ARTICLE pubs.acs.org/Langmuir
Comparison of Sessile Drop and Captive Bubble Methods on Rough Homogeneous Surfaces: A Numerical Study F. J. Montes Ruiz-Cabello,† M. A. Rodríguez-Valverde,*,† A. Marmur,‡ and M. A. Cabrerizo-Vílchez† †
Biocolloid and Fluid Physics Group, Department of Applied Physics, University of Granada, Campus de Fuentenueva, E-18071 Granada, Spain ‡ Department of Chemical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel ABSTRACT: Quasi-static experiments using sessile drops and captive bubbles are the most employed methods for measuring advancing and receding contact angles on real surfaces. These observable contact angles are the most easily accessible and reproducible. However, some properties of practical surfaces induce certain phenomena that cause a built-in uncertainty in the estimation of advancing and receding contact angles. These phenomena are well known in surface thermodynamics as stickslip phenomena. Following the work of Marmur (Marmur, A. Colloids Surf., A 1998, 136, 209215), where the stickslip effects were studied with regard to sessile drops and captive bubbles on heterogeneous surfaces, we developed a novel extension of this study by adding the effects of roughness to both methods for contact angle measurement. We found that the symmetry between the surface roughness problem and the chemical heterogeneity problem breaks down for drops and bubbles subjected to stickslip effects.
1. INTRODUCTION Wetting is ubiquitous in many emerging disciplines such as self-cleaning surfaces, microfluidics, microelectromechanical and nanoelectromechanical systems, and so forth.13 The wettability of a solid surface by a probe liquid is usually monitored with contact angle measurements, from which the surface energy of the solid may be evaluated.48 The wettability of practical solid surfaces is commonly described by a range of observable contact angles, known as the contact angle hysteresis range. The limits of this range are known as the theoretical advancing contact angle (TACA) and the theoretical receding contact angle (TRCA).9,10 However, in practice, TACA and TRCA are rarely reproduced in experimental studies because any wetting system is frequently affected by environmental perturbations, which reduce the experimentally accessible hysteresis range. Instead, the operative limits of the range of measurable contact angles are referred to as the practical advancing contact angle (PACA) and the practical receding contact angle (PRCA), and their values strongly depend on the measurement method and the external conditions. PACA and PRCA are observed when the three-phase contact line is quasi-statically moving on the solid surface, increasing or decreasing the solidliquid area. There are several experimental methods for measuring PACA and PRCA, although their results usually disagree.1113 Contact angle goniometry and Wilhelmy balance tensiometry are the most frequently used methods.1318 It is often difficult to decide whether the disagreement found between contact angle methods reflects varying precisions of measurement or differences in experimental procedures or qualities of surfaces. In the goniometry methods based on the shape of a sessile drop, the motion of the three-phase contact line may be forced either by tilting the r 2011 American Chemical Society
solid surface19 or by changing the drop volume by direct liquid addition or removal.20,21 Drop shape methods enable reciprocal configurations such as a captive bubble (i.e., an air bubble against a solid surface immersed in a liquid). Because the most popular goniometry methods are the sessile drop (SD) and the captive bubble (CB), we focused on them in the current work. Although SD and CB methods are expected to be equivalent because of their symmetry, several authors reported important differences not only in PACA and PRCA10,22,23 but also in the values of TACA and TRCA accessible with both methods under idealized conditions.13 Marmur theoretically studied, using 2D drop/bubble models, the fluctuation of TACA and TRCA on smooth heterogeneous surfaces as the drop/bubble volume changed. Although the approximation of low-pitch solid surfaces or moderately heterogeneous solid surfaces allows treating the effect of roughness and heterogeneity on contact angle in a similar way, we found very different behaviors of drops and bubbles in contact with rough surfaces. The objective of this article is to extend the study of Marmur13 to sawtooth-like rough surfaces using 3D drop/bubble models. The current numerical study allows the identification of, depending on the surface characteristics, the goniometry method (SD or CB) with smaller fluctuations of the contact angle in response to drop/bubble volume variations. This study will be helpful in selecting, in principle, the best-suited goniometry method for the case of rough homogeneous surfaces.
Received: April 5, 2011 Revised: June 2, 2011 Published: June 06, 2011 9638
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Figure 1. Contact angle θ and meniscus angle φ observed with the (a) sessile drop (SD) and (b) captive bubble (CB) methods. The term fluid is indistinctly applicable to a gas or an immiscible liquid with greater or lower density than that of the main liquid (e.g., solid/liquid/oil systems). The term captive bubble refers to a gas bubble or a liquid drop trapped against the surface.
