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Dynamic Dewetting Regimes Explored Renate Fetzer* and John Ralston Ian Wark Research Institute, UniVersity of South Australia, Adelaide, SA 5095, Australia ReceiVed: February 24, 2009; ReVised Manuscript ReceiVed: March 29, 2009
To resolve issues concerning the manner in which substrate wettability and heterogeneity influence contact line motion, we investigate the dewetting process that occurs when a bubble contacts a hydrophobic surface immersed in water. Titania surfaces, partly coated with a self-assembled monolayer of octadecyltrihydrosilane, were used to achieve a wide range of surface composition and wettability. For all surfaces, two distinct dynamic regimes were detected: the initial stage of the dewetting process is captured by hydrodynamics (partly affected by inertia), whereas the regime of low contact line speeds is well described by the molecular kinetic approach, indicating a thermally activated process. In the hydrodynamic regime, physically reasonable model parameter values were obtained where inertia is negligible. In the low speed regime, however, the results contradicted the common expectation that the activation free energy of displacements correlates with the work of adhesion. Rather quantitative agreement with a microscopic pinning energy was obtained. Our data and analysis strongly indicate that pinning controls contact line motion in the low velocity regime of contact line motion on chemically heterogeneous surfaces. Introduction The dynamic process of liquids spreading over and dewetting from solid surfaces is a crucial step in many different technologies, ranging from industrial coatings and lubricants, inkjet printing, drug delivery, plant protection, and minerals processing to microfluidics. Therefore, it is not surprising that the underlying process of wetting and dewetting has attracted major attention in recent decades, cf. recent reviews.1,2 There is no doubt that the dynamic contact angle of a liquid front differs from its static limit, and that this deviation depends on the instantaneous velocity of the moving three-phase contact line. Nevertheless, a comprehensive picture of dewetting dynamics is still missing and there are many unresolved issues concerning the nature and location of energy dissipation1 and the manner in which solid-liquid interactions,3-5 surface heterogeneity6-8 and roughness,9 or substrate viscoelasticity and stiffness10 influence wetting dynamics. Another issue is the static contact angle hysteresis present in most real systems. To date, most theories describe wetting on ideal surfaces but fail to explain contact line motion in a low velocity regime close to static hysteresis. Here we focus on spontaneous dewetting and alter the substrate wettability by varying the chemical surface composition of the solid substrate. This approach allows us to address a wide range of (averaged) solid-liquid interactions in concert with microscopic substrate heterogeneity. By investigating the dewetting process of water on these surfaces we complete part of the puzzle of how energy is dissipated in dynamic wetting phenomena. An experiment, which mirrors many technologically important and natural processes, is used, i.e., the spontaneous dewetting that occurs when a bubble strikes a hydrophobic surface submerged in water is investigated. Using high-speed video microscopy the size of the dry patch as it expands in time is determined; thence the contact angle is extracted and also checked visually. These data are analyzed by using different * To whom correspondence should be addressed.
theoretical models of dynamic contact angles. Spontaneous dewetting has been studied previously by using either bubbles attached to a capillary11,12 or free rising bubbles.13 However, in the studies reported to date the physical parameter values extracted could not be related in any coherent manner to the properties of the underlying surface. We are able to discern distinct velocity regimes and investigate the respective dynamics independently. In concert with the wide range of substrate wettability and heterogeneity, this approach enables us to isolate the impact of specific system properties in either dynamic dewetting regime. Our intention is to relate microscopic model parameter values to these surface properties. Theoretical Models Hydrodynamics. Various theories predict the relationship between the dynamic contact angle and the speed of a threephase contact line.1,2 These models can be classified according to the different energy dissipation mechanisms that balance the driving capillary force. In hydrodynamic theories, first developed by Voinov14,15 and Cox,16 the energy is predominantly dissipated by viscous friction within the flowing liquid. The shear flow in the liquid wedge close to the three-phase contact line leads to a viscous bending of the liquid-vapor interface. Thus, the macroscopic contact angle θ deviates from the microscopic one, θm, and the following relationship holds
