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Anomalous Contact Angle Hysteresis of a Captive Bubble: Advancing Contact Line Pinning Siang-Jie Hong,† Feng-Ming Chang,† Tung-He Chou,‡ Seong Heng Chan,† Yu-Jane Sheng,*,‡ and Heng-Kwong Tsao*,† † ‡
Department of Chemical and Materials Engineering, National Central University, Jhongli, Taiwan 320, R.O.C Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C ABSTRACT: Contact angle hysteresis of a sessile drop on a substrate consists of continuous invasion of liquid phase with the advancing angle (θa) and contact line pinning of liquid phase retreat until the receding angle (θr) is reached. Receding pinning is generally attributed to localized defects that are more wettable than the rest of the surface. However, the defect model cannot explain advancing pinning of liquid phase invasion driven by a deflating bubble and continuous retreat of liquid phase driven by the inflating bubble. A simple thermodynamic model based on adhesion hysteresis is proposed to explain anomalous contact angle hysteresis of a captive bubble quantitatively. The adhesion model involves two solidliquid interfacial tensions (γsl > γsl0 ). Young’s equation with γsl gives the advancing angle θa while that with γsl0 due to surface rearrangement yields the receding angle θr. Our analytical analysis indicates that contact line pinning represents frustration in surface free energy, and the equilibrium shape corresponds to a nondifferential minimum instead of a local minimum. On the basis of our thermodynamic model, Surface Evolver simulations are performed to reproduce both advancing and receding behavior associated with a captive bubble on the acrylic glass.
I. INTRODUCTION The wetting of solid surfaces by liquid droplets (water in particular) is ubiquitous in our daily lives as well as in industrial processes. Wettability is one of the most important properties associated with materials and governed generally by two factors: the chemical composition and the roughness of the solid surfaces. The wettability of an ideal flat solid in terms of the contact angle (CA) between the gasliquid and solidliquid interfaces is described by Young’s equation,13 which relates the interfacial tensions, γij, between the three phases cos θ ¼
γsg γsl γlg
where γsg, γsl, and γlg represent the interfacial tensions of solidgas, solidliquid, and liquidgas, respectively. On a real (nonideal) surface, the CA turns out not to be unique and depends on whether the liquid is advancing over the surface or receding. This phenomenon is known as CA hysteresis, which is generally expressed in terms of the difference between the advancing and receding angle (Δθ = θa θr). The advancing and receding angles are typically obtained by dynamic sessile drop method, i.e., inflating and deflating the droplet volume, respectively. The advancing angle refers to the maximum angle associated with adding volume while the receding angle corresponds to the smallest possible angle upon removing volume. CA hysteresis is accompanied by the pinning of the contact line as the volume of a droplet is withdrawn. Note that, r 2011 American Chemical Society
besides pure liquid, microbead suspensions can exhibit CA hysteresis as well.4 Moreover, pinningdepinning of the contact line can be observed even on nanorough surfaces.5 Pinning of the triple line is what makes it possible to maintain a drop motionless stay on an inclined plane and to capture a liquid column suspended in a vertical capillary. As a result, CA hysteresis can also be characterized by observing the tilting angle above which a drop of a given volume starts moving.6,7 Recently, fabrication of superhydrophobic surfaces with high and low adhesion inspired by various objects found in nature such as rose petal attracts much attention.812 A high adhesive force leads to high CA hysteresis and thus hinders self-cleaning ability. Therefore, understanding the mechanism of CA hysteresis is of great help in the design of superhydrophobic surfaces. There has been an ongoing debate as to the origin of CA hysteresis. In general, two different mechanisms have been proposed: manifestation of adhesion hysteresis3 or mechanical pinning by defects.2,13 In the adhesion mechanism, adhesion hysteresis refers to the restructuring of the interface over some period of time, with the result being that the work required to separate two surfaces (WR) is greater than the energy that was gained by bringing them together (WA). Since the solidliquid interface is not retracting its original path when it recedes, the inflation (adhesion) and deflation (separation) processes are Received: March 13, 2011 Revised: April 22, 2011 Published: May 05, 2011 6890
