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Langmuir 2001, 17, 1338-1348
The Superspreading Effect of Trisiloxane Surfactant Solutions N. V. Churaev,† N. E. Esipova,† R. M. Hill,*,‡ V. D. Sobolev,† V. M. Starov,§ and Z. M. Zorin† Institute of Physical Chemistry of Russian Academy of Sciences, Leninsky Prospect 31, Moscow, 117915, Russia, Dow Corning Corp., Midland, Michigan 48686-0994, and Department of Chemical Engineering, Loughborough University, Loughborough, LE11 3TV, UK Received June 5, 2000. In Final Form: December 12, 2000 The problem of superspreading has recently attracted much attention both from the theoretical point of view and because of the practical use of the phenomenon. However, up to now there is no general agreement about the mechanism of the effect and on the necessary conditions for its realization, in particular regarding different types of surfactants.We report here the results of our investigation of the spreading of vesicular solutions of the trisiloxane surfactant D-8 at a concentration of 0.16 wt %. At this concentration a maximum rate of droplet spreading was observed by Zhu et al. for this surfactant. By including the disjoining pressure isotherm of wetting films, Π(h), into the hydrodynamic equations, quantitative agreement with experimental data was attained. The same approach was used for describing the rate of film climbing over a hydrophobed inclined plate. Rapid spreading was explained by formation of extremely thick wetting films, stabilized by mutual repulsion of vesicles, and, partly, by long-range electrostatic forces. The role of vesicles may also consist of damping capillary waves on the film surface and protecting the thick film against rupture, that is, working as enhanced dilatational (second) viscosity. The proposed form of the Π(h) isotherm agrees with the shapes of the profile of the thick film edge, calculated on the basis of video camera images. Parameters of the Π(h) isotherm are estimated from rates of stationary flow of wetting films over hydrophobed walls between two menisci in glass capillaries of various radii. The critical disjoining pressure at which the film lost its stability is small and corresponds just to the saturated vapor pressure. The proposed approach explains the link between turbidity and superspreading and the requirement for a saturated water vapor atmosphere. The phenomenon of flow and stability of micron thick wetting films containing small interacting particles is a subject of general interest. The problem demands further investigation in more detail with application of different structure-sensitive methods and computer simulation.
Introduction Trisiloxane surfactants have been recognized as effective wetting agents for water-based herbicides on waxy plant leaves.1-3 The unique ability of certain trisiloxane surfactants to promote rapid spreading of aqueous solutions over hydrophobic surfaces such as Parafilm or polyethylene is called superspreading or superwetting. Ananthapadmanabhan et al.4 postulated that the rapid spreading results from the peculiar character of the trisiloxane moietysits wide hydrophobe group. However, it was later shown that molecular geometry is not a critical parameter.5,6 Solution turbidity, that is, the presence of a dispersed phase, was found to be an important parameter. This follows from the experiments with D-8 solutions performed by Zhu et al.,7 which showed that super†
Institute of Physical Chemistry of Russian Academy of Sciences. Dow Corning Corp. § Loughborough University. * To whom correspondence should be addressed. E-mail: r.hill@ dowcorning.com. ‡
(1) Zabkiewicz, J. A.; Gaskin, R. E. Adjuvants Agrochem. 1989, 1, 141-149. (2) Gaskin, R. E.; Kirkwood, R. C. Adjuvants Agrochem. 1989, 1, 129-139. (3) Knoche, M.; Tamura, H.; Bukovac, M. J. Agric. Food Chem. 1991, 39, 202. (4) Ananthapadmanabhan, K.; Goddard, E.; Chandar, P. Colloids Surf. 1990, 44, 281. (5) Hill, R. M.; He, M.; Davis, H. T.; Scriven, L. E. Langmuir 1994, 10, 1724. (6) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1996, 12, 337-344.
spreading occurs only when dispersed particles are present. This work also showed that the rate of droplet spreading over Parafilm surface increases with concentration and reaches a maximum at C ) 0.16 wt %. Sonicated solutions spread faster due to destruction of aggregates of single particles. Addition of formamide reduces the amount of dispersed phase and results in a decrease of the rate and the spread area. The mean particle radius in sonicated D-8 solutions, measured using dynamic light scattering, was r ) 26 nm,7 while without sonication the r value was much larger, r ) 137 nm, probably due to formation of aggregates. The r value for the sonicated solution is close to the value measured using freeze-fracture transmission electron microscopy (r ) 20 nm).8 These values are comparable with the radii of double-tailed unilamellar vesicles.9 Hill and co-workers5,10 concluded that D-8 solutions in the range of concentration from 0.01 to 0.25 wt % contain unilamellar vesicles. Another important factor noted in Zhu et al.7 was the water vapor pressure in the surrounding atmosphere. Superspreading of D-8 solutions was observed only in saturated or supersaturated water vapor. This has given (7) Zhu, X.; Miller, W.; Scriven, L.; Davis, H. Colloids Surf. A 1994, 90, 63. (8) Svitova, T. F.; Hoffmann, H.; Hill, R. M. Langmuir 1996, 12, 1712. (9) Svitova, T. F.; Smirnova, Y. P.; Pisarev, S. A.; Berezina, N. A. Colloids Surf. A 1995, 98, 107-115. (10) He, M.; Hill, R.; Lin, Z.; Scriven, L.; Davis, H. J. Phys. Chem. 1993, 97, 8820.
