Investigation of the Force−Distance Relationship for a Small Liquid

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Investigation of the Force-Distance Relationship for a Small Liquid Drop Approaching a Liquid-Liquid Interface R. Aveyard, B. P. Binks, W.-G. Cho, L. R. Fisher,† P. D. I. Fletcher,* and F. Klinkhammer Surfactant Science Group, School of Chemistry, University of Hull, Hull HU6 7RX, U.K. Received August 8, 1996. In Final Form: October 17, 1996X We describe a novel apparatus for the investigation of the forces between two liquid surfaces. In the configuration described here, an oil drop is formed at the tip of a thin flexible micropipet. The force exerted on the oil drop as it is pressed up to an oil-water interface is determined from the deflection of the pipet shaft. The disjoining pressure in the thin, oil-water-oil emulsion film formed by contact of the drop with the oil-water interface is determined by the hydrostatic pressure applied to the oil contained in the micropipet. The radius and the thickness of the film is derived from the optical interference pattern observed using a microscope. A simple theory is presented for the variation in force and film radius as the drop is moved up into the interface. Experimental results are given for dodecane-water-dodecane emulsion films stabilized by the anionic surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT). Data for the force and film radius as a function of the pipet position relative to the oil-water monolayer show good agreement with theoretical predictions. The variation of disjoining pressure with film thickness for the emulsion films is in accord with electrostatic theory.

Introduction The measurement of forces between surfaces at small separations is of central importance in attempting to gain a fundamental understanding of many aspects of the complex behavior of colloidal systems. In recent decades, important advances in understanding the interactions of solid surfaces across a fluid medium (“solid-fluid-solid” systems) have been made through the development and use of the surface forces apparatus developed by, inter alia, Tabor and Winterton1 and Israelachvili2 and techniques based on atomic force microscopy.3 The interactions between liquid surfaces (either gas-liquid-gas or liquid-liquid-liquid systems) are generally discussed in terms of the normal repulsive force per unit area of surface (denoted the disjoining pressure) which can be measured as a function of separation. This topic has been extensively reviewed.4-9 The vast majority of (these latter) studies have been for air-liquid-air (foam) films although there have been some reports of work on so-called “pseudoemulsion” (i.e., either water-oil-air or oil-water-air) films.10-12 Experimental investigations of various aspects of emulsion films are described in refs 13-28. * To whom correspondence should be addressed. E-mail: [email protected]. † Current address: Department of Interdisciplinary Science, University of West of England, Coldharbour Lane, Bristol BS16 1QY, U.K. X Abstract published in Advance ACS Abstracts, December 15, 1996. (1) Tabor, D.; Winterton, R. H. S. Proc. R. Soc. London, Ser. A 1969, 312, 435. (2) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992. (3) See, for example, Butt, H.-J. J. Colloid Interface Sci. 1994, 166, 109. (4) Scheludko, A. Adv. Colloid Interface Sci. 1967, 1, 391. (5) Clunie, J. S.; Goodman, J. F.; Ingham, B. T. In Surface & Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1971; Vol. 3. (6) Sonntag, H.; Strenge, K. Coagulation and Stability of Disperse Systems; translated by R. Kondor; Halsted Press: New York, 1972. (7) Aveyard, R.; Vincent, B. Prog. Surf. Sci. 1977, 8, 59. (8) Thin Liquid Films. Fundamentals and Applications; Ivanov, I. B., Ed.; Surfactant Science Series 29; Marcel Dekker: New York, 1988. (9) Exerowa, D.; Kashchiev, D.; Platikanov, D. Adv. Colloid Interface Sci. 1992, 40, 201. (10) Sonntag, H.; Buske, N.; Fruhner, H. Kolloid-Z. Z. Polym. 1972, 250, 330. (11) Lobo, L.; Wasan, D. T. Langmuir 1993, 9, 1668. (12) Bergeron, V.; Radke, C. J. Colloid Polym. Sci. 1995, 273, 165.

