Langmuir 1999, 15, 3365-3373
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Dewetting Dynamics on Heterogeneous Surfaces. A Molecular-Kinetic Treatment Jordan G. Petrov,* John Ralston, and Robert A. Hayes Ian Work Research Institute, University of South Australia, Mawson Lakes, SA 5095, Australia Received September 8, 1997. In Final Form: February 12, 1999 When the molecular-kinetic theory of wetting dynamics on ideal surfaces is applied to heterogeneous solid substrates, effective values of its specific parameters λ and K0 are obtained. This paper shows that application of the Blake-Haynes equation to the velocity dependence of dynamic contact angles of water on partially methylated silica yields strong dependencies of λeff and K0eff on static wettability and degree of methylation. Such dependencies can be explained by the molecular-kinetic model of dewetting dynamics on smooth heterogeneous surfaces proposed in this study. Satisfactory agreement between theory and experimental data is observed below 70% methylation, but the values of λeff and K0eff obtained at more complete coverage do not follow the theoretical predictions. At this stage we speculate that this disagreement might be due to formation of a rough coating at higher concentrations of the methylating agent and longer reaction times. Such an interpretation is based on literature data from ellipsometric and AFM studies and elemental analyses of silica surfaces methylated with trimethylchlorosilane.
I. Introduction Real solid surfaces are heterogeneous and rough. A polycrystalline solid exposes different walls, edges, and corners of single crystals at the phase boundary; a surface of an amorphous polymer contains various functional groups or regions of different macromolecular conformation. Scratches, cracks, and pores are almost always present on solid surfaces. In many cases surface heterogeneity is of vital importance. Solid catalysts owe much of their specific functions to the complex chemical composition and physical texture of their surfaces. Lithographic plates reproduce images because of the locally modified wettability. A smooth and homogeneous surface in contact with a gas and a partially wetting liquid possesses a single-valued equilibrium contact angle, Θ0, determined by the specific interfacial free energies γSG, γSL, and γLG according to the Young equation:1
cos Θ0 )
γSG - γSL γLG
(1)
The equilibrium angle on a heterogeneous surface, Θ0,C, composed of smooth homogeneous patches with Young angles Θ0,i depends on their area fractions fi, satisfying the Cassie equation2
cos Θ0,C )
∑i fi cos Θ0,i
(2)
A variety of (metastable) static contact angles ranging between a maximum advancing value, ΘA, and a minimum receding value, ΘR, can be obtained on a real solid surface. Theories relating the static hysteresis, ΘA-ΘR, to surface heterogeneity or roughness have been reported for * Corresponding author. On leave of absence from Institute of Biophysics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 21, 1113 Sofia, Bulgaria. (1) Young, T. Philos. Trans. R. Soc. London 1805, 95, 65. (2) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11.
periodic3-5 and random6-8 nonideality patterns (recent review in ref 9). Many of the existing theories of dynamic contact angles neglect surface heterogeneity or account for it implicitly. The molecular-kinetic theory10,11 and its extensions12-17 consider smooth homogeneous surfaces with single-valued static contact angles. Some hydrodynamic theories18,19 also relate solid surface effects to the Young angle; others recommend using advancing or receding static angles, thus implicitly accounting for surface nonideality through the static contact angle hysteresis.20 Raphael and de Gennes21 considered the interaction of a three-phase contact line moving in a capillary with a single solid surface defect, surrounded by a gradual variation of wettability. Joanny and Robbins22 analyzed the dipping of a plate with periodic heterogeneity. Both papers reported relationships between dynamic contact angle and contact line velocity that hold for a low defect density and almost ideal surfaces. Shanahan23 described the “shading” of the defects by each other, and di Meglio (3) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744. (4) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38, 341. (5) Schwartz, L. W.; Garoff, S. Langmuir 1985, 1, 219. (6) Joanny, J. F.; de Gennes, P. G. J. Chem. Phys. 1984, 81, 552. (7) Pomeau, Y.; Vannimenus, J. J. Colloid Interface Sci. 1985, 104, 477. (8) Robbins, M. O.; Joanny, J. F. Europhys. Lett. 1987, 3, 729. (9) Marmur, A. Adv. Colloid Interface Sci. 1994, 50, 121. (10) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (11) Ruckenstein, E.; Dunn, C. S. J. Colloid Interface Sci. 1977, 59, 135. (12) Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1981, 259, 753. (13) Hoffman, R. J. Colloid Interface Sci. 1983, 94, 470. (14) Blake, T. D. AICHE International Symposium on the Mechanics of Thin-Film Coating, March 6-10, 1988, New Orleans, LA, Paper la. (15) Petrov, P. G.; Petrov, J. G. Langmuir 1992, 8, 176. (16) Ruckenstein, E. Langmuir 1992, 8, 3038. (17) Blake, T. D. In Wettability; Berg J. C., Ed.; Marcel Dekker Inc.: New York, 1993; Chapter 5. (18) Voinov, O. V. Fluid Dyn. (Engl. Transl.) 1976, 11, 714. (19) de Gennes, P. G. Colloid Polym. Sci. 1986, 264, 463. (20) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (21) Raphael, E.; de Gennes, P. G. J. Chem. Phys. 1989, 90, 7577. (22) Joanny, J. F.; Robbins, M. O. J. Chem. Phys. 1990, 92, 3206. (23) Shanahan, M. E. R. C. R. Acad. Sci. Paris 1991, 313, 613.