θ3 ) θm3 ( 9Ca ln(L/LS)
(1)
Here, the plus sign corresponds to advancing contact lines and the minus sign to receding ones. This equation is a very good approximation for contact angles up to about 135°, negligible viscosity of the second fluid (e.g., if it is a vapor phase), and small capillary numbers of the moving liquid, Ca ) Vη/γ, where V is the contact line speed, η the liquid viscosity, and γ the surface tension. L is a typical macroscopic length scale above which viscous shear is negligible and LS denotes a microscopic cutoff length that becomes necessary to remove the diverging shear stress at the contact line evolving from the
10.1021/jp901719d CCC: $40.75 2009 American Chemical Society Published on Web 04/22/2009
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classical no-slip boundary condition. For simple liquids, LS is expected to be of molecular order. Free parameters in the hydrodynamic model are the microscopic contact angle θm, which usually is assumed to equal the static limit θ0, and the ratio of the two length scales, L/LS. Molecular Kinetics. In the molecular kinetic approach, first derived by Blake and Haynes,17 based on the activated rate theory of Henry Eyring, a statistical process of thermally activated local displacements at the contact line is described. Interaction forces between liquid molecules and the solid substrate become important and energy is predominantly dissipated by the solid in terms of contact line friction. In equilibrium, the rate of displacements in either direction is given by
K0 )
{ }
kB T -∆G exp h kBT
(2)
where ∆G is the activation free energy of the displacements, h is Planck’s constant, and kBT is the thermal energy. When the instantaneous contact angle θ differs from the equilibrium contact angle θ0, then the unbalanced capillary force γ(cos θ0 - cos θ) leads to a shift in the respective displacement rates. With λ representing the mean distance of displacements and with the density of adsorption sites approximated by 1/λ2, the three-phase contact line travels with a velocity given by17
{
V(θ) ) 2K0λ sinh
}
λ2γ (cos θ0 - cos θ) 2kBT
(3)
For large arguments, eq 3 reduces to a single exponential. In this approach, the activation free energy ∆G and the final contact angle θ0 are equilibrium quantities and neither these parameters nor K0 and λ depend on the direction of flow. This does not necessarily hold for physically rough and chemically heterogeneous systems, as found elsewhere9 and discussed below. In addition to pure molecular kinetic and hydrodynamic models, various combined models were developed, taking both dissipation mechanisms into account.18,19 An alternative hydrodynamic model was proposed based on the idea of interfacial transformation processes driven by surface tension gradients.20,21 However, whether or not the surface characteristics introduced therein actually describe the processes occurring in the contact line region and how they are connected to molecular forces remains unknown. From comparison of these various models with experimental data, none is generally favored, although a transition from a large shear rate hydrodynamic regime to a regime of small shear rates (large contact angle and/or low velocity) governed by molecular kinetics seems highly likely. In this present investigation of contact line motion in spontaneous dewetting, we used both the classical hydrodynamic and molecular kinetic approaches in our data analysis. The key issue is to understand how surface modification influences the microscopic parameters used in these models. Materials and Methods Materials and Sample Preparation. To access a wide range of surfaces with different properties and wettability, titania surfaces partially coated with a self-assembled monolayer of octadecyltrihydrosilane (OTHS) were used; see ref 5 for further details. Prior to coating, the glass slides were cleaned in piranha solution, rinsed with Milli-Q water, dried with nitrogen, and finally treated with air plasma for 1 min. Then, the slides were sputter coated with an amorphous titanium dioxide layer (DC magnetron sputtering using a titanium target in an argon/oxygen
Figure 1. Schematic of the experimental apparatus.