10.1021/la000789r CCC: $20.00 © 2001 American Chemical Society Published on Web 02/07/2001
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rise to the suggestion that fast spreading may be caused by surface flow of a thin precursor film formed from the vapor phase. However, ellipsometrically measured coefficients of surface diffusion of pure siloxane surfactants, Ds < 10-6 cm2/s,11 are too small for explaining the effect of superspreading. Another possible mechanism of superspreading involves Marangoni flow caused by a difference in surface tension of the precursor water film and a droplet of surfactant solution4 or by dynamic surface tension.8 As was stated by Zhu et al.,7 these effects may take place in any surfactant solution and cannot explain the peculiarities of the trisiloxane surfactants. Lin et al.12 working with D-8 solutions confirmed the concentration maximum using a quartz crystal microbalance to measure spreading rates and showed that there was also a maximum in spreading rate as a function of substrate surface energy. In several subsequent papers6,13,14 an image analysis method was used to measure spreading rates for solutions of trisiloxane and hydrocarbon surfactants vs their concentration, relative humidity, and temperature. They, like Lin et al.,12 also found that the spread area increases linearly with time, and there are maxima in the spreading rate vs concentration and substrate surface energy. They found that the sensitivity of the spreading rate to humidity depended on the type of substratesit is weaker for smoother surfaces. The maximum rate of spreading of trisiloxane surfactant solutions on hydrophobed silicon surfaces was observed for solutions containing vesicles.15 Vesicular solutions spread with nearly constant velocity for an extended time interval. In contrast, micellar solutions at 1 wt % spread slowly and at 0.1 wt % cease to spread after several seconds.16 All the results obtained for rates of spreading of the D-8 trisiloxane surfactant solutions were systematized by Hill.17 The data are represented in the form of a surface plot, which shows a clear dependence of spreading rates both on substrate surface energy and on solution concentration. The maximum rate of spreading (mm2/s) for D-8 solutions occurs at a concentration near 0.4 wt % on substrates with a water contact angle near 70°. A new source for discussion is data obtained for droplets spreading over substrates covered by organosulfur monolayers.6 The measured initial rates of spreading, v0 ) A/t mm2/s, give a linear dependence of the area of spreading A on time t. Maximum values of v0 were observed in the case of intermediate degree of surface hydrophobicity (θ ) 55-60°). In distinction to the previous results,7,8 it was concluded that the presence of a dispersed phase in trisiloxane surfactant solutions and high humidity are not critical to the initial rate of spreading. However, these results must be considered with caution while fast kinetics of the very early stage of spreading may be strongly influenced both by dynamic effects and viscous dissipation inside a droplet. The obtained linear relation A(t) is known for the inertial regime of droplet spreading. According to (11) Tiberg, F.; Cazabat, A. M. Europhys. Lett. 1994, 25, 205. (12) Lin, Z.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1994, 10, 4060-4068. (13) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7270-7275. (14) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7276-7281. (15) Wagner, R.; Wu, Y.; Czichocki, G.; VonBerlepsch, H.; Rexin, F.; Perepelittchenko, L. Appl. Organomet. Chem. 1999, 13, 201-208. (16) Wagner, R.; Wu, Y.; Berlepsch, H. V.; Zastrow, H.; Weiland, B.; Perepelittchenko, L. Appl. Organomet. Chem. 1999, 13, 845-855. (17) Hill, R. M. Curr. Opin. Colloid Interface Sci. 1998, 3, 247-254.
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Khlynov et al.,18 the A/t ratio in this case is proportional to (SR0/F)1/2, where R0 is initial droplet radius, S is the spreading coefficient, and F is the density of the liquid. Therefore, up to now a mechanism responsible for superspreading has not been elucidated. According to Zhu et al.,7 any proposed mechanism must take into account a leading role of water vapor pressure and the presence of a dispersed phase in trisiloxane surfactant solutions. In the present paper, results of an investigation of superspreading of D-8 trisiloxane surfactant solutions (0.16 wt %) over methylated glass surfaces will be discussed in the framework of the Frumkin-Derjaguin theory of wetting that includes the disjoining pressure isotherm of wetting films.19 High rates of spreading of trisiloxane surfactant solutions containing dispersed vesicles are associated with formation of very thick wetting films during spreading. In turn, the stability of such thick films is attributed, at least partly, to mutual repulsion of particles (vesicles) contained inside the film. Similar effects are known for colloidal20-22 and micellar solutions,23-26 lamellar phases,27 and presmectic fluids28,29 under conditions of restricted geometry. Ordered (periodic) structures formed even in bulk colloidal systems (starting with anisotropic tactoids30) were systematically analyzed, for instance, in the monograph of Efremov.31 Possible forms of a disjoining pressure isotherm Π(h) are considered in part 1. Introducing a simple, linearized form of the disjoining pressure isotherm for turbid wetting films of trisiloxane surfactant solutions, it becomes possible to describe quantitatively rates of droplet superspreading (in part 2) and the shape and rates of film climbing over an inclined plate (in parts 3 and 4). In the latter case, profiles of the film edge were calculated on the basis of interference pattern visualized by means of video camera images. Calculated shapes of the profile agree well with the adopted form of the isotherm. The proposed isotherm Π(h) includes regions of thick metastable β-films and of much thinner stable R-films separated by a region of film instability. A spontaneous β f R transition may occur near some critical value of disjoining pressure estimated as Π/ ≈ 400 dyn/cm2 (parts 3 and 5). The proposed form of the isotherm was confirmed experimentally measuring stationary flow rates of β-films between two menisci in vertically oriented methylated glass capillaries of different radii (part 5). The developed approach allows explanation of the role of both dispersed phase as a cause of the thick β-film formation and water vapor pressure, since the critical (18) Khlynov, V. V.; Sorokin, Y. V.; Esin, O. A. Zh. Fiz. Khim. 1967, 41, 1764-1769. (19) Churaev, N. V. Rev. Phys. Appl. 1988, 23, 975-987. Churaev, N. V. Adv. Colloid Interface Sci. 1995, 58, 87-118. (20) Hachisu, S.; Kobayashi, Y.; Kose, A. J. Colloid Interface Sci. 1973, 42, 342-348. (21) Takano, K.; Hachisu, S. J. Colloid Interface Sci. 1978, 66, 124129. (22) Hideki, M.; Koji, K.; Norio, I. Proc. Jpn. Acad. B 1991, 67, 170. (23) Nikolov, A.; Kralchevsky, P.; Ivanov, I.; Wasan, D. J. Colloid Interface Sci. 1989, 133, 13-22. (24) Nikolov, A. D.; Wasan, D. T. J. Colloid Interface Sci. 1989, 133, 1-12. Nikolov, A. D.; Wasan, D. T. Langmuir 1992, 8, 2985-2994. (25) Bergeron, V.; Radke, C. J. Collect. Colloq. Semin. (Inst. Fr. Pet.) 1992, 50, 225-229. (26) Parker, J. L.; Richetti, P.; Kekicheff, P.; Sarman, S. Phys. Rev. Lett. 1992, 68, 1955-1958. (27) Kekicheff, P.; Christenson, H. K. Phys. Rev. Lett. 1989, 63, 28232826. (28) De Gennes, P. G. Langmuir 1990, 6, 1448-1450. (29) Moreau, L.; Richetti, P.; Barois, P. Phys. Rev. Lett. 1994, 73, 3556-3559. (30) Freundlich, H. Kapillarchemie, Bd.2, Leipzig, 1932. (31) Efremov, I. F. Periodic Colloidal Structures; Khimiya, Leningrad Otd.: Leningrad, USSR, 1971.
