Langmuir 2003, 19, 2795-2801
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Dynamics of Partial Wetting and Dewetting of an Amorphous Fluoropolymer by Pure Liquids Jordan G. Petrov,* John Ralston, Matthew Schneemilch,† and Robert A. Hayes‡ Ian Wark Research Institute, University of South Australia, Mawson Lakes, South Australia 5095, Australia Received October 14, 2002. In Final Form: December 17, 2002 We report velocity dependencies of dynamic contact angles for CCl4 and tert-butyl alcohol on the amorphous fluoropolymer AF 1600 and compare them with our previous results for octamethylcyclotetrasiloxane (OMCTS) on the same substrate. The molecules of these liquids have well-defined form and the solid surface is almost smooth and homogeneous, which enables checking the assumptions of the main theories of wetting dynamics. We find that the molecular-kinetic theory well represents the data for CCl4 in the characteristic cos Θ/V scale but qualitatively disagrees with the experiment for tert-butyl alcohol and OMCTS at high velocities. The CCl4 data do not fulfill the requirement of the hydrodynamic theory for symmetric advancing and receding branches. For the other two systems, the hydrodynamic theory works better at high velocities, but the nonlinear initial sections of the Θ3/V plots require consideration of the velocity dependence of the microscopic dynamic contact angle Θc(V). This is done by the molecularhydrodynamic theory accounting for both contact line and bulk viscous friction. This theory shows agreement with the experimental data in the entire velocity range for all systems compared. It fits the receding and advancing Θ/V branches simultaneously, yielding unique values of the microscopic dynamic parameters λ, Ko,s, and Lc,max that do not depend on direction of the contact line motion. Only this theory describes the wetting-dewetting asymmetry observed for tert-butyl alcohol and OMCTS. However, it yields inadequately large values of λ and too small hopping frequencies Ko,s for CCl4 and tert-butyl alcohol. Such results, also reported previously, might mean that not λ and Ko,s but their combinations better serve for description of wetting dynamics. Here we analyze the characteristic molecular-kinetic velocity Vo,s ) λKo,s and the coefficient of friction of a unit contact line length ξc ) kT/Ko,sλ3 and find a significant difference between the studied systems.
I. Introduction The advance or retreat of a liquid on a solid surface is a rate-determining step for many natural and industrial processes. To optimize this step, the velocity of the fluid interface needs to be controlled, facilitated by a thorough understanding of wetting and dewetting dynamics. Gravitational, capillary, and van der Waals forces, opposed by viscous friction in the advancing or receding liquid, constitute the force balance of a gas-liquid-solid system under dynamic conditions.1,2 At the moving contact line, adsorption-desorption processes and forwardbackward hopping of the molecules of the liquid3 give rise to a contact line friction. The latter is determined by the activation energy of the corresponding event according to the Eyring theory for absolute reaction rates.3,4 Hydrodynamic models5-10 assume that the viscous friction is * Corresponding author. Institute of Biophysics of the Bulgarian Academy of Science, 1 Acad. G. Bonchev Str., Block 21, 1113 Sofia, Bulgaria. E-mail:
[email protected]. † Present address: Department of Chemistry, Imperial College, Kensington Str., London, SW7 2AY, U.K. ‡ Present address: Philips Research, 4 Prof. Holstlaan Str., 5656 AA Eindhoven, NL. (1) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (2) Kistler, S. F. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993; p 311. (3) Blake, T. D. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, p 251. (4) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (5) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1969, 30, 323. (6) Hansen R. J.; Toong, T. Y. J. Colloid Interface Sci. 1971, 36, 410. (7) Dussan, E. V. J. Fluid Mech. 1976, 77, 665. (8) Voinov, O. V. J. Fluid Dyn. (English Transl.) 1976, 11, 714. (9) Tanner, L. H. J. Phys. D: Appl. Phys. 1979, 12, 1473. (10) Cox, R. G. J. Fluid Mech. 1986, 168, 169.