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Langmuir clearly not thermodynamic reversible and involve dissipation of energy. In the local defect mechanism, the CA hysteresis is believed to originate from chemical blemish that is more wettable than the rest of the surface.2,13 During retraction, the contact line tends to get trapped at certain position on the surface because real solid surfaces are actually not homogeneous on a microscopic scale and contain hydrophilic defects. While the triple line retracts and encounters the blemish, it will be pinned locally and forced to stretch and warp. As the force exceeds a threshold value, the line breaks off the blemish and the energy is dissipated. That is, the defects swept by the line dissipate energy. According to the adhesion mechanism, the amplitude of CA hysteresis is related to the hysteresis in the adhesion energy per unit area, WR þ WA > 0.3 On the other hand, based on the defect mechanism, the amplitude of the hysteresis is proportional to the number of defects and the “snapping” energy, which is lost as viscous dissipation in the fluid. For strong, sparse defects, the dissipated energy is related to the maximum pinning force, which depends on the defect’s wettability, size, and shape.12 While the defect mechanism considers the pinning behavior of receding contact line as the origin of CA hysteresis, it is inadequate to explain the pinning behavior of advancing contact line. In this paper, on the basis of adhesion hysteresis mechanism, we shall develop a simple thermodynamic model and show that contact line pinning is a consequence, instead of the origin, of the CA hysteresis. In section III, we shall demonstrate the CA consistency between the advancing pinning behavior of a captive bubble and the receding pinning behavior of a sessile drop on an acrylic glass. In section IV the simple thermodynamic model of CA hysteresis is proposed based on the hysteresis of the solid liquid interfacial tension. In sectionV numerical simulations based on γsl hysteresis are performed by using Surface Evolver. The comparisons among experiment, theory, and simulation are made.
II. EXPERIMENTAL AND SIMULATION METHOD A. Materials and Experimental Method. The advancing and receding CA measurements were performed on the surface of pretreated poly(methyl methacrylate) substrate. The acrylic glass slices purchased from Kwo-Yi Co. (Taiwan) were used as the substrates and cleaned by sonication with alcohol in an ultrasonic cleaning tank for 5 min. Note that the surface of the substrate is highly smooth and possesses a rms roughness about 2 nm determined by AFM. After sonication, the acrylic glass slide was rinsed with deionized water, sonicated again in deionized water for 5 min, and then dried in a stream of nitrogen gas. The CA measurements were conducted at room temperature under the open-air condition with a relative humidity of 4550% by using a CA goniometer, drop shape analysis system DSA10-MK2 (Kr€uss, Germany).13 The static contact angles were determined for sessile droplets and captive bubbles. With the images taken from the droplet and bubble, the value of contact angles were determined by the tangent method (fitting by the general conic section equation) offered by the drop shape analysis system. Both base diameter and volume of droplet or bubble can be acquired as well. For sessile droplets, the advancing contact angles can be measured by the advancing inflation process. Initially, a water droplet about 1.0 μL was placed on the acrylic glass substrate surface, and a picture of the droplet was taken after 10 s by the CCD video camera of drop shape analysis system. Another water droplet about 1.0 μL was added through inserting the syringe to the previous droplet every 30 s to inflate the droplet. This advancing inflation procedure was repeated for more than 10 times. When the droplet is large enough, the receding deflation
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Figure 1. Variation of the contact angle (CA) and base diameter (BD) with the bubble volume on the surface of an acrylic glass for (a) a deflating bubble (liquid advancing) and (b) an inflating bubble (liquid receding). process was then adopted to determine the receding contact angles. About 1 μL water volume was withdrawn by the syringe from the previous droplet every 30 s, and the pictures were captured after water removal. The receding deflation procedure was also repeated for more than 10 times. For captive bubbles, the acrylic glass slide was immersed in water, and the inflation and deflation processes are similar to those for sessile drops. The syringe was kept inside the captive bubble all the time while it was withdrawn from the sessile drop each time for image recording. The outer diameter of the syringe is 0.5 mm, and the distortion of the bubble shape by the syringe was avoided. All experiments have been repeated a few times, and the results are essentially the same. B. Surface Evolver Simulation. In order to examine our thermodynamic model and make a comparison with experimental results, numerical simulations by public domain finite element Surface Evolver (SE) package are performed for captive bubbles.15,16 The basic concept of SE is to minimize the energy of a surface subject to constraints, and the minimization is done by evolving the surface down the energy gradient. In addition to surface energy, the total energy can also include gravitational energy. Therefore, the gravity effect can be easily incorporated into the simulations. SE models surfaces as unions of triangles with vertices that are iteratively moved from an initial trial shape until a minimum energy configuration is obtained.16 Note that in typical SE simulations CA hysteresis is not taken into account. In our simulations, however, the hysteresis effect is incorporated simply by changing the solidliquid tension before and after wetting.
