3161
Langmuir 1992,8, 3161-3167
Formation of Soap Films from Polymer Solutions Robijn Bruinsma Department of Physics, University of California, Los Angeles, California 90024
Jean-Marc di Meglio,’ David QuM, and Sylvie Cohen-Addad Laboratoire de Physique de la Mati2re Condens6e,+Coll2ge de France, 75231 Paris Cedex 05, France Received May 7, 1992. In Final Form: September 28,1992
The classical Frankel law describing the formation of soap films was recently found to break down for polymer solutions. We discuss in this paper how this could be due to the effects of normal stresses and slip of the polymer along the surfactant-coveredsurfaces. We also study the consequences of a detailed description of the surface layer including the Marangoni effect, in the case of soluble surfactants (using the classical Gibbs theory) as well as in the case of insolubleSurfactants. These predictions are compared to experimental results.
I. Introduction Pulling liquid films out of a reservoir (and the reverse process of film drainage) has been a problem of interest for many years. In a celebrated paper, Landau and Levichl theoretically derived the law relating the thickness d of a film entrained on a flat plate to the pulling velocity. Franke12calculated a similar law for a soap film entrained on a frame (see section 11): d/Kil = 1~39(Ca)~/~ Here Ca is the capillary number:
Ca = qV/yo
(1.1)
(1.2)
q is the viscosity of the solution, V the upward velocity of
the frame, yo the liquid surface tension, and capillary length
~0-l the
(1.3) p is the density of the liquid and g the gravitational acceleration. This law is valid when van der Waals interactions can be neglected (Le., when film thicknesses are larger than 0.1 pm) and when the gravitationalpressure can be neglected with respect to the capillary pressure (Le., for Ca > ~ 0 - 9 , h(z) approaches a constant value, d / 2 , so
-
(11.1)
lim h(z) = d / 2 z--
Since for z 0, h(z)must describe the reservoir surface, we also demand that lim h(z) = m 2
(11.2)
4
The film is pulled up by a wire frame moving at velocity V. The surface of the film is covered by surfactants which are assumed to form an inextensible flexible sheet. The sheet is attached to the wire frame and thus moves also with a velocity V. Three regions can be distinguishedduring the formation of the film (Figure 1): a static meniscus at the bottom, then a “dynamic”region where the profile is controlled by large pressure gradients, and finally, for z a,the film itself of constant thickness d . a. Static Meniscus. The profile h(z) is here determined by the balance of the Laplace pressure (ro/Rwith R the radius of curvature of the interface) with the
0743-746319212408-3161$03.00/0 0 1992 American Chemical Society
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Bruinsma et al.
3162 Langmuir, Vol. 8, No. 12, 1992 A
”‘
a
d / t 2= K~
A‘
(11.7)
The dimensional analysis of eq 11.6 gives qV/d2= ( ~ / € ) ( ‘ Y ~ K ~ )
3
(11.8)
Eliminating 4 gives ~~d= ( C a P
(11.9)
which is the Frankel law.
Figure 1. Scheme showing the profile of a soap film vertically pulled out of a reservoir with a velocity V. The z axis is vertical. h(z)is the half f i i thickness;the fluid velocity in the z direction is called u(x). Region 1 isthe staticmeniscus,region 2 corresponds to the dynamic regime, and region 3 is the entrained film.
gravitational pressure pgz, i.e., the competition between surface tension-favoring flat interfaces-and gravity-promoting horizontal interfaces:
(11.3) From eq 11.3, it is simpleto get the height zo of the meniscus, zo = 2 1 / 2 ~ ~and - 1 , the curvature at the top of the meniscus which is 21/2Ko. b. Flat Film. Well above the reservoir, for z >> zo the
-
film is flat and h(z) d/2 as mentioned. Inside the film there are no flow gradients so there are no viscous stresses, and since the film is flat, there is no capillary pressure either. The pressure inside the film is thus equal to the atmospheric pressure. c. Dynamic Regime. Separating the static meniscus and the asymptotic film,there is a crossover region with a rapid increase of pressure from -pgKg-l to zero. The large pressure gradient attempts to squeeze liquid out of the film. In the film interior, the pressure gradient must be balanced by compensating forces. In the Frankel theory, one assumes that the pressure gradient is balanced by a gradient of the viscous stress tensor:
111. Non-Newtonian Effects Induced by the Polymer We now can examine the assumptions made in view of possible violations under polymer addition. Polymer solutionsare well known to exhibit unusual hydrodynamic behavior. Flow gradients create additional stresses (normal stresses)which must be included in the Naviedtokes equation. These non-Newtonian effects appear when the Deborah number De (the product of shear rate and relaxation time) is larger than 1. The shear rate in the experiments4ranges from 1to lo2s-l while the relaxation time of semidilute PEO solutions (with M = 10s Da) is of order 10-LIO-l so De is of order 1. Moreover, an entangled polymer network resists shear and would tend to slip over the surfactanbcoveredfilm,provided that there is no strong attraction between the polymer chains and the surface. In this section we will examine whether these effects will alter the profile and the dependence of d on Ca. a. Normal Stresses. We will include non-Newtonian flow into the framework of the Frankel theory by adding to the usual viscous stress tensor the “normall stresses which we will denote by giiN. Normal stresses are well known to lead to the swelling of streams emerging from an orifice: and we expect that for the present case normal stresses will lead to the swelling of the film and thus to an increased dependence of d on Ca. In our problem, the flow is directed along the z axis and the flow gradient is normal to the film,Le., along the x axis (see Figure 1).In the limit of low shear rate (&/ax 01,the normal stress differences can be expressed as6