the only dissipative force, while the molecular-kinetic approach considers only the nonhydrodynamic resistance.11,12 Combined molecular-hydrodynamic theories take into account both types of dissipation.13-15 All theories introduce microscopic dynamic parameters that are specific for the given system, because they depend on the interactions between the particular liquid, solid, and gas. The hydrodynamic solutions operate with cutoff lengths (Lc) or slip lengths (Ls) characterizing the dimension of the region, where continuum hydrodynamics or its “no-slip” boundary condition fail. The molecular-kinetic theory defines the frequency Ko of molecular oscillations between potential wells at the wetting perimeter and the distance λ between them. Lc, Ko, and λ appear also in the combined molecular-hydrodynamic theory. Each of these parameters has a clear physical meaning, and its value should not contradict common sense. However, comparison of theory and experiment for systems where the liquid does not completely wet the solid often gives physically unreasonable results. One finds subatomic slip lengths Ls, frequencies Ko much below 1 Hz, and values of λ that considerably exceed molecular dimensions.16-20,34 Faced (11) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (12) Ruckenstein, E.; Dunn, C. S. J. Colloid Interface Sci. 1977, 59, 135. (13) Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1981, 259, 753. (14) Petrov, P. G.; Petrov, J. G. Langmuir 1992, 8, 1762. (15) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 39, 1. (16) Petrov. J. G.; Sedev, R. V.; Petrov, P. G. Adv. Colloid Interface Sci. 1992, 38, 229. (17) Gribanova, E. V. Adv. Colloid Interface Sci. 1992, 39, 235. (18) Cazabat, A. M. Adv. Colloid Interface Sci. 1992, 42, 65. (19) Hayes, R.; Ralston, J. J. Colloid Interface Sci. 1993, 159, 429. (20) Hayes, R. A.; Ralston, J. Langmuir 1994, 10, 340.
10.1021/la026692h CCC: $25.00 © 2003 American Chemical Society Published on Web 02/12/2003
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with such values, inconsistency of the theoretical models or inappropriate application of the theory is implied. These problems have motivated us for new experiments with well-defined solid surfaces and pure liquids. Amorphous fluoropolymers offer an attractive choice,21-24 because they can be cast as smooth amorphous films of low surface energy γsl ) 12.7 mN/m.24 The DuPont AF1600 coating on mica shows21 atomic force microscope images with root-mean-square roughness of 0.3-0.4 nm per 1 µm2 and peak-to-valley heights of less than 2 nm over a 10 × 10 µm area.24 AF 1600 coated glass exhibits small contact angle hysteresis with polar and nonpolar liquids, which suggests an almost homogeneous surface. We found ΘA - ΘR ) 4° for water and 6° for hexadecane,22,23 while on a conventional poly(tetrafluoroethylene) surface the same liquids give hysteresis of 22° and 18°, respectively.25 In a previous paper,23 we studied the dynamic contact angles of hexadecane and octamethylcyclotetrasiloxane (OMCTS) on AF 1600. In the present investigation, we report dynamic contact angle-velocity data for nonpolar carbon tetrachloride and polar tert-butyl alcohol. These liquids and OMCTS have well-defined molecular form, remaining the same irrespective of their location in the bulk liquid or at the solid surface. Such a feature reduces the molecular-kinetic reasons for asymmetry of the wetting and dewetting modes, which is a central matter of discussion in this paper. The molecular volumes and the dimensions of the molecules of CCl4 and tert-butyl alcohol are the same, but significantly less than the corresponding values for OMCTS. The data for the two new liquids, together with the previous results for OMCTS, should give a more reliable basis for interpretation of the microscopic dynamic parameters obtained from comparison of theory and experiment. II. Theoretical Section Molecular-Kinetic Theories. Blake and Haynes11 and Ruckenstein and Dunn12 developed molecular-kinetic theories, postulating that the entire energy dissipation is located at the moving contact line. Under such conditions, the entire fluid interface obeys the Laplace equation and the dynamic contact angle is defined rigorously as a boundary condition of the quasi-static meniscus profile. At equilibrium, the molecules of the liquid at the contact line oscillate between adjacent adsorption centers, which are randomly distributed at a mean distance λ at the solidgas and solid-liquid interfaces. The equilibrium oscillation frequency Ko ) (kT/h) exp(-∆go/kT) is determined by the activation free energy of the oscillation ∆go and the temperature T. Under dynamic conditions, the unbalanced Young force γ(cos ΘY - cos Θ) decreases ∆go in the forward direction and increases it in the backward direction, changing the number of corresponding oscillations per second to K+ and to K-, respectively. The mean velocity of a unit contact line length is then given by V ) (K+ K-)λ. The theory assumes that the positive and negative change of ∆g0 is the same and relates the change in ∆g to the driving force |∆g| ) γλ2(cos ΘY - cos Θ). The same value of ∆g for advancing and receding implies unique (21) Drummond, C. J.; Georgaklis, G.; Chan, D. Y. C. Langmuir 1996, 12, 2617. (22) Schneemilch, M. Ph.D. Thesis, University of South Australia, Mawson Lakes, SA, Australia, 1999. (23) Schneemilch, M.; Hayes, R. A.; Petrov, J. G.; Ralston, J. Langmuir 1998, 14, 7047. (24) Quinn, A.; Sedev, R.; Ralston, J. J. Phys. Chem. B., submitted. (25) Johnson, R. E., Jr.; Dettre, R. H. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993; p 1.