10.1021/la971012+ CCC: $18.00 © 1999 American Chemical Society Published on Web 04/03/1999
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and Shanahan24 showed that taking into account this phenomenon could explain some dynamic contact angle data on hydrophilic surfaces with nonwetting patches. The effect of the microscopic properties of a smooth homogeneous surface on wetting dynamics has been explicitly considered by the molecular-kinetic theory of Blake and Haynes.10 It relates the dynamic contact angle, Θd, to contact line velocity, V, Young angle, Θ0, equilibrium oscillation frequency, K0, of a molecule between adjacent adsorption centers on both sides of the wetting perimeter, and distance between the centers, λ:
cos Θd ) cos Θ0 -
( )
2kT V arsinh h 2 γλ 2K0λ
(3)
The oscillation frequency K0 depends on the activation free energy of the oscillations, ∆g*, which is constant if the centers are monoenergetic:
K0 )
kT -∆g*/kT e h
(4)
When this theory is applied to dewetting or wetting of real surfaces with contact angle hysteresis, the minimum static receding angle, ΘR, and respectively the maximum advancing one, ΘA, are usually substituted for Θ0 in eq 3 (e.g. refs 25-29). Some authors30 prefer the mean value (ΘR + ΘA)/2, adhering to earlier ideas31-33 that it can be considered as an effective Young angle which better characterizes the static wettability of a real surface.64 In another publication34 two of the present authors examined the possibility of leaving Θ0 as a third free parameter which can be obtained from the simultaneous fit of eq 3 to both the advancing and receding dynamic contact angles. Thus a single valued static angle is obtained, but it also has an effective meaning for a heterogeneous surface. Moreover, this procedure yields physically unreasonable values of K0 for surfaces with significant contact angle hysteresis.34 The single-valued static contact angle, Θ0, and the specific microscopic parameters of the molecular-kinetic theory, λ and K°, characterize an ideal solid surface. If eq 3 is applied for real surfaces, λ and K0 become effective quantities that are expected to depend on the degree of surface nonideality similarly to the effective Young angle. This paper deals with the relationship of λeff and K0eff to solid surface heterogeneity. We consider recession of a pure liquid (water) against a pure gas (nitrogen) on partially methylated silica with different hydrophobicity. Previous experimental dependencies of receding dynamic angles versus contact line velocity35,36 and data on the relationship between the static contact angles and the (24) di Meglio, J.-M.; Shanahan, M. E. R. C. R. Acad. Sci. Paris 1993, 316, 1543. (25) Schwartz, A. M.; Tejada, S. B. J. Colloid Interface Sci. 1972, 38, 359. (26) Gribanova, E. V.; Molchanova, L. I. Colloid J. 1978, 40, 30 and 217 (in Russian). (27) Hopf, W.; Stechemesser, H. Colloids Surf. 1988, 33, 25. (28) Petrov, J. G.; Petrov, P. G. Colloids Surf. 1992, 64, 143. (29) Hayes, R. A.; Ralston, J. J. Colloid Interface Sci. 1993, 159, 429. (30) Redon, C.; Brochard, F.; Rondelez, F. Phys. Rev. Lett. 1991, 66, 715. (31) Adam, N. K.; Jessop, G. J. Chem. Soc. 1925, 127, 1863. (32) Adam, N. K. Adv. Chem. Ser. 1964, 43, 53. (33) Wolfram, E.; Faust, R. In Wetting, Spreading and Adhesion; Padday, J. F., Ed.; Academic Press: London, 1978; p 213. (34) Hayes, R. A.; Ralston, J. Langmuir 1994, 10, 340. (35) Newcombe, G. Wetting Dynamics and Aqueous Film Drainage on Silica Surfaces. MAppSci-Thesis, South Australian Institute of Technology, 1989. (36) Newcombe, G.; Ralston, J. Langmuir 1992, 8, 190.
degree of methylation of silica37,38 are reanalyzed on the basis of eqs 3 and 4. Strong correlations between λeff, K0eff and the static contact angle, respectively, degree of methylation are obtained. This behavior does not follow from the homogeneous model,10 where λ, K0, and Θ0 should be independent parameters. The extension of the molecular-kinetic theory of wetting dynamics for smooth heterogeneous surfaces presented in this study explains this dependence. II. Experimental Section 1. Hydrophilic Silica Surface. Infrared spectra of completely hydroxylated silica surfaces39-41 show a sharp peak near 3750 cm-1, indicating freely vibrating (“free”) hydroxyls and a broad band with a maximum near 3520 cm-1 attributed to (“bound”) OH groups connected by mutual H bonds. The total density of surface silanols for such surfaces is usually 4.6 ( 0.5 OH/nm2, although the scatter of the reported data is considerable.41,42 This number is claimed to be insensitive to the origin, particle size, and structural characteristics of silica but depends on the method of preparation and the pretreatment temperature of the sample. These factors affect also the ratio of the densities of “free” and “bound” hydroxyls on the silica surface. The silica samples used to obtain the experimental data reanalyzed in this study were Suprasil disks with a diameter of 1 cm. They were polished to a λ/4 degree of smoothness and did not exhibit roughness at the limit of optical resolution. Tubes of the same material with internal radii between 1.25 and 4.5 mm were used in parallel experiments. The cleaning procedure was of crucial importance to ensure good wettability. The latter was checked through a “steam test” in which condensation of water as a film with uniform interference fringes indicated the cleanliness of the pure silica surface.43 Fully hydroxylated silica is covered by a thick water film formed atop of a physically adsorbed water layer. 2. Methylated Silica Surfaces with Varying Hydrophobicities. After a chemical reaction with trimethylchlorosilane (TMCS) the silica surface becomes hydrophobic due to replacement of the “free” and some of the “bound” surface hydroxyls by large umbrella-shaped (CH3)3 groups. Sagiv44 proposed the mechanism for this reaction when it occurs in organic solvent at room temperature. TMCS is first hydrolyzed by dissolved or adsorbed water
HO-H + Cl-Si-(CH3)3 f HO-Si-(CH3)3 + HCl and the obtained alkylsilanol condenses with the surface silanols Sis-OH: (37) Crawford, R.; Koopal, L. K.; Ralston, J. Colloids Surf. 1987, 27, 57. (38) Blake, P.; Ralston, J. Colloids Surf. 1985, 15, 101. (39) Davydov, V. Ya.; Kiselev, A. V.; Zhuravlev, L. T. Trans. Faraday Soc. 1964, 60, 256. (40) Kiselev, A. V.; Lygin, V. I. Infrared Spectra of Surface Compounds; Wiley-Interscience: New York, 1975. (41) Iler, R. K. The Chemistry of Silica; Wiley-Interscience: New York, 1979. (42) Vansant, E. F.; Van der Voort, P.; Vrancken, K. C. Characterization and Chemical Modification of the Silica Surface; Elsevier: Amsterdam, 1995. (43) Vig, J. R.; LeBus, J. W.; Filler, R. L. Proc. Annu. Symp. Freq. Control 1975, 29, 220. (44) Sagiv, J. J. Am. Chem. Soc. 1980, 102, 92.