atmosphere). Titania layers of thicknesses between 40 and 50 nm were deposited, providing smooth, coherent (no exposed silica), and optically transparent samples. To hydrophobize the initially hydrophilic titania surfaces, the slides were coated with an organic self-assembled monolayer of OTHS (97%, obtained from Sigma-Aldrich and used without further purification), which is known to form a cross-linked network anchored to the underlying metal oxide.22 The bare titania samples were again cleaned in air plasma for 1 min and then directly immersed in a 1 mM OTHS solution in cyclohexane. After a certain immersion time the samples were rinsed thoroughly with ethanol. Depending on the immersion time (30 min up to 18.5 h), different surface wettabilities (advancing water contact angles between 45° and 110°) were achieved, with a static contact angle hysteresis of about 25° for all surfaces. Surface characterization by atomic force microscopy (AFM) revealed uniform surfaces (i.e., without detectable patches) with an rms roughness that did not exceed 0.4 nm on a 1µm2 spot. The surface coating is stable when in contact with water. Experimental Arrangement. A schematic of the experimental arrangement is shown in Figure 1. A liquid container was used on top of which the sample of interest was placed upside down. Using a fine glass capillary, small bubbles (typically limited to diameters between 150 and 350 µm to avoid bouncing and deformation, in contrast to bubbles with diameters of up to 1.5 mm used in other studies13,23) were blown into the liquid below the substrate. Single bubbles detached from the capillary and rose freely, due to buoyancy, until they contacted the substrate. After a certain time of film thinning (typically about 70 ms), rupture and dewetting of the liquid film between the bubble and the surface took place. The dewetting process was monitored from above with an optical microscope (Olympus BXFM) with a high-speed camera (Photron 1024 PCI, 256 × 256 pixels at 10 000 frames per second) attached. For each sample the experiment was repeated several times and the dewetting dynamics were monitored for several bubbles on different spots, with excellent reproducibility (see below). Data Processing. From the optical images the radius r of the circular dewetted patch between the bubble and surface was obtained (called contact radius below) as a function of time, cf. Figure 2 for typical results. For further data analysis, the instantaneous contact angle for each “snapshot” is of interest. To calculate the respective contact angle from a given contact radius, we assume that the volume of the gas bubble stays constant during the dewetting process. This approximation is reasonable, because the gas pressure inside the bubble (atmospheric plus Laplace pressure) changes by less than 1% during the dewetting process, although the Laplace pressure drops by up to 6%. The volume of each bubble was obtained from additional images focused on the maximum extension of the bubble (i.e., the bubble diameter) after its relaxation to equilibrium. Further, we confirmed by side-view experiments that
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Figure 2. Dewetting of water from OTHS surfaces of varying hydrophobicity. The three-phase contact area, given by the contact diameter, increases monotonically in time. The experimental error is typically (5% of the shown value.
Figure 3. Contact line speed of water on OTHS surfaces as a function of instantaneous dynamic contact angle. The data points were calculated from the experimental results shown in Figure 2. The solid lines represent the best fit of the hydrodynamic model to the data in the initial stage of dewetting. Fit parameters are given in Table 1.
TABLE 1: Derived Parameters for Dewetting on OTHS Coated Titania Surfaces expt no. VB [nL] θm [deg] ln(L/LS) θ0MKT [deg] K0 [MHz] λ [nm] 1 2 3 4 5 6
9.90 55.91 11.18 10.73 26.79 7.75
23.4 25.7 39.4 46.0 52.3 55.5
12.7 13.6 22.9 28.2 30.5 29.2
25.6 30.3 45.0 48.9 55.3 58.9
0.206 0.278 1.125 1.457 0.856 0.667
3.06 2.42 1.67 1.62 1.89 1.79
the bubbles remained spherical during the dewetting process and also checked the calculated dynamic contact angles. For constant bubble volume VB and spherical bubble shape, the following relationship between the contact radius r and the instantaneous contact angle of the surrounding liquid, θ, holds: 13 3 r ) (3VB/π) sin3 θ/(2 + 3 cos θ - cos3 θ). Knowing VB, this relationship allows us to compute θ for each snapshot. Furthermore, the dewetting speed is obtained by taking the time derivative of the contact radius (polynomial fit over 5 points). In Figure 3, the dewetting speed is plotted versus the respective calculated contact angles for the data sets shown in Figure 2. Fitting Procedure. For fitting of the hydrodynamic model eq 1 to the data we used standard procedures (least-squares) to obtain the best fit. Because there are only two free parameters, the microscopic contact angle θm and the constant ln(L/LS), the fit is straightforward and the results are reliable. When fitting
Figure 4. Velocity data, plotted in logarithmic scale versus cos θ. The solid lines show the best fit of the molecular kinetic theory eq 3 to data in the low velocity regime. Fit parameters are given in Table 1. For large arguments, eq 3 reduces to a single exponential (dotted lines).