S0743-7463(96)00786-X CCC: $12.00

Most of the thin film studies to date have been made using capillary cells in which the thin film is formed in the contact region between two (nearly) hemispherical liquid surfaces. In such an apparatus, the thin film radius is typically of the order of 100 µm, far larger than the contact film radius likely to be formed when two micrometer-sized emulsion drops approach. We describe a novel apparatus in which a small oil drop supported in an aqueous phase on a thin capillary can be moved in a controlled manner toward an oil-water interface. The force exerted on the oil drop and the radius and thickness of the thin emulsion film formed at “contact” can be determined directly. The instrument, which contains a number of features in common with an apparatus developed to examine the adhesion forces between biological cells and solid surfaces,29 enables the variation of disjoining pressure with film thickness to be measured for emulsion (13) Van den Tempel, M. J. Colloid Sci. 1958, 13, 125. (14) Sonntag, H.; Netzel, J.; Klare, H. Kolloid Ze. Ze. Polym. 1966, 211, 121. (15) Netzel, J.; Sonntag, H. Abh. Dtsch. Akad. Wiss. Kl. Chem., Geol. Biol. 1966, 6A, 589. (16) Platikanov, D.; Manev, E. Proceedings of the IVth International Congress on Surface Active Substances, Volume II, Physics and Physical Chemistry of Surface Active Substances; Overbeek, J. Th. G., Ed.; Gordon and Breach: New York, 1967. (17) Sonntag, H.; Netzel, J.; Unterberger, B. Spec. Discuss. Faraday Soc. 1970, 1, 57. (18) Manev, E. D.; Sazdanova, S. V.; Wasan, D. T. J. Dispersion Sci. Technol. 1984, 5, 111. (19) Fisher, L. R.; Parker, N. S. Faraday Discuss. Chem. Soc. 1986, 81, 249. (20) Ivanov, I. B.; Chakarova, SV. K.; Dimitrova, B. I. Colloids Surf. 1987, 22, 311. (21) Muller, H. J.; Balinov, B. B.; Exerowa, D. R. Colloid Polym. Sci. 1988, 266, 921. (22) Fisher, L. R.; Mitchell, E. E.; Parker, N. S. J. Colloid Interface Sci. 1989, 128, 35. (23) Velev, O. D.; Gurkov, T. D.; Borwankar, R. P. J. Colloid Interface Sci. 1993, 159, 497. (24) Velev, O. D.; Nikolov, A. D.; Denkov, N. D.; Doxastakis, G.; Kiosseoglu, V.; Stalidis, G. Food Hydrocolloids 1993, 7, 55. (25) Velev, O. D.; Gurkov, T. D.; Chakarova, Sv. K.; Dimitrova, B. I.; Ivanov, I. B.; Borwankar, R. P. Colloids Surf. A: Physicochem. Eng. Aspects 1994, 83, 43. (26) Velev, O. D.; Gurkov, T. D.; Ivanov, I. B.; Borwankar, R. P. Phys. Rev. Letts. 1995, 75, 264. (27) Velev, O. D.; Constantinides, G. N.; Avraam, D. G.; Payatakes, A. C.; Borwankar, R. P. J. Colloid Interface Sci. 1995, 175, 68. (28) Koczo, K.; Nikolov, A. D.; Wasan, D. T.; Borwankar, R. P.; Gonsalves, A. J. Colloid Interface Sci. 1996, 178, 694.

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focus of a laser reflecto-optical device. The laser and detector of the reflected intensity are mounted on a second piezo (piezo 2). The laser is maintained at a vertical height such that the laser beam impinges on the bottom edge of the mirror attached to the pipet. In this position, the reflected laser intensity is approximately half the maximum reflected intensity obtained when the laser spot is positioned centrally on the mirror. This measured intensity is kept constant by continual electronic adjustment of the vertical height of the laser using a feedback circuit. By these means, the laser height accurately (within approximately 200 nm) follows the vertical movement of the mirror arm of the micropipet. When the drop is far from the interface and experiences no force, the movement of the pipet and drop (controlled by piezo 1) is matched by the movement of the laser (piezo 2). When the drop approaches the oil-water interface, the force experienced causes the horizontal arm of the micropipet to deflect vertically. The difference in movement (measured using position transducers attached to both the micropipet and laser) between piezo 1 and piezo 2 yields the deflection and thus the force according to eq 1

force ) deflection × force constant of micropipet (FC) (1)

