics is able to adjust to the experimental data. A way to validate the fitted parameters would be to evaluate them by some independent measurements and to compare these values with the fitted ones. If the agreement were good, the model would be validated for the liquid and solid under study. Unfortunately, every model includes some molecular or at least microscopic parameters like the jump length, the jump frequency or the slip length32 which can not be measured directly. Some numerical simulations such as the molecular dynamics are able to measure the jump frequency directly and therefore compare the fitted and measured parameter with success,33-35 but no experimental method has so far been devised to measure such quantities. Therefore, it is not straightforward to reject or accept the results of one fitting analysis. Moreover, although the HD and MKT approaches are physically different, macroscopic and molecular, their mathematical expression leads to behavior which is very similar, as shown by the scaling laws. Without a rigorous treatment of both the experimental data and the fitting procedure, it may be possible to interpret the results with either theory. This is one of the reasons (31) Vasilchina, H.; Tzonova, I.; Petrov, J. Colloids Surf. A 2004, 250, 317–324. (32) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H. J.; Craig, V. S. J. Rep. Prog. Phys. 2005, 68, 2859–2897. (33) De Ruijter, M. J.; Blake, T. D.; De Coninck, J. Langmuir 1999, 15, 7836– 7847. (34) Blake, T. D.; Clarke, A.; De Coninck, J.; De Ruijter, M. J. Langmuir 1997, 13, 2164–2166. (35) Seveno, D.; Ogonowski, G.; De Coninck, J. Langmuir 2004, 20, 8385–8390.

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Figure 3. Scheme of the experimental setup.

why these models still coexist in spite of their differences. Another problem is a tendency to measure the dynamics of the contact angle (in spontaneous spreading) or the variation of the dynamic contact angle as a function of the contact-line velocity (in forced wetting) over a too narrow range of time or velocity. For instance, if the number of experimental points does not cover, at least, several decades of time, it may be possible to fit the experimental data with any of the theoretical approaches.

Figure 4. Relaxation of the contact angle versus time for the liquids under study. The error bars are omitted in the early stages (inertial regime) and only 1 point in 10 points is shown for clarity. The error associated with each measurement was taken to be the larger of either the standard deviation or the error due to the magnification.

Experiments Setup. We measure the relaxation of a droplet deposited on top of a flat surface from the moment of deposition until near equilibrium, combining images obtained using two cameras that operate over different time scales: • A high speed camera: from the first milliseconds to a half or one second. • A CCD camera: from a half second to several seconds. The deposition procedure consists in forming a droplet from the tip of a needle and estimating the distance (by trial and error method) between the bottom end of the tip of a needle and the surface so that when the droplet detaches from the tip, the bottom of the droplet is in close vicinity of the surface so that the droplet velocity is minimized when the spreading starts to avoid impact pressure effects.36 We always use the same kind of needle so that the droplets detaches from the tip of the needle at the same volume. Of course, depending on the liquid properties (surface tension and density), the droplet detaches from the needle at different volumes for different liquids. Impact or deposition velocities are low enough and viscosities (Table 2) are high enough to prevent any break up of the drop37,38 and guaranty conservation of the volume during the dynamics. The two cameras capture the droplet profile from the side and enable us to measure contact angles down to 10-8°. Smaller contact angles can not be accurately measured with this technique because of magnification limitations. Examples of the captured images are given in Figure 2. Cameras. The high-speed camera (C-MOS. Vossk€uhler HCC1000) was here used at a frequency of 923 i/s with a resolution of 1024  512 pixels. The droplet was back illuminated with a LED flash, synchronized with the camera. For longer times, images of the spreading drops are captured with a CCD camera via a stereo binocular microscope (8, Zeiss Stemi SV 11) and video recorded. The drops were back-illuminated with a cold light. This method enabled us to capture up to 50 i/s (Figure 3). Materials. Experiments were performed by spreading Squalane and two poly(dimethylsiloxane)s, with trimethyl ends (36) Schiaffino, S.; Sonin, A. A. Phys. Fluids 1997, 9, 3172–3187. (37) Rioboo, R.; Marengo, M.; Tropea, C. Exp. Fluids 2002, 33, 112–124. (38) Rioboo, R.; Adao, M.; Voue, M.; Coninck, J. D. J. Mater. Sci. 2006, 41, 5068–5080.