the molecular kinetic theory eq 3, however, the procedure is more complicated. Instead of two independent parameters, we are now dealing with three fitting parameters which are correlated with each other. Therefore, we have to critically test the fit results we obtain from standard fitting procedures. To gain independent and reliable fitting results for the molecular kinetic model, we plotted the velocity V in logarithmic scale versus the cosine of the instantaneous contact angle, cos θ, cf. Figure 4. For this plot, the MKT eq 3 predicts a straight line with slope of log(e)λ2γ/(2kBT) for large arguments where the sinh function reduces to a single exponential. From the slope of the respective data in Figure 4 we directly obtain the jump distance λ; the values were compared with the results obtained from standard fitting procedure (least-squares) of the full MKT to the relevant data. Results and Discussion Qualitative Behavior. As shown in Figure 2, on all surfaces the contact radius r initially increases fast, but then the rate of increase continuously slows until r reaches a plateau value. The initial slope is mainly determined by the substrate hydrophobicity (which increases with the number of the respective data set), while the final contact radius also depends strongly on bubble size (bubbles were significantly larger for cases 2 and 5, cf. the respective bubble volume VB in Table 1). To eliminate these variations the r(t) data were normalized by calculating the corresponding contact line velocity V as function of calculated instantaneous angle θ, as explained above. In Figure 3 the results for V(θ) are given for the same data sets presented in Figure 2. One can clearly see that the impact of bubble size is eliminated; the data sets are nicely ordered following the wettability of the surface. Data for different sized bubbles on the same substrate collapse on to a single V(θ) curve, indicating that the final contact angle does not depend on the bubble size in the range studied here. To extract quantitative information about the dewetting process, data were first fitted by using the hydrodynamic model, eq 1. As shown by the solid lines in Figure 3, the initial dewetting process is apparently well described by hydrodynamics, cf. Table 1 for respective fit parameters. In addition to eq 1 more accurate models were used, where corrections due to the geometry of the bubble or due to the motion of the coordinate system are included,23 to fit the data, without any observable change in the results. The experimental data in the
Dynamic Dewetting Regimes Explored final low velocity regime, on the other hand, deviate strongly from hydrodynamic behavior. Here, the molecular kinetic model, eq 3, was found to fit the low speed regime of the V(θ) data very well, cf. Figure 4. Respective fit parameters are given in Table 1. As mentioned above in the section on methods, in addition to standard least-squares fitting procedures, in Figure 4 the velocity on a logarithmic scale was plotted versus cos θ in order to obtain an independent and reliable value for λ. From this type of plot, the transition between the regime dominated by hydrodynamics and the low velocity regime governed by the MKT was found to be quite sharp (at velocities of about 2 to 5 cm/s) and thus we are able to locate this transition for each data set. The choice of data relevant to the low velocity regime is crucial to obtaining consistent and reliable fitting results. So far, two distinct dynamic regimes that govern capillary driven bubble spreading were detected. Initially, dewetting of water proceeds rapidly and hydrodynamic models describe qualitatively the dynamics. In a subsequent stage, when the contact line motion slows down, the dynamics strongly deviates from hydrodynamic behavior but is well-captured by the molecular kinetic theory. These observations are entirely reasonable, for in the initial stage of high speeds and low contact angles, viscous shear is the dominant dissipation mechanism, and is described by hydrodynamic approaches. As dewetting proceeds, contact line motion slows down and viscous dissipation becomes less relevant. At the same time, nonhydrodynamic dissipation at the contact line resulting from solid-liquid interactions and, possibly, contact line pinning becomes significant. These statistical processes are described by the molecular kinetic model. Note that the combined molecularhydrodynamic approach by Petrov and Petrov,18 which allows for both dissipation mechanisms at the same time, also predicts a transition from an initial regime of large speeds controlled by hydrodynamics to a final stage of slow dynamics governed by molecular kinetics. Applying the combined model to the entire dewetting process, however, does not yield further information.23,24 Hydrodynamic Regime. The two parameters that tell us something about friction and dynamics in the hydrodynamic regime are the logarithmic factor ln(L/LS) and the microscopic contact angle θm, cf. Table 1. θm is systematically smaller than the final contact angle. This indicates that the final contact angle relaxation is nonhydrodynamic in nature. For ln(L/LS), values between 12 and 30 are typical. Taken literally, these results translate to a typical macroscopic length scale that is between 5 and 13 orders of magnitude larger than the microscopic cutoff length scale. Assuming a cutoff length LS in the nanometer range results in macroscopic scales up to kilometres, taking the capillary length as a typical macroscopic scale yields cutoff lengths up to 6 orders below the angstrom range. Since these values are not physically reasonable at all, we conclude that our results cannot be taken literally and the contact line motion is not purely hydrodynamic in nature. One reason might be that even in the hydrodynamic regime nonviscous friction plays a significant role and has to be taken into account, although the dynamics is dominated by hydrodynamic features, cf. for instance the combined model by de Ruijter et al.19 Since the free parameters in the combined models cannot be fit independently and cannot be fixed, no further information is gained by using these combined approaches. Inertial effects which resist the initial acceleration of the contact line and, thus, slow down contact line motion in the very beginning of the dewetting process may contribute. For inertia to affect capillary flow, Hoffman proposed a critical
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Figure 5. Fitting parameter ln(L/LS) plotted versus the respective microscopic contact angle θm of the hydrodynamic regime. The error of the fitting is less than the scatter in the experiments.
