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Stochastic Relaxation of the Contact Line of a Water Drop on a Solid Substrate Subjected to White Noise Vibration: Roles of Hysteresis Srinivas Mettu and Manoj K. Chaudhury* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015 Received November 24, 2009. Revised Manuscript Received January 6, 2010 Relaxation of the three phase contact line of a sessile drop of water on a low energy surface is studied by subjecting it to a white noise vibration. While a spring force acts on the contact line whenever the contact angle deviates from its equilibrium value, it is opposed by hysteresis. The drop, therefore, remains pinned at a metastable state. With an appropriate amount of vibration, the drop can reach a global equilibrium state irrespective of its initial state, be it advanced or retreated. While the end state is free of hysteresis, the current study sheds light on the dynamics of relaxation that is analyzed in conjunction with a modified Langevin equation. Instead of exhibiting a smooth relaxation as predicted by the Langevin equation with a smooth background potential, stepwise relaxation is observed in most cases. These stepwise relaxations can be explained if the background potential is made slightly corrugated that signifies the existence of metastable states of a drop on a surface. The fluctuation of the displacement of the contact line is highly non-Gaussian. It is shown that an exponential distribution of the displacement fluctuation arises due to the nonlinear hysteresis term in the Langevin equation. The observations of stick-slip motion, the large time of relaxation, and the anomalous displacement fluctuation suggest that hysteresis is present during the relaxation process of the drop even though the final state reached by the drop is free of hysteresis. Finally, we compare the displacement fluctuations of the contact line on two different surfaces: a silicone rubber and a fluorocarbon monolayer. Although the displacement fluctuation is exponential in both cases, the later surface exhibits a greater variance of the distribution than the former plausibly due to differences in hysteresis. This result indicates that the fluctuation of displacement may be used as a tool to study the surface property of a low energy substrate.
Introduction Wetting of a solid substrate by a liquid is encountered in various natural and technological settings, including the currently developing fields of microfluidics1 and superhydrophobicity.2 However, in spite of the intense studies in the field since the early days of Young,3 Gibbs,4 and others,5-11 much remains to be done in understanding the energetics, dynamics of spreading, as well as how they are affected by the nonidealities that give rise to wetting hysteresis.12-20 The equilibrium contact angle (θe) of a liquid drop on an ideally smooth and homogeneous solid substrate is given by the famous Young’s equation (Figure 1) γlv cos θe ¼ γsv - γsl
ð1Þ
*To whom correspondence should be addressed. E-mail: mkc4@ lehigh.edu. (1) Daniel, S.; Chaudhury, M. K.; de Gennes, P.-G. Langmuir 2005, 21, 4240. (2) Genzer, J.; Efimenko, K. Biofouling 2006, 22, 339. (3) Young, T. Miscellaneous Works; Peacock, G., Ed.; Murray: London, 1855; p 1. (4) Gibbs, J. W. Collected Works; Longmane, Green: New York, 1906; Dover: New York, 1961; Vol. 1. (5) Fox, H. W.; Zisman, W. A. J. Colloid Sci. 1950, 5, 514. (6) Girifalco, L. A.; Good, R. J. J. Phys. Chem. 1957, 61, 904. (7) Good, R. J.; Girifalco, L. A. J. phys. Chem. 1960, 64, 581. (8) Fowkes, F. M. Ind. Eng. Chem. 1964, 56, 40. (9) Zisman, W. A. Adv. Chem. Ser. 1964, 43, 1. (10) Wenzel, R. N. J. Ind. Eng. Chem. 1936, 28, 988. (11) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (12) Good, R. J. J . Am. Chem. Soc. 1952, 74, 5041. (13) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38, 341. (14) Eick, J. D.; Good, R. J.; Neumann, A. W. J. Colloid Interface Sci. 1975, 53, 235. (15) Johnson, R. E.; Dettre, R. H. In Contact Angle, Wettability and Adhesion; Advances in Chemistry Series; Fowkes, F. M., Ed.; American Chemical Society: Washington, DC, 1964; Vol. 43, p 112. (16) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744. (17) Dettre, R. H.; Johnson, R. E. J. Phys. Chem. 1965, 69, 1507. (18) Marmur, A. Adv. Colloid Interface Sci. 1994, 50, 121. (19) Marmur, A. J. Colloid Interface Sci. 1994, 168, 40. (20) Brandon, S.; Marmur, A. J. Colloid Interface Sci. 1996, 183, 351.