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Figure 5. Droplet radius dynamics for the DBP, Squalane, and PDMSOH1. In the inertial region, the full lines have respectively a slope of 0.596, 0.542, and 0.547. The intersections between the full and the dotted lines give the crossover times between inertia and capillarity. The error are omitted for clarity. Table 3. tI, θ, Ca, We, Re, and Bo when t = tI liquid

solid

squalane PDMSOH1 PDMSOH2 DBP

silicon wafer silicon wafer silicon wafer Estar

tI (ms) θ (deg) 17 27 27 19

70 64 87 70

Ca

We

Re

Bo

0.028 0.02 0.05 0.01

0.063 0.029 0.013 0.038

2.25 1.45 0.26 3.8

0.37 0.36 0.66 0.22

(PDMSOH1, and PDMSOH2) on top of silicon wafers. In other experiments, drops of n-dibutyl phthalate (DBP) were spread on poly(ethylene terephthalate) (PET) film (Estar, Kodak Limited) (Table 2). All the experiments were carried at a temperature of 25 °C. The liquids were provided by the Aldrich Chemical Co. and were used as received without further preparation. The viscosity of the liquids were measured using a Brookfield DV-II þ PRO digital viscometer. The surface tensions were measured via a pendant drop method.39 The liquids were chosen because of their low volatility. We used monocrystalline (100) silicon wafersprovided by ACM. To prepare reproducible substrates, we followed a rigorous cleaning procedure.26 It consisted of five steps. First, the wafers were sonicated for 3 min in two successive chloroform baths. They were then placed under a UV lamp in a reactor (39) De Ruijter, M. J.; Kolsh, P.; Voue, M.; De Coninck, J.; Rabe, J. Colloids Surf. A 1998, 144, 235–243.

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Table 4. Results of the Fitting Procedures for the MKT, HD, Petrov, Linear MKT, Linear Petrov, De Ruijter Models, and Shikhmurzaev’s Formulaa model

parameters

DBP

squalane

PDMSOH1

1.24 ( 0.02 7.61 ( 0.49 0.16 ( 0.02 K0 (MHz) λ (nm) 1.07 ( 0.04 0.79 ( 0.01 1.79 ( 0.06 error 0.70 ( 0.01 0.45 ( 0.02 0.79 ( 0.02 17.75 ( 0.02 10.59 ( 0.23 13.89 ( 0.16 HD ln(L/LS) error 1.94 ( 0.01 0.53 ( 0.02 0.65 ( 0.02 1.47 ( 0.08 3.57E14 ( 3.96E15 0.09 ( 0.05 Petrov K0 (MHz) λ (nm) 1.03 ( 0.01 0.006 ( 0.01 2.97 ( 0.26 0.81 ( 0.43 11.73 ( 8.05 10.76 ( 0.7 ln(L/LS) error 0.67 ( 0.02 0.44 ( 0.02 0.52 ( 0.05 linear MKT ζ (Pa 3 s) 2.47 ( 0.02 0.95 ( 0.01 2.55 ( 0.01 error 1.60 ( 0.02 0.53 ( 0.22 3.03 ( 0.03 linear Petrov ζ (Pa 3 s) 1.71 ( 0.02 0.69 ( 0.07 6.32 ( 8.90 7.07 ( 0.13 3.80 ( 0.73 13.61 ( 0.30 ln(L/LS) error 0.96 ( 0.01 0.48 ( 0.02 0.65 ( 0.02 De Ruijter ζ (Pa 3 s) 1.75 ( 0.02 0.83 ( 0.08 0.01 ( 0.02 ln(L/a) 2.79 ( 0.06 0.86 ( 0.43 8.22 ( 0.09 error 0.96 ( 0.01 0.48 ( 0.02 0.62 ( 0.01 33.78 ( 7.7 10.95 ( 2.24 46.46 ( 14.51 Shikhmurzaev’s formula Sc -0.06 ( 0.49 0.60 ( 1.04 -0.28 ( 2.90 σsg error 1.24 ( 0.08 0.60 ( 0.04 3.34 ( 0.29 a The figures are the average values and the standard deviations calculated from the distributions of the parameters. MKT