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Sis-OH + HO-Si-(CH3)3 f Sis-O-Si-(CH3)3 + H2O Using a thin film IR technique enabling spectra between 1300 and 200 cm-1 to be obtained, Tripp and Hair45,46 gave convincing evidence in support of this mechanism.65 The degree of methylation of the silica surface, φ, is defined as the ratio of the actual number of trimethyl groups bonded to a unit area, Γ(CH3)3, to the maximum number, Γ(CH3)3,max achieved at saturation. This quantity depends on the surface population of different hydroxyls (the “free” silanols are much more reactive than the “bound” ones) and on the cross-sectional area of a trimethyl group, Acs, which determines Γ(CH3)3,max if enough reactive silanols are present. Morrow and McFarlan48 found that a maximum of 1.4 (CH3)3 groups/nm2 could be attached to the silica surface after preheating at 150 °C and 1 h of reaction. This number was independent of the initial silanol density; it was the same for pyrogenic aerosil silica exchanging 2.5 OH/nm2 and a nonporous precipitated silica exchanging 5.6 OH/nm2. In both cases 1.1 “free” OH/nm2 and 0.3-0.4 “bound” OH/nm2 have been substituted by trimethyl groups. Other literature sources41,42 give higher values of Γ(CH3)3,max up to 2.8 (CH3)3 groups/nm2. Morrow and McFarlan48 explained this difference by the fast initial uptake observed by Snyder and Ward49 and Sindorf and Maciel.50 Thus, for the conditions of methylation used in refs 3537 one could expect Γ(CH3)3,max ≈ 1.4 ( 0.1 (CH3)3 groups/ nm2. After overnight preheating in air at 140 °C to remove the adsorbed water,41 the samples were methylated in dry N2 atmosphere with cyclohexane solutions of TMCS or tert-BDMCS. Afterward they were rinsed five times with cyclohexane, dried at 100 °C overnight in an oven, cooled in a vacuum desiccator, and used immediately. Surfaces with different hydrophobicities were obtained by varying the solution concentration from 0.5 to 10 v/v % and the reaction time from 30 to 60 min. The static wettability of Suprasil plates optically polished and methylated in the same way has been studied in a separate paper.37 The dependence of the static receding and advancing contact angles on degree of methylation, φ, was experimentally determined. The latter quantity is a measure of surface heterogeneity similar to the area fractions of bare and methylated silica surface, f1 and f2, entering the Cassie equation. 3. Dewetting Experiments on Partially Methylated Silica. Effective Parameters of the MolecularKinetic Theory. The results of two types of dewetting experiments will be analyzed here. The first concerns a spontaneous expansion of the three-phase contact line formed when a nitrogen bubble touches a methylated silica disk in a cell filled with water (conductivity less than 8 × 10-7 Ω-1, pH ) 5,6 and γ ) 72 mN/m at 25 °C). In the second, nitrogen displaces water in a methylated Suprasil tube under a pressure gradient. Profiles of the fluid interface were recorded photographically, and static and dynamic contact angles were measured goniometrically.35,36 It was shown36 that the velocity dependencies of the receding dynamic contact angles obtained in the above (45) Tripp, C. P.; Hair, M. L. Langmuir 1991, 7, 923. (46) Tripp, C. P.; Hair, M. L. Langmuir 1995, 11, 149. (47) Menewat, A.; Henry, J., Jr.; Sirawadane, R. J. Colloid Interface Sci. 1984, 101, 110. (48) Morrow, B. A.; McFarlan, A. J. Langmuir 1991, 7, 1695. (49) Snyder, L. R.; Ward, J. W. J. Phys. Chem. 1966, 70, 3941. (50) Sindorf, D. W.; Maciel, G. E. J. Phys. Chem. 1982, 86, 5208.
Figure 1. Examples of experimental dependencies of the receding dynamic contact angle on the contact line velocity for water receding against nitrogen on partially methylated silica surfaces with different hydrophobicities. The static receding angles are indicated in the figure. The solid lines represent the nonlinear least-squares fits of the Blake-Haynes equation to the experimental data. Table 1. Effective Parameters of the Molecular-Kinetic Theory10 Determined from Eq 3 Substituting Θr for Θd and ΘR for Θ0 ΘR (deg)
λeff (nm)
K0eff (s-1)
34 49 55 59 63 67 70 76
2.2 ( 0.2 1.6 ( 0.1 1.5 ( 0.1 1.4 ( 0.1 1.3 ( 0.05 1.1 ( 0.05 0.96 ( 0.1 0.84 ( 0.05
(6.6 ( 8.0) × 104 (2.2 ( 1.8) × 105 (5.1 ( 1.2) × 105 (1.8 ( 1.5) × 106 (9.3 ( 3.4) × 105 (5.2 ( 1.4) × 106 (1.2 ( 1.0) × 107 (2.4 ( 0.7) × 107
Table 2. Effective Parameters of the Molecular-Kinetic Theory10 Determined from Eq 3 Substituting Θr for Θd and Θ0,eff ) (ΘR + ΘA)/2 for Θ0 ΘR (deg)
Θ0,eff (deg)
λeff (nm)
K0eff (s-1)
34 49 55 59 63 67 70 76
37.7 54.7 59.3 63.5 66.8 71.6 74.1 81.8
2.2 ( 0.2 1.7 ( 0.1 1.5 ( 0.1 1.5 ( 0.1 1.4 ( 0.05 1.2 ( 0.05 1.0 ( 0.05 1.0 ( 0.05
(1.3 ( 1.7) × 104 (5.5 ( 5.2) × 104 (1.2 ( 0.6) × 105 (3.5 ( 2.1) × 105 (2.6 ( 1.0) × 105 (9.6 ( 3.0) × 105 (4.1 ( 1.8) × 106 (3.5 ( 1.3) × 106
experiments were independent of the type of experiment and the geometry and dimensions of the system. No difference was found between surfaces methylated with TMCS and tert-BDMCS if the static contact angles were the same. For this reason all Θr(V) data which correspond to the same static wettability were plotted together. Figure 1 illustrates these Θr(V) dependencies, showing four of them obtained at different static contact angles. The solid lines represent the fit of eq 3, substituting Θr for Θd and ΘR for Θ0. Table 1 collects the effective parameters λeff and K0eff resulting from this fit for all surfaces studied. Table 2 gives λeff and K0eff obtained when eq 3 was fitted to the same Θr(V) dependencies, but substituting Θ0,eff ) (ΘR + ΘA)/2 for Θ0. Both sets of data show a strong correlation between λeff, K0eff, and the static contact angle, ΘR or Θ0,eff, which implies that the free parameters in eq 3 are not independent when the BlakeHaynes theory is applied to partially methylated silica surfaces with varying heterogeneity. The physical basis of such a correlation seems to be related to the systematic dependence of λeff, K0eff, ΘR, and Θ0,eff on the degree of methylation φ. The λeff(φ) and K0eff(φ) dependencies were obtained combining the data of Table 2 with the ΘR(φ) and ΘA(φ) data from ref 37. The dependence λeff(φ) shown in Figure 2 saturates at about
3368 Langmuir, Vol. 15, No. 9, 1999
Figure 2. Correlation between the effective values of λeff obtained from the fits and the experimentally determined degree of methylation of silica, φ, defined in the text. Data from Table 2.