Petrov et al.
values of λ, Ko, and ΘY for the given system. Hence, this model describes a system without static contact angle hysteresis, with values of λ and Ko that do not depend on direction of the contact line motion. This yields the symmetric sinh function in the final Blake-Haynes equation11 and the symmetric arsh function in the reverse relationship:
cos Θ ) cos ΘY -
2kT V arsh 2 2K γλ oλ
(1)
Later, Blake3,26 developed his theory introducing an activation free energy of the wetting-dewetting fluctuations at equilibrium, ∆go,w, that is a sum of a surface and a viscous component. Following Cherry and Holms,27 he related the activation free energy of the viscous flow to viscosity µ and molecular volume v of the liquid, ∆go,v ) kT ln(µv/h). The new complex wetting-dewetting frequency Ko,w was then expressed through µ, v, and the hopping frequency at the contact line Ko,s:
Ko,w )
(
)
( )
∆go,w h kT exp ) Ko,s h kT µv
(2)
Assuming that λ is the same at the solid surface and in the bulk of the liquid, Blake formulated a more sophisticated version of the molecular-kinetic theory:3,26
cos Θ ) cos ΘY -
(
)
2kT µνV arsh 2Ko,sλh γλ2
(3)
Equations 1 and 3 can be expressed in the same mathematical form,
cos Θ ) cos ΘY - A arsh
(BV)
(4)
with A ) 2kT/γλ2 and B ) 2Ko,s(1)λ for the simple theory. For its more general version, B ) 2Ko,s(2)λ/(µv/h). Fitting eq 4 to the experimental data gives A and B, which enables calculation of λ and Ko,s. For ideal surfaces, the advancing and receding branches start at cos ΘY, but for systems exhibiting static contact angle hysteresis they commence at cos Θo ) (cos ΘA,max + cos ΘR,min)/2, where ΘA,max is the maximum static advancing contact angle and ΘR,min is the minimum static receding contact angle.28 Thus, the symmetry of the cos Θ/V dependence for wetting (V > 0) and dewetting (V < 0) is a common feature of the simple and the more complex molecular-kinetic theory, that could be used as a qualitative test of the model on ideal as well as on real solid surfaces. Hydrodynamic Theories. The hydrodynamic models consider two or three regions of the moving meniscus. The inner region is located in the closest proximity of the contact line. Because of the singularity arising from the no-slip boundary condition,5,7,10 either this zone is ad hoc excluded8 or the liquid is allowed to slip there.7,10 The outer region is quasi-static and obeys the Laplace equation. The intermediate region separates the inner and outer zones. Dussan7 matched the inner and outer regions for small capillary numbers Ca ) µV/γ and obtained a two-region (26) Blake, T. D. AIChE International Symposium on the Mechanics of Thin Film Coating, New Orleans, LA, March 6-10, 1988; Paper 1a. (27) Cherry, B. W.; Holms, C. M. J. Colloid Interface Sci. 1969, 29, 174. (28) Adam, N. K. Adv. Chem. Ser. 1964, 43, 53.