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where γsv, γsl, and γlv are the interfacial tensions of solid-vapor, solid-liquid, and liquid-vapor interfaces, respectively. However, when the solid substrate is neither ideally smooth nor homogeneous or has other imperfections,12-20 the observed contact angle is not that predicted by Young’s equation. This can be seen when a sessile drop on a solid substrate is inflated by adding some liquid. Here, the three phase (vapor-liquid-solid) contact line is initially pinned to the solid substrate while the contact angle increases until a critical angle (θA) is reached, beyond which the contact line moves freely. Similarly, when such a drop is deflated by removing some liquid, the contact line recedes after another critical angle (θR) is reached. With θA > θR, the difference of the two angles is called the contact angle hysteresis. A sessile drop can indeed subtend a range of contact angles from θA to θR on a solid substrate. It has been pointed out by several authors12-20 that roughness and other imperfections of a solid substrate modify the ideal parabolic free energy of wetting to a corrugated profile that has metastable states. In the absence of hysteresis, if the contact angle is increased or decreased beyond its equilibrium value (θe), the drop is readily restored to its equilibrium like a spring with a spring constant being proportional to the surface tension of the liquid (γlv). However, in the presence of hysteresis, the drop can get stuck to any of the metastable states that prevents it from relaxing to global equilibrium state. Johnson and Dettre15-17 suggested that if some adequate amount of vibrational energy is supplied to a drop, it would relax to the global equilibrium state. More recent analysis carried out by Marmur and Brandon18-20 showed that the advancing contact angle would decrease and the receding angle would increase thus causing the drop to relax to the equilibrium contact angle if sufficient energy is available. The above concept has also been investigated in various experimental settings. Smith and Lindberg21 conducted the first (21) Smith, T.; Lindberg, G. J. Colloid Interface Sci. 1978, 66, 363.
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DOI: 10.1021/la9044094
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Figure 1. Schematic of a drop on a solid substrate subjected to white noise vibration in the vertical direction. (left) Drop on a solid substrate in advancing mode. The radius of the drop in this case aA is less than the equilibrium radius of drop ao. (right) Drop on a solid substrate in receding mode. In this case, aR > ao.
experimental study in this direction, in which they showed that the acoustic vibration is capable of relaxing the drop toward equilibrium. Andrieu et al.22 also studied the effect of vibration on a sessile liquid drop on a solid substrate by subjecting it to a harmonic vibration. They found that, at a high enough amplitude of vibration, both the advancing and receding angles relax to the same equilibrium value. Garoff and co-workers23,24 studied the effect of pulsed vibration on the capillary height that is formed when a large diameter cylinder is immersed in a liquid pool. They reported that the capillary height reaches an equilibrium value only after the application of several vibration pulses. In a recent experimental study, Meiron et al,25 established optimal conditions for measuring equilibrium contact angle by applying vertical sinusoidal vibrations to a drop. Similar experimental studies carried out by Bormashenko et al.26-28 deal with the transition of a drop from the Cassie-Baxter state to the Wenzel state on a superhydrophobic substrate. Noblin et al29,30 studied systematically the transition of the contact line of a drop from a pinned to a moving state when subjected to vertical harmonic vibrations of high enough amplitudes. The conclusion of their study is that the contact line is in the pinned state at low amplitude and shows a stick-slip regime at intermediate amplitude, finally reaching a freely moving state at high amplitude of vibration. All the above studies clearly pointed out that vibration of sufficient power assists a drop to overcome the metastable energy barriers thus to reach the globally minimum energy state. In other words, the end state is free of hysteresis. What is not clear is how the nonidealities of the solid surface that give rise to hysteresis play their roles in the kinetics of the relaxation of such a process. This point is highly pertinent in understanding the dynamics of spreading in general, but it is crucial to understanding the motion of a drop on a surface assisted by vibration. Daniel and coworkers31,32 studied the motion of various liquid drops on a (22) Andrieu, C.; Sykes, C.; Brochard, F. Langmuir 1994, 10, 2077. (23) Nadkarni, G. D.; Garoff, S. Langmuir 1994, 10, 1618. (24) Decker, E. L.; Garoff, S. Langmuir 1996, 12, 2100. (25) Meiron, T. S.; Marmur, A.; Saguy, I. S. J. Colloid Interface Sci. 2004, 274, 637. (26) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Langmuir 2007, 23, 6501. (27) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Langmuir 2007, 23, 12217. (28) Bormashenko, E.; Pogreb, R.; Whyman, G.; Bormashenko, Y.; Erlich, M. Appl. Phys. Lett. 2007, 90, 201917. (29) Noblin, X.; Buguin, A.; Brochard, F. Eur. Phys. J. E 2004, 14, 395. (30) Noblin, X.; Buguin, A.; Brochard, F. Eur. Phys. J. Spec. Top. 2009, 166, 7. (31) Daniel, S.; Chaudhury, M. K. Langmuir 2002, 18, 3404. (32) Daniel, S.; Sircar, S.; Gliem, J.; Chaudhury, M. K. Langmuir 2004, 20, 4085.