Figure 1. Schematic diagram of the liquid surface forces apparatus. The lower diagrams show the isolated micropipet with mirror extension arm and mirror attached and a close up view of the tip of the pipet supporting an oil drop close to the oil-water interface.

films of radius of only a few micrometers, close in size to films formed in actual emulsion systems. The novel design of the instrument allows independent control of the movement of the oil drop relative to the oil-water interface and thus may be used to investigate dynamic effects associated with drop movement. Furthermore, as will be discussed later, a solid particle can be mounted on the pipet tip in place of the oil drop enabling the interactions in solid-liquid-liquid systems to be studied. Substitution of the oil-water interface by a transparent solid surface will enable additional system types (solid-liquid-solid and liquid-liquid-solid) to be investigated. In this paper we describe the use of the instrument to determine the equilibrium forces between an oil drop approaching an oil-water interface. Description of the Apparatus and Principles of Operation Description of the Apparatus. The apparatus is shown schematically in Figure 1. A thin glass micropipet, shaped as shown in Figure 1, is attached to a piezo translator (piezo 1) capable of vertical movement of 40 µm. Coarse X, Y, Z micrometer translators allow positioning of the oil drop in the center of the field of view of the microscope objective lens at an initial vertical position a few micrometers below the oil-water interface. A mirror mounted on a horizontal extension attached to the horizontal portion of the micropipet is positioned at the (29) Francis, G. W.; Fisher, L. R.; Gamble, R. A.; Gingell, D. J. Cell Sci. 1987, 87, 519.

The accuracy of the force measurement is mainly determined by stray vibrations and is estimated to be approximately 5 nN. The oil drop was held on the end of the micropipet as follows. The thick end of the micropipet was attached to a flexible tube connected to a syringe barrel as shown in Figure 1. Oil was allowed to drain through the syringe barrel and tube into the micropipet. Using this arrangement, an equilibrium is established between the hydrostatic pressure P (determined by the height h between the oil level and the drop position) and the Laplace pressure of the curved drop surface. For a drop far from the oilwater interface (i.e., when the drop surface experiences no external force), the radius of curvature of the drop ro is controlled by the height h according to

hFg ) 2γ/ro

(2)

where F is the oil density, g is acceleration due to gravity, and γ is the oil-water tension. It was checked by independent microscopic observations that eq 2 is obeyed and that the oil drop pins at the inner diameter of the micropipet as shown in Figure 1. The maximum hydrostatic pressure (and hence h) that can be maintained at equilibrium arises when the drop radius is equal to the inner radius of the micropipet, rc. Thus, the maximum value of h (hmax) that can be reached before oil flows from the micropipet is given by

hmax ) 2γ/Fgrc

(3)

It should be noted that hmax can be exceeded without loss of the oil drop from the micropipet if there is an additional downward force on the drop resulting from interaction with the oil-water interface. For a drop far from the oil-water interface, the drop height b (the distance from the end of the pipet to the drop apex) depends on P and is given by

2ro - x4ro2 - 4rc2 b) 2

(4)

where ro is the drop radius of curvature and rc is the inner radius of the pipet. Independent microscopic measure-

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distance from the drop position) is calculated as follows. The meniscus shape around a vertical cylinder has been considered by Carroll.30 Under conditions when gravity can be neglected, i.e., when the cylinder radius is small relative to the capillary length of the system, the magnitudes of the two principal radii of curvature of the meniscus are both equal to the cylinder radius divided by cos θ. For the systems considered here the capillary length is typically of the order of millimeters whereas the cylinder radius (equal to the film radius r in this context) is of the order of micrometers, and so the approximation is justified. Using this result, the distance z is given by Figure 2. Geometry of the drop in contact with the oil-water interface. The circular film region forms a spherical cap of radius r and radius of curvature rf.

