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Theoretical Contact Angles on a Nano-Heterogeneous Surface Composed of Parallel Apolar and Polar Strips Changpeng Fang and Jaroslaw Drelich* Department of Materials Science and Engineering, Michigan Technological University, Houghton, Michigan 49931 Received December 27, 2003. In Final Form: May 8, 2004
Neumann-Good’s parallel strip model (J. Colloid Interface Sci. 1972, 38, 341) was used to analyze the contact angle hysteresis for a liquid on a heterogeneous surface composed of alternatively aligned horizontal apolar (θ ) 70°) and polar (θ ) 0°) strips. The critical size of the strip width, below which the contact angle hysteresis disappears, was determined on the basis of the analysis of the activation energy for wetting to be from 6 to 12 nm. This calculated value of the critical strip size is 1 order of magnitude smaller than that of 0.1 µm, which has been commonly considered as the limit of heterogeneity size causing the appearance of the contact angle hysteresis.
1. Introduction There is a general belief that solid surfaces with features of heterogeneity or roughness having dimensions smaller than 0.1 µm are free of contact angle hysteresis (see, for example, a review paper by Andrade et al.;1 definition of contact angle hysteresis is provided as a note in the reference list).2 On the contrary, the reports on substrates in which only one contact angle is measured (advancing contact angle equal to receding contact angle) are extremely rare. In our previous contributions, we demonstrated experimentally that surface roughness with asperities as small as several nanometers has a profound effect on wetting characteristics of the substrate.3,4 We also could not eliminate the contact angle hysteresis effects on such good quality substrates as self-assembled monolayers of thiols formed on gold-coated silicon wafers, with a roughness less than 1-2 nm and heterogeneity that was only caused by imperfections of the molecular monolayer and boundaries formed between domains of aligned thiols.5 Therefore, we believe that surface heterogeneity at a level that is smaller than 100 nm influences the contact angle hysteresis in many solid-liquid systems. In this paper, we use Neumann and Good’s model,6 with certain modifications of the calculation methods, to predict the contact angle hysteresis for liquid on a nano* To whom correspondence should be addressed. E-mail:
[email protected]. Phone: 906-487-2932. (1) Andrade, J. D.; Smith, L. M.; Gregonis, D. E. In Surface and Interfacial Aspects of Biomedical Polymers; Andrade, J. D., Ed.; Plenum Press: New York, 1985; Vol. 1. (2) The contact angle hysteresis is the difference between the advancing and receding contact angles. The advancing static contact angle is measured for the liquid recently advanced over the solid surface whereas the receding static contact angle refers to the contact angle measured for the liquid recently retreated from the solid surface. The advancing (receding) dynamic contact angle is measured for the liquid advancing over (retreating from) the solid surface and depends on the velocity of the liquid movement. (3) Miller, J. D.; Veeramasuneni, S.; Drelich, J.; Yalamanchili, M. R.; Yamauchi, G. Polym. Eng. Sci. 1996, 36, 1849. (4) Veeramasnuneni, S.; Drelich, J.; Miller, J. D.; Yamauchi, G. Prog. Org. Coat. 1997, 31, 265. (5) Drelich, J.; Miller, J. D.; Good, R. J. J. Colloid Interface Sci. 1996, 179, 37. (6) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38, 341.
heterogeneous surface. Also, using this model, we determine the critical size of heterogeneity that causes the appearance of contact angle hysteresis. In this paper, we first review briefly the NeumannGood model6 on the determination of contact angles on a heterogeneous surface composed of parallel strips. Next, we present our theoretical analysis of the contact angle hysteresis on the heterogeneous surface composed of parallel and alternating strips of varying hydrophobicity. Our analysis consists of the following parts: (1) the total free energy changes, in a conventional liquid-gas-solid system, are calculated for a smooth and heterogeneous surface composed of hydrophobic and hydrophilic strips; (2) advancing and receding contact angles are defined on the basis of an energy barrier intrinsic to the system; and (3) the modified model is used to analyze the effects of the area fraction of the hydrophobic strip and the surface strip width on the contact angle hysteresis. As the most important result of this contribution, we calculated the critical size of heterogeneity below which the contact angle hysteresis disappears. This size is significantly different than that discussed in the previous report.6 Our modeling also leads us to another critical size of heterogeneity above which the contact angle hysteresis remains constant. 2. Description of the Model 2.1. Homogeneous Surface. Neumann and Good6 developed the thermodynamic model for analysis of contact angles on a vertical plate immersed into a liquid (Figure 1). In such a system, as shown in Figure 1, θ ) 90° is the reference state, and the total free energy change (∆G) is caused by the capillary rise or depression of liquid. The total free energy change according to the change in the position of the three-phase contact line at the immersed plate comes from three components:6 (1) ∆G1, interfacial energy change due to a change in solid-vapor interfacial area and a corresponding change in the solid-liquid interfacial area; (2) ∆G2, interfacial energy change as a result of the increase in liquid-vapor interfacial area; and (3) ∆G3, interfacial energy change caused by the work that has to be done against gravity.
