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Wetting Transition of n-Alkanes on Concentrated Aqueous Salt Solutions. Line Tension Effect Anne Dussaud† and Miche`le Vignes-Adler* Laboratoire des Phe´ nome` nes de Transport dans les Me´ langes du CNRS, SP2MI, Bd 3 Te´ le´ port 2, BP 179, F-86960 Futuroscope Cedex, France Received December 4, 1995. In Final Form: November 18, 1996X The spreading of n-octane droplet at the surface of sodium chloride aqueous solutions was investigated as a function of the salt concentration in a saturated closed cell. As long as the salt concentration is lower than 1.2 M, the macroscopic drop takes the form of a lens in a stable coexistence with its adsorbed thin film. A wetting transition occurs for a salt concentration of about 3 M: spreading is considerably enhanced and the lens becomes a metastable pancake with a contact angle smaller than 1°. After a few hours, it breaks up into a multitude of stable microdroplets which disappear 24 h later. Drop size dependent contact angles reveal a salt concentration dependent positive line tension. The values of line tension are very near the theoretical value. It is confirmed that the positive line tension stabilizes the microdroplets. The discussion of the results is based on the specific behavior of the interaction free energy ∆F of the oil due to the very high salt concentration of the liquid substrate.
I. Introduction The understanding of the spreading of liquid layers on the surface of an immiscible supporting liquid is of utmost importance in numerous industrial applications such as environmental, oil, and chemical industries to name a few. In a three-phase system, one of them being the atmosphere, the spreading properties are generally ruled by the well-known Harkins spreading coefficient S. The liquid spreads when S > 0 and forms a lens in the opposite case. Major practical difficulties may occur when S approaches zero since the system comes close to the wetting transition and shows up many fluctuations. The present study is mainly motivated by oil recovery which explains several choices. The actual oil-water mixture immersed in a siliceous matrix is usually modelled by three fluid phasessn-alkane/water/gassflowing in fracture networks. Experimental studies pointed out the importance of favorable spreading coefficients to improve the efficiency of the flow and to recover more oil; usually surfactants are used to decrease the surface free energy of the spreading liquid to get a continuous oil phase rather than a highly dispersed one. Although the aqueous phase is a brine in oil reservoir, much less attention has been paid to the exact role of salt in the oil-spreading properties, and it is the purpose of this work to fill in this gap and to study the influence of large addition of salt on n-alkane spreading. Salts such as sodium chloride are known to increase close to saturation by about 10-15% the surface tension of the water. In other words, salt increases the surface energy of the water substrate and facilitates the spreading of n-alkanes. Actually, the physics is quite intricate. For light n-alkanes which are volatile, one must also consider the possible adsorption of n-alkane molecules on the aqueous surface concomitantly with spreading. If much work has been devoted to adsorption of n-alkanes on water, less experimental studies have addressed the effect of sodium chloride (NaCl) on n-alkanes spreading. In a remarkable experimental study, Hauxwell and Ottewill1 have investigated the adsorption of aliphatic † Present address: Department Chemical Engineering, Princeton University, Princeton, NJ 08544. X Abstract published in Advance ACS Abstracts, January 15, 1997.
hydrocarbons (n-alkanes) vapors onto water surface, over a range of partial pressures of the vapor, until the saturation vapor pressure was reached. They found that the short n-alkanes, n-pentane, n-hexane, and n-heptane form multilayers at saturation pressure while n-octane forms microlenses connected to an adsorbed film. This transition from multilayers to lenses with the chain length of n-alkanes was related to the dispersion forces contribution to the surface free energy of water. Later, the predominant role of dispersion forces on the hydrocarbon adsorption on water surface was investigated in a theoretical study based on the Lifshitz theory of molecular forces by Richmond et al.2 They calculated the disjoining pressure for thin liquid hydrocarbon films on aqueous bulk substrates. This approach using the dielectric properties is global and does not take into account the possible structuration of water near the surface involved by the presence of n-alkane. However, it could predict the formation of stable multilayers for short-chain n-alkanes with n < 6. This confirmed the Hauxwell and Ottewill study. Moreover, the authors have used their theoretical model to investigate the influence of salt on the stability of hydrocarbons layers on aqueous substrates. If sufficient NaCl is dissolved in water, their calculations predict that both n-octane and n-dodecane will form wetting films. The aim of the present work is to further investigate the influence of salt on the wetting transition of n-alkanes on water. Among several n-alkanes, n-octane was chosen. The experiment consists of depositing an n-octane droplet on the surface of an aqueous solution in an n-octanesaturated atmosphere at 20 °C, for several salt concentrations (NaCl, 0-5 M). The difference between spreading in the absence and in the presence of salt is observed and analyzed. On pure water and at low salt concentrations, the n-octane droplet lays as a lens on the aqueous surface for more than 15 h. Although not precisely studied, its shape was much less flattened than at high salt concentration. For a high enough salt concentration, experiments show that n-octane spreads in a first stage, gets a pancake shape, and stops spreading. After a time varying between 2 and (1) Hauxwell, F.; Ottewill, R. H. J. Colloid Interface Sci. 1970, 34, 4, 473. (2) Richmond, P.; Ninham, B. W.; Ottewill, R. H. J. Colloid Interface Sci. 1973, 45, 1, 69.