ments of the variation of b with P have confirmed the validity of eq 4. Drop Shape, Film Formation, and Force as the Drop Approaches the Oil-Water Interface. When the liquid drop supported by the pipet approaches the oil-water interface, a thin emulsion film is formed at the apex of the drop as shown in Figure 2. The shape of the oil drop is, in general, determined by a combination of gravity and capillary forces. However, for the systems considered here, the capillary length ()(γ/∆Fg)1/2, where ∆F is the density difference of the two liquid phases) associated with the oil drop is of the order of millimeters whereas the drop radius is of the order of micrometers. Under these conditions gravity forces can be neglected and the radius of curvature of the drop surface not in the film region formed by contact with the oil-water monolayer remains constant at ro. For the film region at the apex of the drop, the total tension of the film is simply 2γ since the film contains two oil-water interfaces. Thus, since the hydrostatic pressure P must be balanced by the Laplace pressure, the radius of curvature of the film rf is twice ro. Therefore the disjoining pressure in the film Π is given by Π ) P/2. In principle, there is a small excess tension present in the emulsion film due to the disjoining pressure. However, this can be estimated to be of the order of 0.01 mN m-1 and is neglected here. The shape of the drop in contact with the oil-water monolayer can be calculated using simple geometry. The radius r of the circular emulsion film is

r ) x(-b2 - x2 + 2rob - 2rox + 2bx)

(5)

The angle θ made by the oil-water monolayer to the vertical at the contact line with the emulsion film is

θ ) arccos(r/ro)

(6)

In principle, for films in which there is a net attractive free energy of interaction between the two oil-water interfaces, a nonzero contact angle will be present between the film and adjoining meniscus. For such films, the value of this contact angle must be added to θ. For the films investigated in this study, the interactions across the films are all repulsive, and hence the film contact angles are all zero. In this case, θ is given by eq 6. The downward force F exerted on the pipet by the interfacial tension is given by the product of the emulsion film perimeter and the vertical component of the tension.

F ) 2πrγ cos θ

(7)

The vertical distance z between the film perimeter and the flat level of the oil-water interface (i.e., at a long

z)

2r/cos θ - x(2r/cos θ)2 - 4r2 2

(8)

Experimentally we measure the force F and the film radius r as a function of the vertical pipet displacement relative to a zero position taken to be when the apex of the drop first contacts the oil-water interface, i.e., when the first force is detected. The overall movement of the pipet (denoted as pipx) takes account of the displacement of oil-water interface (z), the deflection of the pipet due to the force ()F/FC where FC is the pipet force constant), and the change in length x.

pipx ) (b - x) + z + F/FC

(9)

The series of eqs 5-9 together with knowledge of P, γ, and rc enable us to calculate the complete profile of the drop in contact with the oil-water interface together with the variation of F and r with pipx. As will be shown later, the calculated forces and film radii can be directly compared with measured values of F and r using no adjustable parameters. Additionally, as mentioned, the disjoining pressure in the emulsion film is simply obtained as half the applied hydrostatic pressure. Emulsion Film Thickness Determined by Optical Interferometry. The thickness of the water film as a function of radial position was determined using reflectance interference microscopy. In the configuration used in this work, the 546 nm line isolated using an interference filter from the output of a mercury arc lamp was directed downward through the microscope objective to the oil drop. The interferogram resulting from the interference between the reflections from the upper oil-water interface and the drop surface was collected by the objective lens and captured using a video camera coupled to a frame grabber and stored as a digital image record. For each interferogram, four separate frames were measured under identical illumination and amplification conditions in order to allow normalization of the interference intensity and correction for the background intensities. In the context of this discussion, the term “intensity” is used to denote the optical power impinging on a unit area of the detector. The four frames were (i) the “raw” interference intensity II, (ii) the intensity resulting from reflection from the upper oilwater interface only (i.e., with the drop and pipet removed from the field of view) IR, (iii) the background intensity recorded either by placing a matt black surface under the objective or by closing the shutter IB1, and (iv) the background intensity recorded with the objective immersed in oil with the pipet (containing no drop) positioned approximately 10 µm below the focal plane IB2. The intensity IB1 was found to arise mainly from electronic noise in the camera system whereas IB2 contains a small additional intensity arising from light reflected or scattered from the end of the pipet. The normalized, back(30) Carroll, B. J. J. Adhes. Sci. Technol. 1992, 6, 983.