10.1021/la0364587 CCC: $27.50 © 2004 American Chemical Society Published on Web 06/30/2004
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Figure 1. Capillary rise of a liquid along a vertical plate.
Thus, the total free energy change in the three-phase system can be written as
∆G ) ∆G1 + ∆G2 + ∆G3
Figure 2. ∆G versus θ curves for homogeneous surfaces with different equilibrium contact angles (γLV ) 70 mJ/m2, ∆g ) 1000 kg/m3).
(1)
where
x
2γLV x1 - sin θ ∆Fg
(2)
2γLV (x2 - x1 + sin θ) ∆Fg
(3)
∆G1 ) -LyγLV cos θY
x
∆G2 ) LyγLV
x
1 ∆G3 ) LyγLV 3
2γLV [(2 - sin θ)x1 + sin θ - x2] (4) ∆Fg
Here, Ly is the length of the three-phase contact line in the y direction; γLV is the liquid-vapor interfacial tension; ∆F is the density difference between the liquid and vapor; g is gravitational acceleration; θ is the instantaneous contact angle characterizing the particular configuration of a liquid on a solid surface; and θY is the equilibrium contact angle defined by Young’s equation:7
γSV - γSL ) γLV cos θY
(5)
where γSV and γSL are the solid-vapor and solid-liquid interfacial free energies per unit areas, respectively. In our first step, eqs 1-5 were used to calculate the relationship between the free energy change and the instantaneous contact angle for a liquid on a smooth and homogeneous surface. The results are shown in Figure 2. It should be pointed out here that when the contact angle is larger than 90°, that is, for capillary depression, eqs 2 and 4 must be modified with ∆G1 and ∆G3 terms attaining the reverse signs. Each curve in Figure 2 passes through an energetic minimum at a characteristic value corresponding to the equilibrium contact angle that is predicted by the Young’s equation (eq 5). In this way, the legitimacy of the model and our calculations is demonstrated. 2.2. Heterogeneous Surface. In this section, we consider the vertical plate consisting of horizontal, parallel, (7) Young, T. Miscellaneous Works; Peacock, G., Ed.; J. Murry: London, 1855; Vol. 1.
Figure 3. Capillary rise of a liquid along a model heterogeneous solid surface.