Petrov et al.
Figure 4. Effective activation free energy, ∆g*eff, evaluated via substitution of the K0eff values from Table 2 in eq 4 versus degree of methylation of the silica surface φ.
extrapolation to φ ) 0 also gives too large a value of λOH ) 2.4 nm for the bare silica. The semilogarithmic plot of the K0eff(φ) dependence shows a deviation from linearity at φ ≈ 0.70, and the linear extrapolation to φ ) 0 gives K0OH ) 1.7 × 104 s-1 in good agreement with the value reported by Gribanova.51 III. Theoretical Analysis
Figure 3. Semilogarithmic plot of the effective oscillation frequencies K0eff obtained from the same fits versus degree of methylation φ. Data from Table 2.
0.85 degree of methylation. The plateau value of λeff ) 1.0 ( 0.05 nm just slightly exceeds the distance of 0.85 ( 0.05 nm between the centers of the maximum 1.4 chemisorbed trimethyl groups per nm-2 (see section II.2). However, a linear extrapolation to φ ) 0 gives for the bare silica surface λOH ) 2.4 ( 0.1 nm, which is much greater than the distance between the centers of all surface silanols (0.47 ( 0.03 nm for 4.6 ( 0.5 OH/nm2). It is twice the value of λOH ) 1.2 nm reported by Gribanova51 for HCl-KCl-KOH solutions of pH 5.6 advancing on fused silica against air. Such a disagreement questions the correctness of the above extrapolation assuming that the λeff(φ) dependence is linear. Figure 3 shows the K0eff(φ) relationship on a semilogarithmic scale. It also exhibits a saturation above 85% methylation and a change of the slope at approximately 75%. The mean plateau value of K0eff gives, for the completely methylated surface, K0 ) 4 × 106 s-1; the linear extrapolation of the initial part of the dependence to φ ) 0 yields K0OH ) 3 × 103 s-1 for the pure silica. This value is slightly below the value of K0OH ) 1.1 × 104 s-1 found by Gribanova51 for aqueous electrolyte solutions of pH 5.6 on fused silica. Figure 4 plots the effective activation free energy, ∆g*eff (formally calculated via substitution of the K0eff values of Table 2 in eq 4), versus degree of methylation. As one would expect, this dependence follows the trend of the K0eff(φ) relationship in Figure 3 with a change of the slope at φ ) 0.70-0.75 and a saturation above φ ) 0.85. The dependencies λeff(φ) and K0eff(φ), based on the data from Table 1, show similar features. The lowest value of λeff ) λCH3 ) 0.85 ( 0.05 nm obtained at φ ) 1.0 is equal to a distance of 0.85 ( 0.05 nm between the centers of the trimethyl groups at maximum methylation, and the linear (51) Gribanova, E. V. Adv. Colloid Interface Sci. 1992, 39, 235.
1. Molecular-Kinetic Theory of Wetting Dynamics on an Ideal Solid Surface. For the sake of comparison with the following consideration of wetting dynamics on a smooth heterogeneous surface, we will first briefly repeat the derivation of the Blake-Haynes expression for an ideal solid surface with n randomly distributed monoenergetic adsorption centers on 1 cm2. The number of centers per unit length of the wetting perimeter is n1/2. Under dynamic conditions the frequency of molecular jumps between adjacent centers, K, is the difference of the number of jumps per second in the direction of contact line motion, K+, and that in the opposite direction, K-:
K ) K + - K-
(5)
The mean transverse velocity of a unit length of the contact line, V, can be obtained by dividing the sum of the displacements occurring per second on such a length, 1/2 ∑n1 Kiλi, by the number of adsorption centers on it, n1/2. This definition is a discrete analogue to the definition of the mean velocity of the hydrodynamic flow in a flat slot, V h ) 1/h∫h0 v(y) dy. Since the centers are monoenergetic and randomly distributed, Ki ) K ) constant and λi ) λ ) n-1/2 ) constant, that yields n1/2
V)
∑1 Kiλi n1/2
)
n1/2Kλ n1/2
) Kλ
(6)
The motion of the wetting perimeter is driven by the additional force that acts on a unit length under dynamic conditions:
F ) γ(cos Θd - cos Θ0)
(7)
The force applied at each center located on the contact line, F/n1/2, multiplied by the length of the molecular jumps, λ ≈ n-1/2, gives the work done to overcome the free energy barrier, ∆g*. The corresponding change of the free energy per center,
∆g ) -
Fλ ) -γλ2(cos Θd - cos Θ0) n1/2
(8)
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Langmuir, Vol. 15, No. 9, 1999 3369
bilities, so that the portion of its length following their borders is negligible. Thus, a molecule of the liquid located at the stationary wetting perimeter oscillates between adjacent (left and right) adsorption centers of the same type. The equilibrium frequency of these oscillations per center is K01 for the “1-1” and K02 for the “2-2” jumps, and the jump lengths are λ1 and λ2, respectively.66 The mean velocity of a unit length of the composite contact line (see eq 6) is given by n1/2
V)
Figure 5. Sketch of the model smooth heterogeneous surface. The area 1 is hydrophilic, and the patches 2 are hydrophobic. The motion of the three-phase contact line (CL), indicated by arrows, corresponds to dewetting of the solid.