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Figure 1. The experimental setup.
10 h, the n-octane lens systematically breaks up into a multitude of microdroplets whose diameters vary between 10 and 500 µm. After 24 h, the microdroplets disappear. Moreover, the equilibrium surface and interfacial tensions in the presence or in the absence of n-octane vapor were measured in order to relate the spreading behavior to the values of spreading coefficients. The contact angles of the microdroplets formed after the breakup of the initial droplet were shown to depend on their radius. This has been attributed to a measurable positive line tension that contributes to the microdroplet stabilization. The experimental results show that the wetting transition induced by high salt concentrations likely involves complex long-range molecular interactions in the condensed thin film that exhibit oscillations. II. Materials and Methods 2.1. Fluids. The n-octane is purchased from Aldrich (purity 99%). However, it is thoroughly cleaned because its interfacial tension with water is always too low. Hence, n-octane is shaken with alumina and then washed with milliQ water for 8 h. The sodium chloride (NaCl) is purchased from Aldrich. Its purity is controlled by the value of the surface tension of the aqueous solutions which must depend linearly on the salt concentration.3 Then, the n-octane and the NaCl aqueous solution are mutually saturated by mechanical stirring for 6 h at the experimental temperature; they are designated below as “n-octane” and “solution”. The NaCl concentration is ranging between 1.2 and 5 M. 2.2. Experimental Setup. Two experimental methods are used. The first provides microscopic observations of the droplet. The setup (Figure 1) consists of a circular Petri dish with a diameter of 60 mm filled with the aqueous solution and placed inside a larger Petri dish. This cell is placed in a closed temperature-regulated cell. The larger Petri dish contains an n-octane layer to saturate the cell atmosphere. The presence of blotting paper on the walls improves saturation. All connections of the cell with the outside are achieved through holes closed with septa. The n-octane drop is formed at the tip of a glass capillary and carefully deposited on the substrate. Its volume varies from 3 to 8 mm3. The cell is placed in the field of a microscope and is illuminated by a reflected monochromatic light. Visualization of the phenomena is made by a CCD camera connected to a tape recorder. The videotapes are processed afterward using an image analysis system. The second method provides a macroscopic observation of the n-octane droplet deposited on the aqueous surface. The setup is nearly the same as the first one, but the cell is illuminated from below by a parallel beam issuing from a distant mercuryvapor lamp. The cell is placed in a temperature-controlled room (20 ( 1 °C). The shadow of the lens is formed on a screen and is recorded by a CCD4 camera. The shadowgraphy was used because the refractive index of water and n-octane are very close which hinders a clear visualisation by reflected light. (3) Bikerman, J. J. Physical Surfaces; Academic Press: New York, 1970. (4) Bekki, S.; Vignes-Adler, M.; Nakache, E.; Adler, P. M. J. Colloid Interface Sci. 1990, 140, 492. (5) Bruhat, G. Optique; Masson & Cie: Paris, 1965; p 119.