and alternating strips (Figure 3). Each type of strip has a different wetting characteristic that is defined by equilibrium contact angles θY1 and θY2, respectively. In the next part of this paper, we call these strips apolar and polar, or θY1 and θY2 strips. In our calculations, the following parameters were used: θY1 ) 70°, θY2 ) 0°, g ) 9.8 N/kg, Fwater ) 1000 kg/m3, γLV ) 70 mJ/m2, and ∆F ) 1000 kg/m3. As shown in Figure 3, the spacing of the strips is termed as ∆h, which is equal to the sum of the widths of apolar and polar strips, ∆h1 + ∆h2. The term f ) (Ly∆h1)/(Ly∆h) ) ∆h1/(∆h1 + ∆h2) is the fractional area of the surface with the equilibrium contact angle of θY1, and the fractional area of the surface with the equilibrium contact angle of θY2 is 1 - f. The total free energy change due to the capillary rise of liquid at the heterogeneous surface shown in Figure 3 can be described by equations similar to eqs 1-5. From the Neumann-Good model, we know that both the ∆G2 and ∆G3 terms depend only on the instantaneous contact
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angle θ and not on the properties of the two types of solids. Therefore, these two terms remain the same as in eqs 3 and 4, respectively. However, in changing from the reference state to other configurations, the three-phase contact line now passes a number of θY1 and θY2 strips, and, therefore, the ∆G1 term should be modified. In Neumann and Good’s original treatment, they considered only equal widths of the two types of strips and used the constant increment of θ (2 and 2/3° in their cases) to represent the strip width. However, this leads to at least two drawbacks that limit the application of their model. First, their model can only treat the situation when f ) 0.5. Importantly, second, if there is a large difference between the intrinsic contact angles of the two component materials, a constant increment of θ may not ensure constant strip width. To analyze the contact angle hysteresis for different f values and different strip widths efficiently, we modified the calculation of the ∆G1 term in eq 1 as follows:
∆G1 ) ∆G11 - ∆G12
(6)
∆G11 ) LyγLV cos θY1h1
(7)
∆G12 ) LyγLV cosθY2h2
(8)
Figure 4. Free energy ∆G as a function of the instantaneous contact angle on a heterogeneous surface composed of horizontal, parallel strips characterized by equilibrium contact angles of 70 and 0°. Other parameters of the system include γLV ) 70 mJ/m2 and ∆g ) 1000 kg/m3.
where
Here, h1 + h2 ) h is the total capillary rise, defined as follows:
h)
x
2γLV x1 - sin θ ∆Fg
(9)
From eqs 6-9, we know that to calculate ∆G1, h1 and h2 must be calculated first for a given θ.
h ) K∆h + h(θ)
(10)
where K is an integer so that 0 e h(θ) < ∆h. When 0 e h(θ) < f∆h,
h1 ) Kf∆h + h(θ)
(11a)
h2 ) K(1 - f)∆h
(11b)
and when f∆h e h(θ) < ∆h,
Figure 5. Total free energy change (∆G) as a function of the instantaneous contact angle for homogeneous polar (0°) and apolar (70°) surfaces.
3. Contact Angle Hysteresis
h1 ) Kf∆h + f∆h
(11c)
h2 ) K(1 - f)∆h + h(θ) - f∆h
(11d)
On the basis of eqs 1-11, a program code was written for the calculation of the free energy change with the contact angle for a heterogeneous surface consisting of alternatively aligned strips. The examples of correlation between ∆G and θ, for θY1 ) 70°, θY2 ) 0°, and f ) 0.5, ∆h ) 0.01h0 (here, h0 ) (2γLV/∆Fg)1/2, are shown in Figure 4. Similar to those in the literature,6,8 the ∆G versus θ curve in Figure 4 is not smooth but has a number of local minimum and maximum points between 0 and 70°. This fluctuation on the ∆G versus θ curve is crucial to the analysis of contact angle hysteresis and is discussed in the next section. (8) Li, D.; Neumann, A. W. Colloid Polym. Sci. 1992, 270, 498.
3.1. Fluctuation of the ∆G versus θ Curve. Why does the fluctuation occur in the ∆G versus θ curve? When undergoing capillary rise, the three-phase contact line encounters the strip boundary and moves from one type of strip to another. During this transition, the free energy changes. Therefore, by intuition, one easily determines that the minimum and maximum points on the ∆G versus θ curve are associated with the strip boundaries. It is the case when θ falls in the range of 0-70°. However, when θ > 70°, neither minimum nor maximum points in the curve in Figure 4 appear. Let us first analyze the nature of the fluctuation using the ∆G versus θ curve for homogeneous materials. Figure 5 shows the ∆G versus θ curves for homogeneous θY ) 70° and θY ) 0° surfaces. We can see that the θY ) 70° curve attains its minimum value at the exact value of 70°, and the θY ) 0° curve attains its minimum value at 0°. In the θ range from 0 to 70°, ∆G decreases with an increasing
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Figure 6. Schematic model for a metastable state.