III. ANOMALOUS CONTACT ANGLE HYSTERESIS OF A CAPTIVE BUBBLE The CA hysteresis is often manifested by inflation and deflation of a sessile drop on the surface. The contact line pinning associated with the deflation of the sessile drop is always observed in CA hysteresis and referred to as receding pinning. If one applies the same process to a captive bubble on the same surface, however, the inflation of a bubble corresponds to receding of the liquid and the deflation of a bubble corresponds to advancing of the liquid. Consequently, it is anticipated that pinning of the contact line will be observed during the inflation of the bubble. However, instead of receding pinning, the contact line pinning is only observed during the deflation of the bubble. This phenomenon can be called advancing pinning. Figure 1 shows the variation of the advancing and receding contact angles with the bubble volume for a captive bubble on the acrylic glass. 6891
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of a newly created droplet on the acrylic glass is θa ≈ 79°. Nevertheless, both hysteresis phenomena satisfy the statement that invasion of the liquid phase takes place only when θ = θa and liquid phase retreat occurs only as θ = θr. This consequence reveals the similarity between CA hysteresis and liquid phase movement across a wettability separation boundary17 or a corner boundary.1820
Figure 2. Variation of the contact angle (CA) and base diameter (BD) with the droplet volume on the surface of an acrylic glass for (a) an inflating droplet (liquid advancing) and (b) a deflating droplet (liquid receding).
The change of the base diameter of the bubble, which characterizes the circular shape of the contact line, with its volume is plotted as well. The initial bubble volume and base diameter are 8.1 μL and 2.3 mm, respectively. As the volume declines gradually, the base diameter of the captive bubble remains the same until the volume is reduced to about 4.1 μL. Since the contact line does not move due to pinning, the CA measured in the liquid phase grows accordingly. When the base diameter starts to shrink, the CA reaches its maximum value, which is the same as the advancing angle (θa) measured by the sessile drop method as shown in Figure 2. The advancing CA remains essentially unchanged as the bubble shrinks furthermore. On the contrary, as the bubble volume is increased gradually, the base diameter grows accordingly and the CA remains essentially the same. That is, liquid receding is associated with contact line movement and the CA is kept the same as that the receding CA (θr) determined by the sessile drop method. Evidently, advancing and receding of the liquid phase associated with a captive bubble are opposite to those associated with a sessile drop. The former displays advancing pinning while the latter exhibits familiar receding pinning. The defect model states that CA hysteresis occurs on a nonideal surface, which is marred by defects. This model can successfully describe continuous invasion of liquid phase caused by an inflating drop and pinning of contact line associated with liquid phase retreat driven by a deflating drop. When the liquid phase retreat is driven by an inflating bubble, however, the defects that are located in the liquid phase and more wettable than the substrate seem unable to pin the contact line based on the same scenario. As a result, the defect model which is successful in explaining receding pinning of a sessile drop fails to describe continuous retreat of the liquid phase driven by an inflating bubble and advancing pinning associated with a deflating bubble. From the viewpoint of a sessile drop, the hysteresis behavior of a captive bubble is anomalous. Besides the contrast behavior in CA hysteresis, there is a fundamental difference between a sessile drop and a captive bubble: a newly created drop shows a CA close to θa while a newly created bubble exhibits a CA close to θr. As shown in Figure 1 for a deflating process with the initial volume V0 = 8.1 μL and an inflating process with V0 = 2.4 μL, the initial CA of a newly created bubble is the same as the receding CA on the acrylic glass, θr ≈ 50°. As illustrated in Figure 2, the initial CA