Figure 2. A liquid droplet at the liquid surface (a) pancake shape and (b) when gravity is neglected. 2.3. Measurements and Experimental Procedure. 2.3.1. Surface and Interfacial Tensions. The surface and interfacial tensions are monitored by a du Nou¨y type apparatus using a stirrup instead of a ring in a temperature-regulated cell set to 20 °C. The surface tension of the aqueous solution is determined in the absence and in the presence of n-octane vapor. The precision of our device is (0.1 mN‚m-1. γoct, γoct/wat, and γwat denote the surface tension of n-octane, the interfacial tension at the oil/water interface, the surface tension of the aqueous substrate measured in an n-octane-free atmosphere, respectively. γswat means that the measurement is made in an n-octanesaturated atmosphere. 2.3.2. Geometrical Characteristics of the Macroscopic Droplet. The diameter Φ of the initial macroscopic droplet is measured as a function of time by shadowgraphy. It is read on a translucent millimetric paper. The thickness is too large to be measured by interferometry. It can only be approximated from the volume when the droplet is very flat as a pancake (Figure 2a). Hence, the thickness h can be calculated from the relation
V ) πΦ2h/4
(1)
As long as the contact angles are large, they cannot be measured by interferometry. However, for flat lenses, the contact angles R, β (Figure 2a) are small, and the observation of the interference fringes of the lens edge gives an evaluation of the contact angle by considering a linear profile in the edge. In this case, tg(R + β) = R + β = h/d, where h is calculated from the number N of fringes by h ) λN/n at the distance d determined in the digitized image. λ is the light wavelength and n the n-octane refractive index. 2.3.3. Geometrical Characteristics of the Microscopic Droplet. The microdroplets formed after the breaking-up of the initial macroscopic droplet are observed by interferential microscopy. The lens diameter is measured on the digitized image. The analysis of the interference pattern provides the microdroplets thickness h or more precisely an interval for h, e.g. h ( λ/8n (5). Three wavelengths (λ ) 480, 546, 650 nm) were systematically used for each measurement to improve the precision. The total number of dark fringes N, at a given light wavelength λi (i ) 1, 3), provides the interval for h
for dark center droplets λi λi λi λi 1, the disjoining pressure becomes negative while ∂Π(h)/∂h < 0, and the film tends to thicken by condensation of vapor or by an n-octane flow (10) Hough, D. B.; White, L. R. Adv. Colloid Interface Sci. 1980, 14, 3. (11) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface forces; Consultants Bureau: New York, 1987.
Figure 10. A typical interaction free energy and disjoining pressure isotherm of an n-octane thin film adsorbed on pure water.
from the droplet that acts as a reservoir. It is stabilized by the action of the disjoining pressure. The thickening is limited because it cannot exceed the thickness h1 beyond which the film is unstable (∂Π/∂h > 0). Indeed, the attractive forces between both interfaces tend to thin the film. This is why the excess of n-octane forms a lens of a given contact angle with the film. Actually, because of the possible thickening of the film connected to the droplet, we can reasonably suppose that the contact angles correspond to an interaction free energy ∆F(h) ranging between ∆F(he) and ∆F(h1) and that this situation is a metastable thermodynamic equilibrium (Figure 10a). This is known as pseudo partial wetting.25 Clearly, the values of the contact angles depend upon the depth of ∆Fmin. According to the typical sketch of ∆F(h) profile (Figure 10a), the enhancement of spreading with salt is expected as a result of the increase of the repulsive forces with respect to the attractive forces. This point can be verified by the evaluation of ∆F for several salt concentrations from the experimental surface tensions and contact angles. The interaction free energy can be explicited as
∆F(h) ) Fwat/oct/air(h) - Fwat/oct/air(∞)
(15)
Fwat/oct/air(∞) ) γoct + γoct/wat
(16)
Moreover
In the absence of the droplet, if γswat, the surface tension of the water surface covered by an n-octane-condensed monolayer is considered as the total free energy of the water/n-octane/air system at equilibrium with n-octane vapor, we have
∆Fmin ) γswat - (γoct + γoct/wat)
(17)
where γswat is obtained from the surface tension measurements (Table 1, column 3). Comparison of (9) and (17) yields
∆Fmin ) Sf Data are given in Table 1, column 9.