θ value for the θY ) 70° surface. For the θY ) 0° surface, ∆G increases when θ increases. For the liquid at a heterogeneous surface, when the three-phase contact line moves on the θY ) 0° strip, ∆G decreases. When the three-phase contact line reaches the strip boundary and passes over to the θY ) 70° surface, ∆G begins to increase, and, consequently, ∆G reaches a local minimum point at the strip boundary. Likewise, if the three-phase line passes over the boundary from the θY ) 70° strip to the θY ) 0° strip, ∆G reaches a local maximum point. Therefore, between 0 and 70°, local minimum and maximum points appear in the ∆G versus θ curve (Figure 4). 3.2. Metastable State. A general consensus is that the metastable states that appear on the ∆G versus θ curves are responsible for advancing and receding contact angles6,9,10 (as well as intermediate contact angles). It is not clear, however, what metastable position corresponds to the advancing or receding contact angles. Either the issue often is ignored, or angles for the metastable positions located at the closest vicinity to the boundary conditions are attributed to advancing and receding contact angles.6 This second approach often leads to the result that the advancing and receding contact angles at heterogeneous surfaces correspond to the equilibrium contact angles describing the components made of the heterogeneous surface. Apparently this is not the case in many practical situations, and both advancing and receding contact angles are recorded somewhere between the equilibrium contact angles of the components of the heterogeneous material.9 Local energetic minima at the ∆G versus θ curve are not always trapping positions for the three-phase contact line unless certain conditions are met. This is due to the internal energy of the liquid that can be used by the system to overcome some of the energetic barriers associated with the ∆G versus θ relationship during wetting of a heterogeneous surface by a liquid. Figure 6 and the following discussion serve as an example. In Figure 6, position 1 is a minimum ∆G point, and positions 2 and 3 are the maximum ∆G local points adjacent to position 1. For position 1 to be a metastable state, ∆Gb* and ∆Ga* must both be larger than a certain critical energy ∆G* available in the system (we define this term later) so that the three-phase contact line can jump neither backward over position 2 nor forward over position 3. Here, ∆Gb* is the energy barrier for the threephase contact line to move from position 1 back over position 2, and ∆Ga* is the energy barrier for the threephase contact line to move forward from position 1 over position 3. Therefore, if either ∆Gb* or ∆Ga* or both are smaller than ∆G*, position 1 cannot be considered as a metastable position, because the three-phase contact line can jump over the small energy barrier and move to another position. Then what is ∆G*? ∆G* is the largest energy barrier that the three-phase contact line can jump over without (9) Johnson, R. E., Jr.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744. (10) Schwartz, L.; Garoff, S. J. Colloid Interface Sci. 1985, 106, 422.
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an extra external energy supply. We define it as the activation energy of wetting (a critical energy barrier) that is available internally in the system due to thermal energy and vibrations. Thus, it depends on the properties of the liquid and temperature as well as conditions of the environment. 3.3. Advancing and Receding Contact Angles. For the purpose of the discussion in this paper, we define advancing and receding contact angles as follows: advancing the largest θ value that corresponds to contact angle the highest metastable state on the ∆G versus θ curve receding the smallest θ value that corresponds to contact angle the lowest metastable state on the ∆G versus θ curve
To explain the meaning of the above definitions, we return to the ∆G versus θ curve and identify the energetic states corresponding to receding and advancing contact angles. As we defined before, ∆Gb* is the energy barrier against the three-phase contact line moving backward (decreasing θ), and ∆Ga* is the energy barrier against liquid moving forward (increasing θ). Through calculations, we found that between the two equilibrium contact angles (in this case, 0 and 70°), ∆Gb* decreases and ∆Ga* increases as θ increases. This is shown in Figure 4. Both advancing and receding contact angles are marked in Figure 4 at positions 5 and 1, respectively. According to our definition, we know that position 1 and position 2 are metastable state positions, and position 3 and position 4 are not metastable state positions even though these two positions correspond to local minima on the ∆G versus θ curve. If the three-phase contact line is at position 3, it can jump forward to position 1 because ∆Ga* is smaller than ∆G*. The three-phase contact line will stay at position 1 as long as no additional (external) energy is supplied to the system. Position 2 is also a metastable state, but the corresponding angle is not the receding contact angle because, according to our definition, the receding contact angle is the smallest detectable contact angle. Similarly, θA is the advancing contact angle because there is no metastable state on the ∆G versus θ curve for θ > θA (to the right of position 5). At θR < θ < θA, all the minima in the ∆G versus θ curve are metastable or stable states that can correspond to intermediate contact angles. In summary, to predict the advancing and receding contact angles, we need to (1) determine the ∆G versus θ correlation; (2) find out ∆Gb and ∆Ga for each local minimum point on the ∆G versus θ curve; and (3) determine the metastable states among those minima and calculate the advancing and receding contact angles according to the above definitions. 4. Application of the Model For the calculations that follow, the major parameters used are θY1 ) 70°, θY2 ) 0°, g ) 9.8 N/kg, Fwater ) 1000 kg/m3, γLV ) 70 mJ/m2, and ∆F ) 1000 kg/m3. The selection of these values was dictated by our interest in wetting characterization of minerals; that is, hydrophobic coal contaminated with hydrophilic inorganic inclusions11 and hydrophilic minerals modified with hydrophobic collectors.12 The value of molar activation energy for wetting, ∆G*, which equals 49 kJ/mol, is taken from the literature (11) Gosiewska, A.; Drelich, J.; Laskowski, J.; Pawlik, M. J. Colloid Interface Sci. 2002, 247, 107. (12) Drelich, J.; Atia, A. A.; Yalamanchili, M. R.; Miller, J. D. J. Colloid Interface Sci. 1996, 178, 720.