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solution correct to order Ca0:
G(Θm,o) ) G(Θc) + G(Θ) ≡
()
Lh µV ln γ Ls
(5)
∫0Θβ - 2sinsinβ βcos β dβ
The parameter Lh represents a characteristic dimension of the flow in the outer region, and the slip length Ls gives an estimate of the size of the inner region. The slope of the fluid interface in the inner region determines the microscopic dynamic angle Θc. In this model, the viscous deformation of the fluid interface is confined within the inner region, so that the whole macroscopic dynamic meniscus is quasi-static (Ls is about 2-3 molecular diameters29). Under such conditions, the macroscopic dynamic contact angle Θm,o is defined as a boundary condition of the Laplace equation at the solid wall. Matching the asymptotic solutions for the inner, intermediate, and outer regions, Cox10 obtained a threeregion solution correct to Ca+1. When one of the fluids is a gas, it reads
[()
]
Qin Qout Lh µV ln + G(Θm) ) G(Θc) + γ Ls f(Θc) f(Θm)
[() ]
(6)
(7)
His dynamic contact angle θext is defined by the slope at the solid wall of the extrapolated quasi-static part of the meniscus. Lh is the drop height, half-width of the slot, or capillary radius, and C ) 1.0, 1.5, and 1.8, respectively. Voinov8 found a good approximation of G(Θ) ≈ Θ3/9, which simplifies eqs 5 and 6 within 1% accuracy if Θ < 135°. For such angles, eqs 5-7 obtain the same form:
Θ3 ) Θc3 + 9HV
γH µ
)
(9)
Values of Ls,max or Lc,max below the diameters of CCl4, tertbutyl alcohol, and OMCTS molecules will then indicate deficiencies of the theory or its application. Combined Molecular-Hydrodynamic Theories. A velocity dependence of the microscopic dynamic contact angle θc/V implies friction located at the contact line. Voinov8 analyzed the energy conservation during the wetting process and found the following relationship between the friction force Fc and Θc:
Fc ) cos ΘY - cos Θc γ
(10)
Brochard-Wyart and de Gennes15 compared the viscous Fv and the contact line Fc components of the total dissipative force F. They evaluated Fv from the lubrication theory valid for small contact angles and Fc from the linearized Blake-Haynes theory that holds for small contact line velocities:
F ) Fv + Fc )
3µV ln(Lh/Lc) kTV + Θ K λ3
(11)
o,s
Petrov and Petrov14 combined Voinov’s hydrodynamic solution for large contact angles and the nonlinearized Blake-Haynes expression for Fc,
Fc )
(8)
For eq 5, Θ ) Θm,o, for eq 6, Θ ) Θm, and for eq 7, Θ ) Θext. For eq 5, H ) (µ/γ) ln(Lh/Ls), and for eqs 6 and 7 H represents the corresponding expression in the brackets multiplied by µ/γ. Since Lh and Ls should not significantly differ for V > 0 and V < 0, H and dθm,o3/dV in eq 5 should be (practically) the same for wetting and dewetting. This requirement holds also for eq 7, because C is a numerical constant that does not depend on the direction of the contact line motion. All of the above theories assume that Θc does not depend on contact line velocity and take Θc ) ΘY. For nonideal (29) Thompson, P.; Robbins, M. Phys. World 1990, 3, 35.
(
Ls,max ) Lh,max exp -
Here the macroscopic dynamic angle Θm represents the boundary condition of the quasi-static fluid interface in the outer region. Qin is an integration constant of the profile of the inner region depending on Θc, µ, and the particular slip law. Qout is the integration constant for the outer region profile; it is a function of Θm, µ, and the geometry of the system. Voinov8 excluded the tip of the meniscus at a cutoff distance Lc from the contact line, that should be close to the dimensions of the liquid molecules. He matched the solutions for the intermediate and outer regions for small drops, narrow slots, and capillaries, where gravity is negligible, and obtained an equation valid for contact angles from 30° to 135°:
Lh µV ln -C Θext3 ) Θc3 + 9 γ Lc
surfaces, one could use the maximum static advancing ΘA,max and the minimum static receding ΘR,min contact angles.10 Thus, independent of the choice of Θc, eqs 5 and 7 predict Θ3/V plots that are symmetric for V < 0 and V > 0 and linear if Θc ) constant. A Θc(V) dependence would yield nonlinear initial sections of the Θ3/V plots. The most general hydrodynamic theory of Cox10 cannot explicitly predict the theoretical Θ/V trend, because the integration constants Qin and Qout in eq 6 are unknown. If Qin(Θc)/f(Θc) is independent of Θc and Qout(Θm)/f(Θm) does not depend on Θm, that is, if the second and the third term in the brackets do not vary with V, the requirement for symmetric linear Θ3/V plots at V < 0 and V > 0 would apply also for the Cox solution. Determining H from eq 8 enables evaluation of Ls or Lc if Lh is known for the given system. Since such specification is difficult, values of H are usually reported. However, we can estimate the maximum possible slip length Ls,max or the maximum possible cutoff distance Lc,max substituting Lh with the maximum dimension of the outer region, as maximum meniscus or drop height, capillary radius, halfwidth of the slot, and so forth.