PDMSOH2 0.37 ( 0.04 1.14 ( 0.03 0.99 ( 0.001 12.22 ( 0.67 0.74 ( 0.1 0.67 ( 7.90 1.88 ( 0.66 10.21 ( 0.8 0.42 ( 0.07 3.81 ( 0.13 5.48 ( 0.18 7.36 ( 12.22 11.84 ( 0.60 0.60 ( 0.03 0.78 ( 0.15 6.74 ( 0.29 0.45 ( 0.05 24.91 ( 2.91 1.25 ( 0.94 1.02 ( 0.34

saturated with ozone to remove all organic contamination from the surface. Next, inorganic contaminants were removed by immersion in a 7:3 mixture of sulphuric acid and hydrogen peroxide at 150 °C, rinsing with Milli-Q water and drying under a flux of nitrogen. Finally, the wafers were placed for a second time in the UV-ozone reactor under a flux of oxygen saturated with water vapor. The wafers were always cleaned just prior to the experiments. In Voue et al.40 it has been shown by complete spreading experiments that the wetting properties of wafers prepared using this procedure do not change within the time scale of our experiments. The PET surface was simply unpacked and used immediately for the experiments. The packaging of the PET surface protects it from the atmosphere. Contact angle measurements showed that this industrially prepared material had consistent wetting properties from point to point and over time.

Extracting the Contact Angle and Droplet Radius from the Images For each liquid, experiments were repeated four times in order to check reproducibility. For each image, thanks to the high contrast between the image of the droplet and the background, edge detection41 could be used to identify precisely the liquid/air interface. The part of the profile close to the solid was fitted by part of a circle during the very early stages of the droplet spreading (when the spherical cap assumption is not valid due to inertia) and later by the Laplace equation. Biance et al.21 have shown that when a droplet is deposited on a solid surface the early stages of the spreading are dominated by inertia; the Laplace equation is then not appropriate to fit the liquid/air interface due to significant deformation of the droplet. Fitting the part of the interface close to the solid substrate by part of a circle was found to be a better procedure. Once the effects of inertia had disappeared, we assumed that the interface was in local equilibrium, so that use of the Laplace equation was then fully justified. In any case, this equation could be taken as simply the best mathematical function to fit the interface. To handle the large number of images (several hundred), an algorithm was written to automatically treat successive images, so that a full analysis of the output from the high-speed and CCD cameras could be completed within approximately 1 h on an office

Figure 6. Error in velocity between the MKT and its linearized version as a function of the contact angle for PDMSOH1, DBP, PDMSOH2, and squalane (from left to right). The error is calculated from data given in Table 4.

(40) Voue, M.; Valignat, M. P.; Oshanin, G.; Cazabat, A. M.; De Coninck, J. Langmuir 1998, 14, 5951–5958. (41) Canny, J. IEEE Trans. PAMI. 1986, PAMI-8:6, 679–698.

As inertia is not included in the models, we must first determine how long its effects last in order to remove data not driven solely

PC. We took care to ensure that the contact angles and the drop radius obtained from fitting the interface with part of a circle or the Laplace equation and that obtained from the two cameras all overlapped. The number of images analyzed ranged from 495 for the PDMSOH2 up to 933 for the PDMSOH1. We used a magnification of 10 μm/pixel and 13 μm/pixel for the high speed and CCD cameras, respectively. The images provided an accuracy of 2.1° for the contact angle and 13.8 μm for the droplet radius. The dynamics were then averaged over four repeat experiments and at each time the standard deviation was calculated. The error associated with each measurement was taken to be the larger of either the standard deviation or the error due to the magnification. In the early stages of the dynamics the standard deviation is larger than the error due to the magnification. Later, the standard deviation become smaller and the error is then given by the magnification. To reduce the error bars, it is then crucial to improve the optics used in the experiments. The contact-angle dynamics over four decades of time are given in Figure 4.

