Langmuir 1992,8,3083-3092
3083
Dimple Formation and Behavior during Axisymmetrical Foam Film Drainage Jean-Luc Joye and Clarence A. Miller' Department of Chemical Engineering, Rice University, P.0. Box 1892, Houston, Texas 77251
George J. Hirasaki Shell Development Company, P.O.Box 481, Houston, Texas 77001 Received May 7,1992.In Final Form: September 4, 1992
Draining foam or emulsion f h are generally of nonuniform thickness. A thick region or 'dimple" forme in the central part of a circular film. It is separated from the Plateau border by a thinner 'barrier ring". We have developed a new numerical model to simulate the entire drainage process, including the film formation. The model assumes that drainageis arieymmetricand that the fluid interfacesare immobile. The initial conditions are a pair of static hemispherical menisci. Fluid is withdrawn at a constant rate for a specified time to form a film. The condition for the transition from a nearly "plane-parallel" f i to a dimpled film in the absence of disjoining pressure was determined. The ratio of the minimum to maximum thickness in the film and a dimensionless rate of drainage are correlated with the ratio of the maximum possible curvature in the dimple to the curvature in the meniscus. The rate of drainage is always less than that given by the Reynolds theory for drainage between a pair of disks that is pressed by a pressure equal to the capillary pressure. When the film is approximatelyplane-parallel, the pressure drop from the center of the film to the Plateau border is less than half of that predicted by the Reynolds theory and there is a significant pressure gradient beyond the nominal film radius. When a dimple forms, most of the resistance to flow is in the thinbarrier ring. The presence of disjoining pressure makes a qualitative difference in film drainage. Low electrolyte concentrations in a film containing ionic surfactant produce a repulsive disjoining pressure that inhibits formation of the thin barrier ring and thus of the dimple itaelf. The film drains rapidly to ita equilibrium thickness. For high electrolyte concentration, the disjoining pressure is dominated by van der Waals attraction. As a result a thin annular fiIm forms that forces the dimple into a lens with a finite contact angle. These types of behavior are observed experimentally.
Introduction Thin liquid films were first observed in the form of soap bubbles. The qualitative observations made by Newton' and Gibbs2 revealed that the walls of the soap bubbles grow thinner in time and pass through thicknesses of the order of visible light wavelength. Then different colors appear due to the interference of light reflected from the two interfaces of the film. When the thinning process is advanced,thin black spota are formed which are sometimes very unstable. The interest in thin liquid films has continued to grow due to their importance in the understanding of dispersed fluid systems, such as emulsions and foams. The coalescence of the dispersion is directly related to the drainage time and the stability of the film. A thin liquid film consists of two surface layers separated by a liquid, Surface-activematerial, when present in the liquid phase, is preferentially adsorbed and forms the surface layers. The main driving force for film drainage is the capillary pressure, the pressure difference between the center and the periphery of the film. Studies of film drainage indicatethat when a film reaches a thickness of about 100 nm, other forces besides gravity and capillary forces begin to influence drainage. Such forces have a molecular origin and operate within the film. Their influence is expressed as a pressure, II, called by Derjaguin3 the disjoining pressure. By definition, II is assumed positive when it resists film thinning. In an
* To whom correspondence should be addressed.
(1)Newton, I. Optics; London, 1704,Book I, Part 2,Experiment 4; Book 11, Part 1. (2)Gibbe, J.W.Trans. Conn.Acad. Arts Sci. 1878,3,108-343(collected works). ( 3 ) Deryagin, B. V. Colloid J. USSR (Engl. Trans[.)1955,17,191.
equilibrium film, the disjoining pressure II is equal to the capillary pressure. There are two major long-range components in the disjoining pressure. The first results from the Londonvan der Waals attraction, which acts to thin the film.&' In the case of an aqueous surfactant solution, the second stems from electrical double-layer repulsion. Derjaguin, Landau, Verwey, and Overbeek*pg elaborated a theory describing the electrical double-layer repulsion and the van der Waals attraction, often referred to as the DLVO theory. With the combined effect of the capillary pressure and disjoining pressure, one finds that a film can reach a thick equilibriumthickness if electricalforces are strong enough to balance capillary pressure and the van der Waals forces. Such films are called "common black films".l0J1 In contrast, if electrical forces are weak, the equilibrium thickness of very thin liquid films, 'Newton black films" or "Perrin films", is determined by the short-range repulsive disjoining pressure, which prevents further This short-range repulsive pressure is called the structural or hydration disjoining pressure. Recently, in the case of an aqueous surfactant solution above the critical micelle concentration, it has been found that (4)London, F.Trans. Faraday SOC.1937,33,8. (5)Boer, J. H. Trans. Faraday SOC.1936,32,10. (6)Scheludko, A. Adu. Colloid Interface Sci. 1967,1, 391. (7)Hamaker, H. C.Physica 1937,4,1058. (8)Deryagin, B. V.; Landau, L. Zh. Eksp. Teor. Fiz. 1941,11,802. (9)Verwey, E. J. W.; Overbeck, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (10)Scheludko, A. R o c . K. Ned. Akad. Wet. 1962,B65,97-108. (11)Jones, M. N.; Mysels, K. J.; Sholten, P. C. Trans. Faraday SOC. 1966,61, 583. (12)Perrin, J. Ann. Phys. 1918,10 (9),165.
