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Flow in Thin Liquid Films
by J. Lyklema, P. C. Scholten, and K. J. Mysels Chemistry Department, University of Southern California, Los Angeles, California 90007 (Received June 19, 1964)
An improved experiniental method for studying vertical soap films slowly pulled out of a solution is described, and the interpretation of the optical thickness measurements is discussed. The behavior of mobile films presents complications which can be explained but prevent a quantitative interpretation. Rigid films, on the other hand, give results for films thicker than 800-1100 A., which agree very closely with Frankel’s law relating film thickness to the velocity of pullout (eq. 1). Since Frankel’s law is derived on the assumption that the viscosity is constant up to the monolayer, this finding is not compatible with the existence of any thick, rigidified water layers (or of slip) in the neighborhood of the surface.
Whether the properties of liquids change in the neighborhood of a phase boundary and, if so, to what depth does any effect extend are old problems. In particular, there is considerable disagreement about the viscosity of water near a charged surface. The generally accepted present-day picture of dilute solutions of electrolytes and macromolecules is based on the assumption that the viscosity is constant to within a few molecular diameters from the particle considered. The possibility of an increased viscosity in a very thin layer of the order of 5 8. caused by a high, local, electric field intensity near a charged surface has been examined by Lyklema and Overbeek. I n contrast to the preceding approach which depends on underlying structural assumptions, the macroscopic approach based on direct observations of flow along surfaces provides much evidence for the existence of thick, rigid layers of water as shown, for example, in the review of Henniker. However, macroscopic approaches involving solid walls are always made uncertain by the disturbing effect of any surface roughness and of a contaminating dust or microbial growth. For example, the peculiar results obtained by one of us3 were certainly due mainly to microorganisms growing in the pores of sintered glass. Much more promising seems to be the study of water near a monolayer surface because of the relatively well-understood structure of the latter, along with its perfect smoothness, its easy renewal, and its ability to conform to any contaminating dust. For such systems the postulate of absence of slip and of norThe Journal o f Physical Chemietry
mal viscosity, i e . , the applicability of standard hydrodynamics up to the surface monolayer, was made explicitly by Harkins and Kirkwood4 in 1938. Crisp5 confirmed it experimentally, but the precision of his measurements could not preclude drastic changes of viscosity within the last few tenths of a millimeter next to the surface. More recently Goodrich6 found evidence of slip. In both cases trough techniques involving a single monolayer surface were used. A further step in this direction is the study of flow between two monolayers. This type of flow determines the thickness of a soap filni pulled out of bulk solution. Measurements of -1lysels and COX’confirmed the postulaote of constant viscosity down to within about 100 A. of the surface of both mobile and rigid films.* Soap films are of particular interest because they have also been usedg-” for direct measurements of
(1) J. Lyklema and J. Th. G. Overbeek, J . Colloid Sci., 16, 501 (1961). (2) J. C. Henniker, Rev. Mod. Phys., 21, 322 (1949). (3) K. J. Mysels and J. W. hlcBain, J . Colloid Sci., 3, 45 (1948). (4) W. D. Harkins and J. G. Kirkwood. J . Cfiem. Phys.. 6 , 53, 298 (1938). ( 5 ) D. J. Crisp, Trans. Faraday SOC.,42, 619 (1946). (6) F. C. Goodrich, J . Phys. Chem., 66, 1858 (1962). (7) K. J. Mysels and M.C. Cox, J . Colloid Sci., 17, 136 (1962). ( 8 ) . Rigid films have a high surface shear viscosity or a yield value which is absent in mobile films. (9) B. V. Deryagin and A. S. Titievskaya, Discussions Faraday Soc.. 18, 27 (1954).
