Flow in Thin Liquid Films


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 Fran...
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J. LYKLEMA, P. C. SCHOLTEN, ASD K. J. MYSELS

116

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

118

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.

119

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

J. LYKLEMA, P. C. SCHOLTEN, AND K. J. R ~ Y S E L S

120

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 (<30°) r is givenz2by (4)

The Journal of Physical Chemietry

For the air-water interface its value is 0.1427 so that r2 = 0.0204 and r4 = 4 X The factor F is given by

F =

+

+

1 2r2 4r4 1 - 2r2 cos 2(6/1) r4

+

(5)

Taking F into account changes the calculated value of 6 by -8.5Oj, if 6 = 0 and by 0% if 6 = 1. For green light the effect is, a t most, minus 11-15 8. for film thicknesses in the range of 300-700 A. (it is less in violet light). A soap film, however, is clearly not homogeneous but presumably formed by two surfaces with an intralamellar solution having essentially bulk concentration as already postulated by Gibbs and recently demonstrated by Corkill and c o - w o r k e r ~ . ~To ~ this layered structure must correspond a symmetrical but complicated variation of the refractive index. Hence, as a second approximation we can consider the film to be a sandwich of three homogeneous layers, the two outer ones having a small constant thickness and a higher refractive index than the central core of variable thickness (see Figure 4). Reflection from any three transparent layers has been calculated explicitly by CrookJZ4but the implicit expressions given by Va&iEekZ5 are more useful. Reduced to our symmetrical case, as outlined in the Appendix, they show that the outer layers of small and fixed thickness introduce a constant phase change which results in an apparent change of the thickness of the film by a constant amount, whereas the change in the reflectivity of each side of the film is quite small. In order to estimate the magnitude of this effect we consider two realistic models. One, based on fully extended Clz chains plus a sulfate group, assumes a thickness dl of 16 8. for each outer layer and is applicable to rigid films. The other, based on an area of 52 8./ molecule (as estimated by van Voorst VaderZ6)and a density of 1.0, assumes a corresponding thickness of 8.5 8. and is applicable to mobile films. In both models the refractive index nl is assumed to be 1.45. This is somewhat higher than that corresponding to normal hydrocarbons in order to take into account the effect of the sulfate group. In both cases the core of varying thickness dz is assumed to have the refractive index of water, 1.333. (22) C. J. Vas‘ic’ek, “Optics of Thin Films,” North-Holland Publishing Co., Amsterdam, 1960, p. 49. (23) J. W. Gibbs. “Collected Papers,” Longmans Green and Co., New York, N. Y., 1931, p. 300; J. M. Corkill, J. F. Goodman, D. R. Glaisman, and S. P. Harwood, Trans. Faraday Soe., 57, 821 (1961). (24) A. W. Crook, J . O p t . SOC.Am., 38, 954 (1948). (25) See ref. 22, p. 185. (26) F. van Voorst Vader, Trans. Faraday SOC.,56, 1067 (1960).

121

FLOWIN THISLIQUIDFILMS

no= I

AIR

MONOLAYER

WATER

n, = 1.45

d, (:I6 or 8.5A)

T

I

200c

n, = 1.45 AIR

a,i

no = I

Figure 4. Assumed sandwich structure of film.

/

IO M

Applications of eq. 12 (Appendix) show that the real thickness of the film is less than that compuoted, neglecting the effect of the surface layers, by 7.25 A. for mobile films and by 13.5 A. for rigid films. There is no significant difference between measurements in green and in violet light. These results are consistent with those obtained using different approaches by Duyvisl’ and by FrankeL2’ Thus, a realistic sandwich structure produces a correction which is significant, especially for thinner films. In view of the fact that the t y o extreme models give corrections differing by only 6 A., it is likely that each differs from reality by less than this amount. As our picture of the structure of soap films becomes more refined, it should be possible to narrow down this uncertainty further and to improve somewhat the interpretation of experimental data.