Eliminating Inertial Effects from the Data

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Figure 7. DBP on Estar, MKT. Top left: distribution of K0. Top right: best fit obtained from G-Dyna. Bottom left: correlation between K0 and λ. Bottom right: distribution of λ.

Figure 8. Squalane on silica, De Ruijter. Top left: distribution of ζ. Top right: best fit obtained from G-Dyna. Bottom left: correlation between ζ and ln(L/a). Bottom right: distribution of ln(L/a).

by capillarity. This can be done by scaling law analysis of the growth of the droplet radius with time. When inertia predominates, the droplet radius increases as t1/2. This is true both for droplet impact and simple deposition at nominally zero speed.21,38 In our experiments, the droplets touch the solid at a velocity of the Langmuir 2009, 25(22), 13034–13044

order of 0.1 m/s which is significant. We therefore assume that the droplet is subject to a small impact rather than just deposition, yielding a correspondingly longer period in the inertial regime. Figure 5 shows that the dynamics of the radius at the start of the experiments is actually composed of two regimes. As expected, in DOI: 10.1021/la901125a

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Figure 9. Squalane on silica, linear Petrov model. Top left: distribution of ζ. Top right: best fit obtained from G-Dyna. Bottom left: correlation between ζ and ln(L/Ls). Bottom right: distribution of ln(L/Ls).

the first milliseconds inertia dominates.38 It is then followed by a transition zone. At longer times, capillarity becomes the controlling driving force. To estimate the time when the inertial effects are negligeable compared to the capillary ones, straight lines were fitted to each range of data and the point of intersection (tI) was calculated for each liquid.21 PDMSOH2 is not included in this calculation because, in the early stages of the dynamics, very large error bars were found. It was therefore not possible to accurately estimate the regime which drives the dynamics. The crossover time for this liquid was set equal to that found for PDMSOH1. Indeed, Biance et al.21 and Rioboo et al.37 showed that the duration of the inertial regime depends weakly on the liquid properties but more strongly on the impact velocity and initial droplet radius. As the impact velocity and droplet size are comparable (R0=0.9 mm) for PDMSOH1 and PDMSOH2, we believe that the assumption of a similar behavior over time is valid. Moreover, at the intersection point, the value of the dynamic contact angle, Ca, Weber number We=FV2D/γ (with D the dynamic droplet diameter), Reynolds number Re=We/Ca, and Bond number Bo=H2/CL2 with H the droplet height and CL the capillary length (CL = (γ/Fg)1/2, with F the density of the liquid and g the acceleration due to gravity) were calculated for each liquid. The results are given in Table 3. The small values of Ca, We, and Re confirm that we can use the theoretical models described earlier and that inertial forces are negligible compared to capillary forces. In this way, we ensured that most of the experimental data were in the capillary regime even for the milliseconds time scale. The fact that the duration of the inertial regime was not simply a decreasing function of the viscosity was due to variations in the velocity of deposition. Bo is always below unity which suggests that the spherical cap assumption is valid for t>tI. Nevertheless, we found that fitting the interface by the Laplace equation is more stable and accurate than fitting it by a circle especially when the contact angle is low and therefore the curvature of the interface low. 13040 DOI: 10.1021/la901125a