0743-7463/92/240&3083$03.00/0 Q 1992 American Chemical Society
3084 Langmuir, Vol. 8,No.12,1992
stepwise drainage, which involves two or more black films of different thicknesses, occurs. The reason for this phenomenon is that the micelles of ionic or nonionic surfactant in the aqueous solution acquire an ordered structure, which provides an additional disjoiningpressure term.13 Numerous researchershave developed models describing the transport processes that take place in a thin film as the liquid drains from it. Reynolds,'* a century ago, developed a solution for a Newtonian fluid flowingbetween two parallel-plane disks forced toward one another. The Reynolds theory has been applied to describe the drainage of thin liquid films having immobile surfaces with varying degrees of success by several researchers.lsJ6 Wasan and Malhotra17extended the application of the Reynolds model by accounting for flow in the Plateau border as well as the effect of van der Waals forces. Other researchers, such as Barber and Hartland,18 I v a n ~ v , 'and ~ Tambe and Shamq20developedhydrodynamicmodels describingfilm drainage based on the Reynoldsmodel but accounting, in various degrees of detail, for the surfaceproperties. These include surface tension, surface tension gradient, surface composition, surface diffusion, and surface shear and dilational viscosities. Observations have shown that most foam or emulsion films are of nonuniform thickness. The central part of the film is thicker than its periphery. Thus, a "dimple" is formed, entrapped by a thinner "barrier ring". The first hydrodynamictheory for the profile and evolution of a dimple was developed by Frankel and Mysels,2l who presented expression for the thickness at the center of the f i i and at the barrier ring. The calculatedrate of thinning at the barrier ring was nearly equal to the one predicted by the Reynolds model, whereas the rate of thinning at the center of the film was much lower. The results were in reasonable agreement with subsequent experiments carried out by Platikanov.z2 Hartland= presented a solution for symmetrical film drainage. He assumed that the center of the film was a spherical cap and that the profile outside the film border was independent of time. Later, Hartland and Robinsonz4 developed an improved solution, assuming that the film interface was parabolic. Jain and 1van0v~~ introduced a very simplifiedmodel where the barrier ring was considered to be a layer of small but uniform thickness hb bounded by two thicker regions representing the dimple, which was assumed to be a spherical cap, and the outer meniscus. Numerical solutions for the drainage of a dimpled film have been carried out by Lin and Slattery.% The results are in good agreement with Platikanov's observationsz2 when the effects of disjoining pressure are negligible. Chen (13)Nikolov, A. D.; Waaan, D. T. J. Colloid Interface Sci. 1989,133, 1.
(14)Reynolh, 0.Philos. Trans. R. Soc. London 1886, 177, 157. (15)Allen, R. S.;Charles, G. E.; Maeon, S. G. J. Colloid Sei. 1961,16, 150.
(16)Mackay, G.D. M.; Maeon, S. G. Can. J. Chem. Eng. 1963,41,203. (17)Malhotra, A. K.;Wssan, D. T. MChEJ. 1987,3.9(91,1533-1641. (18)Barber, A. D.; Hartland, 5.Can. J. Chem. Eng. 1976,54,279. (19)Ivanov, I. B., Ed. Thin Liquid Films; Marcel Decker: New York. (20)Tambe, D. C.; Sharma, M. M. J. Colloid Interface Sci. 1991,147 (l),137. (21)Frankel, 5.P.; Myaels, K. J. J. Phys. Chem. 1962,66,190. (22)Platikanov, D. J. J. Phys. Chem. 1964,68,3619. (23)Hartland, S.Chem. Eng. Prog., Symp. Ser. 1969,65,82. (24)Hartland, S.;Robineon, J. D. J. Colloid Interface Sci. 1977,60, 72. (25)Jain, R. K.;Ivanov, I. B. J. Chem. SOC.,Faraday Trans. 2 1980. 76,260. (26)Lin, C.-Y.; Slattery, J. C. AIChE J. 1982,28,787. (27)Chen, J. D.; Slattery, J. C. AIChE J . 1982,28,955. (28)Chen, J. D. J. Colloid Interface Sei. 1984,98,329.