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Figure 8. Effect of strip width on contact angle hysteresis (f ) 0.95).
Figure 7. Contact angle hysteresis for a heterogeneous surface composed of parallel and alternating apolar (70°) and polar (0°) strips as a function of coverage with apolar material. (a) Advancing (the curves above that of θC) and receding (the curves below that of θC) contact angles; (b) contact angle hysteresis.
report.13 Similar values were calculated from experiments for a number of different systems.14,15 4.1. Effect of Strip Fraction on the Contact Angle Hysteresis. Figure 7 shows the effect of the area fraction of the apolar strip on the contact angle hysteresis for different strip widths. From the curves in Figure 7a, it can be seen that the advancing and receding contact angles are always located between the equilibrium contact angles characteristic to components of the heterogeneous surface (in the case discussed here, between 0 and 70°). Furthermore, advancing and receding contact angles increase with the increasing fraction of the 70° strip. Results in Figure 7b indicate that the contact angle hysteresis attains its maximum value at the position where f is roughly equal to 0.5, and both increasing and decreasing f make the contact angle hysteresis smaller. In other words, for a given ∆h, the equal width of the two component strips leads to the maximun contact angle hysteresis. This result is consistent with the values of contact angle hysteresis measured for water drops on hydrophilic minerals modified with adsorbed layers of hydrophobic collectors.16 It is shown in Figure 7 that, for the same value of f, the contact angle hysteresis decreases with decreasing strip (13) Blake, T. D. In Wettability; Berg, J. C., Ed.; Mercel Dekker: New York, 1993. (14) Hayes, R. A.; Ralston, J. J. Colloid Interface Sci. 1993, 159, 429. (15) de Ruijter, M.; Kolsch, P.; Voue, M.; De Coninck, J.; Rabe, J. P. Colloids Surf., A 1998, 144, 235. (16) Drelich, J. Miner. Metall. Process. 2001, 18, 31.
Figure 9. Minimum strip width that can cause contact angle hysteresis for a model heterogeneous surface composed of parallel and alternating apolar (70°) and polar (0°) strips as a function of coverage with apolar material. Curve 1, ∆h* versus f; curve 2, ∆hi versus f.
width. It can be predicted that, if the strip width is sufficiently small, the hysteresis does not occur, that is, both the advancing and receding contact angles are equal to θC, the contact angle defined by the Cassie equation:17
cos θC ) f cos θY1 + (1 - f) cos θY2
(12)
We discuss this aspect in the next section. 4.2. Effect of Strip Width on Contact Angle Hysteresis. Figure 8 shows the effect of strip width on the advancing and receding contact angles. f ) 0.95 is used for the related calculations, and the other parameters remain the same as used in the previous section. As shown in Figure 8, the contact angle hysteresis decreases with decreasing the strip width. When ∆h decreases to a critical value of ∆h* (121 nm in this case), there is no hysteresis observed. However, at this critical point, the width of the polar strip is only about 6 nm. In Figure 9, curve 1 represents the relationship between ∆h* and f, and curve 2 is the related minimum strip width (either of the apolar strip when f < 0.5 or of the polar strip when f > 0.5) corresponding to ∆h*. It can be seen, from curve 2, that this critical minimum strip width always falls into the range of 6-12 nm for different f values. This is significantly smaller than demonstrated in the previous studies.6 The discrepancy is a result of both the analysis of different system and the adaptation of a different (17) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11.