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disjoining pressure corresponds to a relative vapor pressure p/ps near saturation. The experiments were performed at a concentration of 0.16 wt %, which corresponds (according to ref 7) to the maximum rate of spreading of D-8 solution. The mean distances between vesicles may be in this case commensurable with the range of action of interparticle forces. At lower concentrations, vesicles participate in free Brownian motion and do not interact. At higher concentration aggregates are formed, which decreases the number of kinetic entities in the dispersion. Finally, some speculations about the forces stabilizing thick β-films containing vesicles are considered in part 6. Theoretical Approach Part 1: Isotherm of Disjoining Pressure of Wetting Films. After spreading of a microdroplet of D-8 solution, a flat wetting film of finite area and several microns thick is formed on a hydrophobic substrate.7,8 According to de Gennes,32 formation of such a “pancake” may occur when the coefficient of spreading S ) γsv - γ sl - γ is small and positive. Here γsv and γ sl are the surface free energies of solid-vapor and solid-liquid interfaces, and γ is the surface tension. The de Gennes theory relates the value of S with the isotherm of disjoining pressure of a wetting film Π(h) introduced by Derjaguin:33
S ) Π(h)h +
∫h∞Π(h) dh
(1)
Here h(Π) is the thickness of a wetting film which depends on the capillary pressure Pc of the meniscus being in equilibrium with the film. In turn, the capillary and disjoining pressures depend on the relative vapor pressure in the state of equilibrium:
Pc ) Π ) -(RT/vm) ln(p/ps)
(2)
where vm is the molar volume of a liquid and R is the gas constant. When S < 0, the liquid forms a finite contact angle, θ > 0, with a solid substrate. The value of an equilibrium contact angle θ may be predicted on the basis of the Frumkin-Derjaguin theory using the Π(h) isotherm.19,34 In this case, a transition zone of variable thickness exists between a flat wetting film and the bulk meniscus.35 The condition S > 0 corresponds to complete wetting. No contact angle, formally determined at the point of intersection of an undisturbed profile of the meniscus with a solid substrate, is formed in this case. Taking into account the effect of gravity, eq 1 has been transformed into the following form:32
S ) Π(h)h +
∫h∞Π(h) dh - 21Fgh2
(3)
where F is the density of a liquid and g is the gravity acceleration. Solution of this equation at S ) 0 determines the pancake thickness, equal to hf ) Π/Fg. Positive disjoining pressure in the pancake Π ) Fghf > 0 is equilibrated here by the negative hydrostatic pressure Ph ) -Fghf. An additional small repulsion term equilibrating pancake thickness may arise when S is nonzero, but small and (32) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827-863. (33) Derjaguin, B. V. Acta Phys.-Chim. USSR, 1940, 12, 181. Derjaguin, B. V. Kolloid. Zh. USSR 1955, 17, 207. (34) Churaev, N. V.; Sobolev, V. D. Adv. Colloid Interface Sci. 1995, 61, 1-16. (35) Churaev, N. V.; Starov, V. M.; Derjaguin, B. V. J. Colloid Interface Sci. 1982, 89, 16-24.
Figure 1. Schematic representation of the disjoining pressure isotherm Π(h) in the case of complete wetting at S ≈ 0.
positive. In this case, the disjoining pressure in a pancake Πf is higher than Fgh. In any case, for a pancake of micron size thickness, the disjoining pressure Πf is very small. This means that formation of such a thick pancake is possible only in an atmosphere of nearly saturated vapor pressure. This explains why superspreading was observed only in an atmosphere of saturated vapor.7 Let us analyze possible shapes of the Π(h) isotherm that correspond to the small positive values of S near 0. Because disjoining pressure of the pancake is small, we can neglect the first term in eq 1. The remaining second term must be small and positive, which corresponds to a small difference between areas embraced by the branches of the isotherm in the regions Π > 0 and Π < 0. The qualitative form of such an isotherm is shown in Figure 1. The upper branch of the isotherm corresponds to the metastable state of the β-film. At some value of Π > Π/, where Π/ is the critical disjoining pressure (see Figure 1), the thick β-film may transform spontaneously into a stable state of much thinner R-film. The pancake thickness hf lies in the region of thickness h > h/. Thick metastable films are formed in the course of thinning of a bulk liquid layer that occurs during droplet spreading. Thin R-films may be formed at the edge of the pancake as a result of vapor adsorption or of β f R transition accompanied by consecutive evaporation. This effect was often observed after finishing of the pancake formation. Aggregates of vesicles remain on the substrate after evaporation. Part 2. Rate of Droplet Spreading. For description of droplet spreading and pancake formation in the framework of our approach, we need to use only the β-branch of the isotherm Π(h). An analytical expression for the rate of droplet spreading has been obtained on the basis of a simplified form of the Π(h) isotherm. Because of very small values of Π, the part of the isotherm (Figure 1) in the range of thickness between h* and h0 (at Π ) 0) may be represented in a linear form:
Π(h) ) Πf + b(hf - h) at hf < h < h0
(4)
where Πf is the disjoining pressure corresponding to pancake thickness hf and the parameter b characterizes the slope of the isotherm. The profile of a droplet during spreading is shown in Figure 2a. Here r is the radial coordinate, h(r,t) is the local thickness of the spreading layer, r0 is the radius of the spreading area, and t is the time. The thickness of a forming pancake is equal to hf, and its radius is rf (Figure
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Langmuir, Vol. 17, No. 5, 2001 1341 2
dr0 4bhf 2 ) [(V/π) - (r0 hf)] 3 dt 3ηr
(10)
0
This equation may be rewritten in a dimensionless form using the following parameters: z ) r0/rf and z0 ) ri/rf, where ri ) r0(0) and rf is the final radius of the pancake (Figure 2b). The dimensionless time τ was represented as 4 τ ) t/τ0, where τ0 ) 3ηV/4πbhf . Substituting these parameters into eq 10, we obtain
dz/dt ) (1 - z2)/z3
Figure 2. Assumed profile of a spreading droplet (a) and its final state (b) in the form of a “pancake”.
2b). The same condition h ) hf is fulfilled at the edge of a flat pancake. It was assumed that the effect of surface curvature of the spreading layer and the action of capillary forces may be neglected (at least during the last stage of spreading considered here). The only driving force is assumed to be the gradient of disjoining pressure dΠ/dr. Under these conditions the equation of droplet spreading derived earlier may be used:36
dh b d dh ) rh3 dt 3ηr dr dr
( ) (
)
(5)
where η is the viscosity and b is the parameter of the isotherm (4). This equation was used for calculation of the time dependence of the spreading radius r0(t) that characterizes the rate of spreading. After integration of eq 5 and some transformations, we obtain
( )( ) 2
bhf dr0 )dt 3η
dh dr
r)r0
(6)
The solution of this equation was represented in the form of a polynomial:
h(r,t) ) B1(t)r2 + B2(t)r + B3(t)
(7)
From the symmetry condition, dh/dr ) 0 at r ) 0, it follows that B2 ) 0. Application of the boundary condition, h(r0,t) ) hf ) constant, results in the expression B1r02 + B3 ) hf, which gives
h ) hf - B1(t)(r02 - r2)
(8)
where the unknown function
B1(t) ) (4/r04){(r02hf/2) - V/2π}
(9)
was found using the condition of volume constancy of the liquid
V ) 2π
∫0∞rh(r) dr ) constant
After substitution of eqs 9 and 8 into eq 6 we obtain (36) Starov, V. M. Kolloidn. Zh. 1983, 45, 1154-1161.