determines the values of K+ and K- decreasing ∆g* in the positive direction by ∆g/2 and increasing it in the opposite direction by ∆g/2. This yields the following expression for K:
K ) K + - K- )
[( [(
(
)] )]
)
Substituting eq 9 in eq 6, one finds the mean velocity of a unit length of the contact line on a smooth homogeneous solid surface:
(
)
γλ2(cos Θd - cos Θ0) VB-H ) 2K0λ sinh 2kT
(10)
2. Molecular-Kinetic Theory of Wetting Dynamics on a Smooth Heterogeneous Surface. We consider a smooth heterogeneous solid surface consisting of an area 1 with adsorption centers of type “1” and patches 2 of different wettability with adsorption centers of type “2” (Figure 5). The mean number of both centers per unit area, n, is a sum of the mean surface densities n1 and n2 of the centers “1” and ”2”, respectively:
n ) n1 + n2
(11)
The mean number of adsorption centers on a unit length of the contact line is n1/2, n1/n1/2 of them being of type “1” and n2/n1/2 of them being of type “2”:
n1/2 ) n1/n1/2 + n2/n1/2
)
(n1/n1/2)K1λ1 + (n2/n1/2)K2λ2
n1/2
n1/2
(12)
The values of n1 and n2 vary at constant n, yielding surfaces with different wettabilities. The latter are in contact with a gas and a liquid that partially wet the solid. The threephase contact line crosses the areas of different wetta-
)
ω1K1λ1 + ω2K2λ2 (13)
where ω1 ) n1/n and ω2 ) n2/n are the molar fractions of the centers “1” and “2”, respectively. If the areas 1 and 2 are smooth and homogeneous, we can use the Blake-Haynes expression for K1 and K2 (eq 9):
[ [
] ]
γλ12(cos Θr,1 - cos Θ0,1) K1 ) 2K 1 sinh 2kT
(14a)
γλ22(cos Θr,2 - cos Θ0,2) 2kT
(14b)
0
K2 ) 2K02 sinh
γλ2 (cos Θd - cos Θ0) kT 2 exp h kT 2 γλ (cos Θd - cos Θ0) ∆g* 2 kT exp ) h kT γλ2(cos Θd - cos Θ0) 2K0 sinh (9) 2kT ∆g* +
∑1 Kiλi
Substituting eqs 14a,b, in eq 13 we find the mean velocity of a unit length of the contact line on the model smooth heterogeneous surface:
(
)
γλ12(cos Θr,1 - cos Θ0,1) + 2kT γλ22(cos Θr,2 - cos Θ0,2) (15) 2ω2K02λ2 sinh 2kT
V ) 2ω1K01λ1 sinh
(
)
If area 1 is completely wettable, Θr,1 ) Θ0,1 ) 0°, and eq 15 becomes
V ) 2ω2K02λ2 sinh
(
)
γλ22(cos Θr,2 - cos Θ0,2) 2kT
(16)
As far as the contact angles on composite surfaces are usually measured, we substitute Θr,2 and Θ0,2 in eq 16 by Θr,c, and Θ0,c. For this reason, we express the mean change of free energy when overcoming the energy barrier on the composite surface, ∆gc, through the free energy changes ∆g1 and ∆g2 on the areas 1 and 2, respectively:
∆gc ) ω1∆g1 + ω2∆g2
(17)
∆gc, ∆g1, and ∆g2 are defined by eq 8 written with the corresponding subscripts. Since the total area of the composite surface, A, consists of the areas A1 and A2, the mean area per adsorption center on the composite surface, λc2 ) A/n, is related to the areas per center in the patches 1 and 2, λ12 ) A1/n1 and λ22 ) A2/n2, by the equation
λc2 ) ω1λ12 + ω2λ22
(18)
Substituting eqs 17 and 18 in eq 16, and bearing in mind that Θr,1 ) Θ0,1 ) 0° and ∆g1 ) 0 for the area 1, we obtain
3370 Langmuir, Vol. 15, No. 9, 1999
V ) 2ω2K02λ2 × sinh
(
Petrov et al.
)
γ(λ12 + (λ22 - λ12)ω2)(cos Θr,c - cos Θ0,c) (19) 2ω2kT
3. Dependence of the Effective Parameters of the Blake-Haynes Theory on Degree of Surface Heterogeneity. When static and dynamic contact angles determined on a heterogeneous surface are used to fit eq 3 or 10, effective (pseudo-homogeneous) jump length, λeff, and equilibrium oscillation frequency, K0eff, are obtained as fitting parameters. Comparison of eq 10 written with λeff and K0eff with eq 19 shows that they coincide if
λeff2 )
λ12 + (λ22 - λ12)ω2 ω2
K0effλeff ) ω2K02λ2
(20a) (20b)
These expressions yield the relationships between the effective parameters λeff and K0eff and the characteristics of the model smooth heterogeneous surface, when one of the areas is completely wettable by the liquid:
λeff )
λ1 ω21/2
K0eff )
[1 - ω2 + (λ2/λ1)2ω2]1/2 ω23/2K02 (λ2/λ1)
[1 - ω2 + (λ2/λ1)2ω2]1/2
(21a)
(21b)
If n1 and n2 vary because of the substitution of centers “1” by centers “2” and they both are considered as material points, λ1 ) λ2 ) λ. The molar fractions ω1 and ω2 are then equivalent to the corresponding area fractions f1 and f2, and eqs 21a,b become
Figure 6. Bilogarithmic plot of λeff versus degree of methylation of silica φ. The slope of the linear fit, -0.44 ( 0.04, coincides well with the theoretical value of -0.50 predicted by eq 22a. The intercept yields λ ) 1.2 ( 0.1 nm, which agrees with literature data and confirms the approximation that λ1 ≈ λ2 ≈ λ.