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surface possessing a gradient of surface energy. A liquid drop should spontaneously move on a gradient surface toward the region of higher wettability provided that the surface is free of hysteresis. However, except for the large drops, the small drops are pinned to the surface as the hysteresis opposes the driving force generated by wettability gradient. The drop does move if it is subjected to a periodic vibration. Daniel and co-workers31,32 argued that hysteresis, which is asymmetric on a gradient surface, rectifies the periodic vibration in such a way that a unidirectional motion of the drop occurs. Dong et al.33 used a similar argument of hysteresis to explain the depinning of a drop moving down on a surface by gravity. Later, in separate publications, Daniel et al.1 as well as Dong et al.33 used an asymmetric periodic vibration to move drops on a gradient free low energy surface but again used the argument of hysteresis as a cause of symmetry breaking of periodic vibration that leads to such a motion. More recently, Brunet et al.34 applied a powerful vibration to a drop on an inclined plane and observed that the drop can climb upward against gravity. These authors discarded the hysteresis-based argument of Daniel et al.1,31,32 and suggested, instead, that a nonlinear friction force operating between the drop and the solid substrate as a cause for the symmetry breaking process. The ratchetlike argument of Daniel et al. based on hysteresis has also been commented upon recently by John et al.,35 who suggested that vibration only depins the drop from its pinned state thus allowing it to move by the available gradient of surface energy. The recent study carried out by Noblin et al.36 uses an argument based on wetting hysteresis in the rectification of drop motion induced by vibration. Here, a sessile drop on a solid substrate was subjected to horizontal and vertical vibrations simultaneously. They proposed that the asymmetric forces acting on the contact line of drop brought about by the contact angle hysteresis are responsible for the ratcheting motion of drops and reinterpreted the experimental results of Brunet et al.34 without invoking the nonlinear friction force between the drop and the solid substrate. The above studies raise important issues regarding hysteresis that need further in-depth studies. The crux of the point, as we pointed out in the beginning of this paragraph, is whether one can say anything about the role of hysteresis during the dynamics of drop relaxation even though the end state may be free of hysteresis. In this paper, we tackle the above issue by subjecting a sessile drop, the contact angle of which deviates from the equilibrium value, to a white noise vibration in the vertical direction. If the drop were to depin from the surface instantaneously, we expect that the characteristic time of the relaxation of the drop would be comparable to the Langevin relaxation time. Second, the kinetics should exhibit a smooth monotonic profile consistent with the solution of Langevin dynamics involving kinematic friction alone. The third point is somewhat subtle. When the dynamics is controlled by kinematic friction, the distribution of fluctuations of the displacements of the contact line is expected to be Gaussian as is the case with the classical Brownian particles.37 A departure from the Gaussian statistics of fluctuation should shed additional light into the role of hysteresis as has been reported38 by us recently for the cases of solid objects drifting on a surface in the presence of Coulombic dry friction or a drop drifting on a (33) Dong, L.; Chaudhury, A.; Chaudhury, M. K. Eur. Phys. J. E 2006, 21, 231. (34) Brunet, P.; Eggers, J.; Deegan, R. D. Phys. Rev. Lett. 2007, 99, 144501. (35) John, K.; Hanggi, P.; Thiele, U. Soft Matter 2008, 4, 1183. (36) Noblin, X.; Kofman, R.; Celestini, F. Phys. Rev. Lett. 2009, 102, 194504. (37) Wang, G. M.; Sevick, E. M.; Mittag, E.; Searles, D. J.; Evans, D. J. Phys. Rev. Lett. 2002, 89, 050601. (38) Goohpattadar, P. S.; Mettu, S.; Chaudhury, M. K. Langmuir 2009, 25, 9969.
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gradient surface with hysteresis when they are subjected to a bias and a white noise vibration. White noise was used in our studies because it is quite common in nature. Other than the randomness of the occurrence of the pulses of a white noise, they are also (albeit remotely) similar to the experiments of Decker and Garoff,24 who observed that repeated square wave pulses are quite effective in depinning a contact line from a surface. Furthermore, when the contact line fluctuation is moderate, the drop remains hemispherical and the stochastic processes underlying the kinetics of the drop motion can be modeled using well established theories. A roadmap of the paper is as follows. After describing the theory to account for the effect of Langevin dynamics of a drop relaxation, the details of the experimental procedure are given. The basic experiment is to place the drop on a surface either in an advancing mode or in a receding mode (Figure 1) and then subject it to vibration of different powers. From the evolution of drop toward the global equilibrium, the relaxation time of the process is estimated. We then model the kinetics of the process using a modified Langevin equation39-41 accounting for hysteresis. After providing some additional details of the relaxation process, we show that the probability distribution function (PDF) of the contact line fluctuations is highly non-Gaussian and explain the observed exponential distribution with a simplified KleinKramers equation. Finally, we compare the displacement fluctuation data on two surfaces of different values of surface energies and wetting hysteresis (PDMS and fluorocarbon) in order to illustrate that these fluctuations may be used as a tool to discriminate the dynamic properties of organic surfaces. The paper then ends with a summary and conclusion section.