(11)
where z ) z(τ), z(0) ) z0, and z(∞) ) 1. The asymptotic form of the solution of eq 11 at τ . 1, when z f 1 and r0 f rf, has the following form:
1 - z ) M exp(-2τ)
(12)
2
where M ) (1 - z0 )/2 exp(1 - z02). Substitution of (1 - z) ) 1 - (r0 /rf) into eq 12 gives the final expression for the rate of spreading expressed as a dependence of the spreading radius r0 on the real time t:
ln[1 - (r0/rf)] ) ln M - 2(t/τ0)
(13)
In Figure 3 the experimental data obtained by Zhu et al. 7 for droplets of D-8 solutions are compared with values predicted by eq 13. The data are plotted in terms of ln[1 - (r0 /rf)] vs the real time t. Values of r0/rf were calculated 2 on the basis of values of the spreading area A(t) ) πr0 (t) 2 and final pancake area Af ) πrf given by Zhu et al. Figure 3 shows that, in accordance with eq 13, the dependence is linear. The slope of the graph gives the characteristic time τ0, which is equal to 187 s. Using droplet volume V ) 0.0078 cm3, solution viscosity η ) 0.001 N s/m2, and final pancake thickness hf ) V/Af, values of the parameter b were estimated. Depending on solution concentrations, they are on the order of 106-107 dyn/cm3. Large values of the derivative b ) -dΠ/dh point to a steep profile of a spreading droplet. This explains also the surprisingly wide range of applicability of the eq 13, namely, for r0/rf values from 0.3 to 0.95 and for the time of spreading from 15 s to 5 min. This shows that the main viscous resistance stems from the peripheral, almost flat part of the spreading droplet. The final state corresponds to a thick flat wetting film or a pancake. Considering droplet spreading, we need to take into account loss of surfactant molecules adsorbed during spreading on the solid surface and at the solution-air interface. A simple calculation estimates the magnitude of the effect. The surface area of a vesicle is approximately twice the surface area of a sphere. For a radius of 20 nm7,8 this gives 1.83 × 106 molecules per vesicle assuming an area per molecule of 55 Å2 (consistent with small-angle X-ray scattering data for the lamellar phase37). The concentration Cs ) 0.16 wt % corresponds to 2.53 mM/L. Using the value above for the number of molecules per vesicle, there are 8.33 × 1014 vesicles per liter and 6.5 × 109 vesicles in a 0.0078 cm3 droplet. Using an area per molecule of 55 Å2, there should be sufficient surfactant in the droplet to cover an area of S ) 66 cm2. A 0.0078 cm3 droplet of 0.16 wt % solution spreads to 2 an area of πrf ) 25 cm2, where rf is the pancake radius.8 The total area to be covered by surfactant is twice this, (37) Hill, R. M., unpublished results.
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Figure 4. Schematic representation of a process of film climbing over inclined plate.
Figure 3. Comparison of experimental data (points) with values predicted by eq 13 (solid line) for droplets spreading of D-8 solutions of different concentration Cs ) 0.078 (1), 0.104 (2), 0.117 (3), and 0.161 wt % (4).
50 cm2. The remaining 16 cm2 corresponds to 16 × 1016/55 ) 3 × 1015 molecules or 3 × 1015/1.83 × 106 ) 1.64 × 109 vesicles. Thus, the vesicle concentration is decreased by a factor of 1/4 to Cs ) 0.04 wt %. This is well above critical aggregation concentration (0.007 wt %) but is quite close to the critical wetting concentration, 0.03 wt %, found by Svitova et al.38 A factor of 4 reduction in concentration results in a factor of (4)1/3 ≈ 1.6 decrease in interparticle distance. The presence of the remaining vesicles may secure stability of a forming pancake, because even low repulsive forces between vesicles are sufficient to compensate the very small disjoining pressure in the pancake (part 1). It should also to be noted that in the case of micron thick films the latter may be stabilized also by forces of electrostatic repulsion. Electrokinetic measurements of the surface potential of a paraffin film covering the inner surface of quartz capillaries have shown that in contact with 10-4 M KCl aqueous solution ζ ) -50 mV.39 Also in hydrophobed (methylated) quartz capillaries in contact with 0.16 wt % D-8 surfactant solution, measured values of the ζ-potential decrease only slightly, from -70 to -50 mV, as compared with a pure 10-4 KCl solution in a hydrophilic capillary, -100 mV. Further, it may be assumed that the surface charge of the solution-air interface is zero, because vesicles are not charged. Under these conditions, electrostatic repulsion forces, which arise in a pancake of thickness hf, may be calculated using the Derjaguin-Landau equation:40 2
Πe ) (π0/2hf )(RT/zF)2
(14)
where 0 is the dielectric constant, R is the gas constant, T is the temperature, z is the ionic valence, and F is the Faraday constant. This equation describes electrostatic repulsion forces in aqueous films at κh , 1 and ζ > 25 mV, where κ is the inverse Debye radius. It follows from eq 14 that a pancake with a thickness of about 3-4 µm may be stabilized by electrostatic (38) Svitova, T.; Hill, R. M.; Smirnova, Y.; Stuermer, A.; Yakubov, G. Langmuir 1998, 14, 5023-5031. (39) Churaev, N. V.; Ershov, A. P.; Esipova, N. E.; Iskandarjan, G. A.; Madjarova, E. A.; Sergeeva, I. P.; Sobolev, V. D.; Svitova, T. F.; Zakharova, M. A.; et al. Colloids Surf., A 1994, 91, 97-112.
Figure 5. Inclined microscope device with a chamber for measuring the position and the shape of the edge of a climbing film during time.
repulsion forces. The role of vesicles may consist not only of additional repulsive forces but also of damping of capillary waves forming on the surface of the thick film, protecting the thick film against local thinning and rupture. The effect of mutual repulsion of vesicles would thus act as an enhanced dilatational (second) viscosity of turbid D-8 solutions. Experimental Section Part 3. Shape of D-8 Films Climbing an Inclined Plate. In contrast to droplet spreading, film climbing over an inclined plate allows study of a slower process and, at the end, to observe the equilibrium state of a thick trisiloxane film. A schematic representation of the method is shown in Figure 4. The plate 1 is immersed into bulk D-8 solution 2, and the length l(t) of the climbed film 3 is measured as a function of time t. At some critical value of hydrostatic pressure at the film edge P/ ) rgH/ and corresponding critical film length l/ the flow stops. The film reaches an equilibrium state when local values of disjoining pressure Π(x) are compensated by local values of hydrostatic pressure P(x) ) FgH(x). The hydrostatic pressure P/ equals the critical disjoining pressure Π/, when a thick β-film becomes unstable. Measuring the film thickness distribution at equilibrium, h(x), it is possible (in principle) to determine the complete disjoining pressure isotherm Π(h) of a wetting film. Such a program was, for instance, realized by Ingram for wetting films of n-alkanes climbing on a vertical quartz plate.41 However, because of the thickness of our films, we are able to use the interference method to measure thickness only near the film edge. The experimental setup is shown in detail in Figure 5. The trough 1 filled with the D-8 solution is placed into the closed chamber 2 fastened to the movable table 3 of an Epival Interphako microscope. The chamber is covered with the plate 4 fixed to the microscope objective 5. The microscope and the chamber are inclined at an angle of φ ) 11° to the horizon. The hydrophobed cover slide 6 (1.5 mm thick) is immersed into the solution by (40) Derjaguin, B. V.; Churaev, N. V. Wetting Films; Nauka: Moscow, USSR, 1984. (41) Ingram, B. T. J. Chem. Soc., Faraday Trans. 1 1974, 70, 868876.