fractions ω1 and ω2 are complementary to one another and ω2 is equivalent to the experimentally available degree of methylation, φ. Such an approximation enables us to directly compare the experimental dependencies from Figures 2 and 3 with eqs 22a,b for λ1 ≈ λ2 ≈ λ or with eqs 21a,b when λ1 * λ2. Another comparison between theory and experiment is possible if λ1 ≈ λ2 ≈ λ, because then ω1 ≈ f1 and ω2 ≈ f2. In this case we can utilize the Cassie equation to evaluate f2 following Lamb and Furlong,53 who used Θ0,eff ) (ΘR + ΘA)/2 to calculate the area fractions of silanols on a partially dehydroxylated pure silica. For semiwettable surfaces, Θ0,1 ) 0°, and eq 2 reads
f2 )
cos Θ0,eff - 1 cos Θ0,2 - 1
(23)
A partially methylated silica surface consists of hydropylic areas 1 containing “free” and “bound” hydroxyls that might be hydrated and hydrophobic methylated parts 2 composed of (CH3)3 groups. Recent AFM investigations52 of the interface between partially methylated silica and water confirmed the existence of TMCS patches with dimensions between 50 and 200 nm, which are much greater than the values of λ ∼ 1 nm discussed in the previous section. This observation supports the assumption in the theoretical analysis that the “1-2” and “2-1” molecular jumps can be neglected when calculating the mean contact line velocity. On such a composite surface we cannot rigorously identify the adsorption centers “1” and “2” of the theoretical model with some of the above surface functionalities. For this reason we assume that n1 represents the mean surface density of the reactive OH groups and n2 the density of trimethyl groups. Because of the substitution character of the methylation reaction, n1 and n2 vary reciprocally with increasing degree of methylation, but their sum n remains constant. Under such conditions the molar
In our system Θ0,eff is the effective Young angle on a given partially methylated silica surface and Θ0,2 is the Young angle on a surface exposing only (CH3)3 groups. Comparison between theory and experiment may then be performed on the λeff(f2) and K0eff(f2) scales. This procedure does not distinguish between “free” OH groups, “bound” hydroxyls, or adsorbed water on the hydrophilic areas 1 because all these species yield Θ0,1 ) 0°. The hydrophobic patches are also characterized macroscopically because the value of Θ0,2 does not specify their molecular structure. 1. Dependence of λeff and K0eff on Degree of Methylation for λ1 ≈ λ2 ≈ λ. Comparison of the theoretical and experimental λeff(φ) relationships is presented in a log λeff versus log φ scale in Figure 6. The points below φ ) 0.70 linearize within the 95% confidence band (R ) 0.989, SD ) 0.086); deviation from linearity is observed at the transition to saturation occurring at φ > 0.85. The straight line has the slope -0.44 ( 0.04, which is very close to the theoretical value of -0.50 predicted by eq 22a. The value of λ ) 1.2 ( 0.1 nm obtained from the intercept agrees well with λOH ) 1.2 nm reported by Gribanova51 for the system fused silica-aqueous electrolyte solutions with pH 5.6 - air and just slightly exceeds the value obtained at maximum methylation, λ(CH3)3 ) λeff,pl ) 1.0 ( 0.05 nm. This coincidence supports the assumption λ1 ≈ λ2 ≈ λ, which gives eq 22a. Figure 7 shows the data from Figure 3 in a bilogarithmic plot of log K0eff versus log φ. The linear dependence required by eq 22b is obeyed up to φ ) 0.70 (R ) 0.989, SD ) 1.9 × 10-6). The linear fit has a slope 2.3 ( 0.2 that is approximately 50% higher than the theoretical value of
(52) Duplock, S. Ph.D. Thesis, University of South Australia, in preparation.
(53) Lamb, R. M.; Furlong, D. N. J. Chem. Soc., Faraday Trans. 1 1982, 78, 61.
λeff )
λ λ ) ω21/2 f21/2
K0eff ) ω23/2 K02 ) f23/2 K02
(22a) (22b)
IV. Comparison between Theory and Experiment
Dewetting Dynamics on Heterogeneous Surfaces
Langmuir, Vol. 15, No. 9, 1999 3371
Table 3. Comparison of the Theoretical Relationships Represented by Eqs 22a,b and the Experimental Dependencies of λeff and K0eff from Table 2 on Degree of Methylation, φ ) Γ(CH3)3/Γ(CH3)3,sat, and on Methylated Area Fraction, f2 ) (cos Θ0,eff - 1)/(cosΘ0,2 - 1), Assuming That λ1 ≈ λ2 ≈ λ dependence
slope
λ or K02 value extracted from the intercept
remarks
log λeff vs log φ log λeff vs log f2 log K0eff vs log φ log K0eff vs log f2
-0.44 ( 0.04 -0.45 ( 0.04 2.3 ( 0.2 2.4 ( 0.2
λ ) 1.2 ( 0.1 nm λ ) 0.96 ( 0.04 nm K02 ) (4.0 ( 1.0) × 105 s-1 K02 ) (1.4 ( 0.5) × 106 s-1
φ ) Γ(CH3)3/ Γ(CH3)3,sat f2 from Cassie eq 23; Θ0,eff ) (ΘR + ΘA)/2 Θ0,2 ) 112° φ ) Γ(CH3)3/Γ(CH3)3,sat f2 from Cassie eq 23; Θ0,eff )(ΘR + ΘA)/2 Θ0,2 ) 112°
Figure 7. Bilogarithmic plot of K0eff versus degree of methylation φ. The slope of the linear fit, 2.3 ( 0.2, is about 50% higher than the theoretical value of 1.5 expected from eq 22b. The intercept gives a reasonable value of K02 ) 4 × 105 s-1 for φ ) 1.0.
1.5. The intercept of the fit gives for the completely methylated surface K02 ) (4 ( 1) × 105 s-1. Similar results are obtained if we compare eqs 22a,b with the data of Table 1 obtained when Θ0 in eq 3 was substituted with ΘR. The log λeff versus log φ dependence (not shown graphically) is linear up to φ ) 0.70 (R ) 0.981, SD ) 0.150) with a gradient -0.50 ( 0.06 that coincides with the theoretical value. The intercept gives λ ) 1.1 ( 0.05 nm, in good agreement with Gribanova’s51 value of λOH ) 1.2 nm and the two lowest values of λeff ) 0.96 ( 0.1 nm and 0.84 ( 0.05 nm, achieved at maximum methylation. This agreement also confirms the reliability of the approximation λ1 ≈ λ2 ≈ λ. The plot of log K0eff versus log φ (also not shown) has a linear part up to φ ) 0.75 (R ) 0.989, SD ) 5.9 × 10-7) with a slope 2.2 ( 0.2 and an intercept giving K02 ) (1.4 ( 0.3) × 106 s-1 at maximum methylation. 2. Dependence of λeff and K0eff on the Area Fraction of the Methylated Surface for λ1 ≈ λ2 ≈ λ. We have calculated the area fraction of the methylated surface, f2, introducing in eq 23 the effective Young angles Θ0,eff from Table 2 and Θ0,2 ) 112°, reported by Adam and Elliot54 for water on (CH3)3C-C(CH3)3, a hydrocarbon “which has nothing except methyl groups to expose”.54 The plots of log λeff versus log f2 and log K0eff versus log f2 are shown in Figures 8 and 9. Their trends are very similar to those in Figures 6 and 7. The linear parts (R ) 0.990, SD ) 0.084 and respectively R ) 0.989, SD ) 2 × 10-6) have slopes -0.45 ( 0.04 and respectively 2.4 ( 0.2, which are close to the theoretical predictions and to the values obtained from Figures 6 and 7 (see Table 3). However, in Figures 6 and 7 saturation begins at φ ≈ 0.85, while in Figures 8 and 9 it starts at f2 ≈ 0.50. According to Laskowski and Kitchener55 this difference is due to the sterically limited chemisorption of trimethyl groups. Using the density of surface silanols reported by Armistead and Hockey56 (1.3 OH/nm2) and the cross(54) Adam, N. K.; Elliot G. E. P. J. Chem. Soc. 1962, 2206. (55) Laskowsk, J.; Kitchener, J. A. J. Colloid Interface Sci. 1969, 29, 670. (56) Armistead, C. G.; Hockey, J. A. Trans. Faraday Soc. 1967, 63, 2549.