Theory Before discussing the experimental details, we briefly outline the theory used to describe the relaxation of the contact line of a drop on a solid substrate subjected to white noise vibration. At equilibrium, the spring force acting on the contact line is balanced by the force of hysteresis that yields the following equation: ks ða - ao Þ ¼ H
ð2Þ
where ks is a spring constant, ao is the equilibrium radius of the drop, and a is the radius of the drop away from the global equilibrium. H is the force of hysteresis which is proportional to γlva(cos θR - cos θA). When a vibration is imposed on the drop, a viscous drag force acts on it, causing it to evolve toward equilibrium. Taking all these forces into consideration, an equation of motion for the drop can be expressed as follows:29,38-43 d2 a 1 da þ ω0 2 ða - ao Þ þ σðVÞΔ ¼ γðtÞ þ dt2 τL dt
ð3Þ
The first term on the left-hand side of eq 3 is due to acceleration, the second term is due to viscous drag force, and the third arises from the force of the drop relaxation with ω0 = (ks/M)1/2, whereas the last term is due to hysteresis (Δ = H/M), with M being the mass of the drop. The term on the right is the acceleration due to white noise vibration. Here, τL is the Langevin relaxation time (ratio of the mass M of the drop and its kinematic (39) (40) (41) (42) (43)
Kawarada, A.; Hayakawa, H. J. Phys. Soc. Jpn. 2004, 73, 2037. Hayakawa, H. Phys. D 2005, 205, 48. de Gennes, P. -G. J. Stat. Phys. 2005, 119, 953. Buguin, A.; Brochard, F.; de Gennes, P.-G. Eur. Phys. J. E 2006, 19, 31. Chaudhury, M. K.; Mettu, S. Langmuir 2008, 24, 6128.
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friction coefficient ζ), and σ(V) = V/|V| is the signum function of velocity (V = da/dt) with σ(0) = 0 when the drop is stationary (V = 0). σ(V) = -1 when V < 0, and σ(V) = þ1 when V > 0. The acceleration γ(t) of white noise is ideally delta correlated with zero mean value. However, in our case, both the width and height of the acceleration pulses are finite. We use the following approximation introduced by de Gennes:41 hγðt1 Þ γðt2 Þi ¼ Γ2 for jt1 - t2 j < τc Æγðt1 Þ γðt2 Þæ ¼ 0 for jt1 - t2 j > τc
ð4Þ
where Γ is the root-mean-square acceleration and τc is the correlation time of the noise. Previously,38,43 in the studies of the drifted motion and spatial dispersion of a liquid drop or a solid object on a surface in the presence of a white noise and a bias, we approximated τc as the time duration of a pulse, which yielded satisfactory results. We use the same method here. The power of the white noise vibration is denoted as K, which is equal to Γ2τc. The contact line moves when the magnitude of hysteresis (σ(V)Δ) is smaller than the applied acceleration (-ω02(a - a0) þ γ(t)), otherwise it remains stuck to the surface. Our eq 3 is similar to a previously derived expression of Noblin et al.29 for the case of a drop subjected to a periodic vibration but ignoring the viscous damping term. We wish to point out that eq 3, which simplifies the hydrodynamics of the drop in the bulk as well at the contact line region in terms of a single Langevin relaxation time, also ignores various prefactors that are absorbed in τL, Δ, and ω0. These are the reasons why some amount of parametric fitting is needed to match with the experimental data. However, we are mainly interested in the trend predicted by eq 3 and do not expect to obtain exact numerical agreements because of the various approximations.
Experimental Section The experimental details are as follows (Figure 1). The substrate was firmly attached to an aluminum platform which was framed on a mechanical oscillator (Pasco Scientific, model no. SF-9324). White noise generated by a signal generator (Agilent, model 33120A) was fed to the oscillator via a power amplifier (Sherwood, model no. RX-4105). The whole setup was placed on a vibration isolation table (Micro-g, TMC) to eliminate the effect of ground vibration. The acceleration of the supporting aluminum plate was estimated with a calibrated accelerometer (PCB Peizotronics, model no. 353B17) driven by a signal conditioner (PCB Peizotronics, model no. 482) and connected to an oscilloscope (Tektronix, model no. TDS 3012B). We used a drop of constant volume (V = 10 μL, M = 10-5 kg) in all the experiments. For most of these studies, the substrate used is cross-linked (PDMS) silicone rubber (Dow Corning Sylgard 184, elastic modulus ∼ 3 MPa), although one particular experiment was performed with a self-assembled monolayer of perfluoroalkylsiloxane supported on a glass slide. Since the details of the preparation of this elastomer were reported previously,44 we only give a brief description of how the elastomer was prepared for our current study. After thoroughly mixing the oligomeric component of the Sylgard 184 with the cross-linker in a 10:1 ratio by weight, it was degassed by applying vacuum for 2 h. The degassed mixture was then cast between two silanized microscopic glass slides for its easy removal after cross-linking. The thickness of the poly(dimethylsiloxane) (PDMS) was uniformly adjusted using spacers of 0.64 mm thickness, which were placed between the slides at (44) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 1013.