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Figure 7. Profiles of climbing film edge calculated from interference patterns: (a) for the film shown in Figure 6; (b) the same, after 6 s; (c) l ) 0.64 cm, v ) 12 µm/s; (d) l ) 0.72 cm, v ) 10 µm/s; (e) l ) 1.2 cm, near the equilibrium position.
Figure 6. Interference pattern of the film edge. The length of the film l ) 0.55 cm; rate of the edge movement v ≈ 10 µm/s. Thin precursor film is seen in the front of the thick wetting film. means of the pusher 7. For the film climbing experiments it was preferable to use the lower side of the plate. In this case saturated vapor pressure in the chamber may be maintained more precisely using a sheet of moist filter paper 8. All the experiments were performed with 0.16 wt % D-8 solutions that had been sonicated for 15 min. Hydrophobed glass slides were prepared by exposure in a closed chamber to trimethylchlorosilane vapor for 2-3 days. Contact angles θA of water droplets on these slides lie in the range 90°-95°. Movement of the film edge was observed using a microscope and video camera Colour CCTV Panasonic WV-CP412 with video card MiroVideo 20 TD live (Miro Computer Products AG), which gives up to 60 frames with frequency 20-0.1 frames/s. Monochromatic light (λ ) 0.59 µm) reflected from the wedge-shaped film creates an interference pattern. The video images of the pattern were recorded in computer memory and visualized on the computer display. The images were recorded at the rate of one or two per second. The actual dimensions of the images were obtained using an object micrometer. The starting time was taken from the moment when the slide contacts the solution. Processing of the interferometric data was performed on the basis of an extended computer program (Sig.Prof) that determines the coordinates x and y of a point on an image. The distance between interference fringes was calculated as ∆ ) (∆x2 + ∆y2)1/2. A shift of one fringe corresponds to a change in the film thickness of ∆h ) λ/2n ) 0.22 µm, where n ) 1.33 is the refraction index of water. One example of interference images of an edge of the climbing D-8 film is shown in Figure 6. Profiles h(x) of the film edge (calculated from the interference fringes) for a set of measurements are shown in Figure 7. The profile h(x) corresponding to the results shown in Figure 6 is shown by curve a in Figure 7. The direction of the cross section is indicated in Figure 6 by the solid line. The spreading front is not strictly linear, reflecting surface heterogeneity. The results for the several runs shown in Figure 7 give a similar edge profile. Thickness of D-8 films, estimated on the basis of the profiles obtained, ranged from 2 to 4 µm. Thin precursor films are clearly seen in Figures 6 and 7 having a thickness of about 100 nm and length from 10 to 30 µm. The
Figure 8. Adopted form of a disjoining pressure isotherm Π(h) of D-8 wetting films (left) and the profile of a transition layer between a thick β-film and thin precursor or R-film (right). microinterference method developed earlier,42 modified by application of a video camera for determination of the intensity of light reflected from the film, gives the mean thickness of equilibrium R-films of the same D-8 solution as about 100 nm. The data were obtained for flat round (r ) 10 µm) films formed in a Teflon cell on a hydrophobed polished quartz plate at the disjoining pressure Π = 800 dyn/cm2, which was higher than the critical value, Π/ ≈ 400 dyn/cm2. This means that at Π > Π/ only thin stable R-films are formed. Therefore, flowing precursor films may be associated with R-films. Such thin films were observed also on the inclined plate near the retreating thick film after the critical height Π/ was reached. The thick film retreats slowly as a result of β f R transition near to the film edge and evaporation of the excess material. The measured β-film profiles (Figure 7) are compared in Figure 8a with the prediction of the theory based on the proposed form of the disjoining pressure isotherm. Between a bulk meniscus and wetting film some transition layer of variable thickness is formed.35 The shape of an equilibrium transition layer h(x) is regulated by the condition of constant pressure inside the layer along the coordinate x. This condition is written as Π(x) + γK(x) ) constant, where Π(x,h) is the local value of disjoining pressure and γK is the local value of capillary pressure. Here γ is the surface tension of a liquid, and K is the local surface curvature of the layer. The same approach was applied later by Dobbs and Indekeu43 for describing a transition zone between a thick metastable aqueous β-film and a thin stable R-film of water. That is just what happens in our case near the thick film edge of the D-8 solution. Near the edge of a stopped film the disjoining pressure Π* is compensated by hydrostatic pressure P/ ) FgH/. In this case the net pressure inside the film is equal to 0. This means that in the (42) Zorin, Z. M.; Churaev, N. V.; Esipova, N. E.; Sergeeva, I. P.; Sobolev, V. D.; Gasanov, E. K. J. Colloid Interface Sci. 1992, 152, 170182. (43) Dobbs, H. T.; Indekeu, J. O. Physica A (Amsterdam) 1993, 201, 457-481.
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Figure 10. Rates of film climbing of D-8 solution, Cs ) 0.16 wt %, over a hydrophobed glass plate inclined on the angle φ ) 11° (curves 1-4). Curves 5-7 show the results obtained when the plate was covered with a thin film of silicon oil.