FLOWIS THIKLIQCIDFILMS
double-layer repulsion and van der Waals attraction. The results of these measurements would be greatly affected if any rigidified layers extended over the whole thickness of the soap film and, thus, could affect (or even determine) its ultimate thickness. Present evidence on the existence of such layers is contradictory. Thus, Dasher and RIabisl* presented X-ray evidence for a hydrous gel structure in rigid films, and Deryagin and Titievskayag postulated a structure of the solvent which prevents complete drainage of thin films. Yet, Riysels, Shinoda, and Frankel13 accounted semiquantitatively for the drainage of rigid vertical films using a constant viscosity, and Sheludko and Exerova’O found experimental agreement with a hydrodynamic approach which applies Reynold’s formula for solid pistons to small horizontal films. There is, however, some question about the validity of the piston model for soap filnis.I4 The present paper reports a considerable experimental refinement of the technique of Mysels and COX,’ which was necessary in order to study the effect of double-layer and van der Waals forces, to be reported e1~ewhere.l~The results presented here deal with a verification of the applicability of the simple hydrodynamic approach for films as thin as 1000 A. or less with a precision which gives an upper limit of some 10 A. for the thickness of the water layer which may be effectively rigidified. This eliminates the possibility that such rigid layers can affoect equilibrium thicknesses exceeding some 50 or 100 A.
117
Figure 1. Film formed by a frame pulled out of a solution.
GLASS KNOBS
THIN W I R E GLASS FRAME
I. Frankel’s Law Let us consider an inverted U-shaped frame being withdrawn a t a constant rate v from a film-forming solution as shown in Figure 1. The experimentally measurable thickness 6 of the film thus formed is related to the velocity v a t which it is formed, the surface tension y of the solution, its viscosity q , its density p, and the acceleration of gravity g. Assuming that the viscosity remains constant up to the surface and that the latter is completely jnextensible, Franke116-1sderived the expression
by purely hydrodynamic considerations. The effect of introducing double-layer repulsion and van der Waals attraction into this simple picture has been discussed by Overbeek. l 9
11. Apparatus and Materials Frame. The construction of the frame used in our measurements is shown in Figure 2. The wire used had
Figure 2. Thin-wire frame. (10) A. Sheludko and D. Exerova, Kolloid-Z., 165, 148 (1959); 168, 24 (1960); A. Sheludko, Proc. Koninkl. Ned. Akad. Wetenschap., B65, 76 (1962). (11) E. M. Duyvis, Thesis, University of Utrecht, 1962. (12) G. J. Dasher and A. J. Mabis, J . Phys. Chem., 64, 7 7 (1960). (13) K. J. MYsels. K. Shinoda. and S. P. Frankel. “Soao FilmsStudies of their Thinning and a Bibliography,” Pergamon Press, New York, N. Y., 1959. (14) S. P. Frankel and K. J. Mysels, J . Phys. Chem., 66,190 (1962). (15) J. Lyklema and K. J. Mysels, to be published. (16) See ref. 13, p. 55. (17) It has recently come to our attention that Derjaguinls has derived a very similar relation for the thickness of a photographic emulsion layer deposited on a film base. In contrast to Frankel’s approach this derivation assumes complete extensibility of the free liquid surface. (18) B. V. Deryagin and S. M. Levi, “Fiaiko-khimiya Naneseniya Tonkikh Sloev na Dviahushchuyusya Podloahki,” USSR Acad. Sci., Moscow, 1959; B. V. Deryagin, Zh. E k s p . i Teor. Fiz.. 15, 9 (1945). (19) J. T h . G. Overbeek, J . Phys. Chem., 64, 1178 (1960).
Volume 69,Xumber 1 January 1965
J. LYKLEMA, P. C. SCHOLTEN, AND K. J. MYSELS
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WORM CONNECTED TO VARIABLE SPEED DRIVE
MICROMETER
Figure 3. Apparatus for measuring the thickness of slowly formed films.