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ground corrected interference intensity R was calculated as

R ) (II - IB2)/(IR - IB1)

(10)

The typical values of the different intensities were as follows. The illumination and gain of the optical system were adjusted to obtain the interference maxima and minima II over the full digital intensity scale of 0-255. Under the same gain conditions, typical values of IR, IB1, and IB2 were 65, 13, and 23, respectively. For two weakly reflecting, identical surfaces, interference theory predicts that R should range from 0 to 4. The uncertainty in R was found to be approximately (0.05. We now discuss the calculation of the water film thickness from the measured, normalized interference intensity R. The light path geometry through the thin film consisting of liquids of refractive index values n0, n1, and n2 is shown in Figure 3. Light emergent from the objective lens immersed in the oil phase strikes the upper interface (interface 1) at angle θ0 where it is partially reflected and the transmitted beam is refracted. The refracted ray is reflected at the lower (drop) interface (interface 2), and this reflected ray combines with the ray reflected from interface 1 to produce the interference pattern from which the thickness of the water film (d1) is deduced. Since the reflectivities of the interfaces are small, only the first reflections at each interface are considered and the interference is thus double beam. As discussed in ref 31, the Fresnel reflectance coefficients for interfaces 1 and 2 and for the resolved light beams either plane or perpendicular to the plane of incidence (s and p components) are given by

n1 cos θ0 - n0 cos θ1 r1p ) n1 cos θ0 + n0 cos θ1 r1s )

n1 cos θ1 - n0 cos θ0 n1 cos θ1 + n0 cos θ0

r2p )

n2 cos θ1 - n1 cos θ2 n2 cos θ1 + n1 cos θ2

r2s )

n2 cos θ2 - n1 cos θ1 n2 cos θ2 + n1 cos θ1

(11)

where the angles θ1 and θ2 are obtained from θ0, n0, n1, and n2 using Snell’s law. The phase difference δ1 between the rays reflected from the upper and lower interfaces is

δ1 )

2πn1d cos θ1 λ

(12)

where λ is the wavelength of light. The reflectance from the two-interface system (containing the interference pattern) is

R ˜I )

r12 + 2r1r2 cos 2δ1 + r22 1 + 2r1r2 cos 2δ1 + r12r22

(13)

where the Fresnel coefficients for either the s or p components are inserted as appropriate. The reflectance of the single, upper interface (which determines the reference frame intensity) is (31) Heavens, O. S. Thin Film Physics; Methuen: London, 1970.

R ˜ R ) r12

(14)

For the unpolarized light used in this study, the overall reflectance values are given by the average of the values for the s and p components. This is shown below for the two-interface (interference) reflectance and is defined similarly for the single interface (reference) reflectance.

R ˜ I ) (R ˜ Ip + R ˜ Is)/2

(15)

As discussed in refs 32-34, the reflectance microscopic visualization of the interference pattern involves incident light rays with a range of incidence angles θ0 varying from 0 to θmax as shown schematically in Figure 3. The value of θmax is determined by the numerical aperture of the microscope optical system and was determined by projection of the light rays through the objective lens to a plane several centimeters below the focal plane of the objective lens followed by measurement of the diameter of the projected circular light spot. The overall interference intensity is the intensity-weighted sum over all incidence angles from 0 to θmax. As shown in ref 32, the intensity weighting factor W is

W ) sin θ0 cos2 θ0

(16)

Thus, the final expression for the reflected intensity normalized with respect to the reflectance intensity from the single upper interface is θmax

R)

∑0 WR˜ I

θmax

(17)