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approach. Neumann and Good limited their analysis to a “poorly-heterogeneous” solid surface composed of strips with contact angles of 30 and 40°. They also assumed that the limit of the strip width affecting the contact angle hysteresis could be predicted from a contortion of the threephase contact line crossing the strips perpendicularly. Neumann and Good determined that there is no difference in the free energy of the three-phase system between the systems with the contorted and the smooth three-phase contact line for strips with dimensions less than about 0.1 µm. On this basis, they assumed that if a contorted threephase contact line cannot occur for the strips with a width of less than about 0.1 µm, no contact angle hysteresis should be expected for such heterogeneous surfaces. This value, unfortunately, has often been used as a rule of thumb for every system with a heterogeneous surface since its publication in 1972, although the authors clearly stated in their paper:6 “This lower limit of 0.1 µ width of strips applies, of course, only for a pair of surfaces with intrinsic contact angles of 30 and 40°, and γLV ) 50. For a larger difference in θY, the limiting strip width will clearly be some smaller dimension than 0.1 µ.” From Figure 8, we can also find another critical strip width, ∆h**, above which the advancing and receding contact angles are just the equilibrium contact angles of the apolar and polar strips, respectively. This means that, when the strip width is larger than ∆h**, the contact angle hysteresis is constant. In the present system, when the width of the polar strip is larger than about 4.6 µm, the receding contact angle would constantly be 0°. 5. Conclusions In this work, the fluctuation of the ∆G versus θ curve was analyzed for the capillary rise of a liquid along a smooth and heterogeneous surface composed of horizontal, alternatively aligned strips of apolar and polar materials. We put forward a redefined concept for the metastable state of the three-phase contact line on the basis of the consideration of the critical energy barrier it can jump over without extra energy supply. Also, advancing and receding contact angles are redefined: the advancing contact angle is the largest θ value corresponding to the highest metastable state, and the receding contact angle
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is the smallest θ value corresponding to the lowest metastable state on the ∆G versus θ curve. It is further supported in this paper that the advancing and receding contact angles fall between the two equilibrium contact angles intrinsic to the two components of a heterogeneous surface. In accordance with previous findings, our modeling indicates that both the advancing and receding contact angles increase with an increase in the fraction of the apolar strip. For a given strip width (∆h), the contact angle hysteresis attains its maximum value when the two component strips have the same width. According to our analysis of the contact angles on the heterogeneous surface composed of parallel and alternating strips, the contact angle hysteresis decreases with a decrease in the strip width. Two critical strip widths (∆h* and ∆h**) were found that below ∆h* no contact angle hysteresis occurs and above ∆h** the hysteresis is constant and equal to the difference between the equilibrium contact angles of the apolar and polar strips. The minimum size of surface heterogeneity, which can cause appearance of the contact angle hysteresis on substrates of practical significance, such as polar solids coated with apolar organic layers, was found to be in the range of a few nanometers. This heterogeneity size limit is 1 order of magnitude smaller than commonly considered by researchers. The analysis presented in this paper does not take into account the thermal energy available in the three-phase systems. It was estimated by Prevost et al.18 and de Gennes et al.19 that thermal energy is sufficient to move the threephase contact line over the surface defects having dimensions less than 10-20 nm. This heterogeneity size limit is only slightly larger than the value calculated in this contribution based on the Neumann-Good model and, therefore, does support our conclusion that the heterogeneous solid surfaces with heterogeneity features smaller than 100 nm still can demonstrate significant contact angle hysteresis. Whether this theoretical prediction is correct requires experimental verification. LA0364587 (18) Prevost, A.; Rolley, E.; Guthmann, C. Phys. Rev. Lett. 1999, 83, 348. (19) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena; Springer-Verlag: New York, 2003; p 84.