(
)
2kT V arsh 2 2K λ o,sλ
(12)
thus obtaining
{
[ ( ) ]}
(13a)
V 2kT arsh 2 2K γλ o,sλ
(13b)
Lh µV ln -C Θext ) Θc3(V) + 9 γ Lc
[
Θc(V) ) arccos cos ΘY -
(
1/3
)]
The above molecular-hydrodynamic theories account for the viscous friction in the intermediate region of the meniscus as well as for the contact line friction. Using the Blake-Haynes theory in both cases implies that Fc(V) is
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a symmetric function for V < 0 and V > 0. The more complex Blake expression for Fc,
Fc )
[
]
V 2kT arsh 2 2Ko,sλ/(µv/h) λ
(14)
yields a more general version of the molecular-hydrodynamic theory. Except the contact line friction and the viscous friction in the intermediate region, it accounts also for the viscous friction in the inner region:
{
[ ( ) ]}
Lh µV ln -C Θext ) Θc3(V) + 9 γ Lc
liquid CCl4 (CH3)3COH OMCTS
surface tension γ density F viscosity µ molecular volume [P] v ) M/FNA [cm3] [mN/m] [g/cm3] 26.7 20.0 17.7
Ko,s )
(15a)
])]
Vo,s )
Both versions can be presented in the same mathematical form:
[
Θc(V) ) arccos cos ΘY - A arsh
(16a)
(BV)]
(16b)
Equation 16b should be read with A ) 2kT/γλ2 and B ) 2Ko,s(1)λ for the simple molecular-hydrodynamic theory or A ) 2kT/γλ2 and B ) 2Ko,s(2)λ/(µv/h) for the more general version. The value of ΘY for an ideal solid surface or Θo for a nonideal one can be determined experimentally, and A, B, and H can be obtained as free parameters of the fit of eqs 16 to the experimental Θext/V data. This directly yields λ and Ko,s(1) or λ and Ko,s(2). If C is specified, one could estimate the maximum possible cutoff length Lc,max from eq 9. Lh,max is substituted with the maximum capillary height of the meniscus Lo(Θo ) 0°), that depends on the fiber radius. Since C varies between 1.0 and 1.8 for small drops, narrow slots, and capillaries, one could use a similar value for thin fibers, where gravity effects are also negligible. Dependence of the Wetting Dynamics Parameters on the Properties of the System. The inadequate values of λ and Ko,s sometimes reported in the literature may mean that these quantities should not be considered separately, because their combinations are representative for wetting dynamics.34 The parameter A ) 2kT/γλ2 in eqs 4 and 16b gives the ratio between the thermal and surface free energy per molecule, and B represents the ratio between the measured contact line velocity V and a characteristic velocity that is defined as Vo,s(1) ) Ko,s(1)λ for the simple version and Vo,s(2) ) 2Ko,s(2)λ for the more general versions of the theory. In eq 11, the second term on the right-hand side, kT/Ko,sλ3, is the coefficient of proportionality between friction force Fc and velocity V and can be designated as the friction coefficient ξc of a unit contact line length. Both the simple and the more general versions of the molecular-kinetic and molecular-hydrodynamic theories imply that λ, Ko,s, and Θo are independent parameters. This feature suggests that Vo,s and ξc should be also independent of Θo. However, one intuitively expects such dependence. The density of adsorption centers and the distance λ between them should affect the solid surface wettability causing variation of λ with work of adhesion Wa ) γ(1 + cos ΘY). Ko,s, Vo,s, and ξc should also vary with the strength of the solid-liquid and liquid-liquid interactions, which determine Wa. Recently Blake and De
1.60 × 10-22 1.56 × 10-22 5.16 × 10-22
0.00941 0.04312 0.0230
(
)
λ2Wa kT exp h kT
(17)
Substituting eq 17 in the definitions of Vo,s and ξc gives
2kT V arsh Θc(V) ) arccos cos ΘY 2 2Ko,sλ/(µv/h) γλ (15b)
Θext ) [Θc3(V) + 9HV]1/3
1.549 0.7887 0.955
Coninck30 quantified this idea:
1/3
([
[
Table 1. Physicochemical Properties of the Liquids
( ) ( )
λ2Wa λkT exp h kT