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Figure 3. Final nondimensional radius (af/ao) of a 10 μL water
Figure 2. Initial and final configurations of a drop (V = 10 μL) on PDMS substrate subjected to white noise vibration in advancing and receding modes. their ends. After cross-linking the elastomer at 75 C, it was removed from the glass slides and Soxhlet extracted in toluene for 24 h in order to remove any uncured monomer. The sample was then dried in vacuum for 48 h. A self-assembled monolayer (SAM) of 1H,1H,2H,2H-perfluorodecyltricholorsilane was prepared on microscopic glass slides using the vapor deposition method.44 The microscopic glass slides (Fisher Scientific) of dimensions of 75 25 1 mm3 were placed in piranha solution (30% hydrogen peroxide and 70% sulfuric acid by volume) for 30 min followed by rinsing with copious amount of distilled deionized (DI) water. After drying with ultrapurified nitrogen gas, the glass slides were treated with oxygen plasma (model PDC-32G; Harrick Plasma) at 0.2 Torr for 45 s. The plasma treated glass slides were immediately transferred to a vapor deposition chamber. The vapor deposition of silane onto glass slide was carried out for 2 h in vacuum. After deposition, the glass slides were rinsed with DI water followed by drying with nitrogen. The roughness of both the PDMS and SAM coated glass was found to be ∼2 nm as obtained from atomic force microscopy. The test liquid used in the experiments was water that was deionized using a Barnstead NanopureII water purifying system. The purified water used in the experiments had a resistivity of 18 M Ω-cm and a surface tension of 72.6 mN/m. The advancing and receding contact angles of water on PDMS are 115 and 70, respectively. The corresponding values for the fluorocarbon monolayer are 116 and 92, respectively. In the advancing mode, a 10 μL water drop was placed on the PDMS substrate using a microsyringe by inflating a small drop until the volume of the drop reached 10 μL. Then the microsyringe was gently removed from the drop. At this stage, the contact angle of the drop is greater than the equilibrium contact angle of ∼90 whereas the radius of drop is less than the equilibrium radius ao (Figure 2). In the receding mode, a water drop of 20 μL was placed on the PDMS using a microsyringe followed by removing 10 μL of the liquid using a microneedle. Here, after removing the needle from the drop, the contact angle of the drop is less than the equilibrium contact angle of ∼90 whereas the radius of the drop is greater than ao. When the substrate is vibrated vertically using white noise, both of the drops evolve toward the equilibrium angle of ∼90 as shown in Figure 2. The stochastic motion of the contact line of drop was recorded with a high speed (1000 frames/s) Redlake Motion-Pro video camera, and the videos were stored in a computer for analysis in leisure. Both ends of the contact line of the drop (Figure 2) were tracked manually in each frame in MiDAS motion analysis software from which the base radius of the drop as well as its fluctuation were obtained as a function of time. 8134 DOI: 10.1021/la9044094
drop on a PDMS substrate subjected to white noise vibration as a function of the power of noise (K). Here, a0 is the final equilibrium radius of the drop corresponding to global equilibrium, which is achieved only with a high power noise. However, the drop does not always reach ao. With a low power noise, it reaches a final value of af, which is less than ao.
Results and Discussion As described earlier, when the drop is not at equilibrium, there is always a spring force (-ks(a - ao)) acting on the contact line of the drop that is balanced by a resistive hysteresis force (H). However, when white noise vibration is applied to the drop, the contact line relaxes toward its equilibrium position. At first, we report the relaxation behavior of the drop starting from its advanced state toward its equilibrium state at various powers of white noise in order to elucidate the effect of the power of noise (K) on the final radius reached by the drop. The radius of the drop reaches a certain final value (Figure 3) only when the power of the noise is above a threshold value (0.15 m3/s3). At powers of noise less than 0.15 m2/s3, the final radius of the drop is lower than the global equilibrium value. This observation is qualitatively in agreement with those reported previously where periodic vibrations were used.22 The dependence of the final radius of the drop on the power of the noise might be a consequence of the metastable equilibrium states in the free energy profile of the drop. At low powers of noise, the energy supplied by the white noise may not be sufficient enough to pull the drop out of a particular metastable state. Thus, the drop remains stuck in a state that is far from the global equilibrium. With the increasing power of the noise, the drop relaxes more readily toward the global equilibrium state. Stepwise Relaxation of the Drop. The signatures of metastable states are present in the stepwise relaxation of the contact line starting from either the advanced (Figures 4 and 5) or the receded state of the drop. Such stepwise relaxations, reminiscent of the stick-slip-like motions in tribology, are present at all powers of white noise. However, they are more evident at low powers and particularly when a contact line is in recession (Figure 5). The stepwise relaxation may be understood by considering the corrugated energy landscapes as originally suggested by Dettre and Johnson.15-17 Figure 6 illustrates such a profile,15-17 in which a global energy minimum (point B) exists in the parabolic energy landscape that is corrugated with multitudes of metastable states. Here, O is the center of the drop. If the drop starts to relax from point A, its eventual destination would be point B, which can be achieved at strong noise strength that is capable of overcoming all the metastable energy states. With the data presented in Figure 5, this seems to have been achieved with a power of noise of 0.3 m2/s3. Langmuir 2010, 26(11), 8131–8140
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Figure 4. Three tracks of the contact line fluctuations are shown for a 10 μL water drop on a PDMS substrate subjected to white noise vibration at K = 0.3 m2/s3. The data illustrate the stochastic and stepwise relaxation behavior of the contact line kinetics. Here, a0 = (a(t) - a(0))/(a0 - a(0)) which varies from 0 to 1 as the drop relaxes from the advanced to the equilibrium state. All three tracks have the same range, but are shifted upward for clarity.