-
a.a.. 1 =a m x i
(II.4) with Qij q(dui/axr + auj/dxj) and ii the flow velocity. TO agood approximation, P is along the z direction and varies predominantly along the x direction. In that limit (the lubrication approximation), eq 11.4 reduces to 81
q(a2u/a2)= -ap/az
(II.5a) o = -ap/ax (11.5b) Since P varies little with x , we can equate it with the Laplace pressure Pc and eq I1.5a becomes for small film curvature
with N1 (>O) and NZthe first and second normal stress coefficients. Recent experiments7have shown that, at y = au,/dx = 100 s-l and for the same system as considered in ref 4, the first normal stress difference in a Couette flow is (Pa) uZZN - uxxN= 0 . O l ~ lin. ~relative agreement with a quadratic behavior. Since NZusually is much smaller than N1,the stress tensor for the present problem will be approximated as
q(e2u/axZ) = ‘ ~ ~ ( d ~ ( h ( z ) ) / d z ~ ) (11.6)
If we combine eq 11.6 with mass conservation, we obtain a differential equation for h(z) which is discussed in the Appendix, section a. The solution of that equation must be matched to the asymptotic regime for z -*m, so h(m) = d/2 while h’(=) = h”(=) = 0, and to the static meniscus ~ . procedure is carried out at 20 where h”(z0) 2 1 / 2 ~ This in the Appendix. A dimensional analysis provides more directly the film thickness dependence on the pulling velocity V. Let be the characteristic height of the dynamic region. Since, and since the film near zo the curvature h” muat be 2ll2~0 thickness is of order d near ZO, we conclude that
-
The trace of the stress tensor has been absorbed into the-still unknown-pressure P. The Navier-Stokes (5) See Bird, R. B.; Curties, C. F. Phys. Today 1984 (Jan),36 for an introduction to non-Newtonian rheology. (6) Ferry, J. D. Viscoelastic properties ofpolymers; J. Wiley BE Sons: New York, 1980. (7) Brackman, J. C. Langmuir 1991, 7,469.
Formation of Soap Films from Polymer Solutions
Langmuir, Vol. 8, No. 12, 1992 3163
10-6
'
/
10-d
1~-7
I
16-6
I
16-5
16-4
$4
Figure 2. Dimensionlessthicknessd/& computed from eq A.14. The first normal stress coefficient N1 is 0.01(SI). The viscosity 7 is (..e) 0.01mPa 8, (- -) 0.06 mPa s, and (+++)0.1mPa s. The full line represents the original Frankel law for comparison.
Figure 3. Dimensionlessthickness d/ ~ 0 - lcomputed from eq A.19. The sliplength b is (- - -) 0.005 pm, (- - -) 0.01pm, (- - -) 0.1 pm, 1 pm, and (--) 10 pm. The full line represents the original Frankel law for comparison.
equations now reduce to
substrate, then k may be much larger.9 If we define the effective length b = q / k , then for the experiments under discussion, b a(q/qo)must range between one and a few hundred angstroms assuming no chemical affinity and using the viscosities (1-100 mPa 8) measured in ref 4. For strong chemical affinity, b would be correspondingly smaller. For the present case the boundary conditions 111.8, in the laboratory frame, become
-
(111.3) (111.4) When non-Newtonian effects become large, we can, to lowest order, neglect the viscous stress in eq 111.3. Balancing the normal stress with the capillary pressure then gives
-
Ml(au/ax)2 To(d2h/dz2) (111.5) Using dimensional analysis on eq 111.5demands that we replace eq 11.8 by (111.6) Nl(VIdl2= Y O K ~ In dimensionless units this leads, together with eq 11.7 to a new scaling relation: d/Kgl = [ ( ~ l r o ~ o ) 1 ~ 2 / ~ l C a (111.7) The factor in brackets is of order 1-10 at higher polymer concentrations. As expected, eq 111.7 predicts a stronger dependence of d/K-l on Ca than that of Frankel's law. A more careful discussion of eqs 111.3 and 111.4 is given in the Appendix, section b, with the results shown in Figure 2. The linear dependence is recovered at higher Ca. b. Slip Boundary Conditions. A polymer liquid strongly resists shear because of entanglement effects at the microscopic level. For this reason, it was proposed8 that a polymer liquid near a solid surface does not obey the standard nonslip boundary conditions but, instead, that it should be allowed to slip with a (relative) velocity V, proportional to the applied viscous stress. If the flow is along the z direction and the surface normal along x , then uz, = kV,
(111.8) with k the friction constant of the liquid sliding over the substrate. If the polymers do not have any chemical affinity for the substrate, then k = qo/awith qo the viscosity of a corresponding monomeric liquid (in the case of a melt) or the solvent viscosity (for a more dilute solution) and a equal to a few angstroms. In either case, qo = 1-10 mPa s. If the monomers have chemical affinity with the (8) de Gennes, P.-G. C. R. Acad. Sci., Ser. 2 1979,288,219.Brochard, F.; de Gennes, P.-G. J. Phys., Lett. 1984,45,I.-597.