(18)
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Langmuir, Vol. 13, No. 3, 1997 587
In the presence of the droplet, if relation (10) is used with a negligible line tension term and γswat is now evaluated from the extrapolated angles values (Table 1, column 11), the corresponding interaction free energy can be expressed as
∆F(h) ) γoct cos R + γoct/wat cos β - (γoct + γoct/wat) (19) ) γoct(cos R - 1) + γoct/wat(cos β - 1)
(20)
From (19), for 3 and 5 M solutions
∆F3M ) (77.7999 - 77.8) ( 2 × 10-4 ) -1 × 10-4 ( 2 × 10-4 mJ/m ∆F5M ) (80.5991 - 80.6) ( 2 × 10-4 ) -9 × 10-4 ( 2 × 10-4 mJ/m (21) It is noticeable that ∆F3M is larger than ∆F5M, and this would mean that the n-octane spreading is more efficient at 3 M than at 5 M. This was also predicted by the extrapolated values of R and β in section 3.3. This order of magnitude of ∆F is consistent with other experimental predictions.12 However, these values of ∆F in the presence of a droplet drastically differ from the values of ∆Fmin in the absence of a droplet evaluated by application of relation 18. Now, for large salt concentrations, this difference between ∆F and ∆Fmin of several orders of magnitude could be explained by an enhanced thickening of n-octane film caused by an intricate behavior of molecular interaction forces in the transition zone between the main droplet and the condensed film. A substantial addition of salt, e.g. electrolyte, increases the water surface tension, but it also modifies the water structure in an intricate manner.13,14 It seems that it can have unexpected consequences on the n-octane monolayer and generate an oscillating behavior of the disjoining pressure isotherm leading to the existence of several minima for ∆F. In Figure 11, we give a possible example to illustrate the discussion. For this profile, at saturation (p/ps ) 1), the n-octane film can have two equilibrium thicknesses (Figure 11b). As the presence of the convex meniscus of the droplet enhances the thickening of the film (p/ps > 1), the n-octane film can undergo a transition from the primary branch (he1 < h < h1) to the secondary branch (he2 < h < h2) while, in the absence of the droplet, it prefers to stay on the primary branch. The lower depth of the secondary free energy minimum corresponding to he2 could explain why smaller contact angles were obtained in these experiments. Measurements of the surface tension near the droplet or ellipsometric measurements providing the film thickness profile could clarify this point. [One must notice that the film cannot explore the positive part of Π(h) although ∂Π/ ∂h is negative there since it would correspond to undersaturation (p/ps < 1), which is not the actual experimental situation.] Actually, the analysis of the line tension data in section 4.3 would tend to confirm the existence of an intricate disjoining pressure isotherm. 4.3. Line Tension. The dependence of contact angles R and β upon the microdroplet radius can be assigned to (12) Mysels, K. J.; Buchaman, J. W. J. Electroanal. Chem. 1972, 37, 23. (13) Derjaguin, B. V.; Gorodetskaya, A. S.; Titiyevskaya, A. S.; Yashin, V. N. Kolloidn Zh. 1961, 23, 535 (cited in ref 11). (14) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolytes Solutions; Reinhold Publ. CoCo.: New York, 1958.
Figure 11. A typical interaction free energy and disjoining pressure isotherm of an n-octane thin film adsorbed on concentrated solution. The thicker portions of lines indicate the physical branches.
the tension line in which the three phases meet. Its mechanical equilibrium is expressed by the relation 10. The positive line tension has an effect of tightening the “collar” that bounds the microdroplets.15 Then, this stabilizes them by maintaining their shape at very low contact angles. The order of magnitude obtained for τ (10-10 N < t < 10-9 N) agrees well with theoretical estimations and other experimental data.16,17 Moreover, τ/γs is ranging between 1.08 and 12.9 nm: it has the order of magnitude of a molecular length at least at the lower bound and it is very near the theoretical value as predicted by Kerins and Widom.18 Positive line tensions are less usual than negative ones; at a three-phase contact line they can only be found in the case of partial wetting and they are only efficient when the droplet volumes are very small. Several approaches exist in the literature to explain the occurence of a positive sign19-21 for τ. The analysis of de Feijter and Vrij19 that was applied to a symmetric planar liquid film in contact with its adjacent meniscus can be generalized to an asymmetric one to provide an explanation for the sign of τ.24 The line tension is considered as the excess tangential force acting on the transition region and rising from the comparison between the actual transition region and an idealized situation where ∆F(h) is neglected and γ(h) ) γ0. The (15) Widom, B. J. Phys. Chem. 1995, 99, 2803. (16) Toshev, B. V.; Platikanov, D.; Scheludko, A. Langmuir 1988, 4, 3, 489. (17) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans. 1996, 92 (1), 85. (18) Kerins, J.; Widom, B. J. Chem. Phys. 1982, 77, 2061. (19) De Feijter, J. A.; Vrij, A. J. Electroanal. Chem. 1972, 37, 9. (20) Churaev, N. V.; Starov, V. M.; Derjaguin, B. V. J. Colloid Interface Sci. 1982, 89, 1, 16. (21) Toshev, B. V.; Avramov, M. Z. Colloids Surf. A 1995, 100, 203. (22) Dussaud, A. Unpublished results. (23) Ross, S.; Morrison, I. D. Colloidal systems and interfaces; John Wiley & Sons: New York, 1988. (24) Toshev, B. Colloid Polym. Sci. 1995, 273, 807. (25) Brochard-Wyart, F.; di Meglio, J. M.; Que´re´, D.; de Gennes, P. G. Langmuir 1991, 7, 335.