Joye et al.
, capillary ring
I
I
syringe Pump b
Experimental Section Esperimente on thin liquid film drainage are conducted using a video microecopy system. Film thicknesees are measured using the interferencemethod,proposed by Deryagina1and Sche1udko.B A specially deeigyd Blase cell used to form the films ie shown Figure 1. It was given to us by A. D. Nikolov and consieta of a horizontal g l w capillary ring, having a radius of 1.8 mm, fused to a capillary tube. The Capillary ring and part of the capillary tube, which is connected to a syringe pump though a rubber tube, are e n c l d in a glaw cell. In the case of liquid-gas system, evaporation can affect the rate of the film drainage and must be avoided. It can be suppressed by pouring a small quantity of water into the bottom of the glaes cell and coveringthe cell. Thin liquid f h are formed in the central portion of the capillary ring by withdrawing an aqueous surfactant solution, placed earlier on the ring (Figure 2). With this procedure f h of different sizes can be formed, depending on the amount of liquid withdrawn. The rate of withdrawal ie controlled by the syringe pump and can be set at any desired value. (29)Babak, V. G. Physicochembtry of Microscopic Liquid F i l m Stabilized by Polymers. Part 2. Modeling of Contact Interaction in Polymer Containing Disperse S y s t e m ; Urd University Press: Sverdlovek, 1988,p 168. (30)Burrill, K.A,; Woods,D. R. J. Colloid Interface Sci. 1978,42,151. (31)Deryagin, B. V.;Tilievskaya, A. S. Proc. Int. Congr. Surf. Act. Subst., 2nd 1917,1,211.
Dimple Formation and Behavior
Langmuir, Vol. 8, No. 12,1992 3085
disperse phase
2R
with 1.1 being the viscosity of the liquid, h the half-thickness of the film, P the pressure in the film, r the radial location, and t the time. The normal stress boundary condition at z = h reduces to i a Pg- P = y - -(r sin a) + II(2h) r ar
(2)
where ahi ar (1 (ah/ar)2)1/2 and y is the equilibrium interfacial tension between the liquid and the dispersed phase. II is the disjoining pressure and is the sum of three different terms:
sin a =
Figure 2. Modelof a film,of radius R, formed in a capillaryring of radius R,.
In order to collect information on f i i thicknessand profile, the cell is placed on a microscope stage (Nikon Optiphot-Pol). Monochromaticlight (wavelength646.1nm) isdirecteddownward toward the fii. Light waves reflected from the top and the bottom surfacesof the film interfere. The interferencepatterns resultingfrom the f i i drainageare recorded on videotape.The thicknessof the f i b at variouspositionscan be determinedusing the formula derived by Scheludko.lo Experiments were conductad on aqueous foam drainage with different surfactants. Deionized and distilled water was used to prepare the surfactant solutions. Several Surfactantsexhibited asymmetric drainage at all times. However, AOS 16-18, a-olefiiulfonate with a carbon chain length of 16-18, drained axisymmetrically for concentrations above 0.01 w t %. Some results of experimentswith this surfactant, which was obtained from Shell Development Co., will be compared with the predictions of the model developed below.