Fitting Procedure For the PDMSOH1, and PDMSOH2 on the silicon wafers, the equilibrium contact angles were fixed at 0.0°,26 for Squalane on silicon wafer at 38.8° (this value was calculated by averaging those obtained when the droplet reached equilibrium), and for DBP on PET at 10.0°42 (Additional and independent static contact angle measurements confirmed that θ0 =10° (1.1°). To ensure statistically reliable results, the fitting procedure generates at least 1000 sets of data from the original experimental set by using the GDyna software. The method implemented in this software consisted in fitting the droplet radius dynamics by a ratio of polynomials that can be easily differentiated to give the velocity of the contact-line versus time. To estimate the error on the velocity, a sampling method, based on the Bootstrap technique,43 is used. From the original droplet radius dynamics data set, 37% (1/e) randomly chosen points are replaced by new points chosen so that their values are within the error bars of the original points. In this way, the original radii are duplicated to generate a large number of new sets of data which are fitted by the same ratio of polynomials and then differentiated to give multiple relationship between the contact-line velocity and time, and therefore the standard deviation for each velocity. From the functions θd(t) and V(t), it is then straightforward to determine the relationship θd (V). Thanks to the successive error calculations, this last relation includes the error on both θd and V. Then, using a combination of Simplex and Levenberg-Marquard fitting procedures which minimize the distance (eq 12) between the experimental data and the theoretical predictions with their free parameters, we obtain an ensemble of acceptable sets of parameters. (42) Blake, T. D. J. Colloid Interface Sci. 2006, 299, 1–13. (43) Press, W.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipies in Fortran, 2nd ed.; Cambridge University Press: New York, 1992.

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Figure 10. PDMSOH1 on silica, HD. Left: best fit obtained from G-Dyna. Right: distribution of ln(L/Ls).

Figure 11. PDMSOH1 (left) and PDMSOH2 (right) on silica, Petrov model. Distributions of K0 and λ (top) and K0 and ln(L/Ls) (middle), and λ and ln(L/Ls) (bottom).

Before starting the fitting procedure, it is helpful to check if every model presented in the theoretical section is applicable for every liquid. Indeed, a distinction can be drawn between the MKT, HD, and Petrov models on one hand and the linear MKT, linear Petrov, and De Ruijter models on the other hand, as these last three models use the linear version of the MKT which is only applicable when dealing with dynamic contact angles close to the equilibrium one. The question is to know how close the dynamic contact angle must be to the equilibrium one for the linear approximation to be valid. Therefore, the four dynamics were fitted by the MKT. The results are given in Table 4. As the quality of the fits is acceptable (the errors are below unity and the distributions narrow), we use the average values of K0 and λ to calculate ζ and also the error in velocity between the MKT and its linear approximation as a function of the contact angle (Figure 6). If we consider that a difference of 20% is acceptable, the linear MKT is applicable for squalane when θ < 74°, for PDMSOH1 when θ < 16°, for PDMSOH2 when θ < 50°, and for DBP when Langmuir 2009, 25(22), 13034–13044

θ < 42°, i.e. it is only applicable for squalane as θd(tI) = 70° (Table 3). For the other liquids, the contact angle is too far from its equilibrium value to consider this linear model. Indeed when fitting the data for the DBP, PDMSOH1, and PDMSOH2 by the linear MKT, the errors are high leading to poor fits. This confirms that the linear MKT model is not adapted for these three liquids. Note, however, that the test is very stringent. For combined models, the observed change in θd is considered to be the results of the dissipation channels. Therefore, the contribution of the contact-line friction term is unknown a priori. It would be quit potential for this contribution to remain within the linear region while the total change due to both contact-line friction and viscous bending suggest otherwise. This will be tested by fitting the DBP, PDMSOH1, and PDMSOH2 data by the linear Petrov and De Ruijter models. We first checked that the error given by the software was low. An error of less than unity means that, on average, the fit is entirely within the experimental error bars. This indicates a good DOI: 10.1021/la901125a

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fit. For each distribution, an average value and its standard deviation are calculated. The widths of the distributions of the output parameters help to determine the utility of the model. When the width is narrow, the standard deviation is small compared to the averaged value and the model is then appropriate to describe the dynamics. When the width is wide, it indicates that the parameters are correlated and that the model is overparameterized and therefore offers multiple solutions with similar probability. The results of the fitting procedures are given in Table 4. DBP on Estar. Here, the best model is the MKT. Indeed, the Linear MKT and HD models do not provide good fits, the errors are larger than unity and the Petrov model gives an aphysical value of ln(L/Ls). The linear Petrov and De Ruijter models give an error just below unity but with a contact-line friction which is not consistent with the one obtained by the MKT (ζ=2.71 ( 0.35).