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Figure 6. Johnson and Dettre’s corrugated free energy (ΔG) profile of a drop showing metastable equilibrium states. Here, O indicates the center of the drop; OB is the radius of the drop corresponding to global equilibrium. A and A0 are the inflection points that correspond closely to receding and advancing contact angles of the drop The figure is not drawn to scale. For example, OA0 , in reality, is much larger than what is shown here.
from the defect.45 The asymmetry would be more pronounced in the case of a strong defect for which the contact line exhibits a nonlinear spring constant. This discussion, albeit speculative, suggests that the energy barriers for the relaxation of the contact line are higher for the receding case than for the advancing case. Numerical Simulations. In order to predict the relaxation behavior theoretically, we carried out a numerical simulation of the modified Langevin equation (eq 3). According to Noblin et al.,29 the threshold acceleration required to overcome hysteresis (Δ) for a receding drop can be estimated using eq 5. Δ¼ Figure 5. Relaxation behavior of the contact line of a 10 μL water drop on a PDMS substrate subjected to white noise vibration at various powers of noise (K). The solid lines are the numerical solutions of eq 3 with τL = 0.01 s and Δ = 48 m/s2. At low powers of noise ( τ*, we generated a large matrix of the jump vectors (x) using eq 10 and randomly selected them to construct a stochastic path over a much longer time scale. The probability distribution of these randomly selected jump vectors is shown in the inset of Figure 9b along with the experimentally obtained jump length distribution at τ = 0.001 s. Although the match is not exact, the discrepancy between experimental and simulation results is small, and, more importantly, the exponential nature of the distribution is clearly evident in the simulation. From the stochastic path created over a large time scale, displacement distributions for various values of τ can be constructed. Such a displacement distribution obtained for τ = 0.01 s (Figure 9b) is found to be exponential, thus indicating the effect of hysteresis.50 In a previous publication38 we studied the motion of a drop on a surface subjected to white noise and bias. The bias was generated by creating a gradient of wettability. This is an experiment of a different kind, in which the entire drop slides on a surface. There, the displacement of the drop exhibited an exponential distribution as well. Furthermore, some asymmetry of the distribution was also noted. Based on the Langevin equation, it was suggested that both the asymmetry and the exponential displacement fluctuations are caused by the nonlinear hysteresis force. The simplified eq 9 only leads to a non-Gaussian nature of the distribution, but it does not predict any other features that are associated with nonlinear friction. Indeed, the experimental data presented in Figure 9a reveal several higher order features, such as the distribution becoming asymmetric as time progresses and a secondary peak appearing at a higher value of the displacement that grows with time as well. In a previous publication,43 we reported the results of a study in which a water drop on an inclined surface was subjected to a white (50) The scope of this method of randomly selecting the jump vectors to construct the stochastic path of the contact line is limited. The method works well for short time scales and not for long time scales. Part of the reason for its failure is that any memory associated with the transition from one vector to the other is lost. We use this simple method here just to show that it is possible to obtain nonGaussian fluctuations when a nonlinear friction is present. A more accurate method would be to solve the Langevin equation (eq 3) by considering metastable energy states.