∑0 WR˜ R

where R has the same meaning as the experimentally determined quantity defined in eq 10. Using known values of θmax and the refractive index values for the oil and water phases, a computer program was used to calculate R as a function of the film thickness thereby allowing the measured R values deduced from the interference images to be converted to film thicknesses. The calculation procedure outlined above yields the equivalent film thickness of a pure water film. The film is, of course, coated with a surfactant monolayer on each side. The measured equivalent water film thickness can be corrected for the optical effects of the monolayer to yield the true thickness of the water film using the procedure described by Frankel and Mysels.35 Calculations for the films investigated here showed that this correction was small (of the order of 0.5 nm) because the refractive index of the tail region of the surfactant monolayer is close to that of the bulk oil phases. Since the correction was less than the estimated uncertainty in the film thickness, it was neglected here. Additionally, it should be noted that the film thickness calculation assumes that the two interfaces are parallel and horizontal. Thus the estimation of the film thickness between the oil drop and the oil-water interface is restricted to positions close to the drop central apex. Experimental Section We first list the main components of the apparatus together with the suppliers. Piezo translators (P830.30) and associated control electronics were supplied by Physik Instrumente as were (32) Gingell, D.; Todd, I. Biophys. J. 1979, 26, 507. (33) Heavens, O. S.; Yuan, Y. F. Phys. Med. Biol. 1979, 24, 810. (34) Gingell, D.; Todd, I.; Heavens, O. S. Opt. Acta 1982, 29, 901. (35) Frankel, S. P.; Mysels, K. J. J. Appl. Phys. 1966, 37, 3725.

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Figure 4. Optical micrograph of an angled view of the tip of a micropipet held in air. The lengths of the vertical and horizontal bars correspond to 20 µm.

Figure 3. (upper diagram) Optical light path of the incident light emerging from the microscope objective. The illumination intensity forms an inverted circular cone of angle θmax. (lower diagram) The geometry of the optical light path through the thin film. the linear variable differential transformer (LVDT) position transducers (E115-21). The laser reflecto-optical device (as used in commercial bar code readers, HEDS-1000) was supplied by Hewlett-Packard. The microscope (Zeiss Universal R) was equipped with epi-illumination from a 50 W mercury lamp. The 546 nm line from the lamp was isolated using a Ealing 35-3664 8 nm bandpass interference filter. The objective lens was a Zeiss Epiplan 100 magnification oil immersion lens. For all the measurements described here the numerical aperture was controlled by variable apertures in the epi-illumination light path and was set to be 1.1 (less than the maximum possible allowed by the objective lens), corresponding to θmax ) 33°. The video camera (Hamamatsu C3077) was controlled using a Hamamatsu C2400 camera controller, the output of which was fed to a Macintosh IIci computer via a frame grabber. Frames were stored digitally as 512 × 512 pixels with 0-255 gray levels. The computer provided full software control of the piezo translators and recorded the digital signals from the position transducers. Data analysis of the digitized images was performed using the image software package NIH Image Version 1.59.64. The liquid system under test was contained in a square cuvette of dimensions 50 × 50 × 50 mm supplied by Chandos. In order to prevent stray vibrations, the whole apparatus was mounted on a JRS active vibration isolation table. The micropipets were made by drawing glass capillaries (101 mm long, 0.76 mm i.d., 1.1 mm o.d., supplied by Plowden & Thompson) vertically through the centers of two heated wire loops. By careful adjustment of the drawing rate and the heater applied voltages, it was found to be possible to produce micropipets with a clean break perpendicular to the capillary long axis. Figure 4 shows a microscope image of the end of a typical micropipet held at an off vertical angle in air. Micropipett with force constants varying from 0.005 to 0.1 N m-1 and with internal diameters varying from 5 to 50 µm can be produced reliably by suitable adjustment of the drawing conditions. The desired bends in the micropipet were produced by carefully moving the micropipet close to a hot wire using a micromanipulator. The