Figure 12. PDMSOH1 contact angle dynamics for θ < 42°. The straight lines have slopes of -0.3 and -0.42, respectively. The error associated with each measurement was taken to be the larger of either the standard deviation or the error due to the magnification.

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It is interesting to observe that there is an inverse correlation between λ and K0: smaller values of K0 yield larger values of λ and vice versa (Figure 7). Such a correlation is predicted theoretically44 and has already been reported in the literature.14 Nevertheless, the range of the correlation is narrow enough to justify the average values listed in Table 4. Squalane on Silica. Here, the errors associated with each fit are below unity, so that it is not possible to use this criterion alone as a way to classify the relevance of the theories. The Petrov model provides parameters that are out of the range of physically acceptable values for K0 and λ. Because of low contact-line velocities and the high number of free parameters, this model is not able to provide a physically acceptable set of parameters. If we consider the Linear Petrov model, with fewer free parameters, the distributions of ζ and ln(L/Ls) are narrow (Figure 9) and the averaged values are physically sensible. The De Ruijter model (Figure 8) gives distributions of parameters broader than for the MKT, HD, or Linear Petrov with a standard deviation for ln(L/a) which represents 50% of the average value. PDMSOH1 on Silica. The errors associated with each fit are all below unity, so that it is not possible to use this criterion as a way to classify the relevance of the theories, but we can compare the values of the output parameters. The MKT and HD models provide values which are physically acceptable and centered around an average value with small standard deviation (Figure 10). The distributions of the parameters given by the Petrov model are much larger especially for K0 (Figure 11) for which the standard deviation represents 55% of the average value. The same comment can be made for the Linear Petrov model. The standard deviation of ζ is even larger than its average value. The values given by the De Ruijter model predict that the viscous channel of dissipation should dominate as ζ is almost null. To check this, it is helpful to plot the scaling law dependence of the contact angle and radius dynamics. This enables us to determine

Figure 13. PDMSOH2 on silica, MKT. Top left: distribution of K0. Top right: best fit obtained from G-Dyna. Bottom left: correlation between K0 and λ. Bottom right: distribution of λ. 13042 DOI: 10.1021/la901125a

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Figure 14. PDMSOH2 on silica, HD. Left: best fit obtained from G-Dyna. Right: distribution of ln(L/Ls).

Figure 15. Squalane on silica, Shikhmurzaev’s formula. Top left: distribution of Sc. Top right: best fit obtained from G-Dyna. Bottom left: correlation between Sc and σsg. Bottom right: distribution of σsg.

whether the asymptotic behavior of either the HD or MKT are recovered. These asymptotic behaviors are theoretically recovered only for small contact angles, where the Taylor expansions of the sine and cosine functions are valid. Here, we consider contact angles below 42°, for which the difference between sin θ and θ is less than 10%. We focus on the measurements of the contactangle rather than the radius, because the difference between the scaling laws associated with the different channels of dissipation is larger. It should be therefore easier to distinguish the MKT from the HD by measuring the slope of the contact angle relaxation in the log scale. The data were fitted by two straight lines with a slope of -0.3 (HD) and -0.42 (MKT) (Figure 12) respectively and the regression coefficients R2 calculated. By comparing the value of the R2, 0.989 for HD and 0.899 for MKT, it would appear that the contact-angle dynamic is better described by the HD at long time. Of course, with this simple analysis we do not know what the (44) Blake, T. D.; De Coninck, J. Adv. Colloid Interface Sci. 2002, 96, 21–36.