IV. THERMODYNAMICS OF CONTACT ANGLE HYSTERESIS There are two goals in this section. First, the adhesion hysteresis of solidliquid contact is used to elucidate the CA hysteresis, which relates to the spontaneous change in solidliquid interfacial tension. Second, we show that the contact line pinning corresponds to the nondifferentiable minimum of surface free energy and thus is a consequence of CA hysteresis. A. Adhesion Hysteresis and SolidLiquid Interfacial Tension. Consider the adhesive contact between solid and liquid
media. Following the Dupre equation, the work of adhesion per unit area is WA ¼ γsl ðγsg þ γlg Þ
ð1Þ
while the work of separation per unit area is given by 0
WR ¼ ðγsg þ γlg Þ γsl
ð2Þ
If the adhesion and separation processes are reversible, one must have WA þ WR = 0 and γsl = γ0sl. Nonetheless, it is often found that ΔW = WA þ WR > 0. The hysteresis in adhesion energy ΔW is therefore related to the decrease in the solidliquid interfacial energy, γ0sl < γsl. Since the surface energy change is less than zero, Δγsl = γ0sl γsl < 0, such hysteresis occurs spontaneously and is associated with the restructuring of the adhesive interface over some period of time. A typical example of interface restructure is the diffusive adhesion associated with polymer-on-polymer surfaces. It takes place when species from one surface penetrate into an adjacent surface while still being bound to the phase of their surface of origin. It is worth mentioning that there exists a subtle difference between the solidliquid adhesion and the wetting of a liquid droplet on the solid surface. While the liquidgas contact is not considered in the adhesion, it plays an important role in the droplet wetting. Since there exist two solidliquid interfacial tensions, one can define two contact angles associated with droplet wetting, θa and θr. According to Young’s equation, one has γlg cos θa ¼ γsg γsl
and
0
γlg cos θr ¼ γsg γsl
ð3Þ
Evidently, the advancing angle θa is corresponding to γsl because the inflation process is adhesion-like. Similarly, the receding angle θr corresponds to γ0sl because the deflation process is separation-like. Note that θa g θr owing to γsl g γ0sl. The aforementioned analysis indicates that the hysteresis of adhesion energy for solidliquid adhesive contact may be obtained from CA hysteresis by advancing and receding contact angles measurements 0
ΔW ¼ γsl γsl ¼ γlg ðcos θr cos θa Þ g 0
ð4Þ
The remaining question is why there exists a family of CA in the range of θa g θ g θr, which is always accompanied by contact 6892
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Figure 3. (a) Liquid advancing is associated with solidliquid energy change (γsl γsg). (b) Liquid receding is accompanied by solidliquid energy change (γsg γ0sl).
line pining. On the basis of the adhesion hysteresis, we shall show that those in-between contact angles reveal the frustration of the surface free energy and can be thermodynamically defined. B. Advancing Contact Line Pinning of a Captive Bubble. When an air bubble with volume V is placed beneath a solid surface, a captive bubble is formed. For a sufficient small bubble, where the change in hydrostatic pressure with height can be negligible, it can be shown that the solutions of the axisymmetric Laplace equation yield a gasliquid interface that has the shape of a spherical cap with the radius R of curvature and CA (π θ). The equilibrium shape of the bubble with radius Rr and CA θr corresponds to the minimum of the surface free energy21
Figure 4. Variation of dimensionless free energy change with the base radius for the first regime V > Vr and the third regime V < Va. In the inset, free energy change is plotted against contact angle. The minimum occurs at θr in the first regime and at θa in the third regime. The “” mark denotes the position of the minimum.