Figure 2. Rough homogeneous surface with a radial sawtooth pattern. The surface is composed of two inclined regions with slope angles of R1 and R2. The width of the descending region is ω, and the width of the ascending region is λ ω. The central region is ascending.
2. THEORY For simplicity, we consider sessile drops and captive bubbles in a gravity-free environment with circular contact lines and a unique contact angle. Under these assumptions, the shape of the liquidfluid interface is a spherical cap. For a spherical cap of volume V, the contact radius rC and the angle φ are related by the expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi 3 p 3 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ cos φ ð1Þ rC ¼ V πð2 þ cos φÞ 1 cos φ As shown in Figure 1, angle φ corresponds to the contact angle θ observed with an SD and the supplementary contact angle π θ observed with a CB. From eq 1, there is just one possible contact angle θ for a drop or a bubble with a given reduced contact radius, rC/(V)1/3. This contact angle, imposed by geometrical constraints (spherical cap model), is usually referred to as a geometrical contact angle.24 The topographically patterned surfaces used in this study were purposely chosen to be radial for consistency with the spherical interface model (eq 1). Furthermore, the symmetry axis of the drop/bubble always coincided with the symmetry axis of the pattern. Thus, the contact line was always circular, and there was only one observed contact angle for each contact radius. 2.1. Rough Homogeneous Surfaces. The 2D model of the sawtooth profile can capture important features of more complicated rough surfaces, and it further enables a close-form expression of the roughness ratio. For this study, we designed rough homogeneous surfaces with sawtooth patterns by concentrically alternating peaks and valleys with a circular shape and with a fixed intrinsic contact angle θi (Figure 2). The period of the pattern is symbolized by λ. The descending regions,
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Figure 3. Apparent contact angles measured with the (a) sessile drop and (b) captive bubble methods on different regions of a rough homogeneous surface with a radial sawtooth pattern (Figure 2). On those surface regions where the contact angle observed with the SD method overestimates the intrinsic contact angle, the contact angle observed with the CB method underestimates it and vice versa. For R1 6¼ R2, we expect to find a dependence of the highest and lowest possible contact angles on the SD and CB methods.
identified with the unsigned slope angle R1, have a horizontal width ω, and the ascending regions (slope angle R2), have a horizontal width λ ω. These parameters are related by the closure equation ω tan R1 = (λ ω) tan R2. A valley is placed at the pattern center. On rough surfaces, the observed contact angle is an apparent contact angle, θap, because the actual local slope may not be macroscopically observable. Taking into account the Young equation, if the drop/bubble attains a (meta)stable/unstable equilibrium configuration on the radial sawtooth pattern described above, then the apparent contact angle should be equal to the concerning intrinsic contact angle plus the local surface slope at the point of contact. From Figure 3, we can illustrate how the contact angle observed with a SD underestimates the intrinsic contact angle on the R2 regions (θap = θi R2), whereas the contact angle observed with a CB overestimates it (θap = θi þ R2). Just the opposite happens in the R1 regions. The apparent contact angle can be described as a function of the polar radius, θap(r), ( θi - R2 , nλ < r < ðn þ 1Þλ ω θap ðrÞ ¼ ð2Þ θi ( R1 , ðn þ 1Þλ ω < r < ðn þ 1Þλ where the upper sign corresponds to the SD and the lower one, to the CB and n symbolizes the boundary index (0, 1, 2...). On the rough homogeneous surfaces used in this study, the configurations allowed by the Young equation must satisfy eqs 1 and 2. Drops/bubbles can reach four equilibrium configurations depending on whether the contact line is over an R2 region, over an R1 region, just at a peak, or at a valley. Stable or metastable equilibrium configurations are those for which the system free energy is at a minimum. The Gibbs free energy, G, of a wetting system is equal to the sum of the interfacial energies associated with the liquidfluid, the solidliquid, and the solidfluid interfaces. The calculation of the Gibbs free energy for rough surfaces is more complex than for smooth surfaces. To preserve the simplicity of the expressions presented in this work, we used some approximations. We intended to identify system configurations corresponding to free-energy minima or maxima rather than to provide an exact 9639
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derivation of the system free energy. Hence, we are aware of the limitations imposed by the use of eq 1 for rough surfaces, where the volume of liquid or fluid entrapped inside the topographical grooves is assumed to be negligible. The dimensionless Gibbs free energy, G*, of the system to study is written as G
