pubs.acs.org/Langmuir © 2010 American Chemical Society
Chemical Nano-Heterogeneities Detection by Contact Angle Hysteresis: Theoretical Feasibility Eyal Bittoun and Abraham Marmur* Department of Chemical Engineering Technion, Israel Institute of Technology, 32000 Haifa, Israel Received July 9, 2010. Revised Manuscript Received September 9, 2010 The theoretical feasibility of detecting chemical nanoheterogeneities on solid surfaces by measurement of contact angle hysteresis (CAH) was studied, using simplified models of cylindrical (2D) and axisymmetric (3D) drops on corresponding models of chemically heterogeneous, smooth solid surfaces. This feasibility depends on the ratio between the external energy input to the drop and the energies needed to deform its liquid-gas interface and move the contact line across energy barriers. A ubiquitous source of external energy is building vibrations, since most contact-angle measurements are done in buildings. The energy barriers that oppose the motion of the contact line were numerically calculated for various parameters of the two systems. The variations of the liquid-gas interfacial energy are discussed in terms of orders of magnitude. By comparing these energies, it is concluded that under regular (“barely perceptible”) building vibrations CAH measurements may detect chemical heterogeneities at the few nanometers scale.
Introduction Real solid surfaces are usually rough and chemically heterogeneous to some extent. As is well-known,1-4 these nonuniformities lead to the phenomenon of contact angle hysteresis (CAH). The range of CAH (the difference between the advancing and receding apparent contact angles) plays an important role in many applications, e.g., coating, antibiofouling,5-7 and microfluidics.8,9 The recent interest in nonwettable surfaces (usually referred to as “superhydrophobic”) has also emphasized the necessity of characterizing surfaces in terms of their CAH.6,10,11 In addition, as will be discussed in the present paper, CAH may serve as a sensitive means for detecting surface heterogeneities.12,13 The phenomenon of CAH stems from the multiplicity of metastable equilibrium states of a solid-liquid-fluid wetting system that is associated mainly with chemical heterogeneity or roughness *Corresponding author. Fax: 972-4-829-3088. E-mail:
[email protected]. (1) Shuttleworth, R.; Bailey, G. L. J. Disc. Faraday Soc. 1948, No. 3, 16-22. (2) Johnson, R. E.; Detrre, R. H. In Contact Angle, Wettability, and Adhesion; Advances in Chemistry Series; Fowkes, F. M., Ed.; American Chemical Society: Washington, DC, 1964; Vol. 43, pp 112-135. (3) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38, 341–358. (4) Joanny, J. F.; de Gennes, P.-G. J. Chem. Phys. 1984, 81, 552–562. (5) Scardino, A. J.; Zhang, H.; Cookson, D. J.; Lamb, R. N.; de Nys, R. Biofouling 2009, 25, 757–767. (6) Marmur, A. Biofouling 2006, 22, 107–115. (7) Schmidt, D. L.; Brady, R. F., Jr.; Lam, K.; Schmidt, D. C.; Chaudhury, M. K. Langmuir 2004, 20, 2830–2836. (8) Chen, J. Z.; Troian, S. M.; Darhuber, A. A.; Wagner, S. J. App. Phys. 2005, 97, 140906–140915. (9) Fang, G.; Li, W.; Wang, X.; Qiao, G. Langmuir 2008, 24, 11651–11660. (10) Li, W.; Amirfazli, A. J. Colloid Interface Sci. 2005, 292, 195–201. (11) Patankar, N. A. Langmuir 2004, 20, 8209–8213. (12) Wolff, V.; Perwuelz, A.; Achari, A. E.; Caze, C.; Carlier, E. J. Mater. Sci. 1999, 34, 3821–3829. (13) Bittoun, E.; Marmur, A.; Ostblom, M.; Ederth, T.; Liedberg, B. Langmuir 2009, 25, 12374–12379. (14) Marmur, A. Adv. Coll. Inter. Sci. 1994, 50, 121–141. (15) Marmur, A. Soft Matter 2006, 2, 12–17. (16) Marmur, A. J. Colloid Interface Sci. 1994, 168, 40–46. (17) Marmur, A. Coll. Surf. A 1998, 136, 209–215. (18) Fang, C.; Drelich, J. Langmuir 2004, 20, 6679–6684. (19) de Gennes, P.-G., Brochard-Wyart, F., Quere, D. Capillarity and Wetting Phenomena; Springer-Verlag: New York, 2003; p 84. (20) Prevost, A.; Rolley, E.; Guthmann, C. Phys. Rev. Lett. 1999, 83, 348–351. (21) Johnson, R. E., Jr.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744–1750.