Figure 9. Interference pattern (a, top) and calculated profile (b, bottom) of the edge of a pancake formed on flat methylated quartz surface after finishing droplet spreading of D-8 solution, Cs ) 0.16 wt %. region of the transition zone, where Π > 0, the γK term must be negative, which corresponds to a convex surface. In the range of film thicknesses where Π < 0, the profile of the transition zone must by concave. Figure 8b shows the theoretical profile of the film edge which corresponds to the shape of the Π(h) isotherm shown on the left. A similar shape of a transition layer was also calculated in ref 43. The theoretical shape of the transition layer, shown in Figure 8, is in qualitative agreement with the measured profiles of the film edge (Figure 7). The reason for the coincidence is the very slow rate of movement of the film edge (of about 10 µm/s), which allows the shape of the edge to be considered nearly in the state of equilibrium. The obtained agreement with theoretical prediction may be considered as an independent confirmation of the proposed form of the Π(h) isotherm for wetting films of vesicular D-8 solutions. The same shape is characteristic of the edge of the pancake film formed after droplet spreading of D-8 solution over a flat hydrophobed quartz plate (Figure 9). The pancake thickness is about 3 µm. On the video image a thin precursor film is clearly seen. Near the front of the film, small (of micron size) droplets of water condensate are formed along some linear scratches on the polished quartz surface, reflecting the relatively high humidity of the air near the edge of the pancake. Part 4. Rate of D-8 Film Climbing over Inclined Plate. Using the device shown in Figures 4 and 5, rates of film climbing were measured. In Figure 10 some results are given showing the dependence of the film length l on time t. The film length l was
determined from the point of the film contact with the capillary meniscus (Figure 4). Curves 1-4 characterize film climbing over a hydrophobed quartz plate, and curves 5-7 show results for a plate covered with a thin (0.1-0.3 mm) film of silicone oil, M ) 20 000. After some time the film edge movement stops, and an equilibrium state is attained. The rate of motion of the contact line was about 10 µm/s at l ≈ 0.5-0.6 cm, rising to about 100 µm/s at much smaller l values. Near to equilibrium, when l approaches l/, rates of front motion decrease to 1 µm/s. Figure 10 shows that l/ values lie in the range from 0.8 to 1.4 cm, which corresponds to the critical height H/ ) (L + l/) sin φ between 3.5 and 5 mm. This corresponds to a critical disjoining pressure Π/ between 350 and 500 dyn/cm2. The same theoretical approach as in part 2 was used for describing the rate of film climbing. First, an equilibrium profile of a meniscus formed between the level of bulk solution and the inclined plate (Figure 4) was calculated assuming that complete wetting takes place. It was shown that the position of the meniscus end and the start of a wetting films equals x ) L ) (γ/Fg)1/2/ sin(φ/2). Here φ is the angle of inclination, γ is the surface tension, F is the density of liquid, and g is the gravity. The hydrostatic pressure at the point x ) L is equal to PL ) FgL sin φ. This value determines the disjoining pressure ΠL(hL) ) PL and the corresponding film thickness hL at x ) L, where l ) 0 (Figure 4). The disjoining pressure isotherm was represented in this case also in a linearized form as shown in Figure 8:
Π ) b(hL - h) at h > h/
and
Π ) 0 at h < h/ (15)
Here the parameter b characterizes the slope of the isotherm, and hL corresponds to the pressure PL ) FgHL, where HL ≈ 0.2 cm. At equilibrium, when the film front reaches the position H/, the film thickness at l ) l/ equals h/. The equation of the film flow is represented in the following form:
dh/dt ) [(b - Fg cos(φ))/3µ](d/dx){h3[(dh/dx) + Λ]}
(16)
where Λ ) Fg sin φ/(b - Fg cos φ) and t is the time. Equation 15 should be solved with the boundary conditions:
h ) hL
at l ) 0 and t ) 0
h ) h/ and l ) l/
at t f ∞
where l/ is the final length of the film at Π ) Π/ at equilibrium. A qualitative form of the solution obtained coincides with the experimental l(t) dependence shown in Figure 10. Analytical expressions were obtained for small values of the ratio l/l/ and for these values near 1. The first approximation is expressed as
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l/l/ ) (4hL3bt/η)1/2
(17)
l/l/ ) 1 - C exp[-(hL3bt/η)]
(18)
and the second as
where C ≈ 3.5, η is the viscosity, and t is the real time. Equation 17 describes the first stage of climbing when the effect of gravity may be neglected. Use of this equation, when l/l/ values are of about 0.6 at t ≈ 40 s, η ) 0.001 N s/m2, and hL ≈ 3 µm, results in b values on the order of 106 dyn/cm3, as in the case of droplets spreading (part 2). In Figure 11 are shown the results of another series of experiments with the same solution. The rate of motion of the film edge was about 100 µm/s at small values of l(t) and decreases to 10 µm/s at l(t) equal 0.5-0.6 cm. Near equilibrium the rates decrease to 1 µm/s. Unlike Figure 10, the graphs are plotted in terms of l(t1/2). The initial part of the graphs is nearly linear in accordance with eq 17. The critical lengths of the films l/ range from 0.6 cm (curves 1 and 2) to 1.1 cm (curves 3 and 4). This corresponds to critical disjoining pressure Π* values between 260 and 400 dyn/cm2, close to values calculated above from Figure 10. Thus, it appears that both droplet spreading and film climbing of thick β-films of D-8 solutions may be described on the basis of the same disjoining pressure isotherm included into hydrodynamic equations. As may be seen, eq 17 corresponds formally to some diffusion kinetics. However, the coefficient of diffusion calculated on the basis of experimental data becomes on the order of 10-2 cm2 /s, too high for molecular diffusion. In the case of film climbing the influence of surfactant adsorption is much lower as compared with droplet spreading. First, a saturated film-air interface is formed here directly from flowing bulk solution. Second, adsorption layers on the solid surface are formed from a standing convective flow of bulk solution accompanied by diffusion of molecules and vesicles. Besides, coverage of methylated glass surface by trimethylsilyl groups at contact angles even near 90° is far from complete. According to Herzberg et al.,44 an advancing contact angle θA ) 90° may be formed at a surface coverage of methyl groups φ > 0.4. Ralston and co-workers45,46 have shown that θA values of about 90° correspond to a coverage φ ) 0.72-0.785. From water vapor adsorption isotherms on methylated silica surface it follows that a contact angle θA ≈ 70° is formed at a surface coverage φ of about 10%.47 The presence of hydrophilic regions on methylated quartz surface causes charge formation as a result of ionization of the remaining hydroxyl groups. Laskovski and Kitchener have shown that ζ-potentials of silica do not change markedly after methylation and are also pH-dependent.48 Our electrokinetic measurements of ζ-potentials in methylated quartz capillaries results in the value -50 mV in 10-4 M KCl solution.49 All these results show that the hydrophobed part of the surface of methylated silica is smaller as compared with Parafilm. Therefore, the surface area on which trisiloxane surfactant molecules prefer to adsorb is diminished.
Discussion Part 5. Direct Determination of the Disjoining Pressure Isotherm for D-8 Solutions. Several attempts have been made to directly determine the disjoining pressure isotherm Π(h) of wetting films of D-8 surfactant solutions. For instance, the thickness of D-8 films spread(44) Herzberg, W. J.; Marian, J. E.; Vermeulen, T. J. Colloid Interface Sci. 1970, 33, 164-171. (45) Crawford, R.; Koopal, L. K.; Ralston, J. Colloids Surf. 1987, 27, 57-64. (46) Horr, T. J.; Ralston, J.; Smart, R. S. C. Colloids Surf., A 1995, 97, 183-196. (47) Alzamora, L.; Contreras, S.; Cortes, J. J. Colloid Interface Sci. 1975, 50, 503-507. (48) Laskowski, J.; Kitchener, J. A. J. Colloid Interface Sci. 1969, 29, 670-679. (49) Churaev, N. V.; Sergeeva, I. P.; Sobolev, V. D.; Derjaguin, B. V. J. Colloid Interface Sci. 1981, 84, 451-460.
Figure 11. Rates of film climbing for a D-8 solution, Cs ) 0.16 wt %, over hydrophobed glass plate inclined on the angle φ ) 11° (curves 1-4).