(-e)
-
(111.9) u(x=h(z))- v = -b(du/ax)l,,,,,, Sinceauldx = V/d,wecansetu(x=h(z))= V(thestandard boundary condition) if b c*, the overlap concentration)-exhibits strong interactions between surfactants and polymers14which lead to the formation of a network of micelles bound by the macromolecules (in equilibrium with unbound micelles). The other systems should not allow any interaction between surfactant and polymer; the hydrophilic part of the nonionic surfactant has the same chemical structure as the polymer skeleton (ClZEdPEO), and the zwitterionic surfactant should not interact by Coulomb interaction with the polysaccharide. The deviations from the Frankel law also vanish for the SDS/PEO system when using a low MW (lo6) PEO. The results obtained with high MW PEO are shown in Figure 4 and reveal increasing deviationsfrom the Frankel law on polymer addition. First it must be emphasized that the viscoelaetic properties of PEO solutions are greatly enhanced by addition of SDSe7Furthermore, in the shear rate range of the experiments, the viscosity decreasesonly slightly with the shear rate,l6 and the first normal stress difference increases nearly as the square of the shear rata7 So our experimentscould in principle follow the theoretical description of section 111;Le., the film thickness should increase as Ca, and for large capillary numbers and for a (12) Dimitrov, D.; Pnnniotov, I.; Richmond, P.; Terminnssinn-Sarnga,
L.J. Colloid Interface Sci. 1978, 65, 483.
(13) To be published in a forthcoming paper. (14) Cabnne, B.; Duplessix, R. J. Phys. (Paris) 1982,43,1529. Cabane, B.; Duplessh, R. J. Phys. (Paris) 1987,48,651.
(15) Axelor, M. A. V. Private resulta to be published. Note that a simple shear-thinning effect should not account for the observed deviationr, an shown in ref 4.
given Ca, the smaller the viscosity (or the polymer concentration), the larger the discrepancy with Frankel's law. Neither of these effects are observed in the experiments. Hence, normal stresses probably present for our shear rates do not explain the results. Turning to slip boundary conditions, we saw in section II1.b that this can be described by introducing a slip length b: according to the ratio of the film thickness to b, two regimesareexpectedwithd =(Ca)2/3andd=(Ca)2.1ndeed the results qualitatively follow this behavior. If we fit the data with the (Ca)2/Land the (Ca)Lregimes, we can extract from the crossover length the values for b. These values are displayed in Figure 5 as a function of the PEO concentration. Unfortunately the measured b values are far too large (about of order micrometers) compared to the predicted value8 a(q/qo) estimated in using the measured viscosity 1,which in turn assumes that the PEO concentrationinside the soap film is equal to ita bulk value. Clearly, we have not yet succeeded to explain the observed deviations from the Frankel law. The SDS/PEO system exhibits a very strong polymer/surfactant interaction, and a recent X-ray reflectivity measuremenP shows that the PEO chains penetrate the surfactantlayers of a soap f i i this should give birth to effects (such as unexpected surface elasticity, or surface viscosity enhancement~'~) which are not accounted for here and appeal for more experimental investigations. A possible source of the failure of the existing models could be due to the existence of a gradient of polymer concentration inside the film which would give in turn a gradient of osmotic pressure and thereby would alter the velocity profile.
Acknowledgment. We thank D. Durian, P.-G. de Gennes, C. Knobler, K. Mysels, E. Raphalsl, and S. Troian for fruitful discussions and Atochem for financial support. Appendix a. Frankel Theory. For the classical theory, the velocity profile is symmetric and obeys the boundary condition u(x=*h(z)) = V
(A. 1)
Solution of eq 11.7 yields the following velocity profile:
Conservation of mass along the film imposes J-lhu(x) dx = Vd or Vd
27 d3h 2Vh + -h331, dz3
This differential equation is transformed to2 d3p l - p -=dX3 p3 by introducing the dimensionless variables p = 2h/d and X = %/[with[ = d(24(Ca))-1/3(tis the characteristic length (16) Cohen-Addnd, 5.;di Meglio, J.-M.; Ober, R. C. R.Acad. Sci.,Ser.