G ALF rC 2 ¼ r π cos θ W i λ2 λ2 γLF λ2
ð3Þ
where the sign “” corresponds to the SD and the sign “þ” corresponds to the CB, rW symbolizes the Wenzel factor, and ALF is the actual liquidfluid area. The first term in eq 3 refers to the energy contribution of the liquidfluid interface, and the second term, the contribution of the solidliquid interface (SD) or the solidfluid interface (CB). The Wenzel factor for the sawtooth patterns is ω/λ sec R1 þ (1 ω/λ)sec R2. The apparent area of the liquidfluid interface as a function of the contact radius rC and the observed contact angle θ is ALF, ap ¼
2π rC 2 1 ( cos θ
ð4Þ
where the sign þ or is applicable to the SD or the CB, respectively. When the contact line is in an R2 region or an R1 region, the apparent area of the liquidfluid interface is equal to the actual one, on average. Otherwise, when the contact line rests at a peak or at a valley, the actual liquidfluid area might be roughly estimated as ALF ALF, ap ( 2πrC Æzæ ¼ ALF, ap ( πωrC tan R1
ð5Þ
where þ refers to the situation in which the contact line is at a valley and , at a peak. In these calculations, we assume that the rough surface has an average amplitude, Æzæ, that is roughly estimated as ω tan R1 ð6Þ 2 2.2. StickSlip Phenomena. One of the most popular techniques for measuring the values of PACA and PRCA on real surfaces is the growingshrinking drop/bubble, where the solidliquid area is slowly increased or decreased by the addition or removal of liquid/air volume accordingly.22,25,26 During these experiments, a jerky motion of the contact line is occasionally observed when the boundaries between the surface chemical domains or the topography asperities are distributed parallel to the three-phase contact line.27 For the radially patterned surfaces described in section 2.1, the stickslip behavior is maximized because the entire contact line finds the same feature at once during its quasi-static motion. The stickslip phenomena produce an oscillating behavior in the contact angle (namely, their theoretical and practical values) as the drop/bubble volume changes. This fluctuation is higher for the range of tiny drops/bubbles, and it decreases as the drop/bubble volume increases. Because of the stickslip behavior of contact lines, highest and lowest possible contact angles (HPCA and LPCA, respectively) for a wetting system do not always correspond to the values of TACA and TRCA. The convexity of the liquid meniscus plays an important role in the fluctuation amplitude in the contact angle responses reproduced with the SD or CB method. Because the liquidfluid interface is more convex, the fluctuation in the contact angle is greater. Hence, as Marmur reported13 for smooth heterogeneous surfaces with very Æzæ
Figure 4. Apparent contact angle (continuous line) and geometrical contact angle (dotted lines) of the (a) sessile drop or (b) captive bubble methods for several reduced volumes (V* = V/λ3) as a function of the dimensionless contact radius (rc/λ). The surface was rough homogeneous with θi = 80, R1 = 60, and R2 = 30 (ω/λ = 0.25). Stable and unstable equilibrium configurations allowed by the Young equation for a reduced volume V* = 5000 are shown with circles (intersections between both curves).
hydrophobic chemical patches (where the sessile drops are highly curved and the captive bubbles have a small slope), the fluctuation in TACA observed with the SD method is markedly greater than the contact angle uncertainty observed with the CB method on the same surface. Otherwise, when the intrinsic contact angle is lower than 90 (small-slope drop versus highly curved bubble), the uncertainty in TRCA observed with the SD method is much lower than with the CB method.
3. RESULTS AND DISCUSSION In this section, we present the results for the rough homogeneous surfaces described in section 2.1. We are aware of the restrictive character of the sawtooth-like pattern, even though this choice is intended only to illustrate the topography effects of more complicate patterns described by two maximum and minimum slopes. For each surface and for a particular volume of a drop or bubble, we previously found the system configurations allowed 9640
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Figure 5. Dimensionless Gibbs free energy, G*, for every configuration allowed by the Young equation as a function of the contact angle observed with the sessile drop method (open symbols) or the captive bubble method (closed symbols) on a rough homogeneous surface with θi = 80, R1 = 60, and R2 = 30 (ω/λ = 0.25, rW = 1.36). We used the reduced volume V* = 5000. The contact angle calculated with the Wenzel equation for this sawtooth pattern is shown by a vertical dashed line. Triangles correspond to configurations where the contact line was resting at peaks, and squares, at valleys. Circles correspond to drop configurations where the contact line was on either the ascending or the descending slope. Unlike the SD method, for the chosen volume no configuration of CB attained on the ascending or descending slope areas was allowed by the Young equation.