bubble chooses to expand its circular base, i.e., rB g r* as depicted in Figure 3b, the surface free energy of the system associated with liquid phase receding becomes 0
ΔFR ðR, θÞ ¼ ðγsg γsl Þπf½R sinðπ θÞ2 ½Rr sinðπ θr Þ2 g þ γlg 2πfR 2 ½1 cosðπ θÞ Rr 2 ½1 cosðπ θr Þg,
ð8Þ
0
FðR, θÞ ¼ ðγsg γsl Þπ½R sinðπ θÞ2 þ γlg 2πR 2 ½1 cosðπ θÞ
for rB g r
ð5Þ
subject to the constant bubble volume constraint V ðR, θÞ ¼ ðπR 3 =3Þ½1 cosðπ θÞ2 ½2 þ cosðπ θÞ ð6Þ Since the spreading film in the vicinity of the contact line is neglected, the Kelvin equation is not involved.22 Note that before the formation of a bubble the surface has been wetted by liquid and adhesion hysteresis has occurred. Thus, it possesses the solidliquid tension γ0sl. The base radius of the bubble relates to the radius of curvature by rB = R sin(π θ). For a newly created bubble at equilibrium, the volume is V* = V(Rr,θr), and the base radius is r* = Rr sin(π θr). Upon inflation or deflation, V = V* ( ΔV, the bubble is able to adjust its shape by changing rB or θ from its original equilibrium shape to reach the surface energy minimum. If the bubble chooses to shrink its circular base, i.e., rB e r* as shown in Figure 3a, the surface free energy of the system associated with liquid phase advancing is given by ΔFA ðR, θÞ ¼ ðγsl γsg Þπf½Rr sinðπ θr Þ2 ½R sinðπ θÞ2 g
þ γlg 2πfR 2 ½1 cosðπ θÞ Rr 2 ½1 cosðπ θr Þg, ð7Þ for rB e r The first term on the right-hand side of eq 7 depicts the gain of solidliquid contact due to liquid phase invasion. The solid liquid tension inside the bubble has been recovered to γsl because the reverse process of adhesion hysteresis has already occurred after the bubble formation. The second term represents the change of liquidgas contact. On the other hand, if the
Note that liquid phase retreat is accompanied by the exposure of the wetted surface, on which adhesion hysteresis happened and the solidliquid tension is γ0sl. When R = Rr (rB = r*) and θ = θr, ΔFA = ΔFR = 0. Evidently, the free energy change of the advancing process is different from that of the receding process. If γ0sl = γsl, however, one has ΔFA = ΔFR and the hysteresis effect disappears. For a bubble with initial volume V* and base radius r*, the volume change leads to surface free energy change, which is a function of CA and base radius (or radius of curvature). Note that ΔF(θ,rB) is continuous but not differentiable at rB = r*. Therefore, the minimum cannot be determined simply by calculus. Under the constraint of constant volume, ΔF(θ,rB) can be expressed as ΔF(rB) or ΔF(θ), and the minimum can be decided accordingly. For specified interfacial tensions (θa = 79° and θr = 51°), the plots of the dimensionless free energy (ΔF* = ΔF/2πRr2γlg) against the dimensionless base radius (rB/r*) or CA (θ) are illustrated in Figures 4 and 5. Dependent on the final bubble volume, three regimes can be identified: (i) V > Vr, (ii) Vr g V g Va, and (iii) V < Va. The two characteristic volumes are defined as Vr = V(Rr,θr) and Va = V(Ra,θa), where Ra sin θa = Rr sin θr = r*. Note that the volume of a newly formed bubble is V* = Vr. In the first regime associated with an inflating bubble, the minimum dimensionless free energy takes place at rB > r* as shown in Figure 4 for V = 1.1V* and 1.5V*. Note that ΔF*(rB) is differentiable at the minimum. Moreover, the variation of ΔF* with θ is shown in the inset as well and indicates that the minimum occurs at θ = 51°. In other words, for V > Vr associated with liquid receding, the bubble expands its base radius by moving the contact line outward with the receding angle θr. 6893
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Figure 5. Variation of dimensionless free energy change with contact angle for the second regime Va < V < Vr. The contact angle varies with the bubble volume while the base radius is pinned at rB = r*. The “” mark denotes the position of the minimum.