Model for Axisymmetrical Film Drainage Our objective is to simulate the film formation and drainage according to the experimental procedure. As discussed in the previous section, an aqueous solution is placed inside the ring. The thin liquid film is formed by withdrawing fluid at a constant rate Q, during a time t l . Then the liquid withdrawalis stopped, and the film drains due to the capillary suction. Statement of the Problem. To keep the problem as simple as possible, we make the following assumptions: ( 1 ) The drainage is axisymmetric. The thickness of the film is considered to be 2h. (2) The liquid is an incompressible Newtonian fluid with constant viscosity. (3)The pressure P g in the gas (or in the disperse phase) is independent of time and position. (4) The effect of mass transfer on the velocity distribution is neglected. ( 5 ) All inertial effects are neglected. (6) The effect of gravity is neglected. (7) There is enough surfactant present in the system so that the resulting interfacial tension gradients are sufficiently large to maintain the tangential components of the velocity at the surface equal to zero. (8) The flow obeys the lubrication approximation. (9) The governing equations derived from the lubrication approximation are still valid in the meniscus region because the flow in this region is negligible. (10)The interface forms a nonzero contact angle with the capillary wall. When the liquid wets the capillary wall, we assumethe contact angle to be small. The use of a nonzero contact angle avoids the numerical complicationsof an infinite slope at the capillary wall. (11) In the initial state the liquid is bounded by a pair of static hemispherical menisci of radius R,, separated by a distance 2hi (see Figure 5). Governing Equations. From assumptions 1, 2 , and 5-8,the NavierStokes and mass conservation equations can be simplified into the following equation:23
+
= %dW +
+
&dW is the van der Waals attraction term and can be expressed as4-' &dw
-(A/6d2h)3)
(3)
A is the Hamaker constant.' ne1 is the pressure due to electrostatic repulsion and can be expressed
ne,= 64nkT42eXp(-~(2h- 26)) qj = tanh
(4)
(ze$/4k7')
= (2e2z2n/tkT)-1/2
K - ~
where n is the number of counterions per cubic centimeter in the bulk solution, 6 the thickness of the adsorbed monolayer, z the valence, e the electronic charge, t the dielectric constant for the solvent, $ the electrical potential at the interface, and K - ~the Debye-Hiickel characteristic length. Her is the short-range repulsion term and is expressed as IIsr= C, exp(-C2(2h))
(5)
CIand CZ are constants and are chosen such that the disjoining pressure curve has almost an infinite slope at a thickness 2h of 30 A corresponding to the thickness of a Newton black film. The substitution of eqs 2-5 into eq 1 yields a partial differential equation in h. Due to the natural symmetry of the system, we have at r = 0
According to assumption 10, the interface forms with the capillarywall a nonzero contact angle, independentof time: ah/& = s = constant slope
at r = R,
(8)
A typical value for s is 5. During the time of withdrawal t l , the withdrawal rate Q is constant and is equal to the flux at the capillary wall. For 0 < t < tl we have
When withdrawal is stopped, a no flow boundary condition
Joye et al.
3086 Langmuir, Vol. 8, No. 12, 1992
_I 25 B e r m withdrawal
..............
0 ~., 0.00
1=0
+**.
0.03
0.02
0.05
0.04
radial position m m
Figure 3. Typical film profile as a function of time: R, = 1.8 mm, Q = 2.5 x lo-" m3/s, 3:= 30 dyn/cm,disjoining pressure not included. Points are predictions of the Frankel-Mysels theory.
I
I
R
Figure 4. Thin film features. at the capillary wall is imposed. For t
aPJar = 0
I > 11
Figure 5. Schemefor approximatesolution for the film radius.
. " ~ ~ " I " " , " I ' " " " ' ' , " ' ' l ' ~ ' ' ~ ' " ~ " " ' l .
0.01
After Withdmwd
> t l we have
at r = R,
(10)
In order to integrate eqs 1and 2, an initial condition is needed. According to assumption 11, the initial profile is given by r2 + (h- (R,+ hi))' = R: (11) hi is the initial half-thickness at the center. R,is the radius of the hemispherical menisci and is chosen so that the slope at the capillary wall corresponds to the slope in eq 8. It can be expressed as
7) +
R, = Rc( 1 s2
Numerical Method. The set of partial differential equations for thin liquid film drainage derived above can be solved numerically by employing a finite difference
method. We adopted an automatic time-step sue selection such that the maximum relative change in the thickness per time step did not exceed a given value (Ahlh), (typically 5%). A grid system in the r direction with uneven grid spacing was used. Half of the grid blocks were in the thin film region, the other half in the meniscus region. The total number of grid blocks was 200. The stability of the method depends on the time level at which the flux terms (right side of eq 1)are evaluated. We used the so-called 'semi-implicit" procedure, where the spatial differences for the flux terms were evaluated with a linear approximation to the new time level using unknown values of the dependent variable h. The governing equations (eqs 1-10) can be transformed into a system of linear equations, where the unknowns are the change in half-thickness at each grid block. The resulting equations are pentadiagonal. The system was solved using an LU (lower and upper) decomposition and then an upper and lower back-substitution. The convergence of the solution was checked by varying The flow in the meniscus space intervals and (Ah/h),. region was found to be negligible, so that assumption 9 was satisfied. Sufficientlysmall values of the contact angle at the capillarywall did not affect the solution significantly.