by the local Young equation from the equality between the geometrical contact angle (eq 1) and the apparent contact angle (eq 2). Next, we identified the (meta)stable and unstable configurations from the minima and maxima of the system free energy evaluated at every allowed configuration. This procedure was repeated for increasing or decreasing drop/bubble volumes to mimic an ideal experiment of a growingshrinking drop/ bubble. The maximal stickslip response, in terms of TACA and TRCA values, on the surfaces designed in section 2.1 was reproduced using the SD method and the CB method separately. 3.1. Equilibrium Configurations. In Figure 4, we plot the apparent contact angle (continuous line) and the geometrical contact angle (dotted lines) observed with (a) SD and (b) CB as functions of the dimensionless contact radius for different values of the reduced volume (V* = V/λ3 = 1, 100, 1000, and 5000). We selected a rough homogeneous surface with the following parameters: θi = 80, R1 = 60, and R2 = 30 (ω/λ = 0.25, rW = 1.36). It is noticeable that the values of HPCA (θi þ R1 or θi þ R2) and LPCA (θi R1 or θi R2) were different for the SD method (140 and 50, respectively) and the CB method (110 and 20). Unlike smooth heterogeneous surfaces, the contact angle measured on rough homogeneous surfaces usually depends on the measuring method. This fact is very important in analyzing the main differences found with the SD and CB methods. From a suitable choice of the surface parameters of the radial patterns (θi, R1, and R2) and a fixed ω/λ value, we were able to design a heterogeneous surface that was phenomenologically equivalent to a rough surface (in the sense of predicting the same contact angle response) using either the SD method or the CB method. Only for ω/λ = 0.5 (R1 = R2), a chemically heterogeneous surface will be phenomenologically equivalent to a rough homogeneous surface using both methods.
Figure 6. Theoretical advancing contact angle and theoretical receding contact angle values provided by the sessile drop (triangles) and captive bubble (circles) methods in terms of the cubic root of reduced volume for two different rough homogeneous surfaces: (a) θi = 70, R1 = 5, and R2 = 38.5 (ω/λ = 0.9) and (b) θi = 70, R1 = 38.5, and R2 = 5 (ω/λ = 0.1). We show the value of intrinsic contact angle θi with a horizontal line.
Graphically, the equilibrium configurations allowed by the Young equation for a given drop/bubble volume on rough surfaces correspond to the intersections between the geometrical contact angle given by eq 1 and the apparent contact angle described by eq 2. It is worth pointing out that the intersections are mostly found at vertical asymptotes of the apparent contact angle curve (case V* = 5000 in Figure 4). These configurations reveal that the contact line “rests” at a slope transition (peak or valley). Again, the values of HPCA and LPCA do not usually agree with the values of TACA and TRCA for a fixed drop/ bubble volume. The agreement will happen only when the geometrical contact angle curve intersects the apparent contact angle curve at horizontal branches (free-moving contact line). 3.2. (Meta)Stable and Unstable Equilibrium Configurations. Once the configurations allowed by the Young equation 9641
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Table 1. Contact Angle and Method with a Lower Measurement Uncertainty for the Very Different Rough Homogeneous Surfaces Studied in This Worka R1 = R2
a
high hysteresis θi f 90
TACA by CB TRCA by SD
low hysteresis θi f 90 low hysteresis θi . 90
TACA by CB
low hysteresis θi , 90
TRCA by SD
R2 f 0
R1 f 0
TACA by CB, TRCA by SD
TACA by SD,TRCA by CB
As expected, the symmetrical case in rough surfaces (R1= R2) is fully equivalent to case ω/λ = 0.5 in heterogeneous surfaces.13
for a known drop/bubble volume were obtained, we computed the system free energy from eq 3 at each configuration. In Figure 5, we plot the system free energy as a function of the observed contact angle, evaluated at all of the configurations allowed by the Young equation for parameters θi = 80, R1 = 60, R2 = 30, and V* = 5000 (intersections marked in Figure 4). In Figure 5, we identify the system configurations where the contact line “rests” at a peak (triangles), at a valley (squares), and over an R1 region or an R2 region (circles). The last two configurations are found with the CB for V* = 5000. We observe in Figure 5 that the global energy minimum of the CB-surface system is located at the contact angle predicted by the Wenzel equation28 for this surface (vertical dotted line). However, we notice that the peaks correspond to (meta)stable configurations (stick) of drop and bubble whereas the valleys correspond to unstable configurations (jump) of drop and bubble. This is a fundamental difference observed for smooth heterogeneous surfaces. 