2 1992, 516, 39. (17) de Gennes, P.4. J. Phys. Chem. 1990,94, 8407.
3166 Langmuir, Vol. 8, No. 12, 1992
Bruinsma et al. d/
...
K,'
..
1
10-51
j
** *
I
10-5
/
3
d/K;'
10-3:
++ +
10-4,
+
10-5
. 10-8
d)
Ca -3
i
d/
** '
KY1
10-3
/ lo-!
-3
Figure 4. Experimental results of the dimensionless film thickness d l ~ 0 - as l a function of the capillary number Ca, for SDS solutions with PEO at different concentrations c: (a) c = 0 (no polymer); (b) c = 0.05% ;(c) c = 0.1 % ;(d) c = 0.3 % ;(e) c = 0.5% (the concentration being the ratio of polymer mass to the total mass of the solution). The surface tension (independent of c ) is equal to 33.0 f 0.2 mN/m, and the viscosities are, respectively, 1, 3.4, 6.7,62,and 154 mPa s. The full line represents the Frankel prediction. The discussion * ( d / 8 < 5 X 10-9,van der Waals forces make the film thinner; is restricted to 'intermediate" capillary numbers: at small d / ~ -values at highest d/K-l values, one observes a plateau due to an effect of the finite size of the frame.
of the dynamic meniscus). It is solved with the following limits:
-
I.
We now match the curvature CY of the solution of eq A.5 for X 0 with the curvature a t the top of the static meniscus which is 21/2~-1.The entrained thickness is then (A.7) A standard numerical integration yields a = 0.643, and thus d l ~ g - l =1.89(Ca)2/3. This is the well-known Frankel law.2
0
0
Figure
.2
.4
.L
Variation of the slip length b with the PEO concentration c. 5.
b. Normal Stresses. The boundary conditions remain eq A.l. Solution of eq 111.4 with boundary condition P = P, (the capillary pressure) at x = h(z) and
Langmuir, Vol. 8, No. 12, 1992 3167
Formation of Soap F i l m from Polymer Solutions
using eq 111.2 give
we saw in section 111,a dimensionalargument shows that dlK0-l is proportional to Ca. Between this linear regime and the Frankel regime, the thickness d depends on Ca through the function a(r)according to
Using eq A.8 in eq 111.3 gives
d/K[' = 21/232/3 a(r)(Ca)2/3 (A.14) This is represented in Figure 2 for Nl = 0.01 (SI)and for different viscosities ranging from 1 to 60 mPa 8. c. Effect of Slip Boundary Conditions. The polymer solution slips on both walls, and the boundary conditions are (see eq 111.9)
If the normal stress is zero, the velocity profile is parabolic. We treat the normal stress as a small perturbation around this state. Hence, we write u as V- ( ( x 2 - h2)/2)K,where the coefficient K is a function of z only. Identification with eq A.9 yields
u(x=*h(z)) = V - V, (A.15) with V, = *b(au/ax)lxr*h. The velocity profile is given by u=
v---'d3h(x2 - 2hb - h2) 27 dz3
To insure self-consistently that K does not depend on x , we replace x2 by ita spatial average across the film: ( x 2 ) = h2/3. This gives a quadratic equation for K
The total flux in terms of K is Vd = 2Vh + 2Kh3/3 (A.12) We again introduce the reduced variables I.L and X defined below eq A.5 and the dimensionless parameter l?= 2N1V/ ~ 5 By . eliminating K from eqs A.ll and A.12, we obtain the differential equation
(A.16)
The total flux is Vd = 2Vh + 2 -(I
P
T
-)-3bh d3h dz3
(A,17)
From eq A.17, we can again find two limiting cases: (i) the film thickness is very large compared to b, and we are back in the Frankel regime and (ii) the film thickness is much smaller than b, and then our dimensionalargument of section I11 shows that d goes as Gal2. From the reduction of eq A.17, we obtain (A.18)
where e = b/d and p and X are the dimensionless width and height. The thickness of the film is given by which can be solved numerically as before. At low capillary numbers is small; hence, the contribution of the second term in eq A.13 is negligible, and we recover the Frankel regime. Conversely at high Ca or large r,the normal stress contribution is comparable to the capillary pressure. As
d/K['
21/232/3a(€)(Ca)2/3
(A.19)
In Figure 3 we have plotted dl~0-lversus Ca (on a log-log scale) for a slip length b ranging from 5 X low3to 10 Pm. Registry No. SDS,151-12-2.