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authors have proposed the following simplified relation for the line tension (eq 28 in ref 19)
τ)
∫h∞γ(h) sin θ dh - ∫h∞γ0 sin θid dhid e
e
(22)
where γ0 and γ(h) are respectively the equilibrium surface tension and the surface tension in the transition zone, and θid and θ are the slope angles of the liquid surface with the horizontal in the idealized and real situation, respectively. The first integral is the net tangential force acting on the profile of the actual Plateau borders and the second integral is the net force on the idealized Plateau border. The surface tension in the transition zone can be expressed as a function of the interaction free energy ∆F(h)
2γ(h) ) 2γ0 + ∆F(h)
(23)
Then relation (22) becomes
τ)
∆F(h) sin θ dh 2
∫h∞γ0 sin θ dh + ∫h∞ e
e
∫h∞γ0 sin θid dhid e
(24)
sin θ and sin θid are both positive. If ∆F(h) is negative for all h g he and its contribution to the force balance in (24) dominates over the two other ones, then τ is negative. Figure 10 illustrates such a case. Now, positive values of τ can be explained if there exists a region in the interaction free energy where ∆F(h) > 0 or γ > γ0 and such that
∆F(h) sin θ dh > 0 2
∫h∞ e
This can be achieved only if ∆F(h) possesses at least one maximum in the positive half plane (Figure 11a). The higher values of τ obtained at 3 M indicate that the area bounded by the plot ∆F(h) in the positive ordinates region with respect to the area bounded by the plot ∆F(h) in the negative ordinates region is higher at 3 M than at 5 M. We have not fully understood why this happens, but this result has recently been confirmed in our laboratory on a very different experimental setup. Moreover, the oscillatory behavior of the disjoining pressure could explain the occurence of two different values for τ at 3 M, corresponding respectively to two different equilibrium thicknesses. Moreover, the positive line tension, specially at 3 M, could play a role in the phenomena of instabilities of the macroscopic lens. Indeed, Widom15 has analyzed the effect of the line tension on the contact angle between a small sessile drop and a solid substrate. He found that, in the domain of sufficiently high positive line tension, a critical droplet size of diameter Dcri and volume Vcri exists beyond which the contact angle jumps discontinuously to 180°. In other words, for a critical size, the droplet is detached from the substrate which remains in contact only with the vapor. This is called a line tension driven wettingdrying transition.15,17 Widom has shown that this critical size value depends upon τ and upon the contact angle θ0 that would have the macroscopic droplet when the line tension effects are negligible. For θ0 ) 0, the transition occurs when
τcri )
τ ) 0.76 (3Vcri/π)1/3γoct
(25)
Widom’s calculations can be extended to the liquidliquid interface.22 For θ0 ) 0, τcri can be shown to be 0.66. For a value of τ equal to 10-9 N, the highest value obtained for 3 M, (25) yields Dcri ) 83 nm for the critical diameter of a spherical droplet at the drying transition. During the expansion of holes, the formation of submicronic droplets like a “spontaneous emulsion” was observed at the solution surface before the disappearance of the initial droplet (section 3.2). We could not measure their contact angle values which, according to Widom,15 should be rather large. However, if this drying transition were to occur, the dispersion of the liquid at the surface followed by the formation of a mist of n-octane above the surface could explain some of our observations (section 3.2). It could explain the sudden acceleration of the growth of holes due to an increased loss of mass and then the concomitant disappearance of the the n-octane droplet. More sophisticated devices to measure the formation of a mist of n-octane above the liquid substrate surface should confirm this point. 4.4. Onset of the Instability. The difficulty of the present experiment and analysis is related to the proximity of the wetting transition, and minor changes in the external constraints may induce dramatically different behaviors, in particular if temperature gradients exist. We have not clearly understood why the initial large droplet is destabilized on salt and not salt-free or dilute solution surfaces. Presently, some experimental evidence can only be emphasized. After reaching a maximum, the lens diameter decreases strongly in concentrated solutions and not in the other case. As the contact angle does not change significantly during this stage, this is not related to a retraction of the droplet under the action of the capillary