ξc )
(18)
λ2Wa h exp kT λ3
(19)
According to eqs 17, 18, and 19, the semilogarithmic plots of Ko,s, Vo,s, and ξc versus Wa should be linear if λ does not depend on Wa. If λ ) λ(Wa), one should observe linearity of the plot lg Ko,s versus λ2Wa/kT with a slope of -0.434 and an intercept lg (kT/h) ) 12.8. III. Experimental Section Materials. Optical silica fibers with a radius R ) 60 µm and a length of 4 cm were dip-coated from a 1% w/v solution of AF 1600 in FC75 (3 M fluorocarbon solvent). One minute after immersion, they were withdrawn at a speed of 0.5 cm/s, and the solvent was allowed to evaporate in a laminar flow cabinet for 1 h. Carbon tetrachloride with 99.9% purity from Aldrich and tertbutyl alcohol with 99.5% purity from Merck were used as received from the suppliers. Their properties are listed in Table 1 together with those of OMCTS studied in our previous publication.23 The static advancing and receding contact angles are shown in Table 2. The values of ΘA and ΘR were read 1 min after stopping the motion of the solid substrate. The static angle hysteresis ΘA ΘR that characterizes the nonideality of the solid surface is small and the same for all liquids, but the effective equilibrium contact angle Θo and the work of liquid-solid adhesion Wa significantly differ from system to system. Experimental Methods. The profiles of the liquid menisci were recorded as described previously.23 Their Z, X coordinates were fitted by a numerical solution of the Laplace equation, using the algorithm detailed by Huh and Scriven.31 The analytical solution of Derjaguin,32 valid with 1% accuracy for thin fibers with R/(γ/Fg)1/2 < 0.1, was also applied, as done previously:33
[
(
Z ) r cos Θext 0.462 + ln
(γ/Fg)1/2
)]
X + xX2 - r2 cos2 Θext
(20)
Under the experimental conditions of this study, the two procedures yield fits that overlap within the pixel resolution over distances up to 300 µm from the solid wall.
IV. Results and Discussion Comparison between Theory and Experiment. Figure 1 shows the dependence of the extrapolated (30) Blake, T. D.; De Coninck, J. Adv. Colloid Interface Sci. 2002, 96, 21. (31) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85. (32) Derjaguin, B. V. Dokl. Akad. Nauk SSSR (in Russian) 1946, 51, 517. (33) Petrov, J. G.; Sedev, R. V. Colloids Surf. 1993, 74, 233. (34) Scarpe, M. R.; Peterson, I. R.; Tatum, J. P. Langmuir 2002, 18, 3549.
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Table 2. Static Wetting of AF 1600 by the Liquids Used
liquid
static advancing contact angle, ΘA [deg]
static receding contact angle, ΘR [deg]
static contact angle hysteresis, ΘA - ΘR [deg]
effective equilibrium contact angle, Θo [deg]
work of adhesion γ(1 + cos Θo) [mJ/m2]
CCl4 (CH3)3COH OMCTS
60.4 53.0 42.0
55.8 48.2 37.3
4.6 4.8 4.7
58.1 50.6 39.7
40.8 32.7 31.3
Figure 1. Dependence of the extrapolated dynamic contact angle Θ on contact line velocity V for CCl4, tert-butyl alcohol, and OMCTS advancing (V > 0) or receding (V < 0) on an AF 1600 coating on thin silica fibers.
Figure 2. Presentation of our data in the characteristic molecular-kinetic plot cos Θ/V requiring symmetry of the advancing and receding branches (see eq 4). Such a symmetry is observed for CCl4, but tert-butyl alcohol and OMCTS exhibit strongly asymmetric cos Θ/V dependencies. The solid lines show the fit of eq 4 to the whole set of the data for CCl4 or to their low-velocity parts for tert-butyl alcohol and OMCTS. Table 3. Parameters of the Molecular-Kinetic Theory Obtained from the Best Fit of Equation 4 to the Data in Figure 2a
Figure 3. Hydrodynamic Θ3 vs V plots. If Θc ) constant as all analyzed hydrodynamic theories assume, the Θ/V data should linearize in this scale giving the same dΘ3/dV at advancing and receding.