This theoretical result is consistent with experimental observation: no pinning associated with liquid receding by bubble expansion. In the second regime associated with a deflating bubble, the minimum dimensionless free energy takes place at the CA θ*, which is dependent on the bubble volume and in the range θr e θ* e θa, as shown in Figure 5 for V = 0.5V* or 0.7V*. Note that ΔF*(θ) is not differentiable at the minimum. The variation of the dimensionless free energy change with the base radius is also demonstrated in the inset. It is evident that the minimum always occurs at rB = r* regardless of the bubble volume. This consequence indicates that as the bubble volume is shrunk to the volume Vr g V g Va, the contact line of the bubble is pinned and therefore the CA rises accordingly for liquid advancing. When the bubble is further deflated and reaches the third regime, the position of the minimum happens at rB < r* as shown in Figure 4 for V = 0.5Va and 0.9Va. Now ΔF*(rB) is differentiable again at the minimum. From the viewpoint of the CA, the dimensionless free energy minimum is located at the advancing angle θ = 79° as illustrated in the inset. This result discloses that for V < Va associated with liquid advancing, the bubble reduces its bare radius by moving the contact line inward with the advancing angle θa. The above analysis clearly indicates that upon bubble deflation the experimental observation of apparent CA increment from θr to θa, and contact line pinning associated with liquid advancing can be explained by the free energy minimization based on adhesion hysteresis.
V. SURFACE EVOLVER SIMULATIONS BASED ON ADHESION HYSTERESIS Since SE involves only the minimization of surface energy, the hysteresis effect cannot be seen and the simulation result always follows Young’s equation if the solidliquid tension is unique on the surface. Adhesion hysteresis can be introduced into SE simulations simply by changing γsl into γ0sl when the surface is wetted. Therefore, two physical parameters, i.e., advancing and receding contact angles, must be specified for CA hysteresis simulations by SE. Initially, a captive bubble with a given volume V0 reaches equilibrium based on γ0sl because the surface is wetted originally. Then the inflation or deflation process is preceded gradually through adding ΔV to or removing ΔV from the
Figure 6. Comparison between experimental (left) and simulation (right) results. The captive bubble is inflated first and then deflated. Advancing pinning for a captive bubble, instead of receding pinning for a sessile drop, is clearly observed. The agreement is quite well.
previous equilibrium bubble by maintaining the same base region and number of vertices. As a result, the evolution of the bubble surface toward surface energy minimum for each new volume is based on γsl for the previous bubble base and γ0sl for the previous wetted region. After the equilibrium bubble shape is obtained, new CA and base radius can be determined. Their base and wetted regions with corresponding solidliquid interfacial tensions are used for further bubble volume change. In order to compare with experimental data, the gravitational contribution is included, but the effect is weak due to small bubble volume. Figure 6 shows the comparison between experimental and simulation results from bubble inflation (liquid receding) to bubble deflation (liquid advancing). Consistent with experimental observations, continuous bubble expansion in terms of base diameter increase is clearly seen in SE simulations. When the bubble is deflated, however, contact line pinning is observed, and the base diameter remains the same during liquid advancing. As the bubble volume is smaller than a critical value, further bubble deflation leads to continuous bubble shrinkage in terms of base diameter decrease. During the inflation and deflation processes, the essential bubble characteristics for both experiments and simulations are very similar to each other. The quantitative features of the bubble in the simulation can be expressed in terms of CA and base diameter as well. Figure 7 illustrates the variation of the CA and base diameter with the bubble volume for θr = 51° and θa = 79°. The basic features associated with hysteresis are undoubtedly observed: (i) during bubble inflation (liquid receding) as shown in Figure 7b, the CA remains at θr and the base diameter continues growing; (ii) during bubble deflation 6894
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Figure 7. Simulation results of the variation of the contact angle (CA) and base diameter (BD) with the bubble volume for (a) a deflating bubble (liquid advancing) and (b) an inflating bubble (liquid receding).