Results and Discussion The resulta are presented and discussed in two parts. The first part treats the hydrodynamics of thin film drainage and dimple formation in the absenceof disjoining pressure. The second part showsthe influence of dijoining pressure on drainage once the dimple has formed. Drainage without Disjoining Pressure. Figure 3 showsthe f i b half-thicknessas a function of radial position and time for one set of conditions representative of those employed experimentally. Although the entire drainage process was simulated, film profiles are shown only for times after fluid withdrawal has been stopped. During the early stages of drainage, no dimple is formed. Then the half-thickness at the center, denoted by ho, becomes larger than that at the barrier ring, denoted by h-, and a dimple forms. As the dimple drains, the barrier ring reaches a final radial location R,which we define as the film radius (see Figure 4). Estimation of Film Radius. The film radius, R, is the solution to the Young-Laplace equation subject to eq 8, a zero contact angle with zero thickness at R,and a known volume equal to the difference between the initial volume and the volume withdrawn. However, with these conditions the solution for the film radius can only be obtained numerically. Using the following scheme, we develop an approximate analyticalsolution that predicts the film radius for various conditions. Before withdrawal, the initial hemispherical menisci, of radius R,,are separated from the midplane (z = 0) by hi, as shown in Figure 5. The curvature of the
Dimple Formation and Behavior
Langmuir, Vol. 8, No. 12,1992 3087
5oorl
1
I
450
01
I
4
i............i.........1......4
:..; ....e-" "*L "C
*......-1
i.....I........;.,I
R/R, 0.01
0.001 0.01
0.1
1
R P S
Figure 6. Film radius prediction. Ranges of variables investigated 10 pm IR I300 wm, 1 mm IR, I10 mm, 30 dyn/cm Iy I72 dyn/cm. menisci is assumed to remain constant duringwithdrawal. If enough liquid is withdrawn, the hemispherical menisci intersect the midplane and determine a film radius (see Figure 5). This approximate scheme yields a film radius Ra given by R, = (R:
- (R, + hi - (Qtlr R,21) 21112
1I I I I I 1 I 1 I I I I 1 I I , I 1 I I I I I I I , I I I I I I I I 1 l 1 I 1 1 I I I I I l ' I ' I (
0 0.00
0.02
0.04
0.06
0.10
0.08
radial position m m
Figure 7. Film profile during withdrawal: R, = 1.8 mm, Q = 2 X 1O-l1 m3/s,y = 30 dyn/cm.
16o 5oo{
\\\
(13)
Qt represents the volume of liquid withdrawn. For a given rate of withdrawal Q the minimum time of withdrawal required to form a film is
R, gives only an approximate value of the film radius R. However, we have found that Ra can be correlated with the film radius R obtained from the numerical simulations
....
for a wide range of parameters. Figure 6 shows the plot of RIR, versus RJR,. A good representation of this correlation is given by RIR, = 0.72(Ra/R,)'.26
(14)
Equations 13 and 14 allow the film radius to be estimated for various conditions without detailed simulations and numerical calculations. Estimation of Film Thickness during Liquid Withdrawal. It is also useful to have an approximate value of film thickness immediately following liquid withdrawal. During withdrawal, one finds that the meniscus advances radially at a certain velocity U and leaves a film of liquid behind (see Figure 7). If we approximate the radius of curvature of the meniscus as a constant equal to R., the initial radius of curvature, and the velocity U as the derivative of R with respect to the time of withdrawal, the problem becomes equivalent to a long bubble moving at the same velocity U in a capillary tube of radius R,. Bretherto@ developed an approximate asymptotic solution for the thickness of the liquid film formed between the front and rear menisci of a long advancing bubble. The thickness given by the Bretherton theory corresponds to the half-thickness of the film in our case and can be expressed as hg, = 0 . 4 6 R , ( 3 ~ U / r ) ~ with / ~ U = dR/dt
I
I
I
I
I
I
I
I
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
I
radial position mm
Figure 8. Half-thickness prediction during withdrawal: R, = 1.8 mm, y = 30 dyn/cm. and its radial position as predicted by our detailed simulations are compared in Figure 8 with those given by eq 15. For the latter case the radial position R and velocity dRldt are obtained using eq 14. The agreement is reasonable. The faster the rate of the withdrawal, the thicker the film formed. Transition from a Nearly Uniform to a Dimpled Film. Once withdrawal of liquid is stopped,the film drains by capillary suction. In order to describe the transition from a nearly uniform or plane-parallel film to a dimpled film, a dimensionless parameter CR is introduced. CR represents the ratio of the maximum possible curvature in the dimple to the curvature in the meniscus. The former, which we designate R d , is the radius of a section of a sphere tangent to the film surface at the center and intersecting the midplane at the film radius R (see Figure 9). From geometry one finds that Rd can be expressed as
(15)
The minimum half-thickness of the film during withdrawal (32) Bretherton, F. P.J. Fluid Mech. 1961, 10, 166.