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noise vibration. The drop exhibited a stochastic fluctuation with a net downward drift. When a PDF was constructed from the displacement fluctuation, which is also a measure of the fluctuation of the gravitational work on the drop, it exhibited a secondary peak as well. As time progressed, the secondary peak grew with time while the primary peak diminished, which is somewhat similar to our current result of contact line fluctuation. More studies are required to understand the origins of these higher order features. Displacement Fluctuation to Study Surface Chemical Properties. Analysis of displacement fluctuation, as outlined in the above paragraph, is within the scope of modern nonequilibrium statistical mechanics.51 The detailed analysis of such fluctuations provides insights into the work, entropy and heat fluctuations of a system that characterize it far from equilibrium. The analysis that we adopted here is somewhat pedestrian compared to what can be done in detail. Nevertheless, here, our main interest lies in surface chemical phenomena and in that regard we are encouraged that analysis of fluctuation, even at some semiquantitative level, captures some interesting physics of a dynamic surface chemical process. In order to illustrate our point, we carried out an experiment of relaxation of the contact line of a 10 μL water drop on a perfluorinated glass slide in the advancing mode following the same method as that used for PDMS. The contact angle hysteresis (24) of water on this perfluorinated glass slide is much smaller than that (45) on PDMS (Sylgard-184). The data summarized in Figure 10a show that the contact line fluctuation is much larger on the fluorocarbon SAM than on PDMS, which is consistent with the differences of hysteresis observed with the two surfaces. The drop also reaches faster (∼0.25 s) on the perfluorinated surface from the advanced to the equilibrium state when compared to PDMS (∼1.5 s), providing further evidence of the effect of hysteresis on the relaxation process. Figure 10b compares the probability distribution functions of the displacement fluctuation of the contact line for both surfaces. The kurtosis of the displacement PDF on the fluorocarbon surface is about 3.6, whereas that of the PDMS surface is about 4.5. These values of the kurtosis, which are greater than the value of 3 expected for a Gaussian distribution, prove that the displacement PDFs in both cases is (51) Ritort, F. Poincare Semin. 2003, 2, 195.
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Figure 10. (a) Relaxation kinetics of the contact line of a 10 μL water drop on PDMS (red) and perfluorinated glass slide (blue). (b) Probability distribution functions of the displacement fluctuations (xτ = Δa(τ)) of the contact line on both the surfaces: PDMS (square) and perfluorinated glass slide (triangle). These PDFs have distinctive exponential features. The dotted lines indicate fits attempted by Gaussian functions, and solid lines are exponential fits.
non-Gaussian. The data in each case can be fitted with an asymmetric double sigmoidal function, the center of which is much like a Gaussian function but has exponential tails. These features of the PDFs of the two distributions strongly suggest the eminent presence of hysteresis during the process of the contact line relaxation on the fluorocarbon surface as is the case with PDMS.
Summary and Conclusions We showed that a liquid drop subjected to white noise vibration of sufficient power reaches the same equilibrium state irrespective of whether the drop is initially at the advanced or in a receded state. At low powers, the drop cannot fully relax to the same state as that of high powers. Furthermore, the drop does not relax smoothly, but does in a stepwise stick-slip manner. While the average path of the relaxation can be modeled approximately using a modified Langevin equation with an average background hysteresis, the proper description of the stepwise relaxation processes warrants the consideration of metastable states. The relaxation time as observed experimentally and that predicted by the modified Langevin equation is much larger than that expected of a relaxation governed by kinematic friction alone. Ignoring the detailed stepwise relaxation processes, we find that average relaxation kinetics is exponential at high powers, reaching asymptotically to the Langevin relaxation time, at very high acceleration pulses. If such high accelerations are indeed achieved in practice, the hysteresis would have no role to play in the overall relaxation kinetics. We also showed from an approximate steady state solution of the Klein-Kramers equation that the displacement distribution of the contact line has an exponential component with Δ 6¼ 0, which is demonstrated experimentally. Numerically obtained distribution of the displacement fluctuations from the stochastic paths constructed of the jump vectors exhibits exponential tails as in experiments. We expect that a complete time dependent solution of eq 7 would provide more in-depth information about the relaxation and the fluctuation processes, which are the subjects of our future research. The experimental results also demonstrate some other higher order features, such as the distribution becoming progressively asymmetric and a secondary peak appearing at longer times. New models are now being developed to understand the origin of these higher order features for organic surfaces of various surface energies and hysteresis. In summary, the observations based on the attainment of the same final state of a vibrated drop starting from either the Langmuir 2010, 26(11), 8131–8140
advanced or receded state, stepwise relaxation of the contact line, τR being much larger than τL, and the exponential displacement distribution strongly suggest that the convergences of the contact angles of a drop starting either from an advanced or from a receded state to the same value does not necessarily guarantee that the hysteresis is spontaneously eliminated during vibration. However, one can safely state that the hysteresis is mitigated at the end state.