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of the solid surface.14-22 Each metastable equilibrium state (a local minimum in the curve of the Gibbs energy versus the geometric CA) is represented by its corresponding apparent contact angle (CA) within the hysteresis range.14,15 The reason for the multiplicity of minima points in the Gibbs energy is that at multiple locations along the solid surface the local geometric CA equals the local Young CA.16,17 The lowest minimum point refers to the global minimum in the Gibbs energy and is represented by the most stable CA. In-between two successive local minima, there must exist a local maximum. The height of this maximum defines the energy barrier that has to be overcome in order to move from one local minimum to the next. The magnitude of the energy barrier between two successive minima increases as the system approaches the global minimum.14,16 For a given wetting system, the theoretical and effective (experimentally measured) CAH range may be different. The highest CA for which there exists a local minimum in the Gibbs energy defines the theoretical advancing CA. Similarly, the lowest CA for which there exists a local minimum defines the theoretical receding CA. Practically, however, the wetting system is always exposed to some external energy source, such as vibrations. This energy source may enable the overcoming of some of the energy barriers in the Gibbs energy as proposed by Johnson and Dettre.2,21 Since the energy barriers are largest near the global minimum and lowest near the advancing and receding CAs, the external energy source affects the effective CAH range: the effective advancing CA must be lower than the theoretical one, and the effective receding CA must be higher than in theory. Therefore, if the external energy source is sufficiently strong, the effective CAH may be eliminated and the surface may appear to be chemically homogeneous despite its actual heterogeneity. CAH has the potential to serve as a sensitive measure of chemical heterogeneity. For example, it has been experimentally demonstrated that it is possible to identify uniformly spread, nanometric “holes” in a monolayer by CAH measurements.13 Therefore, an important general question is: what is the order of magnitude of the scale of chemical heterogeneity that can be detected by CAH (22) Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277–1281.
Published on Web 09/28/2010
DOI: 10.1021/la102757t
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measurements under regular laboratory conditions? This question was partially answered by Fang and Drelich,18 who theoretically showed that the critical strip width, below which CAH cannot be detected on a surface made of periodic strips of two different types of chemistries is of the order of magnitude of 10 nm. The problem was also analyzed with regard to thermal vibrations by de Gennes et al.19 and Prevost et al.,20 who concluded that an energy of the order of magnitude of 20kbT (where kb is the Boltzmann constant) is required to overcome hysteresis caused by surface defects of about 20 nm in size. Since almost all CA measurements are performed in laboratories within buildings, the objective of the present paper is to answer this question with regard to liquid drops on chemically heterogeneous, smooth surfaces under typical, natural building vibrations. It will be shown that under regular conditions of barely perceptible building vibrations CAH measurements can detect chemical heterogeneity at the nanoscale.