IV. The Behavior of Mobile Films Our first measurements were made on mobile films of a solution about M in alkylbenzene sulfonates and 9 X M in lithium chloride. A typical result is shown in Figure 5 by open circles, along with the straight line based on Frankel’s. theory. At high velocities the agreement is good, but below about 5 p/sec. the deviations become large. At very low velocities the thickness is essentially constant, which shows that it is not determined by hydrodynamic and rate-dependent factors but is an equilibrium value. This will be discussed in another article. l5 At intermediate velocities, films are much thicker than predicted by Frankel’s law. Visual observation of these films showed that the very thin equilibrium “black” film was always forming spontaneously just above the field of view of the photomultiplier. As has been described qualitatively in the past,l3
1

I

/

/

/

I

I

I

I

1

(VFP’3 ---*

Figure 5. Mobile films: variation of observed thickness with rate of pullout. Open circles, experiments beginning with submerged frame; filled circles, experiments beginning with a black film.

the influence of gravity this welt then flows down. Thus, in the deviation region it was the thickness of the welt that was being measured. At higher velocities the measurements were more representative of the real thickness of pulled out film because the black film formed later in the thicker films, and the rapidly rising film carried the welt out of the field of view. To verify this explanation another series of measurements was made. It is represented by filled circles in Figure 5 . In this series the frame was first raised about 1 mm. above the meniscus, and the film was allowed to thin in this position until all of it became black. The frame was then raised a t the desired uniform velocity. This permitted immediate formation of the welt, whereas in the original measurements, in which’ the frame was submerged at the beginning, this occurred only after 9 certain delay required by the thinning of the upper part of the film. As may be seen, the two series agree a t lower velocities where the delay is insignificant compared to the time needed to produce a film, but the large deviations from Frankel’s law extend (27) S. P. Frankel, unpublished.

Volume 69, Number 1

January 1966

J. LYKLEMA, P. C. SCHOLTEN, AND K. J. MYSELS

122

Table I : Frankel’s Law Results Obtained with Solutions Giving Rigid Films” NaLS. M x 103

1.59 2.83 3.7 4.0

3.58 0.87

Solution LiC1, M X 108

100 100 16 1.92 ...

...

-79

dynes om.-’

--

No. of observations

21.3j 21.3 24.6 27 26.8 32.1

Numerical constant

Least-squares Frankel’s law line--Intercept, Std. pev., .I. A.

20

1.78

18

28

9 17 7 7

1.83 1.86 1 93 1.75 1.83 f 0.055

50

31 18 29 19 24

Av.

46 - 35 83 32 =k 16

0 Many of these solutions were clearly supersaturated with respect to the alkyl sulfate-alcohol compound (adduct). When this precipitated after some time, the films became mobile and the solutions had to be replaced. In all cases the dodecyl alcohol content was 2.5y0 by weight of the dry sodium dodecyl sulfate (NaLS).

to much higher velocities in the absence of the delay. At the highest velocities, when most of the welt is carried upward by the rising film and does not affect the measurement, there is again agreement between the two series. An attempt to obviate these difficulties by bringing the field of view still closer to the meniscus (0.05 to 0.15 mm.) did not change the results appreciably. Thus, the results a t higher frame velocities confirm that mobile films conform closely to Frankel’s law, but the complications due to formation and flow of the welt distort the results at lower velocities and prevent the use of these mobile films in a more refined test. All the results reported below were therefore obtained with rigid films.

V. Results with Rigid Films Figure 6 shows the results obtained for rigid films made from a series of solutions of varying ionic strength. Table I gives the exact composition of each solution. The solid lines of the figure are fitted to the points through which they pass by the method of least squares. The dotted parts are extrapolations. As may be seen from Table I, the slopes of these lines are very close to the value 1.88 predicted by Frankel’s law, whereas the intercepts are close to zero. When the thickness of the film is below 500-1100 8., definite deviations from Frankel’s law appear in qualitative agreement with Overbeek’s prediction. l9 At low ionic strengths double-layer repulsion between the ionized surfaces predominates and causes deviations toward larger thicknesses. A t higher ionic strengths the double layers are thinner so that van der Waals forces can become important and cause gradual or abrupt deviations toward thinner films. In all cases an equilibrium thickness is reached a t sufficiently low speeds. The Journal of Physical Chemietry