(liquid advancing) as shown in Figure 7a, the base diameter remains unchanged while the CA grows from θr. After the CA reaches θa, the base diameter starts to decline but the CA remain at θa. Figure 6 reveals the qualitative agreement between experimental observations and SE simulation results. Quantitatively, the results of our SE simulations as shown in Figure 7 agree very well with those of experimental measurements as depicted in Figure 1. The aforementioned consequences confirm the validity of the simple thermodynamic model based on adhesion hysteresis, and the contact line pinning associated with the bubble shrinkage is caused by thermodynamic frustration in surface free energy instead of pinning by defects.
VI. DISCUSSION In this study the same advancing and receding CA are obtained for both sessile-drop and captive-bubble techniques. This result agrees with the previous study when the surface is smooth enough.23 Compared to a sessile drop, however, CA hysteresis of a captive bubble is anomalous because contact line pinning occurs at liquid advancing instead of liquid receding. This experimental observation cannot be explained by the well-known defect model. Beside the defect model, another popular mechanism for CA hysteresis has the same origin as adhesive hysteresis. On the basis of adhesion hysteresis, both analytical analysis and SE simulations are able to reproduce the hysteresis behavior associated with a captive bubble. The theoretical results agree quite well with experimental measurements. Although the adhesion hysteresis can be introduced quantitatively by changing the solidliquid interfacial tension on a wetted surface, its physical origin and working mechanism are discussed in this section. The comparison between adhesion hysteresis and defect model is given as well. The orientation of surface molecules or chemical groups often depends on the phase (liquid or gas) they are exposed to. After the surface is wetted by the liquid phase, the rearrangements of surface molecules such as the molecular reorientation may take place. As a consequence, there are two solidliquid interfacial free energies, γsl before rearrangement and γ0sl after rearrangement. In general, it takes time for the rearrangement process, including the reorientation and diffusion, to end. As a consequence, there exist two time scales for the wetting or dewetting
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process: the surface relaxation time (τR) and the characteristic time associated with the formation of the equilibrium shape (τS). Let us consider the equilibrium process of the substrate wetted by the liquid phase. When τR . τS, there exist two equilibrium stages. During the first stage, the surface is just wetted with γsl, and the surface rearrangement occurs very slowly. In addition to γsg and γlg, the shape of the interface is thus determined by γsl, which gives the advancing angle θa according to Young’s equation. Owing to liquid wetting, the surface rearrangement takes place and finally reaches the second equilibrium stage. The solid liquid interfacial energy then becomes γ0sl. Because the process of spontaneous rearrangement depicts the evolution from the state with higher surface free energy to that with lower surface free energy, the condition γ0sl < γsl must be satisfied. When the surface arrangement is completed, the solidliquid tension is reduced to γ0sl, and thus one would expect that the shape of interface is changed accordingly. For example, the angle of a sessile drop would be lowered to a new value θr = (γsg γ0sl)/γlg with θa > θr based on Young’s equation. However, such a phenomenon is not observed experimentally, and it is generally attributed to contact line pinning at the boundary, which seems to be a mechanical equilibrium. After the formation of a bubble, the surface exposed to air encounters the reversal of surface rearrangement. For a sufficient time period (>τR), the solidliquid tension is recovered to γsl. As a result, it is anticipated that the CA (or shape) of the bubble would be altered accordingly by moving the contact line in order to satisfy Young’s equation which is corresponding to the local minimum of the surface free energy. Unfortunately, the bubble state is frustrated after the reversal of surface rearrangement, and the surface free energy is actually at the nondifferential minimum instead of local minimum. Note that after the reversal of surface rearrangement two domains with different wettability are formed: the wetted domain with γ0sl and the rest of the domain with γsl. They are separated by the wettability boundary as depicted in Figure 3. Under the condition of constant bubble volume, the invasion of the contact line into the wetted domain with lower wettability leads to the decline of the CA, i.e., θ < θr. However, the surface free energy rises with decreasing the