Figure 4. Experimental and theoretical Θ/V dependencies according to the molecular-hydrodynamic theory. The solid lines represent the fit of eqs 16 following the experimental trend in the entire velocity range studied. Table 4. Comparison of the Parameters of the Hydrodynamic Plot Θ3 versus V Obtained from the Linear Sections in Figure 3 Lc,max (receding) [cm]
Lc,max (advancing) [cm]
CCl4 5.4 ( 0.2 (1.0 ( 2.6) × 102 (2.3 ( 4.8) × 103 58.9 ( 0.2 (CH3)3COH 5.0 ( 0.4 (1.1 ( 1.7) × 103 (1.1 ( 1.7) × 106 51.2 ( 0.4 OMCTS 5.0 ( 0.4 (7 ( 4) × 103 (1.2 ( 0.7) × 107 39.5 ( 0.3
CCl4 0.109 ( 0.014 0.070 ( 0.008 2.6 × 10-17 (CH3)3COH 0.253 ( 0.010 0.277 ( 0.024 5.6 × 10-8 OMCTS 0.128 ( 0.004 0.148 ( 0.008 4.2 × 10-7
5.7 × 10-12 1.6 × 10-8 7.6 × 10-8
a For tert-butyl alcohol and OMCTS, the best fit was obtained consecutively omitting the final left and right points until a minimum χ2 value was achieved. b λ ) [2kT/γA]1/2; Ko,s(1) ) B/2λ; Ko,s(2) ) Ko,s(1)(µv/h).
branches as required by the theory. Equation 4 (the solid line) simultaneously fits all points at V < 0 and V > 0. The dependence for tert-butyl alcohol is qualitatively different. It is strongly asymmetric with respect to V ) 0, and its receding and advancing branches cannot be fitted by a single curve in the entire velocity range. A partial fit was obtained via consecutive skipping of the final left and right points until a minimum χ2 value was found. The solid line representing eq 4 starts deviating from the experimental data at lower velocities for the receding mode than for the advancing mode. Similar asymmetry and disagreement between eq 4 and the experimental data at high velocities can also be seen for OMCTS. The fits in Figure 2 give the free parameters A and B, which enable calculation of the values of λ, Ko,s(1), and
liquid
λ
[nm]b
Ko,s(1)
[s-1]b
Ko,s(2)
[s-1]b
Θo [deg]
dynamic contact angle Θ on contact line velocity V for CCl4, tert-butyl alcohol, and OMCTS. The latter was taken for comparison from our previous investigation.23 Each point is an average of 20 measurements corresponding to subsequent positions of the wetting perimeter along the fiber. The small standard deviations at the low velocities illustrate the good quality of the fluoropolymer surface. Figure 2 presents the data in the characteristic molecular-kinetic plot cos Θ/V. The carbon tetrachloride shows a symmetry of the receding and the advancing
liquid
dΘrec3/dV
dΘadv3/dV
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Table 5. Parameters of the Molecular-Hydrodynamic Theory Obtained from the Best Fit of Equations 16 to the Data in Figure 4
a
liquid
λ [nm]a
Ko,s(1) [s-1]a
Ko,s(2) [s-1]a
Lc,max [cm]a
Θo [deg]
CCl4 (CH3)3COH OMCTS
8.1 ( 0.2 9.3 ( 0.3 7.0 ( 0.5
0.050 ( 0.002 0.076 ( 0.015 707 ( 580
11.3 ( 0.4 77.1 ( 15.2 (1.27 ( 1.04) × 106
2.7 × 10-12 7.9 × 10-8 3.3 × 10-6
59.1 ( 0.2 50.8 ( 0.3 40.0 ( 0.2
λ ) [2kT/γA]1/2; Ko,s(1) ) B/2λ; Ko,s(2) ) Ko,s(1)(µv/h); Lc,max ) Lo,r exp(γH/9µ).
Figure 5. Dependence of the equilibrium oscillation frequency Ko,s (a), characteristic molecular-kinetic velocity Vo,s (b), and coefficient of friction of the unit length of the contact line ξc (c) on the work of liquid-solid adhesion Wa. The squares correspond to the simple theory (1), and the circles to its more general version (2).