Weber number, We > 0.015.25 The Weber number We ) FV2R/γ (the dimensionless quantity that estimates the relative importance of inertial over capillary forces) with R estimated through the bubble radius indeed reaches values up to 0.04 on the most hydrophobic surfaces, while We stays below 0.003 on the more hydrophilic samples (final contact angle below 30°). Thus, it is reasonable to expect inertial effects (higher apparent friction) in the initial stage of dewetting only on the more hydrophobic samples. This is in agreement with the trend observed for the results ln(L/LS): the values increase with substrate hydrophobicity. Because η ln(L/LS) can be interpreted as friction coefficient or resistance, this trend means that the resistance to contact line motion increases for an increasing driving force and faster initial dewetting, which is exactly what is expected for inertia. If this explanation holds, ln(L/LS) should be unaffected by inertia and remain constant below a critical substrate hydrophobicity. In addition to the data shown in Figures 2 and 3 and discussed so far, we repeated our experiments on further OTHS coated titania samples and processed the data in the same way as reported above. The fit parameter ln(L/LS) of the hydrodynamic regime is shown in Figure 5 for all performed experiments, plotted against the respective second fit parameter, θm. In addition to the increase with substrate hydrophobicity, as discussed above, we find that the ln(L/LS) values remain constant (around 10) below a specific substrate hydrophobicity, i.e., a microscopic contact angle of about 35°. The dewetting process on these samples might be unaffected by inertia and ln(L/LS) could represent an intrinsic value. With LS assumed to be 1 nm, the value ln(L/LS) ) 10 leads to a macroscopic length scale of L ) 22 µm, above which viscous shear is negligible, a result that is physically quite reasonable. Molecular Kinetic Regime. The molecular kinetic model described the qualitative run of the data in the low velocity regime very well, indicating that a thermally activated process of displacements controls the contact line motion. To obtain more insight as to whether single liquid molecules or molecular clusters are displaced and what activation free energy is necessary for these displacements, the model parameters K0 and λ are now examined and compared with various theoretical predictions and expectations. In Figure 6, the parameters λ and K0 extracted from the best fit of the MKT to the low velocity regime of all data sets captured on OTHS covered surfaces are shown, plotted against the respective third free fit parameter, θ0MKT. For the mean displacement distance λ, results between 1 and 3 nm are found; the displacement rate K0 is of the order of 1 MHz. While λ decreases with increasing surface hydrophobicity, the data for
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Figure 7. Activation free energy ∆G/λ2 as calculated from the displacement rate K0 (Figure 6b) with eq 2 and normalized to the respective values of λ2 (Figure 6a). The data are compared to the work of adhesion (dashed line), the hysteresis energy (dotted line), and an estimate of the local pinning energy (solid line), eq 4.
Figure 6. Parameters λ (a) and K0 (b) gained from the best fit of the molecular kinetic theory to all experimental data sets of spontaneous dewetting in the low velocity regime. The error of the fitting is less than the scatter in the experiments. The dashed line in panel b gives the proposed trend for λ ) 1 nm.