Theory When a drop on a solid surface vibrates (e.g., due to building vibrations), the gained kinetic energy may be consumed by a few processes, mainly oscillatory variations in the liquid-gas interfacial area and motion of the contact line. These processes are very complex and may occur either independently or simultaneously. Therefore, a relatively simple analysis is desirable in order to identify the orders of magnitude of the energy consumed by each process. In addition, energy dissipation takes place, due to either viscous flow of liquid inside the drop or motion of the contact line. The latter is assumed here to be included in the energy needed to overcome the energy barriers between metastable states, and dissipation due to internal flow is assumed negligible. First, the energy barriers that oppose the motion of the contact line are estimated using simple models of a drop on a chemically heterogeneous but smooth solid surface. Then, the variations in liquidgas interfacial area are discussed at an order of magnitude level. These estimates enable the discussion of the feasibility of chemical heterogeneity detection by CAH. Two simplified models of wetting systems are studied in the present paper: a cylindrical (two-dimensional) liquid drop and an axisymmetric drop, both on a solid surface in a gaseous atmosphere (Figure 1a). The solid is smooth but chemically heterogeneous. In the cylindrical system, the chemical heterogeneity is characterized by a periodic variation of the Young CA along the length of the solid surface (Figure 1b). The axisymmetric system is characterized by a periodic radial variation of the Young CA (Figure 1c). It is of interest to analyze both systems, in order to find out whether the results are effectively independent of the particular system. The periodic variations in the local Young CA are assumed sinusoidal as a zeroth-order approximation of the chemical heterogeneity 2πx cos θY ðxÞ ¼ cos θ0 þ j cos ð1Þ λ where θY is the local Young CA, θ0 is the average CA, j is the amplitude of heterogeneity, x is the coordinate along the solid surface, and λ is the wavelength of the chemical heterogeneity. The Gibbs energy for a cylindrical drop (per unit depth) is Z r2D G2D ¼ σ l A2D þ 2 ½σ sl ðxÞ - σs ðxÞ dx ð2aÞ l 0
and for an axisymmetric drop Z r ½σ sl ðxÞ - σ s ðxÞx dx G ¼ σl Al þ 2π
Figure 1. Schematic description of the model systems: (a) A side view of two-dimensional and axisymmetric wetting systems; (b) the sinusoidal variations of the local Young CA along the surface in the two-dimensional model; (c) the sinusoidal variations of the local Young CA along the surface in the axisymmetric model. solid-liquid interfacial tensions, and r is the base half-length or base radius of the drop. The first term in the right-hand-side of these equations represents the surface energy of the liquid-air interface and the second one stands for the change in surface energy due to wetting (or dewetting) of the solid surface. In the absence of gravity, which is assumed here for simplicity, the shape of the drop must be circular in the 2D case and spherical for axisymmetric systems. Therefore, Al is given by ¼ A2D l Al ¼ 2πr2
2r2D θ sin θ 1 - cos θ sin2 θ
ð3aÞ ð3bÞ
where θ is the geometric CA, which, at equilibrium, becomes identical with the local Young CA. The base half-length or radius of the drop relates to the drop volume, V, (or volume per unit depth, V2D) by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2V 2D ¼ sin θ 2θ - sin 2θ
ð4aÞ
1=3 3V sin θð2 - 3 cos θ þ cos3 θÞ- 1=3 π
ð4bÞ
r
ð2bÞ
2D
0
In these equations, the superscript 2D stands for cylindrical drop, Al is the interfacial area of the liquid-air interface, σl and σs are the surface tensions of the liquid and solid, respectively, σsl is the 15934 DOI: 10.1021/la102757t
r ¼
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Figure 2. Dimensionless Gibbs energies (a-c), and energy barriers (d-f) vs the geometric contact angle for two-dimensional drops. Upper row: j = 0.25, R = 102, and θ0 = (a) 50°, (b) 75°, and (c) 100°. Lower row: the values of j are given by the numbers in the figures, the values of R are 1102 (Δ) and 1105 (o), and θ0=(d) 50°, (e) 75°, and (f) 100°.
Figure 3. Dimensionless Gibbs energies (a-c), and energy barriers (d-f) vs the geometric contact angle for axisymmetric drops. Upper row: j = 0.25, R = 102, and θ0 = (a) 50°, (b) 75°, and (c) 100°. Lower row: the values of j are given by the numbers in the figures, the values of R are 1102 (Δ) and 1105 (o), and θ0 = (d) 50°, (f) 75°, and (f) 100°.