0.03
-
(16)
If the radius of curvature of the meniscus is approximately
3008 Langmuir, Vol. 8, No.12, 1992
Joye et al.
4 I I I I
P
I
i
8
P c
------
--
0.01
1
0.1
2
10
1 00
R, I R'
Figure 10. Transition from a plane-parallel f i i to a dimpled f i i . Values of variables are the same aa those of Figure 6.
Figure9. Radiusof curvature in the dimple and in the meniscus.
equal to R,, CR can be expressed as
Note that CR is proportional to the half-thickness at the center, ho, and therefore does not remain constant during drainage of the film. A physical interpretation can be made for the curvature ratio CR as follows. The maximum possible curvature in the dimple 1/Rd represents the driving force for the liquid to flow from the center to the periphery of the film. The curvature in the meniscus l/Rc represents the driving force for the liquid to flow from the periphery of the film into the meniscus region. If 1/Rd is less than l/Rc, not enough liquid can be "pushed" out of the film to satisfy the amount of liquid "pulled" out by capillary suction. Therefore, the ilmgets depleted and a dimple fluid at the periphery of the f forms. If I/& is larger than l/Rc, liquid in the film can be provided at a high enough rate to satisfy the capillary suction and no dimple forms. We found from our simulations that for values of CR larger than about 0.7 no dimple is formed. The film is slightly concave, and the transition from the film region to the Plateau border is gradual (see Figure 3). The departure from a plane-parallel film can be characterized by the ratio of the half-thickness at the film radius hR to the half-thickness at the center ho (see Figure 4). The ratio hdho is correlated to CRin Figure 10 (CR> 0.7). It is greater than unity but does not exceed 1.2. Under these conditions the film is approximately plane-parallel. When CR becomes less than 0.7, a dimple is formed. The halfthickness at the center, ho, becomes larger than at the barrier ring, hmh, and the dimple can be characterized by the ratio hmidho (see Figure 4). The latter is plotted as a function Of CR in Figure 10 (CR< 0.7). As one can observe, the dimple becomes more pronounced as CR decreases. The reason for the discontinuity between the two curves is that, at CR = 0.7, hmi, is equal to ho and is located at the center of the film. However, since the film is slightly concave, ha is larger than hmin and ho. Rate ofThinning. The rates of thinning at the center of the film and at the barrier ring as predicted by the simulations were correlated with the curvature ratio CR. Figure 11 shows the results for the rates of thinning
normalized by the Reynolds velocity V&l4 for drainage between a pair of disks of radius R separated by a liquid layer of thickness 2 h
V , = 8h3Pc/3pR2 (18) Figure 12 shows the pressure profile during the film drainage as obtained by the numerical simulations for various CR and that given by the Reynolds theory %l4 PRe - Pg= Pc(l- 2(r/R)2)
(19)
Figure 11 shows that the rate of drainage is always less than the Reynolds velocity of thinning. The reasons for this are (1) The pressure drop from the center of the film to the Plateau border, as obtained from the simulations, is less than half of that predicted by the Reynolds theory. (2) There is a significant pressure gradient beyond the nominal film radius for large CR. Therefore, the rate of thinning is significantly reduced due to the resistance to flow that extends into the Plateau border region. Wasan and Malhotra" recognized the influence of the flow in the Plateau border on the rate of film thinning and developed
Langmuir, Vol. 8, NO. 12, 1992 3089
Dimple Formation and Behavior
7 1
'"1 350
300
i r
I-C,.0€4 --Cc.=04
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
4 s time of withdrawal
-I
0.00
0.1 5
radial position m m
Figure 12. Pressure profile.
Figure 13. Film profile: R, = 1.8 mm, Q = 10-l0 ma/s, y = 30 dyn/cm, disjoining pressure not included.