K0 are randomly distributed and do not show a clear dependence yields the static on substrate wettability. The free parameter θMKT 0 receding contact angle θ0r rather than an equilibrium value θ0. In the original molecular kinetic approach, single molecules are proposed to exchange from one adsorption site to another through adsorption-desorption steps.17 On titania surfaces fully covered by a closely packed OTHS layer, a mean distance between adsorption sites of 4.5 to 5 Å was found.22,26 Thus, our results for λ ranging from 1 to 3 nm are larger than expected. Further, they show a clear tendency: the more hydrophobic the surface the smaller the displacement length λ. When we compare our data to results obtained for advancing liquid fronts on the same titania-OTHS system,5 however, opposing trends for λ are observed. This discrepancy is rather unexpected if molecular displacements are interpreted by means of adsorption-desorption events, for these should be independent of the direction of flow.17 The equilibrium displacement rate K0 is correlated with the activation free energy ∆G, cf. eq 2. Considering that displacements of molecules occur via adsorption-desorption processes, ∆G (and K0) is expected to correlate with the hydrophobicity of the substrate.27 In these experiments, however, no systematic dependence of K0 on substrate wettability is observed. One might argue that the scatter in the experimental data of K0 is larger than the expected trend. However, if there is a 1:1 correlation between the activation free energy (multiplied by the density of adsorption sites) and the work of adhesion (per unit area),27 the surface wettability is expected to impact K0 by at least 7 orders of magnitude over the range of studied surfaces. As indicated by the dashed line in Figure 6b, this clearly exceeds the experimental scatter of the data. From the K0 values, the respective activation free energy ∆G was calculated from eq 2. This energy, normalized by the
respective data for λ2, is shown in Figure 7. On the basis of the proposed adsorption-desorption steps, the quantity ∆G/λ2, i.e., the activation free energy per unit area for displacements at the contact line, may be related to the work of adhesion,27 ∆G/λ2 ≈ Wad ) γ(1 + cos θ0). This correlation (or at least a lower friction for higher substrate hydrophobicity) was observed for advancing contact lines in experiments3-5,7 and simulations.8,28 The dashed line in Figure 7 represents the work of adhesion of water for our surfaces. The latter exceeds substantially the activation free energy found for the slowly receding contact lines. Furthermore, opposing trends are observed for the work of adhesion and the results for the activation free energy of displacement as a function of substrate wettability. From the asymmetric results for λ found for advancing5 and receding contact lines and the fact that K0 and ∆G/λ2 do not depend on substrate wettability in the proposed way, we conclude that molecular displacements are not due to adsorption-desorption steps. Rather we suspect that (chemical and/ or topographical) surface heterogeneity on a nanometer scale substantially influences contact line motion in a low velocity regime. Substrate Heterogeneity and Contact Line Pinning. In earlier attempts to model surface heterogeneity, the displacements at the contact line were assumed to be based on the original adsorption-desorption event model. Depending on the surface composition of a chemically heterogeneous substrate, effective values for λ and K0 were achieved by using a Cassietype average. Although these approaches were found to successfully describe patches large compared to λ6 as well as molecular-scale heterogeneity,7,8 they fail to predict static contact angle hysteresis and asymmetric dynamics close to the static limit, i.e., in the low velocity regime. To achieve the latter, another molecular mechanism of displacements at the threephase contact line, quite apart from adsorption-desorption processes, is required. Such a possible mechanism is the local pinning and depinning of the contact line due to substrate heterogeneity on a microscopic (nanometer) scale.9 Contact line pinning is a well-known phenomenon that gives rise to, for instance, static contact angle hysteresis, and it is reasonable to consider its possible impact also in dynamic wetting situations, especially in a low velocity regime. Great effort was undertaken to quantify dynamic contact line pinning by means of a critical behavior and dynamic phase transition, described by a universal exponent, cf. ref 29 for an overview. Instead of collective pinning, our approach is to
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reduce the complex collective problem to single local pinning and depinning events. We assume that these events are independent and each follows Boltzmann statistics. Thus, the overall contact line dynamics is described by the molecular kinetic model, where the distance of jumps of molecular clusters, λ, correlates to the typical dimension of the chemical or topographical features and, therefore, might easily exceed a few angstroms. The displacement rate K0 is controlled by the “strength” of the pinning sites, i.e., a microscopic pinning energy. A good correlation between the activation free energy (mean values from advancing and receding dynamics) and the static contact angle hysteresis H ) γ(cos θr0 - cos θa0) was found in recent studies on liquid hydrogen wetting cesium substrates (where the hysteresis was changed by means of surface roughness),9 indicating that the pinning strength might be approximated by the hysteresis energy. Here, θ0a,r denote the static advancing and receding contact angle, respectively. In the present study, the static contact angle hysteresis was about 25° and remained quite constant over the range of samples. The associated hysteresis energy, H ) γ(cos θ0r - cos θ0a), is represented by the dotted line in