Appendix A. Langevin Relaxation Time. In this section, we outline some of the approximate methods used to estimate the Langevin relaxation time of a 10 μL water drop on the PDMS substrate. The first method is hydrodynamic, the second one is based on a molecular kinetic theory, and the third one is obtained from a somewhat different experimental setting. Subramanian et al.52 provided an approximate solution of the hydrodynamic equation of Cox53 in a graphical form. For contact angles close to but still less than π/2, the ratio of the mass of the drop and the contact line resistance (i.e., M/ζ) are obtained in two ways: wedge approximation and exact lubrication approximation. The first approximation yields a value of M/ζ as 0.04 s, whereas the second approximation yields a value of 0.13 s. The value of M/ζ = 0.01 s as used in our calculations is smaller than that estimated using the hydrodynamic method. Let us now estimate ζ from the molecular kinetic theory of Blake and Haynes,54 which is based on the following relationship between the contact line velocity V and the molecular kinetic parameters: V ¼ Vo sinhðF=2nkB TÞ
ðA1Þ
where Vo is the molecular velocity, F is the driving force, n is the number density of adsorption sites on a surface, kB is the Boltzmann constant, and T is the temperature. Using the values provided for the various parameters of the above equation by Blake et al.,55 that is, Vo ∼ 108 m/s and γlv/(2nkBT) ∼ 0.2, we can estimate ζ = F/V = 2nkBT/Vo from a linear approximation of eq A1. The ratio M/ζ is thus estimated to be about 0.003 s for our (52) Subramanian, R. S.; Moumen, N.; McLaughlin, J. B. Langmuir 2005, 21, 11844. (53) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (54) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (55) Blake, T. D.; Clarke, A.; De Coninck, J.; de Ruijter, M. J. Langmuir 1997, 13, 2164.
DOI: 10.1021/la9044094
8139
Article
Mettu and Chaudhury
B. Spring Constant of the Contact Line of a Drop. The total surface energy of a spherical cap on a solid surface is contributed by the liquid-vapor, solid-vapor, and solid-liquid interfaces as follows: 2πγlv a2 þ π γsl - γsv a2 1 þ cos θ
U¼
ðB1Þ
By expanding U in Taylor series up to two terms with respect to a, one obtains eq B2. Figure A1. Velocity of the sliding of a 10 μL water drop on a silanized (decyltrichlorosilane) silicon wafer as a function of the angle of inclination R. Here, g is the gravitational acceleration.
system. The value of M/ζ = 0.01 s as used in our simulation is about 3 times the value predicted by the molecular kinetic theory. A direct experimental method of estimating the Langevin relaxation time is to place a drop on an inclined substrate and measure the sliding velocity as a function of the angle of inclination. τL can be estimated from the gradient of the sliding velocity with respect to g sin R. Although we attempted to perform this experiment on PDMS, a 10 μL water drop did not slide down on this surface because of hysteresis. However, the experiment was successful on a less hysteretic hydrocarbon (decyltrichlorosilane) SAM supported on a silicon wafer. The surface of this SAM is mainly populated with methyl groups, as is the case with PDMS. Indeed, both surfaces have similar surface energies (∼22 mJ/m2). What is observed in the experiment is that the drop does not slide down the wafer until a critical angle is reached, beyond which the drop slides with a velocity increasing linearly as g sin R (Figure A1). The slope of the V versus g sin R plot yields the value of τL as 0.005 s. We may expect that this value would be representative for PDMS, because both of the surfaces have similar surface chemistry. However, it is difficult to make a direct comparison, as the relaxation of the contact line of a spherical cap of water on a surface occurs radially, which is not the case for a sliding drop. The first point of this Appendix is to point out that the value of τL used in our simulation to fit the experimental data is within the range of the values estimated from other methods. The second point is based upon the fact that the overall kinetic relaxation time (∼1.5 s) of the drop in going from either a receded or an advanced state to the equilibrium is much larger than the Langevin relaxation time estimated by other methods. This supports the suggestion made here that the underlying process of the kinetic relaxation of the drop is not due to kinematic friction but is due to hysteresis.
8140 DOI: 10.1021/la9044094
DU 1 D2 U 2 U ¼ U0 þ ða - ao Þ þ ða - ao Þ þ ::::: Da 2 Da2 υ
ðB2Þ
υ
Here, the volume (υ) of the liquid drop is used as a constraint. In order to evaluate the first and second derivatives of U, a relationship between a and θ is needed, which can be obtained from the volume of a spherical cap and setting its derivative with respect to a as zero. One thus obtains a
dθ ¼ - ð2 þ cos θÞsin θ da
ðB3Þ
Equation B2, in conjunction with eqs B1 and B3, now becomes eq B4. U ¼ U 0 þ 2πaðγlv cos θ þ γsl - γsv Þða - ao Þ þ 2πγlv ðsin2 θð2 þ cos θÞÞða - ao Þ2 =2
ðB4Þ
At equilibrium (∂U/∂a = 0), the term (γlv cos θ þ γsl - γsv) vanishes, which leads to Young’s equation. The spring constant is given by eq B5: ks ¼ 2πγlv sin2 θð2 þ cos θÞ
ðB5Þ
Since the equilibrium contact angle of water on PDMS is ∼90, ks is estimated to be 4πγlv, which is twice the value of the spring constant (2πγlv) used in the simulations. The derivation used in this Appendix is however very approximate, as we assume a minor perturbation of the drop from the equilibrium value. The drop shape is also assumed to be perfectly that of a spherical cap, which may not be the case due to vibration and gravity. Furthermore, the free surface of the water drop may have small scale corrugations during vibration that will involve additional surface energy. In reality, many of these assumptions underlying the derivation of eq B5 may be violated.
Langmuir 2010, 26(11), 8131–8140