heat of the wire causes the thin pipet to bend away from the heat. With careful control of the movement, the micropipet can be successfully bent to the desired shape. The mirror extension arm was prepared by gluing (using epoxy glue) a small square of aluminum foil (approximately 2 mm × 2 mm) to a thin glass capillary. The mirror extension arm was then glued to the horizontal section of the micropipet. The force constant of the micropipet in its final assembled state was measured by determining the deflection (typically of a few millimeters) using a traveling microscope following the placing of calibration weights on the micropipet. It was checked that the deflection was linearly proportional to the applied weight. Finally, the internal diameter of the micropipet was determined microscopically with the use of a calibrated reference graticule supplied by the National Physical Laboratory. n-Dodecane (>99%, Aldrich) was passed twice over an alumina column prior to use to remove trace, polar impurities. The anionic surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT) was supplied by Sigma and used without further purification. Water was purified by reverse osmosis and passed through a Milli-Q reagent water system. NaCl (AR grade) was from Prolabo. Interfacial tension measurements were made using either a Kruss K10 du Nouy ring tensiometer or a Kruss Site 04 spinning drop tensiometer. All experiments were carried out at room temperature, 20 ( 1 °C. A typical experiment was performed as follows. The internal radius and force constant of the pipet were first determined. The pipet was then mounted in position and the sample cell filled with the aqueous phase to the desired level above the pipet end, using degassed solution (necessary to avoid bubble formation on the pipet). The pipet was filled with the oil phase by addition through the syringe mounted on the end of a flexible tube attached to the thick end of the pipet and allowed to drain into the pipet and the required pressure P set by adjustment of the height of the end of the flexible tube. A layer of oil a few millimeter thick was then added to the aqueous surface of the sample. The vertical position of the pipet was adjusted under computer control until the position at which the apex of the drop first contacted the oil-water interface was found and noted. The height of the pipet was then increased in increments of a few micrometers. At each position eight frames of the interference pattern were captured (for subsequent averaging) and the signals from the position transducers were noted. It was found that steady position transducer and interference intensity signals were obtained at each position rapidly (less than a few seconds) after the motion was stopped. Thus it was concluded that the measurements refer to equilibrium states of the emulsion film (i.e., free from effects due to hydrodynamic forces associated with film drainage) and that film drainage is complete within a few seconds. For each sample, measurements were made at various hydrostatic

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Figure 5. A typical video image of the interference pattern formed as the oil drop is pressed to the oil-water interface. The lengths of the vertical and horizontal bars correspond to 4 µm. pressures in order to obtain the variation of disjoining pressure with film thickness.

Results and Discussion Measurements were made for emulsion films consisting of dodecane-water-dodecane stabilized by the anionic surfactant AOT at concentrations above and below the critical micelle concentration (cmc) which is 2.3 mM.36 The films were formed by contacting aqueous solutions of AOT with pure dodecane. Under the conditions used in these experiments, it is known that the AOT resides virtually exclusively in the aqueous phase at equilibrium,36 and thus the results are not perturbed by any transfer of material across the interfaces resulting from nonequilibrium distribution. In this first study aimed primarily at testing the performance of the apparatus, three AOT concentrations were investigated: 0.3, 1.0, and 3 mM for which the interfacial tensions were 18.0, 8.1, and 2.5 mN m-1, respectively. The densities of water and dodecane were taken to be 0.9983 and 0.74875 g cm-3 and the refractive indices of water and dodecane at a wavelength of 546 nm were taken to be 1.334 and 1.423, respectively.37 Figure 5 shows a typical video image (average of eight frames) of the interference pattern produced as an oil drop is pressed against the oil-water interface. The central gray region corresponding to the emulsion film is seen at the center of concentric rings of interference maxima and minima corresponding to the curved outer regions of the drop. Figure 6 shows the radial intensity profiles for two interferograms, one corresponding to a pipet position close to initial drop contact and the second corresponding to an increased upward movement of the pipet. It can be seen that upward movement of the drop causes the radius of the film region r to increase. The intensity of the central film region remains constant indicating that the film thickness remains constant as the pipet moves upward. The decrease in intensity amplitude of the interference fringes with increasing distance from the drop center is a consequence of both the range of incidence angles caused (36) Aveyard, R.; Binks, B. P.; Mead, J. J. Chem. Soc., Faraday Trans. 1 1986, 82, 1755. (37) Selected Values of Properties of Hydrocarbons and Related Compounds; Thermodynamics Research Center, Texas Engineering Experiment Station, Texas A&M University: College Station, TX, 1978.