an approximate theory. (3) When the dimple is formed, most of the resistance to flow is in the thin barrier ring. Frankel and Mysels2' developed quasi-steady-state solutions for the thicknesses at the center and at the barrier ring, neglecting accumulation at the barrier ring. They found that thinning at the barrier ring can be described by the Reynolds formula. However, thinning a t the center is much slower than predicted by the Reynolds theory, the half-thickness ho at the center being given by the following equation:
ho = (0.0096R6p/rR,t)"4
0.10
0.05
(rlR)*
l'OI
0.5
I
7 low elecvolytc concentration
lShon range repulsion]
(20)
Equation 20 was obtained assuming that the initial thickness at t = 0 is infinite. Differentiating eq 20, one obtains the rate of thinning at the center VF-M.When VF-Mis normalized by the Reynolds velocity VR,and then expressed in terms of CR,one finds
jl /\
"5i
-1.0
I" 0
I 20
I
(equilibnum thickness at 90 nml
I
I
I
40
60
80
i 100
thickness nm
The rate of thinning given by eq 21 is plotted in Figure 11. Figure 3 compares our calculated film profiles with the thicknesses at the center and at the barrier ring predicted by the Frankel-Mysele theory. For CR larger than 0.45, the Frankel-Mysels theory predicts higher raks of thinning and smaller thicknesses at the center of the f i i . For CR less than 0.45, the rate of thinning at the center from our simulations is in agreement with VF.M (see Figure 11). However, the thickness at the center line, from our calculated film profiles, reaches that predicted by the Frankel-Mysels theory only for CR less than 0.3 (see Figure 3). A similar observation can be made for the barrier ring, except that the rate of thinning and the thickness at the ring, from our simulations, reach those predicted by the Reynolds theory only if CR is less than or equal to about 0.1. The reason for this is shown in Figure 12. One sees that for small CR(50.1) the pressure gradient at the location of the barrier ring is the same as that given by the Reynolds theory. Platikanov22measured experimentally the thicknesses at the center and at the barrier ring during the drainage of a dimpled film and found reasonable agreement with the Frankel-Mpels theory and therefore with our simulations as well. However, Platikanov did not report experimentaldata before and during the dimple formation.
Figure 14. Disjoining pressure isotherms. Low electrolyte concentration: Ce1= 6 X lo4 moUL, Hamaker constant 7.5 X J, electrical potential 94 mV. High electrolyte concentration: C d = 2 X mol/L, Hamaker constant 7.6 X 10-" J, electrical potential 26 mV.
Influenceof Disjoining Pressure on Film Drainage. The presence of disjoining pressure makes a qualitative difference in f i i drainage. Figure 13 shows the halfthickness as a function of radial position and time in the absence of disjoining pressure for a liquid withdrawal rate 4 times that of Figure 3. The radius of the film formed in 140Mm, almost 5 times that of Figure 3. As one can see, the drainage of the dimplein Figure 13 ultimatelybecomee very slow due to the high resistance to flow in the thin barrier ring. At low electrolybe concentration the repulsive electrostatic component of the disjoining pressure is dominant and inhibits the formation of the barrier ring. A disjoining pressure isotherm for one such case is shown Figure 14. Figure 15shows film profiles obtained from the simulations for this disjoining pressure isotherm and the same conditions of initial film formation as those of Figure 13. The drainage is faster than in the absence of disjoining pressure. During the later stages of the drainage, the repulsive disjoining pressure stops the thinning and the film rapidly
Joye et al.
3090 Langmuir, Vol. 8, No.12, 1992 700
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Figure 15. Film profile: R, = 1.8 mm, Q = 10-10 m3/s, y = 30 dyn/cm, disjoining pressure included. Low electrolyte concentration: Ce1 = 6 X 10-1 mol/L.
Figure 16. Film profile: R, = 1.8 mm, Q = lO-"J m3/s, y = 30 dyn/cm, disjoining pressure included. High electrolyte concentration: Ce1= 2 x mol/L.
reaches its equilibrium thickness. Similar behavior was simulated by Chen and Slattery27128 and is in qualitative agreement with experimentalobservations of Platikanov.22 We also observed similar behavior for a film containing 0.02 wt % of AOS 16-18. At high electrolyteconcentration the disjoiningpressure is dominated by van der Waals attraction forces, as shown in Figure 14. The resulting film profiles as a function of time are shown in Figure 16. Early in the drainageprocess, the barrier ring thins very quickly due to the van der Waals attraction forces. When the short-range repulsion forces become dominant, a very thin annular film forms that forces the dimple into a lens with a finite contact angle. The drainage of the lens is then very slow because of the high resistance to flow in the thin annular film. The contact angle can be calculated from the disjoining pressure isotherm using the Frumkin-Derjaguin theory:33
(Figure 17f)is smaller and thicker than that of Figure 17c with a larger contact angle, measured to be 2.5O. A more complex disjoining pressure isotherm accounting for micelle ordering in the film would be required to extend our model to describe the observed stepwise thinning beha~i0r.l~~~~