Ko,s(2) shown in Table 3. When the static contact angle Θo is set as a free parameter, the obtained values coincide within the error limits with the effective equilibrium angle (cf. Tables 2 and 3). Such a coincidence supports the suggestion to substitute ΘY in eq 4 by Θo, when nonideal surfaces are studied. No difference (within the error limits) was found between the values of λ for CCl4, tert-butyl alcohol, and OMCTS, although the molecular diameters considerably differ from each other (v1/3 ) 0.54 nm for CCl4 and tert-butyl alcohol vs 0.80 nm for OMCTS). Moreover, λ exceeds the molecular dimensions by almost an order of magnitude. For all compared systems, Ko,s(2) . Ko,s(1), which shows that application of the more complex molecular-kinetic theory might increase the very small values of Ko,s(1) reported in the literature. Figure 3 presents the data in the characteristic hydrodynamic scale Θ3 versus V. All plots are linear for V > 0
Figure 6. Presentation of Ko,s (a), Vo,s (b), and ξc (c) versus λ2Wa/kT. The squares correspond to the simple theory (1), and the circles to its more general version (2).
and V < 0 except for the initial sections. The latter suggest velocity-dependent microscopic dynamic angles Θc(V) for all systems studied. The difference between the intercepts of the linear parts and the static contact angles ΘA and ΘR (shown by full points) supports the conclusion for contact line friction dominating at low velocities. For CCl4, the receding and advancing linear parts have different slopes and the maximum cutoff lengths Lc,max calculated from eq 9 are unrealistic (Table 4). In contrast to this system, tert-butyl alcohol and OMCTS exhibit close values of dΘ3/dV at receding and advancing that give reasonable values of Lc,max. Summarizing the conclusions from Figures 2 and 3, one finds that the wetting-dewetting dynamics of CCl4 is well represented by the molecular-kinetic theory, but it cannot describe the receding and advancing of tert-butyl alcohol and OMCTS on AF 1600 at high velocities. For these two liquids, the hydrodynamic theory better agrees
Dynamics of Partial Wetting and Dewetting
with experiment at high velocities, but the nonlinear initial sections of the Θ3/V plots require consideration of the dependence Θc(V). Figure 4 checks the molecular-hydrodynamic theory which accounts for the Θc(V) dependence. The solid lines represent the simultaneous fit of eqs 16 to the advancing and receding branches of the Θ/V data. They yield unique values of A, B, and H for each system and λ, Ko,s, and Lc,max that do not depend on direction of the contact line motion. Good correspondence between the theoretical and experimental Θ/V dependence is observed in the entire velocity range. Moreover, the molecular-hydrodynamic theory describes the wetting-dewetting asymmetry. All these features satisfy the basic assumptions of this theoretical model. Table 5 lists the values of microscopic dynamic parameters, calculated from the fits in Figure 4. Setting Θo as a free parameter again gives values that are very close to the effective equilibrium angles from Table 2. The values of λ again significantly exceed the molecular dimensions, being even higher than those in Table 3. The smallest λ is found for the largest OMCTS molecule, while CCl4 and tert-butyl alcohol having the same dimensions show different values of λ. These facts suggest that λ has a more complex meaning as defined by the molecular-kinetic theory. Unreasonably small values of Ko,s(1) are obtained for CCl4 and tert-butyl alcohol, but the more general combined theory gives higher Ko,s(2) values. The maximum cutoff lengths for tert-butyl alcohol and OMCTS are acceptable, while Lc,max for CCl4 is unrealistic. The molecular-hydrodynamic theory describes the trend of the
Langmuir, Vol. 19, No. 7, 2003 2801
experimental data for all considered systems, but the values of the obtained dynamic parameters are plausible only for tert-butyl alcohol and OMCTS. At present, we cannot explain such a difference. Dependence of the Microscopic Dynamic Parameters on the Work of Adhesion. Figure 5 presents the semilogarithmic plots of Ko,s, Vo,s, and ζc versus Wa. Only the results from Table 5 are considered, because only the molecular-hydrodynamic theory covers the entire velocity range and gives values of λ, Ko,s, and Lc,max which are independent of the direction of the contact line motion. The more complex version designated with (2) yields more realistic values as compared to the simple (1) theory. It is curious that the characteristics of wetting-dewetting dynamics of OMCTS differ by more than an order of magnitude from those of tert-butyl alcohol and CCl4. The same data are plotted in Figure 6 with the abscissa λ2Wa/ kT. The trend in Figure 6a is nonlinear, but even if an average straight line would be drawn the slopes are much below the expected value of -0.434. The small number of systems compared here does not allow sound conclusions about the precise trend of the variation of Ko,s, Vo,s, and ζc with Wa. The strong difference between the values found for OMCTS and the other two liquids is interesting but should be confirmed for other systems. Such an investigation including four new pure liquids on the same substrate has been performed and submitted for publication in the Journal of Physical Chemistry B. LA026692H