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controlling mechanism is between the end of inertia (θd(tI)=64°, see Table 3) and the time corresponding to θd =42°. PDMSOH2 on Silica. As for the PDMSOH1, the errors associated with each fit are below unity for the MKT, HD, Petrov, Linear Petrov, and De Ruijter models, but a distinction can be made between them. The Petrov model gives the lowest error, the MKT an error which is very close to unity, i.e. our threshold value. The HD approach is in between. However, even though the quality of the fit is excellent for the Petrov model, the distribution of K0 is so large (Figure 11) that the calculation of an average value is not meaninful (The same conclusion can be drawn for the Linear Petrov model). The standard deviation of ζ is even larger than its average value, i.e. there is no unique set of parameters that we can distinguish in term of quality. An arbitrary way to reduce the width of the distribution of K0 is to fix the value of ln(L/Ls) to its average value and rerun the fitting procedure. Then, K0=0.27 ( 0.08 MHz and λ=1.83 ( 0.1 nm with an error of 0.40 ( 0.09. This illustrates the overparameterization of the model for data such as these. The MKT provides a fit of DOI: 10.1021/la901125a

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moderate quality (see Figure 13) with small standard deviation for K0 and λ. Like in the PDMSOH1 case, the De Ruijter model suggests that the viscous dissipation should dominate as the logarithmic term is much higher than ζ. This is confirmed by the HD model as it described the data very well and therefore seems to be the best model to fit the data (see Figure 14). The scaling law analysis also somewhat favors viscous dissipation as the regression coefficient is higher for the HD than for the MKT (0.981 and 0.941 respectively). Shikhmurzaev’s Formula. To be complete, let us describe the results obtained by fitting the contact angle dynamics by the Shikhmurzaev’s formula. We fixed the value of FsG to 0.6 and let Sc and σsg as free parameters.17 As the contact angles are well below 180°, σsg could also have been fixed to zero. Nevertheless, to check the physical consistency of the values obtained from the fit, as we can not directly compare them to other values coming from other approaches, σsg was also considered as an adjustable parameter. The errors associated with each liquid are close to unity except for PDMSOH1 (Table 4). Note that the formula given is a simplified version of a more complex and general theory. A bad fit is then questionable: does the failure come from the simplification involved in the equation or from the theory itself? We do not try to answer to this question as it is beyond the scope of this work. The values of σsg present the same trend for every liquid and is illustrated in Figure 15. A wide range of values are accessible (that is the reason why the standard deviations of σsg are large compared to the average values) but the distributions of points shows that the values between 0 and 1 are the most frequent ones. Moreover, it seems that there is an correlation between Sc and σsg: smaller values of Sc yield smaller values of σsg and vice versa. This analysis is very brief and would require much more works to be done, but we believe that it shows the usefullness of the Shikhmurzaev’s formula which has been little applied elsewhere.

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Conclusion Thanks to our experimental approach of using both high-speed and standard cameras taking pictures at respectively 923 and 50 i/s, we have been able to follow the contact angle and radius dynamics of four liquids on top of the smooth silica surface silicon wafers or PET over almost five orders of time from the first milliseconds to several seconds. Analysis of the images allows us to make several hundred contact angle and droplet radius measurements versus time with great accuracy, and to check that our measurements are reproducible. Scaling analysis demonstrated that the early stage of the dynamics is controlled by inertia. The G-Dyna software was then used to fit the data with various theories (MKT, linear MKT, HD, Petrov, linear Petrov, De Ruijter models, and Shikhmurzaev’s formula). The distributions, correlations, and average values of the free parameters were analyzed. The viscous regime for PDMSOH2 and the contact-line regime for DBP were found. For the PDMSOH1, MKT, HD, Petrov, and De Ruijter models are applicable without any possible distinction. For the squalane, HD, MKT, linear MKT, linear Petrov, and De Ruijter models are also applicable. Despite the range of the data and the sophisticated and statistically robust method offered by G-Dyna, these results illustrate the difficulty in defining which model is most appropriate by curve fitting. Supporting Information Available: The methodology implemented in the G-Dyna software including the description of the models, the free parameters associated to each model, the statistical analysis, its validation in terms of contact-line velocity calculation and fitting procedure are given in the form of text, equations, figures, and tables. This material is available free of charge via the Internet at http://pubs.acs. org.

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