Substitution of eqs 1, 3a, and 4a into eq 2a leads to the following expression for the dimensionless Gibbs energy for the 2D drop G2D θ j - cos θ0 - sinð2πr2D Þ ð5aÞ ¼ 2r2D G2D sin θ π λσl where r* r /λ. Similarly, substitution of eqs 1, 3b, and 4b into eq 2b yields the dimensionless Gibbs energy for the axisymmetric drop " # G 1 - cos θ 2 G 2 ¼ πðr Þ 2 - cos θ0 sin2 θ λ σl 1 ð5bÞ - j r sinð2πrÞ þ ½cosð2πrÞ -1 2π 2D
metastable states. The transition between one metastable state to another, more stable one, involves overcoming an energy barrier, defined in a dimensionless form as B2D Glmax - Glmin 2D
ð6aÞ
B Glmax - Glmin
ð6bÞ
2D
2D
where r* r/λ. As is well-known14,16,18,21,22 and demonstrated again in Figures 2 and 3, the Gibbs energy curve has multiple minima that represent Langmuir 2010, 26(20), 15933–15937
where G*1max is a local maximum that follows a local minimum, G*1min, in the direction of the global minimum. Prior to or simultaneously with overcoming energy barriers that enables motion of the contact line, the drop oscillates and its liquid-gas interfacial area, on the average, increases. It is extremely difficult to predict such oscillations, so only a simple order-ofmagnitude estimate is presented here. If a is the amplitude of building vibrations, then the change in the drop surface area is of the order of magnitude of πa for the 2D case and πra for the axisymmetric drop. These expressions can be verified by checking DOI: 10.1021/la102757t
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some simple geometries, such as half a circle that turns by oscillations to half an ellipse (for the 2D case), or half a sphere that turns into half a spheroid (ellipsoid). Thus, a heterogeneous surface would appear homogeneous by CAH measurement if 2D 2D Þ Evib > σ l ΔA2D l þ Bmax ¼ ∼ðσ l πa þ λσ l B max
ð7aÞ
Evib > σ l ΔAl þ Bmax ¼ ∼ðσl πra þ λ2 σl Bmax Þ
ð7bÞ
2D
or
where Evib is the kinetic energy that the drop gains from the building vibrations, Bmax is the highest energy barrier of the system, and ΔAl is the increase in the liquid-gas interfacial area. Obviously, the present treatment of overcoming the energy barriers is only an order of magnitude analysis. Specific characteristics of the vibrations as well as a more sophisticated analysis is required in order to get more accurate results.23,24 To estimate the kinetic energy of the drop it is assumed that the maximum velocity of the building determines the kinetic energy of the drop as a whole. Thus 2D Evib
FV 2D ð2πaf Þ2 2
ð8aÞ
and Evib
FVð2πaf Þ2 2
ð8bÞ
where F is the density of the liquid and f is the frequency of the building vibrations. The use of the maximum velocity makes this a stringent assumption on the “safe” side, since it actually overestimates the available energy.
Results and Discussion Typical Gibbs energy and energy barrier curves are shown in Figures 2 and 3, for 2D and axisymmetric drops, respectively. To enable convenient comparison of the various cases, the dimensionless Gibbs energy is represented by the difference between its actual value and the value at the global minimum ΔG2D G2D - Ggm
ð9aÞ
ΔG G - Ggm
ð9bÞ
2D
and
The values of 50°, 75°, and 100° for the average CA, θ0, were used for the calculations, to account for various degrees of hydrophilicity/hydrophobicity. All of the Gibbs energy curves given in Figures 2a-c and 3a-c were calculated for an amplitude of j = 0.25 and a relative size of the drop of 102. The latter is defined as the ratio of the diameter of the drop before touching the solid surface to the wavelength of the surface heterogeneity22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 4V 2D =πÞ 2D R ð10aÞ λ R
pffiffiffiffiffiffiffiffiffiffiffiffi ð 3 6V=πÞ λ
B B=R
ð11Þ
This form of dimensionless energy barriers is presented in Figure 3d-f. As can be seen in this figure, B** is indeed practically independent of the relative drop size, for a given solid surface. Similarly to the 2D case, the dimensionless energy barriers for axisymmetric drops strongly increase with an increase in the amplitude of heterogeneity (theoretical CAH range). To get an order of magnitude for the maximum dimensionless energy barriers, the following calculation was done. The maxi** mum dimensionless energy barrier for a given system, B*2D max or Bmax, depends on the average CA, θ0, and the heterogeneity amplitude, j. To find out the dependence of the maximum dimensionless energy barrier on these parameters it is necessary to screen all possible combinations of θ0 and j. However, for each θ0 a maximum value of j can be identified, based on the condition that 0 e θY e 120°: it is the lowest of the values that leads to either a theoretical advancing CA of 120° or a theoretical receding CA of 0°. As seen in Figures 2d-f and 3d-f, the maximum dimensionless energy ** barrier increases with j. Therefore, when B*2D max and Bmax are calculated as a function of θ0, assuming the maximum value of j to apply for each θ0, their upper limits are elucidated. The results of these calculations are shown in Figure 4. As can be seen, ** 0 ∼ þ 2 r r2
! for strongly perceptible vibrations ð14Þ
Figure 4. Dimensionless energy barriers for two-dimensional (dashed line) and axisymmetric (solid line) drops vs the average of the average CA, θ0. The peak in each curve refers to highest energy ** barrier, B*2D max or Bmax, for 2D and axisymmetric drop, respectively.