Figure 12. Schematic representation of the method of determination of the disjoining pressure isotherm Π(h) by measuring the rate of an air bubble lifting in hydrophobed glass capillaries of various radii.
ing along the inner surface of hydrophobed quartz capillaries (θA ) 100 ( 3°, r ) 0.7-2.5 mm) and film thickness formed after the receding meniscus were measured. No distinct dependence of film thickness h on capillary radius r was obtained, and film thickness h was only roughly estimated (from 1 to 5 µm). Considerable scatter of the data was caused by longitudinal scratches on the inner surface of the quartz capillaries. The most reproducible data were obtained measuring the steady-state flow of thick β-films over the molecularly smooth surface of glass capillaries (without scratches) between two menisci. The surface of strictly cylindrical capillaries with radii r from 0.35 to 1.3 mm was methylated using trimethylchlorosilane. Advancing contact angles of water were θA ) 102 ( 6°. A vertically oriented glass capillary was filled with a sonicated D-8 solution, Cs ) 0.16 wt %, in such a way that an air bubble, 1-2 mm long, was formed between two menisci (Figure 14). Because the lower end of the capillary was closed by mercury, the capillary pressure is roughly equal to Pc ) 2γ/r, and the disjoining pressure of a corresponding flat film equals (due to the film curvature) Π ) γ/r.40 Film flow occurs due to the difference in hydrostatic pressure FgL, where L is the bubble length, F is the density of the liquid, and g is the gravity. The pressure gradient that causes the film flow is equal to Fg and does not depend on the bubble length.
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Figure 13. Dependence of meniscus displacement ∆l on time t in hydrophobed glass capillaries of different radii r ) 1.18 (1), 1.1 (2), 0.57 (3), 0.71 (4), 0.36, and 0.42 mm (5). D-8 solution, Cs ) 0.16 wt %.
Figure 14. Disjoining pressure isotherm Π(h) of a thick wetting films of D-8 solutions, Cs ) 0.16 wt %, calculated on the basis of measured film flow rates in hydrophobed glass capillaries.
The mean thickness h of a flowing film was determined from the equation of steady-state flow:40
h ) (3ηr∆l/Fgt)1/3
(19)
where η ≈ 0.001 N s/m2 is the viscosity, and ∆l is the change in the position of the meniscus at time t. The ∆l ) l0 - l values were measured using a vertical comparator KM-6 with an accuracy better than 5 µm. Here l0 is the
length of the upper column at t ) 0. The top of the capillary was protected from evaporation. Figure 13 shows the dependence of ∆l on t obtained for a set of capillaries. The flow rates v ) ∆l/t remain constant during a long period of time, up to 5-10 days. This confirms the steady-state flow of the films. It follows from eq 19 that in this case the mean film thickness h also remains constant. Calculating the film thickness h from eq 19 for a capillary of a given radius r, it was possible to obtain the dependence of h on Π ) γ/r - the disjoining pressure isotherm, Π(h), assuming γ ) 20 mN/m and F ) 1 g/cm3. The isotherm obtained is shown in Figure 14. Some groups of experimental points are averaged; the number of averaged data points is designated by the symbol n shown near each point. Despite the scatter of the data, the isotherm Π(h) may be approximated in linear form with the slope b ) 4 × 106 dyn/cm3sthe same order of magnitude as calculated from the droplet spreading and film climbing data discussed in parts 2 and 4. In capillaries having radii less than 0.35 mm no measurable film flow was detected. This value of r* corresponds to the critical disjoining pressure Π/ ) γ/r/ close to 500 dyn/cm2 (Figure 14). At smaller radii and correspondingly larger values of disjoining pressure, formation of thick metastable β-films was not observed. However, it should to be noted that the film thicknesses measured here are at least 2 times smaller than for the pancakes and climbing films. Probably, this may be due to film curvature or much higher degree of hydrophobization (contact angle 100°). An attempt to obtain the disjoining pressure isotherm in hydrophobed quartz capillaries of the same radii was misleading due to the presence on its surface of long scratches oriented parallel to the capillary axis. Mean film thicknesses calculated in the same way are much higher due to uncontrolled flow in channels formed by scratches. In contrast to the millimeter size quartz capillaries, the surface of the glass capillaries used in this study were molecularly smooth. Part 6. Interparticle Forces in Turbid Trisiloxane Surfactant Solutions as a Possible Cause of Thick Wetting Film Stability. The results obtained show that fast spreading and climbing of trisiloxane surfactant solutions over hydrophobed surfaces are caused by formation of extremely thick wetting films of the solution. Stability of such micron thick films may be explained by joint action of repulsive forces between dispersed particles (vesicles) and repulsive electrostatic forces between macroscopic film interfaces. An additional role of the dispersed phase may consist also in damping capillary waves formed at the film-gas interface preventing film rupture. The presence of interacting vesicles thus acts as dilatational (second) viscosity in the thin liquid film. In wetting films, that is, in the conditions of restricted geometry, an organized structure of vesicles may arise. Layered structures formed by colloidal particles, micelles, and lamellar mesophase and in presmectic fluids have been considered as a factor that may stabilize sufficiently thick liquid interlayers.20-29 At first, the effect of layering was observed for molecular liquids near a molecular smooth wall. The number of ordered layers ranges here from 6 to 8.50-54 In contrast to molecular systems where the period of oscillation is of molecular size, in colloidal systems the period is much larger, on the order of interparticle distances. In this case repulsion between organized layers may result in formation of micron thick metastable interlayers and films. In Figure 15 the dependence of a force F acting between vesicles on the distance d between their surfaces is shown
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Unfortunately, it is not known exactly what kind of forces act between vesicles. As a first approximation, we may use an exponential F(d) dependence for repulsive forces and attractive action of molecular forces. This results in the following equation for disjoining pressure that arises between two layers of particles in the lattice:
Π(d) ) F/a22 ) (πr/a22)[kλ exp(-d/λ) - (A/12πd2)] (20) Figure 15. Schematic representation of a dependence of the force F on the distance d between surfaces of interacting vesicles. The distances d1 and d2 correspond to positions of the first and the second potential well.
Figure 16. Cubic-lattice model of an ordered structure of particles inside a wetting film. The initial state of a bulk system at Π ) 0 (a) and the state of a compressed wetting film when Π > 0 (b) are shown.
schematically. The F(d) dependence reflects the joint action of attractive (F < 0) and repulsive (F > 0) forces. At the distance d ) d2 the particles are disposed in the far (second) potential well, where F ) 0. At this distance between particles, formation of an ordered colloidal structure becomes possible, even in a bulk system, when the depth of the second potential well is large enough.31 Let us consider, like Nikilov and Wasan,23 a primitive cubic-lattice model (Figure 16). Figure 16a shows the starting state of a thick wetting film when the distance between particle centers is equal to a2 ) d2 + 2r, where r is the radius of particles. In this case, the force between the particles F and the corresponding disjoining pressure Π are equal to 0, and the particles are disposed in a potential well. Figure 16b shows the state of a compressed layer, when d < d2 and repulsive forces arise. Deformation of the film is assumed to proceed only in the normal direction. The resulting disjoining pressure of the film, Π ) F/a22, may be determined in this case as the ratio of the force F to the cell area a22. Further thinning of the interlayer between particles results at d ) d1 (when particles get into close contact and in a much deeper potential well, when they may coagulate and form aggregates). (50) Horn, R. G.; Israelachvili, J. N. Chem. Phys. Lett. 1980, 71, 192194. (51) Christenson, H. K.; Horn, R. G.; Israelachvili, J. N. J. Colloid Interface Sci. 1982, 88, 79-88. (52) Christenson, H. K.; Gruen, D. W. R.; Horn, R. G.; Israelachvili, J. N. J. Chem. Phys. 1987, 87, 1834-1841. (53) Klein, D. L.; McEuen, P. L. Appl. Phys. Lett. 1995, 66, 24782480. (54) Gao, J.; Luedtke, W. D.; Landman, U. J. Chem. Phys. 1997, 106, 4309-4318.