a diameter of 12.5 p and was made of “Evenohm” chromium-nickel alloy. This elaborate construction of the frame was chosen to reduce the rate of thinning of the films. This in turn reduces the uncertainty about any difference hetween the rate a t which the frame rises and that a t which the film is being pulled out of the solution. The effectiveness of the very thin frames in this respect will be discussed elsewhere.20 The choice of the 12.5-p size was dictated by preliminary experiments on mobile films which showed that there was no observable difference between the thicknesses of films produced in frames made of 5-p and of 12.5-p wires, whereas the thicknesses were very slightly lower when the wire diameter was 50 p and about 200 A. lower for a wire diameter of 200 p. In the following analysis we therefore assume that the velocity of the frame and that of the film are equal. This is additionally justified by the fact that the thickness of rigid films just pulled out was independent of the total height of the film already present. Mechanical Part. (See Figure 3.) The films were made inside a brass box completely submerged in a water bath therniostated to better than 0.01 O and kept a t 25’. The hox had two large glass windows for observation and easy access. Close to the window used for The Journal o,f Physical Chemistry
measurements was a cuvette with surfactant solution. The cuvette could be filled (and the surface renewed by overflow) from outside through a narrow glass tube and a Teflon valve. Surface creep of contaminants (e.g., copper salts) into the cuvette was prevented by placing it in a Teflon cup. Behind the frame were two black glass plates (not shown in the figure) forming a light trap for the transmitted beam. The bottom of the chamber was always covered with a layer of the solution studied, and a slow stream of nitrogen saturated with water vapor maintained a slight excess pressure within the chamber in order to keep out dry air and, thus, prevent evaporation. Since the equilibrium films maintained their thickness for days, we are quite sure that evaporation did not occur. The rod supporting the frame passed through two Teflon collars in a chimney on top of the chamber and was connected to the body of a micrometer. A multiple-speed gear transmission connected to a variablespeed d.c. motor was used to drive the spindle of this micrometer via a fine worm gear. This enabled us to move the frame a t any speed between 40 8./sec. and
(20) P. C . Scholten, K. J. Mysels, and J. Hotchkiss, unpublished.
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FLOWIK THIKLIQUIDFILMS
500 ,/see. A constant voltage transformer assured constancy of speed a t any setting. Optical PaTt. (See Figure 3.) The light source was a fluorescent niercury lanip operated from a stabilized but variable voltage supply. This lamp was screened except for a circular area 3 cm. in diameter. A tenfold reduced image of this area was projected onto the film by means of small lens of 5-cm. focal length. An identical system projected a tenfold enlarged image of the illuminated area of the film onto a photomultiplier tube. A diaphragm in front of the photoinultipljer had an opening of 0.4 X 3 mm. ; thus, out of the 3-mm. illuminated circle on the film, a 0.04 X 0.3-mm. rectangle was selected. Most measurements were made with the lower edge of the observed rectangle 0.2 mm. above the meniscus, but in some experiments on mobile films this distance was reduced to 0.05 nini. I n this case the part of the lamp illuminating the meniscus was screened off in order to avoid stray light. The phototube was mounted in lieu of a camera on a Leitz “focaslide” so that it could easily be replaced by a ground glass for visual inspection, focusing, and adjustment. The intensity of the light source could be measured directly through a bypass in the optical system. I t appeared to be constant after a few hours of operation. Interference filters in front of the phototube allowed selection of the 546- or 436-nip wave length for measurement. Stray light from outside was kept out effectively by shields around the light paths. Inner reflections were reduced by suitably placed diaphragms. The photomultiplier was a lPX1, powered by a stack of dry cells. Its signal was amplified by a HewlettPackard Model 425 A mjcrovolt-ammeter and recorded by a 10-niv. Leeds and Northrup recorder. Materials. Pure dodecyl alcohol was obtained from Applied Science Laboratories, State College, Pa. The preparation of the sodium dodecyl sulfate used has been described. The sodium dodecylbenzene sulfonate was a coniniercial high purity sample, freed of inorganic salts and nonionic materials, obtained through the courtesy of the Colgate Palmolive Peet Co. Inorganic salts were of reagent purity. Distilled water was used for the preparation of the solutions. 111. Thickness Measurements At the beginning of most experiments the horizontal top wire of the franie was submerged. I t was then raised at the desired speed which was checked by timing the motion of the micrometer. For rigid films the record of the photomultiplier current showed a maximum or a shoulder corresponding to the passage of the wire frame and then a constant intensity giving the