Figure 8. Bilogarithmic plot of λeff versus the area fraction of methylated silica surface, f2, calculated from the Cassie equation (see the text). The slope of the linear fit, -0.45 ( 0.04, is very close to the theoretical value of -0.50 predicted by eq 22a. The intercept yields λ ) 0.96 ( 0.05 nm in agreement with literature data and the approximation λ1 ≈ λ2 ≈ λ.
Figure 9. Bilogarithmic plot of K0eff versus methylated area fraction, f2, calculated from the Cassie equation. The slope of the linear fit, 2.4 ( 0.2, is higher than the value of 1.5 predicted by eq 22b. The intercept gives a reasonable value of K02 ) (1.4 ( 0.5) × 106 s-1 for φ ) 1.0.
sectional area (0.42 nm2) of a Si(CH3)3 “umbrella”, estimated by Kiselev et al.,57 they found that the maximum degree of methylation φ ) 1.0 corresponds to a methylated area fraction f2 ≈ 0.54, a value that is much less than 1. A similar number, f2 ) 0.49 ( 0.03, is obtained using the value of Γ(CH3)3,max ) 1.4 ( 0.1 (CH3)3 groups/nm2 found by Morrow and McFarlan48 and Stoeber’s58 cross-sectional area, Acs,(CH3)3 ) 0.35 nm2. Both estimations show that, even when a saturation of methylation is achieved, approximately 50% of the surface remains hydrophilic. For this reason the value of Θ0,eff ) 82° determined on such surfaces is considerably smaller than the value of Θ0,2 ) 112° reported by Adam and Elliot54 for a surface exposing only (CH3)3 groups. 3. Dependence of K0eff and λeff on Degree of Methylation for λ1 * λ2. One can relax the assumption that λ1 ≈ λ2 ≈ λ and apply a nonlinear least-squares fitting procedure to the experimental data, using the more general eqs 21a,b. Figure 10 shows such a fit to the λeff(φ) (57) Kiselev, A. V.; Kovaleva, N. A.; Korolev, A. Ya.; Scherbakova, K. A. Dokl. Akad. Nauk SSSR 1959, 124, 617. (58) Stober, W. Kolloid-Z. 1956, 149, 39.
3372 Langmuir, Vol. 15, No. 9, 1999
Figure 10. Nonlinear least-squares fit of eq 21a (allowing λ1 * λ2) to the experimental dependence λeff(φ). The values of the free parameters obtained, λ1 ) 1.06 ( 0.05 nm and λ2 ) 1.22 ( 0.04 nm, are close to each other and to literature data, thus supporting the assumption that λ1 ≈ λ2 ≈ λ.
dependence in Figure 2, including only the points before saturation. The fit gives λ1 ) 1.06 ( 0.05 nm and λ2 ) 1.22 ( 0.04 nm. Both values are close to each other and to λ ) 1.2 ( 0.1 nm obtained from Figure 6, again supporting the assumption that λ1 ≈ λ2 ≈ λ. However, eq 21b cannot be fitted to the K0eff(φ) dependence from Figure 7 with K02 and λ2/λ1 as free parameters. At present we cannot specify the reason for such a disagreement between the K0eff(φ) data and the predictions of the theoretical model for λ1 * λ2. V. Discussion The experimental dependencies Θr(V) analyzed in this study have been chosen because of the well-defined threephase system consisting of pure fluids (water and nitrogen) and silica methylated with TMCS, one of the most extensively studied solid surfaces. Our analysis shows that a change of static wettability causes a systematic variation of the free parameters λeff and K0eff of eq 3. This correlation implies that some implicit relationships between these parameters and ΘR or Θ0,eff are not taken into account by the Blake-Haynes molecular-kinetic theory of wetting dynamics.10 Similar dependencies were found in the literature for other three-phase systems, but none of them have been commented on or theoretically modeled. Gribanova51 investigated the wetting dynamics of fused silica by aqueous electrolyte solutions of different pH and also reported a decrease of λeff and a weak increase of K0eff with increasing ΘA. These variations reflect the electrochemical change of the OH groups at the solid-liquid interface from a neutral (pH 2) to a charged (pH 12) state. Such a surface dissociation might also follow the theoretical model proposed here. The investigation of Newcombe and Ralston36 contains another set of Θr(V) data for silica surfaces partially esterified with long chain alcohols. Unfortunately, these data were averaged for electrolyte solutions with different pH and ionic strength values and superimpose the effects observed by Gribanova51 and those caused by variation of the hydrophobicity. Nevertheless, this system shows similar but weaker dependences of λeff and K0eff on ΘR. On the other hand, a strong decrease of λeff and a significant increase of K0eff with increasing ΘR were observed for a glycerol-water mixture receding on glass methylated with dimethyl dichlorosilane.59 It is known that this reagent gives a polymeric and rough coating,45,60,61 so that one (59) Petrov, J. G. In preparation. (60) Herzberg, W. J.; Marian, J. E.; Vermeulen, Th. J. Colloid Interface Sci. 1979, 33, 164.
Petrov et al.