where k and λ are parameters of the repulsive forces and A is the Hamaker constant. The values of parameters k and λ were estimated assuming the initial state of the wetting film at Π ) 0 corresponds to d2 ) 1.5 × 10-5 cm. This value is close to the mean interparticle distance in a sonicated D-8 solution at concentration C ) 0.16 wt %. Using the condition Π ) 0 in eq 20, such a combination of the parameters was selected: k ) 105 dyn/cm2 and λ ) 10 nm. Because vesicles contain a water core, the value of the Hamaker constant was chosen to be A ) 10-14 erg, smaller than for polymeric particles in water. More detailed consideration requires inclusion into eq 20 of supplementary terms taking into account undulation forces, osmotic pressure, and finite thickness of the vesicles wall.49 However, such refinement cannot influence markedly our purely qualitative consideration of the forces between vesicles. Equation 20 gives the value of disjoining pressure calculated from the force F of interaction between two layers of particles, the centers of which are disposed at the distance a ) d + 2r. In a multilayered structure having thickness h ) aN, where N is the number of layers, the force F and the disjoining pressure preserve the same values. However, they are related now to the thickness h ) (d + 2r)N. Therefore, plugging into eq 20 the values of d ) (h/N) - 2r, it becomes possible to calculate the isotherm Π(h,N) for wetting films containing N layers of particles. Curve 1 in Figure 17 shows, as an example, the Π(h) isotherm calculated for N ) 7 layers. The isotherm intersects the h-axis at a film thickness h of about 1 µm, which corresponds to the d2 distance (Figure 16). Curve 2 of Figure 17 shows the isotherm of a precursor, or R-film, calculated using eq 14. A possible β f R transition is indicated here by the arrow. The contribution of dispersion forces to the isotherm of R-films is much lower. For instance, at Π ) 800 dyn/cm2 the action of dispersion forces only (at A ) 3 × 10-14 erg for aqueous films on quartz surface) results in a film thickness of about 10 nm. Naturally, curve 1 of Figure 17 is qualitative in character and reflects only a possibility of the existence of metastable states of thick wetting films with different numbers N of ordered layers. Depending on the condition of wetting film formation, a different number N of layers may be formed. This explains the observed scattering of film thickness that may depend, for instance, on the rate of film thinning, vapor pressure, and film curvature. Finally, it needs to be noted that a well-defined layering of particles is not necessary to influence the disjoining pressure. It is more probable that disjoining pressure arises due to formation of an anisotropic structure in a vesicular system under conditions of restricted geometry. In this case, the disjoining pressure equals the difference between the normal and tangential pressure tensors in the film: Π ) PN - PT.56 For instance, a difference in the (55) Bailey, S. M.; Chiruvolu, S.; Israelachvili, J. N.; Zasadzinski, J. A. N. Langmuir 1990, 6, 1326-1329.
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by Wasan,58 must be continued and extended. Repulsive forces acting between vesicles in trisiloxane surfactant solutions should be investigated in more detail. Some contribution to the stability of thick D-8 films may be given by electrostatic forces. Therefore, the proposed explanations of the effect of the stability of thick wetting films of trisiloxane surfactant solutions require further verification. Conclusions
Figure 17. Qualitative form of disjoining pressure isotherms Π(h) for a thick wetting film that contains N ) 7 ordered layers of interacting particles (curve 1) and for a thin R-film stabilized by electrostatic forces (curve 2).
mean distances between particles in directions parallel and normal to the substrate (as shown in Figure 16b) may result in the appearance of a disjoining pressure. In this case, thinning of a wetting film may proceeds not stepwise, but more gradually. This is consistent with the absence of marked jumps or steps on the profiles of the film edges shown in Figures 6, 7, and 9. The proposed approach may explain, in particular, the experimentally observed influence of concentration of turbid trisiloxane surfactant solutions on the rate of spreading.7 At low concentration when interaction between the particles is weak, forming structures may be destroyed by Brownian motion. At higher concentration (when d is near d1, Figure 15), the vesicles approach closer to each other and may coagulate or form multilamellar structures. This hinders formation of a regular structure. Therefore, there exists some optimal concentration for formation of an organized state of the dispersion and consequent stabilization of thick spreading films. Unfortunately, at present it is not possible to calculate more accurately the forces acting in micron thick wetting films. Besides, the presence of an ordered or anisotropic structure of a system of vesicles in a film needs some direct confirmation by means of structure sensitive spectral methods. Bergeron’s recent publication of disjoining pressure measurements for bilayer forming surfactants shows that bilayer structures in thin films can strongly influence the disjoining pressure.57 Computer simulation of colloidal systems in thin interlayers and films, started (56) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Nauka: Moscow, USSR, 1987.
Fast rates of spreading of D-8 solutions over hydrophobic surfaces are explained by formation of micron thick wetting films. Droplet spreading, film flow, and climbing may be described using hydrodynamic equations that include as a driving force the gradient of disjoining pressure of wetting films. Thick films are metastable and rupture at some critical disjoining pressure Π/ of about 400 dyn/cm2. This value corresponds to a relative vapor pressure p/ps close to 1. This explains why superspreading was observed only at nearly saturated water vapor pressure. Thick metastable films arise in the course of thinning of a bulk solution. The metastable film may spontaneously transform into a much thinner equilibrium film with a thickness of about 100 nm, stabilized by electrostatic forces. The observed precursor films may be associated with such thin equilibrium R-films. The form and parameters of the disjoining pressure isotherm of wetting films were estimated from the stationary flow rates of thick films in hydrophobed glass capillaries between two menisci. The experimental isotherm is similar to the one adopted when calculating droplet spreading and film climbing. Stability of the thick films may be explained by the action of repulsive forces between vesicles forming an organized structure under the condition of restricted geometry. Similar effects are known for other systems containing colloidal particles, lamellar phases, or micelles. The proposed mechanism explains why superspreading was observed only for turbid trisiloxane surfactant solutions containing dispersed phase. It is not excluded that long-range electrostatic forces may also contribute to thick film stability. The phenomenon of flow and stability of micron thick wetting films or interlayers containing small repulsive particles is a subject of general interest. This problem demands further investigation, especially with application of structure-sensitive methods and computer simulations. Acknowledgment. This work was supported by Russian Foundation of Basic Investigations, Grant 9803-32770a. The authors thank Dow Corning Corporation for financial support of this work and kindly supplied samples of the trisiloxane surfactants. V. Sobolev and V. Starov acknowledge the support of EPSRC, Grant GR/RO7578/01. LA000789R (57) Bergeron, V. Langmuir 1996, 12, 5751-5755. (58) Chu, X. L.; Nikolov, A. D.; Wasan, D. T. Langmuir 1994, 10, 4403-4408.