thickness of the film. The same was true for very thin mobile films. For thicker mobile films the frame signal was always followed by a minimum, corresponding to a layer of film which had already thinned out greatly in the time required to rise from the meniscus to the field of view. This was followed by a region of small slope corresponding to a film whose thickness was changing slowly by the various processes involved in thinning. This thickness was extrapolated to the moment when the frame left the field of view to give as closely as possible the actual initial thickness of the film. (The first observed film thicknesses, which are often appreciably different, are reported by one of us elsewhere.21) This correction was generally minor and did not affect the over-all result for these mobile films. The determination of the thickness of a film involves two steps. One is the optical measurement which in our case was the determination of the ratio of intensity I of light reflected by the film to that, Io,obtained when the film thickness 6 corresponds to maximum reflection (6 2 X/4). The second step involves the computation of a material thickness from this measurement after taking into account the structure and optical properties of the film. Optical Measurement. I n order to obtain the reflected intensity, the measured light intensity must be corrected for stray light. As stated previously, outside stray light presented no problem. Stray light within the system in the absence of a film was not perceptible. However, once a film was formed, the presence of the meniscus could modify the background and introduce stray light. An upper limit for this factor is found by measuring the intensity of the reflected light at the first reflection minimum (6 = Xj2). This was done repeatedly and found to be of the same order as the noise, which was less than 2% of the reflection from the thinnest film measured, and, therefore, also negligible. Another source of stray light lies in the illuminated surface surrounding the measured area. This was estimated by focusing the slit on a dark portion just below the meniscus and pulling out a film close to the maximum reflectivity. The effect amounted to less than 1% of this maximum. As the thickness of most films measured is quite uniform near the measured area, this last kind of stray light introduces a correction which is not only small but also proportional to the intensity measured and can, therefore, be neglected in the I/Zo ratio. An advantage of the thin wire frames used is that their contribution to the stray light is negligible. For rigid films the maximum reflected intensity 10 (21) J. Lyklema, Ree. tral;. chim., 8 1 , 890 (1962).
Volume 69, Number 1 January 1966
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was generally measured by pulling out films a t a number of speeds around the expected maximum until the maximum value was well defined. For mobile films lo was usually found by producing a film of greater thickness, stopping the frame, and recording the maximum through which the intensity passed. Occasionally a further confirmation was obtained from the second maximum (6 sz 3 X/4), but, here, the more rapid drainage made the measurements less accurate. The measurements depended also on the linearity of the photomultiplier, amplifier, and recorder which were not further tested. The consistency of the data and especially the fact that measurements using two different wave lengths and, therefore, involving very different absolute intensities and different I l l o ratios, agreed to better than 6% indicates that the total accuracy was a t least of that order. It should be noted that because two such sets of measurements a t different wave lengths were always performed on different films and with independent standardization, they really test the over-all experimental reproducibility. All reported measurements were made in green light of 546 mH because of greater precision and convenience. Computation. The conversion of the optical quantity Ill0 into a material thickness 6 is still an incompletely solved problem because of the uncertainty about the optical properties of the film. However, a sufficiently good estimate can be obtained by examining two successive approximations. As a first, approximation, we assume that the film is a homogeneous aqueous structure. A convenient thickness unit for the system is the thickness, 1, of the film which gives maximum reflection, i e . , for which the optical thickness is a quarter wave length. For such a homogeneous film it is given by 1
=
X/4n cos 4
*
(2)
where n is the refractive index of the film, X the wave length under vacuum of the light used, and c$ the refracted angle. The desired relation is
I l l o = F sin2 (8l.l)
(3)
in which the angle is measured in quadrants (i.e., 90’ = 1). F is a correction factor taking into account secondary reflections. If these are neglected, F = 1. Secondary reflections depend on the reflectivity, r2, of the surface of the film where r is the Fresnel reflection coefficient. For unpolarized light going from medium 1 into medium 2 and for angles that are not too large (