Figure 7. Although the agreement is not perfect and the trend in the data is not recovered, the hysteresis is much closer to the activation free energy than is the work of adhesion. However, contact angle hysteresis is a macroscopic effect (potentially resulting from local pinning) and exactly how the microscopic structure of a surface contributes to hysteresis remains a puzzle. In recent studies on the wetting of single chemical defects30 it was shown that the energy (per area of the defect) of a contact line moving over such a defect is given by P ) γ|cos θ(m) - cos θ(d)|, where the contact angles specified denote the wettability of the matrix and the defect, respectively. This relationship holds only for contact lines receding over hydrophilic defects or advancing over hydrophobic defects; in all other situations the associated energy is less. Following this argument, the contact line in these present experiments on receding fronts becomes pinned locally whenever it reaches a microscopic hydrophilic spot. The respective energy barrier (pinning energy) per unit area might accordingly be estimated by
P ) γ(cos θphil - cos θr0)
(4)
where θphil now describes the local wettability of the hydrophilic defect31 and θr0 denotes the static limit of the macroscopic contact angle, reflecting the average surface energy of the matrix. In this approach advancing contact angles are irrelevant for receding dynamics, which is reasonable. For our samples of partially OTHS coated titania surfaces, hydrophilic defects should correspond to bare titania sites, falling in between the hydrophobic OTHS molecular zones. Assuming that the contact angle on these hydrophilic defects, θphil, is quite low (15°) and constant for all investigated surfaces,32 we obtain the solid line in Figure 7 for the estimated pinning energy. In addition to the correct magnitude (already provided through the hysteresis) the pinning energy follows the trend in the data. Note that other reasonable values32 of θphil between 0° and 20° provide quite similar curves. Furthermore, when the amount of OTHS grafted onto the samples is increased to enhance the hydrophobicity, it is reasonable to expect a decrease in the typical size of the hydrophilic defects. This expectation is indeed in agreement with the results for λ, cf. Figure 6a. Extending these arguments on local contact line pinning to advancing contact lines, the pinning energy should be estimated by P ) γ(cos θ0a - cos θphob). Note that, as the wettability of
hydrophobic spots, θphob, approaches 180°, this local pinning energy in fact equals the work of adhesion and one cannot easily distinguish between displacements due to molecular adsorptiondesorption and motion due to local pinning and depinning. The only indication might lie in the values of λ: values in the angstrom range, often found for higher contact line velocities,4 may indicate displacements of single atoms or molecules as predicted by the original molecular kinetic approach while larger displacement distances, as found for the slow contact line motion in our system and other studies,3,5,24 suggest displacement of clusters and pinning. Following this logic, it is reasonable that advancing liquid fronts moving slowly over the same type of surface as investigated here5 show dynamic behavior that is indeed related to the work of adhesion. Different dynamic behavior for advancing and receding contact lines was also found in other similar systems3 and in the experiments of liquid hydrogen wetting cesium substrates.9 The pinning mechanism is the only approach, so far, that can explain these asymmetric results for advancing and receding contact line motion, and any consequent dependence of the activation free energy on the direction of flow. In addition to the quantitative agreement between the respective (estimated) energies, this is a compelling argument that pinning controls contact line motion in the low velocity regime of our present system and, we suspect, in the low velocity of any real system that exhibits significant contact angle hysteresis. Conclusion By investigating the dewetting process of water on partly OTHS coated titania surfaces, we have found two distinct dynamic regimes. In the initial stage of the dewetting process, i.e., at high contact line speeds and small contact angles, hydrodynamic behavior is apparently observed. On hydrophobic surfaces, this regime is additionally affected by inertia, which results in unreasonably large values for the fitting parameter ln(L/LS). For negligible inertia on more hydrophilic substrates, we get physically reasonable values for the ratio of limiting length scales, L/LS ≈ 104. In the low velocity regime, i.e., in the final stage of the dewetting process, contact line motion is governed by a thermally activated process. However, the respective model parameters do not confirm the originally proposed idea of symmetric displacements via molecular adsorption-desorption steps. Instead, we found quantitative agreement between the respective activation free energy and an estimated microscopic pinning energy. In concert with the asymmetric dynamics found for advancing and receding contact lines at low speeds, this provides strong evidence that local pinning controls contact line motion in the low velocity regime of the present OTHS coated titania system. Acknowledgment. Discussions with Terry Blake, Mihail Popescu, Craig Priest, and Rossen Sedev are warmly acknowledged. The financial support of the Australian Research Council Linkage Scheme, AMIRA International, and State Governments of South Australia and Victoria is gratefully acknowledged. Supporting Information Available: Top-view images of the dewetting process and low velocity data with the respective best fit of the MKT (Figure 4 of paper) shown in linear scale. This material is available free of charge via the Internet at http:// pubs.acs.org.
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