3.3. GrowingShrinking Drop/Bubble Experiment. In this section, we calculate the values of TACA and TRCA as in a typical growingshrinking drop (bubble) experiment on a rough homogeneous surface. As expected, the symmetrical case for rough surfaces (R1= R2) was fully equivalent to case ω/λ = 0.5 for heterogeneous surfaces.13 The uncertainty observed during the estimation of TACA and TRCA by each method was strongly dependent on the slope angles. In Figure 6, we show the values of TACA and TRCA provided by the SD and CB methods on two rough homogeneous surfaces: (a) θi = 70, R1 = 5, and R2 = 38.5 (ω/λ = 0.9) and (b) θi = 70, R1 = 38.5, and R2 = 5 (ω/λ = 0.1). For R1 f 0 (Figure 6a), the uncertainty during the estimation of TACA with the SD method is much lower than that observed with the CB method, and the opposite is found for TRCA. Otherwise, for R2 f 0 (Figure 6b), we observe just the opposite behavior to the case of R1 f 0, as expected. In Table 1, we summarize the decision rules for the contact angle and the measuring method with lower fluctuation on different rough homogeneous surfaces. We focused on four kinds of surfaces: high-hysteresis surfaces with intrinsic contact angles close to 90, low-hysteresis surfaces with intrinsic contact angles close to 90, very hydrophobic surfaces, and very hydrophilic surfaces with low hysteresis. For each kind of surface, we studied three different cases as the symmetry degree of pattern: R1= R2 (ω/λ = 0.5), R2 f 0 (ω/λ f 0), and R1 f 0 (ω/λ f 1).
4. SUMMARY AND CONCLUSIONS In this work, we performed a numerical comparison between the SD and CB methods on rough homogeneous surfaces. We selected radially patterned surfaces to produce circular contact lines and to maximize the stickslip effects. We highlighted the
main differences in the criteria applied to identify the equilibrium configurations of drop or bubble, for a given volume, on rough surfaces. Once the equilibrium configurations were found, we distinguished them in (meta)stable and unstable configurations by computing the system free energy. Following this procedure, we were able to predict the values of TACA and TRCA, under ideal experimental conditions, over a volume range typically reproduced in growingshrinking drop (bubble) experiments. We analyzed how the surface properties affect the uncertainty in the measurement of TACA and TRCA using the SD method or the CB method. The main conclusions of this study may be summarized as follows: (1) The highest and lowest possible contact angles observed on rough homogeneous surfaces depend on the measurement method because of the asymmetry between ascending and descending slopes of the surface. This does not happen with smooth heterogeneous surfaces (2) On rough homogeneous surfaces, surface peaks act as potentially (meta)stable positions (local energy minima) of the contact line of sessile drops as well as of captive bubbles. Instead, valleys are potentially unstable positions (local energy maxima) of the contact line of both drops and bubbles. However, the hydrophilic-to-hydrophobic regions of smooth heterogeneous surfaces are potentially (meta)stable positions for drops and potentially unstable positions for bubbles. And the opposite is true for hydrophobic-to-hydrophilic regions. (3) The sessile drop and captive bubble methods on rough surfaces provide different fluctuation in the contact angle (measurement uncertainty), such as varying the drop/ bubble volume, due to two sources: the interfacial curvature of drops and bubbles and the surface characteristics, namely, the asymmetry between ascending and descending slopes on rough homogeneous surfaces.
’ AUTHOR INFORMATION Corresponding Author
*Tel: (34) 958-24-00-25. Fax: (34) 958-24-32-14. E-mail: marodri@ ugr.es.
’ ACKNOWLEDGMENT This research has been supported by the Ministry of Science and Innovation (project MAT2010-14800) and by the Junta de Andalucía (projects P07-FQM-02517, P08-FQM-4325, and P09-FQM-4698.) ’ REFERENCES (1) Sobocki, T.; Jayman, F.; Sobocka, M. B.; Marmur, J. D.; Banerjee, P. Biochim. Biophys. Acta, Gene Struct. Expression 2007, 1769, 61–75. 9642
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