These deviations will be considered in detail elsewhere, l5 but their conspicuous presence justifies confining the quantitative application of our purely hydrodynamic considerations to thicker films in which doublelayer and van der Waals effects are negligible. For the most dilute solution, in which the double layer “thickness,”, 1 / ~is, 111 8.,the limit seems to be about 1100 A.; for the others, about 800 Above this range and up to the limit of our measurements a t about 2200 8. the experimental points lie on straight lines with a standard deviation varying from 18 to 42 8. for the different solutions. From the slopes of the lines fitted to the experimental points by the method of least squares the numerical coefficients of Frankel’s equation (eq. 1) were calculated using the proper surface tension value for each solution. These values range from 1.73 to 1.93 with a weighted average of 1.83 0.055 as compared with the theoretical value of 1.88. This provides strong evidence that the hydrodynamic model used is correct and confirms the older results’ which covered a much wider range but were less accurate. The vertical intercepts of the lin2s vary between the extreme valueos of -33 and +83 A., with a weighted average of 32 A. and a standard deviation of 16 8. This indicates that a layer of only about 1.5 A. on each side of the film is rigid and does not participate in the viscous flow determining the thickness of the film. This corresponds very well to the thickness of the surfactant monolayer in these rigid films as estimated in section 3, i.e., 16 8. It is between these monolayers that flow occurs during pullout. Thus, our results provide an independent confirmation of the generally accepted sandwich structure of these films and of the physical reality of monolayers. One may also conclude that, within the accuracy of our measurements, Le., some 10 A. for each surface,

*

123

FLOWIN THINLIQUIDFILMS

to the importance of the optical corrections. We are also grateful to Dr. Duyvis and Dr. Frankel for making available to us their unpublished optical calculations.

Appendix For our symmetrical model, the general results given by Vai3iEek25reduce to = 2d1/11

51

+ +

r12 r22- 2rlr2 cos x1 rl2rZ2- 2rlr2 cos 21 1

r2 =

tan tan

tl =

{2

=

68

tan q

- 6 - d1[2 -

(6)

=

(12/11)3

(7)

r2 sin x1 rl - r2 cos x1

(8)

r1 sin x1

- r2

+

r1

(9)

cos X I

rlr2 sin XI 1 - r1r2 cos X I

(10)

+ + + {2)/21 12[7

((1

12

i

800 600 400

Figure 6. Rigid films: variation of thickness with rate of pullout for solutions of ionic strengths indicated. For clarity, scales corresponding to different solutions are displaced vertically in steps of 500 b.

there are no rigidified layers of water near the surfaces of our systems. The absence of such thick, rigidified water layers on these highly charged and strongly hydrophilic surfaces suggests strongly that such layers are not generally present in soap films and play no direct role in determining their equilibrium thickness.

Acknowledgment. This work was supported, in part, by the Air Force Office of Scientific Research. We are indebted to Professor Overbeek for stimulating discussions leading to this work and for calling our attention

(11)

where r1 = lrol/ and r2 = lr12!and a replaces 6 / 1 in formula 3. The terms beyond 6 are constant for a given dl and give the correction to be applied (with opposite sign) to the thickness computed by formula 3, neglecting any surface layers. For our model in which z1is small and r1 is moderate, an error of 0.1 8. is made by neglecting the differences between an angle and its tangent or sine, and between its cosine and unity, which gives

The correction term beyond 6 involves only the ratio of the I t values. It depends on the refractive indices but not otherwise on the wave length. Neglecting higher terms, which involves an error of less than 0.5 8.,one obtains simply, in terms of refractive indices a =

6 - dl(nl - n2)(l- n1 - 2n2)/n2(n2- 1 )

(13)

12

The correction term is, therefore, sensitive to the difference of refractive indices between the surface and the bulk, as would be expected, but, otherwise, is not significantly affected by usual variations of refractive index with wave length in the visible. Volume 69. Number 1

Janztarxi 1965