forces at the lens surface but rather to a loss of droplet mass. However, mass balance shows that the n-octane film thickening connected to the lens is not sufficient to explain the total mass loss. Some mass could also evaporate from the main droplet and condense as a wetting film on the glassware of the cell or even escape through extremely small leaks. This effect could be more important for concentrated solutions because the n-octane pancake formed at high salt concentrations exhibits a larger exchange surface for evaporation. To fully understand the process, the vapor pressure inside the cell should be better controlled. Anyway, the onset of the instability can be considered of the n-octane to occur at a critical thickness hpancake c pancake at which surface fluctuations are amplified until the film ruptures. The microdroplets can be stabilized by the action of the line tension as seen below. At the ≈ 30 µm. maximum wetted area, (1) yields hpancake c V. Conclusion The present study on the spreading of an n-octane droplet on the surface of salt solutions surfaces leads to the following conclusions. Whatever the salt concentration, there is partial wetting and the droplet takes the form of a lens in contact with a condensed n-octane film on the solution surface. However, the spreading is drastically enhanced at very high salt concentrations (g3 M); the lens becomes like a pancake with a contact angle smaller than 1°. When the pancake becomes very thin, it bursts into a multitude of microlenses stabilized by a measurable positive line tension. The lens spreading as well as the line tension are larger at 3 M than at 5 M. At salt concentrations larger than 3 M, contact angle data cannot be compared to those deduced from the surface tensions measurements by the Neuman law. Our results have been interpreted by the interaction free energy
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approach. The interaction free energy ∆F(h) (salt concentration dependent) of the condensed n-octane film on the solution surface was assumed to have two negative local minima, the primary minimum corresponding to the thinner film. In the absence of the droplet, the film is probably located in the primary minimum which is a stable thermodynamic equilibrium. In the presence of the droplet, the film is forced to the secondary minimum which is a metastable equilibrium state and its thickness is increased. Moreover, the existence of this secondary minimum is consistent with the observed positive line tension. The two minima originate from the very high salt concentration which induces oscillating structural forces. For salt free or 1.2 M solutions, ∆F(h) has one negative minimum which corresponds to a stable equilibrium state. This explains the stable coexistence of the drop with its thin film on the substrate showing partial wetting. This is in agreement with the predicted behavior from the surface tension data. These results tend to show that the long-range force interaction in the n-octane film on salt water is much more complicated than expected from the analysis of Richmond et al.2 This analysis based on Lifshitz theory can predict an increase of the repulsive contribution due to the van der Waals forces at high salt concentrations, but it cannot account for the attractive forces contribution as shown by these contact angle measurements. The dependence of these forces upon salt concentration is surprising. Indeed, salt enhances n-octane spreading, but it is more efficient at 3 M than at 5 M.
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h1 ) R1(1 - cos R) ) r tg
R 2
h2 ) R2(1 - cos β) ) r tg
β 2
For very small R and β, it becomes
tg
R h1 R ) ≈ 2 r 2
(A.1)
tg
β h2 β ) ≈ 2 r 2
(A.2)
Since h ) h1 + h2
R+β≈
2h r
(A.3)
A second relationship between R and β is obtained from the mechanical equilibrium at the contact line. The Neuman equation giving the balance of surface tension forces and line tension at the contact line can be expressed by the two relations
τ γswat - γoct cos R - γoct/wat cos β - ) 0 r
(A.4)
γoct sin R - γoct/wat sin β ) 0
(A.5)
where r is the droplet radius and τ the line tension. Since R and β are very small, (A.5) becomes
Acknowledgment. This work was supported by the EEC Brite-Euram Contract BRE2-CT92-0191.
γoct sin β β ) ≈ γoct/wat sin R R
Appendix
Finally, (A.3) and (A.6) give (5) and (6). Let us note that the angle derivation does not use the relation (A.4) and that the results are unaffected by τ.
The angles R and β are defined in Figure 2. If r, R1and R2 are respectively the radius and the radii of curvature of the lens, the following relationships hold:
LA951508W
(A.6)