CA, and thereby such an action is energetically unfavorable. On the contrary, if the bubble tends to bulge from θr to θa, then the contact line has to move from low to high wettability domains. Since the surface free energy associated with boundary crossing is increased, the contact line pinning occurs. If one deflates the bubble volume gradually, the same frustrated state remains until the CA reaches θa. Further deflation leads to contact line movement of liquid advancing by keeping the same CA at θa, corresponding to the local minimum of the surface free energy. On the other hand, further inflation on the original bubble volume results in the contact line expanding by keeping the same CA at θr, also corresponding to the local minimum in free energy. Our aforementioned analyses is simply by thermodynamics based on adhesion hysteresis. Contact line pinning is a consequence of frustration of surface energy and not an origin of CA hysteresis. The movement of the contact line is driven by the existence of the local minimum. In the defect model, however, the contact line is mechanically pinned at the local blemish, which is more wettable than the substrate. The line breaks off the blemish when the force exceeds a critical value. Note that both blemish and substrate are ideal so that no hysteresis is associated with either one of them in the defect model. It is difficult to explain pinning of liquid advancing associated with a captive 6895
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Langmuir bubble by the defect model. Evidently, the existence of more wettable blemishes on the surface is the sufficient condition for CA hysteresis for both models. The pinning effect of crossing boundary between different wettability domains is responsible for contact line pinning. Nonetheless, blemishes are induced in the adhesion model but are inherent in the defect model. In the former, the induced defects are uniformly distributed in wetted area and characterized by γ0sl. In the latter, the properties of the local defect including defect size, strength, and density depend on the preparation of the substrate.
VII. CONCLUSION Hysteresis effects are commonly observed in wettingdewetting phenomena. CA measurements for both sessile drop and captive bubble are performed on the acrylic glass. CA hysteresis of a sessile drop on a substrate consists of continuous invasion of liquid phase with the advancing angle (θa) and contact line pinning of liquid phase retreat until the receding angle (θr) is reached. Receding pinning is generally attributed to localized defects that are more wettable than the rest of the surface. On the other hand, CA hysteresis of a captive bubble is opposite to that of a sessile drop. A captive bubble generally forms with the receding angle. Upon inflation of a bubble, the liquid phase retreats continuously with the receding angle. For a deflating bubble, however, the liquid phase tends to invade, but the contact line is pinned until the advancing angle is attained. Consequently, CA hysteresis of a captive bubble is anomalous. Unfortunately, the defect model cannot explain advancing pinning associated with liquid phase invasion driven by a deflating bubble and continuous contact line movement associated with liquid phase retreat driven by the inflating bubble. A simple thermodynamic model based on adhesion hysteresis is proposed to explain anomalous CA hysteresis of a captive bubble quantitatively. It involves two solidliquid interfacial tensions: γsl and γ0sl. Young’s equation with γsl gives the advancing angle θa while that with γ0sl due to surface rearrangement yields the receding angle θr. The hysteresis in adhesion energy is related to the decrease in the solidliquid interfacial energy, γ0sl < γsl. Since the surface energy change is less than zero, Δγsl = γ0sl γsl < 0, such hysteresis occurs spontaneously and is associated with the restructuring of the adhesive interface over some period of time. Our analytical analysis indicates that contact line pinning is a consequence of frustration of surface energy and not an origin of CA hysteresis. For contact line pinning associated with liquid advancing, the equilibrium CA (shape) corresponds to nondifferential minimum instead of local minimum. On the basis of our thermodynamic model, Surface Evolver simulations are performed to reproduce both advancing and receding behavior associated with a captive bubble on the acrylic glass. The simulation results agree quite well with the experimental results.
ARTICLE
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’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected] (Y.-J.S.);
[email protected] (H.-K.T.).
’ ACKNOWLEDGMENT This research work is financially supported by NCU/ITRI Joint Research Center and National Science Council of Taiwan. 6896
dx.doi.org/10.1021/la2009418 |Langmuir 2011, 27, 6890–6896