could expect that surface roughness considerably contributes to the strong dependencies of λeff and K0eff on ΘR registered for this system. We explain the observed correlation between λeff, K0eff, and ΘR or Θ0,eff with the variation of solid surface heterogeneity accompanying methylation of silica. Comparison of the predictions of the molecular-kinetic model of dewetting dynamics on smooth heterogeneous surfaces proposed in this study with the experimental λeff(φ) and K0eff(φ) data supports this interpretation; theoretical and experimental dependencies agree for degrees of methylation below 0.70-0.75. The bilogarithmic plot of the more reliable λeff(φ) data (see the errors in Tables 1 and 2) gives the slope -0.44 ( 0.04, which is practically equal to the theoretical value of -0.50 predicted by eq 22a. The intercept yields λ ) 1.2 ( 0.1 nm, in agreement with the value λOH ) 1.2 nm found by Gribanova51 for pure silica and with our saturation values, λeff,plat ) λ(CH3)3 ) 1.0 ( 0.05 nm. Such a coincidence supports the assumption that λ1 ≈ λ2 ≈ λ used to obtain eq 22a. The same agreement was found between the theoretical and experimental λeff(f2) dependencies, which utilize the Cassie equation to calculate the area fraction of the methylated surface. The nonlinear fit of the experimental λeff(φ) data with the more general eq 21a, which relaxes the requirement λ1 ≈ λ2 ≈ λ, yields close values of λ1 ) 1.06 ( 0.05 nm and λ2 ) 1.22 ( 0.04 nm. This also supports eq 22a and the conclusions based on it. Comparison of theory and experiment on the basis of the K0eff(φ) and K0eff(f2) data is considerably more uncertain, because the values of K0eff are as usually obtained with significant errors (see Tables 1 and 2). The slope found in the bilogarithmic scale is 2.3 ( 0.2, that is, about 50% higher than that theoretically predicted by eq 22b, but K02 ) (4 ( 1) × 105 s-1 and K02 ) (1.4 ( 0.5) × 106 s-1 obtained from the intercepts fall in the range of values obtained for other surfaces with similar static wettabilities.15,28,29 On the other hand, the more general eq 21b that should be valid for λ1 * λ2 could not be nonlinearly fitted to the K0eff(φ) data. If this failure and the higher slopes of the experimental log K0eff/log φ and log K0eff/log f2 plots are not due to the large errors of K0eff, they would reflect some deviation of the real system from the theoretical model or an inconsistency of the latter. More accurate K0eff data and more points in the K0eff(φ) and K0eff(f2) dependencies are obviously necessary to achieve more definite conclusions. At this stage we consider the comparison between theory and experiment on the basis of the λeff(φ) and λeff(f2) dependencies much more reliable and the agreement obtained on this scale representative enough. The deviations between the theoretical and the experimental dependencies at degrees of methylation above 0.70-0.75 or f2 > 0.50 may arise from different causes. Some results found in the literature imply that they could be due to formation of rough coatings at the higher TMCS concentration and/or longer reaction times. Trau et al.61 presented ellipsometric data showing that methylation from hexane solutions of TMCS with concentrations of 10 and 20 v/v % gave up to 10 nm thick coatings, while application of TMCS vapors produced a film 1.4 nm thick. Significant hysteresis of the static contact angle was observed on the surfaces methylated in solution, and a concurrence of ΘR and ΘA was found for the vapor-treated surface suggesting a solid surface uniformity. These results were further confirmed by an AFM study of Biggs (61) Trau, M.; Murray, B.; Grant, K.; Grieser, F. J. Colloid Interface Sci. 1992, 148, 182.
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Langmuir, Vol. 15, No. 9, 1999 3373
and Grieser,62 demonstrating significant roughness of the surface methylated in the 10 v/v % solution of TMCS and no difference to the image of the pure silica surface when vapor methylation was applied. Tsutsumi and Takahashi63 determined the surface density of bound trimethyl groups by elemental analysis and compared silica methylated in cyclohexane solutions and in vapors of TMCS. They showed that at low TMCS concentrations the surface (CH3)3 density was the same for solution and vapor treatments; deviation between them was observed when the degree of methylation exceeded 0.50-0.55. On the basis of the above results, we speculate that the systematic disagreement between theory and experiment observed at methylation degrees φ > 0.70-0.75 may be due to the appearance of significant surface roughness and violations of the limitations of the model of smooth heterogeneous surfaces proposed here. Another possibility is that the bare parts of silica lose their hydrophilicity at more complete methylation. Then the condition Θ0,1 ) Θr,1 ) 0° does not hold, and eqs 21 and 22 do not represent the relationship between K0eff, λeff, and ω2 or f2. On the other hand, the theoretical model proposed here is rather primitive. Collective effects neglected in eqs 6 and 13 or other combining rules, relating the characteristics of the heterogeneous surface to those of the patches of which it is composed, should be also considered. The hydrodynamic approach has already offered some interesting solutions of the complex problem addressed in this study, but there is much more work to be done. Further investigations with solid substrates of defined smoothness and heterogeneity are definitely necessary to check our and other models and to help develop the theory of wetting dynamics on real surfaces.
VI. Conclusions
(62) Biggs, S.; Grieser, F. J. Colloid Interface Sci. 1994, 165, 425. (63) Tsutsumi, K.; Takahashi, H. Colloid Polymer Sci. 1985, 263, 506. (64) If the hystersis is less than 16°, (ΘR + ΘA)/2 ≈ arccos [(cos ΘR + cos ΘA)/2] with an error below 1%; the error does not exceed 10% when ΘA - ΘR < 50°. (65) Much less is known about the methylation of silica with tertbutyldimethylchlorosilane. We accept the assumption of ref 47 that this substance follows the same reaction mechanism. (66) Such a model would be realistic if the dimensions of the areas 1 and 2 are much greater than λ1 and λ2.
The effective parameters λeff and K0eff obtained by application of the molecular-kinetic theory of BlakeHaynes10 to dewetting of partially methylated silica surfaces depend on their static wettability, characterized by the receding static contact angle, ΘR, or the effective Young angle, Θ0,eff ) (ΘR + ΘA)/2. These correlations show that λeff, K0eff, and ΘR or λeff, K0eff, and Θ0,eff are not independent free parameters, and some implicit relationships between them are not taken into account by the theory developed for smooth and homogeneous solid substrates. We explain the above correlation with the heterogeneity of the solid surface which results in dependencies of λeff, K0eff, ΘR, and Θ0,eff on degree of methylation, φ, and on the area fraction of the methylated surface, f2. A molecularkinetic model of dewetting dynamics on a smooth heterogeneous silica surface is proposed that predicts such dependencies. Satisfactory agreement between theory and experiment is observed below φ ) 0.70-0.75 and f2 < 0.50, when λ1 ≈ λ2 ≈ λ. The more reliable λeff(φ) data can be nonlinearly fitted with the more general eq 21a, assuming that λ1 * λ2. The fit gives almost the same values of λ1 ) 1.06 ( 0.05 nm and λ2 ) 1.22 ( 0.04 nm, thus confirming the approximation λ1 ≈ λ2 ≈ λ. The significantly more uncertain K0eff(φ) data cannot be reasonably fitted under such conditions; the reason for this failure cannot be specified at present because of the large errors of determination of K0eff. Deviations of the experimental relationships λeff(φ), K0eff(φ), λeff(f2), and K0eff(f2) from the theoretical predictions are observed above 0.70-0.75 degree of methylation or at methylated area fractions greater than 0.50. Literature ellipsometric, AFM, and elemental analysis data for silica methylated with TMCS imply that these deviations might be due to formation of rough coatings at high TMCS concentrations and/or long reaction times. Such behaviors violate the limitations of the theoretical model of a smooth heterogeneous surface adopted in this study. LA971012+