The disjoining pressure isotherm at high electrolyte concentration, shown in Figure 14, gives a contact angle of 3.1°, accordingto formula22. We measuredthe contact angle of the lens shown in Figure 16 and found a value of 2.5O, which is acceptable agreement considering the limitations on numerical resolution at the contact line. Figure 17 shows the interference patterns at different times during drainage of a film formed for the same conditions as those of Figure 13. A 0.5 wt % solution of AOS 16-18 was used, which corresponds to an electrolyte concentration of 2 X mol/L, including the sodium sulfate impurity. Note that the concentration of surfactant is above the critical micelle concentration. A lens with a small finite contact angle, separated from the meniscus by a uniform annular film, does form (see Figure 17c). The measured contact angle is 0.3O. However,subsequent drainage is complicatedby stepwisethinning of the annular film, a phenomenon not included in the present model. Nucleation of a thinner spot of the annular film is shown in Figure 17d and ita growth in Figure 17e. The final lens (33) Morrow,N. R. Interfacial Phenomena in Petroleum Recouery; Hirasaki, G. J., Ed.;Surface Science Series;Dekker: New York, 1991; V 01. 36, p 23-76.t
Conclusions On the basis of our simulations, we draw the following conclusion for axisymmetrical foam film drainage. Drainage without Disjoining Pressure. (1)If the ratio of maximum possible curvature in the dimple to the curvature in the meniscus CRis lessthan about0.7,a dimple is formed. If CRis larger than 0.7, the film is nearly planeparallel. (2) The rate of drainage is always less than that predicted by the Reynolds theory because (a) the pressure drop across the film is less than half of that predicted by the Reynolds theory, (b) there is a significant pressure gradient beyond the nominal film radius for large CR,and (c) when the dimple is formed, most of the resistance to flow is in the thin barrier ring. (3) If CRis less than about 0.3, the thickness at the center is in good agreement with that predicted by the approximate theory of FrankelMysels. When CRis larger than 0.3, the Frankel-Mysels theory predicts smaller values. A similar statement can be made for the thickness at the barrier ring, except that agreement is only achieved when CRis less than or equal to 0.1. Influence of Disjoining Pressure. (I) At low electrolyte concentration, the repulsive disjoining pressure inhibits the formation of the thin barrier ring. Thus, the film drains relatively fast to a thick uniform equilibrium film. (2) At high electrolyte concentration, the disjoining pressure is dominated by the van der Waals attraction. The result is formation of a thin annular film that forces the dimple into a lens with a finite contact angle. The drainage of the lens is very slow because of the high resistance to flow in the thin annular film. (34)Bergeron,V.; Radke,C. J. Submittedfor publicationtoLanamu.ir.
Langmuir, Vot. 8, No. 12, 1992 3091
Dimple Formation and Behavior
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Figure 17. Drainage of AOS 16-18 (0.5 w t %): R = 140 pm, R, = 1.8 mm at t = 10 a, 30 a, 1 min 37 a, 2 min 27 a, 2 min 50 8,3 min 50 s, respectively. The film drainage started at time code 000:3400.
Acknowledgment. The present work was supported by the Texas Energy Research in Applications Program. The authors gratefully acknowledge A. D. Nikolov who provided us with one of his glass cells to perform the experiments.
List of Symbols A = Hamaker constant, J Ce1= electrolyte concentration,mol/L CR= ratio of maximum possible curvature in the dimple to the curvature in the meniscus
3092 Langmuir, Vol. 8, No. 12, 1992 e=
electron charge, C
h = half-thickness of the f i i , m her = thickness given by the Bretherton theory, m hi = half-thickness separating the two initial menisci, m hmin = half-thickness at the ring, m ho = half-thickness at the f i i center, m ha = half-thickness at the film radius, m k = BoltPnan constant, J/K n = number of counterions per cubic meter P = bulk pressure, Pa P, = capillary pressure, Pa P, = gas pressure, Pa
PR,= pressure given by the Reynolds theory, Pa Q = withdrawal rate, m3/s
r = radial position, m R = f i i radius, m R, = approximate fiim radius, m R, = radius of the capillary wall, m Rd = radius of curvature in the dimple, m R, = radius of the initial hemispherical menisci, m s = slope at the capillary wall t = time, s
Joye et al.
tl = time of withdrawal, s t- = minimum time of withdrawal for film formation, s T = temperature, K U = meniscus velocity during withdrawal, m/s V = rate of thinning, m/s VF-M= rate of thinning give by the Frankel-Myeele theory, 4 s
VR,= rate of thinning given by the Reynolds theory, m/s z = valence Greek Letters a = angle, deg y = interfacial tension, N/m 6 = thickness of the adsorbed monolayer, m e = dielectric constant of the solution, C2J-1 m-1 0 = contact angle, deg K = inverse Debey length, m-1 fi = bulk viscosity, kg m-1 8-1 n = disjoining pressure, Pa 4 = dimensionless group J, = electrical potential at the surface, V