Figure 6. Normalized radial interference intensity profiles for the 3 mM AOT sample with P ) 617 Pa using a pipet of inner radius 12.8 µm. The upper diagram shows the pattern for pipx ) 0.5 µm (close to first contact with the interface) and the lower diagram shows the profile for pipx ) 4.2 µm.

by the finite aperture of the optical system32-34 and the increasing curvature of the drop surface at the edges of the drop. For each system, the interferograms were analyzed to obtain the film radius as a function of pipet position (pipx). Figure 7 shows typical data for the sample containing 0.3 mM AOT at a hydrostatic pressure of 2696 Pa. The inner radius of the pipet was 12.8 µm. As seen from the radial intensity profiles, the film radius increases as the pipet is pushed upward, i.e., with increasing pipx. Using the value measured for the force constant of the pipet (0.031 N m-1 in this case), eqs 5-9 allow the calculation of the variation of r with pipx with no adjustable parameters. The calculated curve shown as the solid line in Figure 7 is in excellent agreement with experiment. Figure 8 shows the measured force versus pipet position for the same conditions, and the calculated curve is again in good agreement with experiment. Similar quality of fits were obtained for different samples, hydrostatic pressures, and pipets. These results provide convincing evidence that the analysis of the droplet and film geometry expressed in eqs 5-9 is correct. In Figure 9 we show the calculated profile of the drop in contact with the monolayer for one particular value of pipx for the experimental conditions pertaining to Figures 7 and 8. It can be seen that the film profile is relatively flat and horizontal in accord with the assumptions made in the optical interference analysis. Figure 10 shows a plot of the calculated normalized interference intensity R versus film thickness which was used to derive d from the measured values of R. The uncertainty in the measurement of R (typically (0.05)

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Langmuir, Vol. 12, No. 26, 1996 6567

Figure 9. Calculated profile of the drop in contact with the oil-water monolayer. The calculation is made using eq 5-9 and refers to a value of pipx of 3.22 µm with all other conditions being the same as in Figures 7 and 8. The vertical and horizontal scales indicate the dimensions in micrometers.

Figure 7. Plot of film radius r versus pipet movement pipx for a sample containing [AOT] ) 0.3 mM, γ ) 18 mN m-1, P ) 2696 Pa, and rc ) 12.8 µm. The solid line is calculated using eq 5.

Figure 10. Calculated variation of normalized interference intensity with water film thickness d for dodecane-water dodecane films. The calculation, made using eqs 11-17, corresponds to a maximum incidence angle of 33°.

Figure 8. Plot of force F versus pipx for the same conditions as Figure 7. The solid line is calculated using eqs 5-9.

translates into an uncertainty in d of approximately (1 nm for d ) 50 nm or so and (5 nm at d ) 10 nm. As discussed earlier, the film thickness is constant with pipet vertical position but varies with hydrostatic pressure since the film disjoining pressure is equal to P/2. Figure 11 shows this variation in the form of plots of disjoining pressure versus film thickness for the three AOT concentrations. Each curve includes data from two separate runs indicating a reasonable degree of reproducibility. Most of the data was obtained using a pipet with inner radius 12.8 µm. As discussed previously (eq 3), the maximum pressure that can be realized experimentally is determined by the value of the interfacial tension and the pipet radius. For the 3 mM AOT sample for which the tension was 2.5 mN m-1, the maximum accessible pressure using the 12.8 µm radius pipet was approximately 400 Pa. The data points at higher pressures were obtained by using pipets of smaller radii (8.3 and 4.8 µm).

Figure 11. Variation of disjoining pressure Π with film thickness d for (from right to left) AOT concentrations of 0.3, 1.0, and 3.0 mM. The solid curves are calculated using eq 19 with the parameters listed in Table 1.

The disjoining pressure Π is the sum of contributions arising from electrostatic forces (Πel), van der Waals forces (Πvdw), and short range forces arising from hydration, monolayer undulations, and other sources (Πsr). Πel and Πsr are repulsive forces corresponding to positive contributions to Π whereas Πvdw is attractive. At small

6568 Langmuir, Vol. 12, No. 26, 1996

Aveyard et al.

Table 1. Surface Potentials and Charge Densities for AOT Monolayers at the Dodecane-Water Interface at 20 °C [AOT]/mM

κ-1/nm

|ψ0|/mV

σo/electronic charges nm-2

0.3 1.0 3.0

17.6 9.6 5.6

90 85 90

0.036 0.059 0.114

separations (