in terms of order of magnitude, eq 12a,b yields the same result, since B*max and B** max are of the same order of magnitude. Thus, the results of the analysis for the 2D case and the axisymmetric case are practically identical, and the conclusions are effectively independent of the system geometry. To enable an estimate of the orders of magnitude of the various ratios in eq 12a,b, it is necessary to know the typical amplitudes and frequencies of building vibrations. For vibrations that are barely perceptible by a human being, typical values are 10-6 m and 100 Hz, respectively.25 For vibrations that are strongly perceptible by human beings, but do not cause damage to the building, typical values are 10-3 m and 1 Hz, respectively.25 Substituting these numbers in, say, eq 12a, assuming the liquid to be water (surface tension of 73 mN/m, and density of 103 kg/m3), eq 12a turns into ! 10-3 102 λ for barely perceptible vibrations ð13Þ 1>∼ þ 2 r2 r
(25) Merritt, F. S., Ricketts, J. T. Building design and construction handbook, 6th ed.; McGraw-Hill: New York, 2001; pp 5.183-5.188.
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It is clearly seen that the first terms in these equations are >1 for any reasonable drop size that is used in CA measurement (usually smaller than about 10 mm). Thus, it is feasible, in principle, that all the kinetic energy will be consumed by the vibration-induced increase in liquid-gas interfacial area. However, since motion of the contact line may occur simultaneously with the interfacial oscillations, it is essential to analyze the order of magnitude of the second terms of the right-hand side of eqs 13 and 14. First, these equations show that the ratio of the maximum energy barrier to the available kinetic energy increases when the drop size decreases. Thus, energy barriers are more easily overcome for larger drops then for smaller ones. For barely perceptible building vibrations, and for a drop of the order of magnitude of 10-3 m, this ratio is 0.1 for a 1 nm heterogeneous wavelength, and 1 for a 10 nm wavelength. Taking into account that at least part of the kinetic energy turns into liquid-air interfacial energy, it is, therefore, safe to conclude that CAH measurement under regular conditions of barely perceptible building vibrations may detect chemical heterogeneities at the few nanometers scale. This observation is indeed supported by some experimental evidence.13 Obviously, good isolation of the measurement system from the building vibrations may enhance the detection sensitivity. It is also interesting to notice that the situation changes when the vibrations are strongly perceptible, an observation that justifies the use of deliberately induced vibrations to measure the most stable CA.22,26-28 Acknowledgment. The authors acknowledge the support from AMBIO (Advanced Nanostructured Surfaces for the Control of Biofouling) project (NMP-CT-2005-011827) funded by the European Commission’s sixth Framework Programme. Partial support from COST Action D43 is also acknowledged by the authors. (26) Decker, E. L.; Garoff, S. Langmuir 1996, 12, 2100–2110. (27) Andrieu, C.; Sykes, C.; Brochard, F. Langmuir 1994, 10, 2077–2080. (28) Meiron, T. S.; Marmur, A.; Saguy, I. S. J. Colloid Interface Sci. 2004, 274, 637–644.
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