760
Langmuir 1987, 3. 760-768
limestone. [These investigators found an apparent surface area of 80 f 8 m2/g(CaO) for vacuum activation at 620 “C.] The impurities in the Franklin limestone (magnesia, aluminosilicates, sulfates, iron, strontia, etc., as shown in Table I) serve as fluxes and/or channel blockers in the activation process. The sintering is enhanced at higher temperatures where “Heating a t 700 OC for various durations leads to higher percentage decomposition, and, as shown by SEM, fragmentation and recrystallization that take place to an advanced stage.“13 In the case of this report, where the vacuum activation was carried out below ca. 600 “C, there is little or no fragmentation, and virtually all of the nitrogen sorption “is due to the layer structure of the adsorbent and to the penetration of the adsorbed phase between the 1ayers,”13as noted for c1ays,l6gels,17and related systems.lg The mechanism of thermal activation of limestones depends upon several factors: 1. Temperature of the Sample. If the activation is carried out in vacuo, one is not limited to the case where “...CaC03 does not decompose below 500 oC...”,Zoand the thermodynamically limiting behavior is obtained. When (17)Fuller, E. L.,Jr. J. Colloid Interface Sci 1967,33,47. (18)Bishop, A. C. An Outline of Crystal Morphology; Hutchison, 1967. (19)Everet. D. H. In T h e Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker: New York, 1967. (20) Glasson, D. R.; O’Neil, P. In Characterization of Porous Solids; Greg, S. J., Sing, K. S. W., Stoeckli, H. F., Eds.; The Society of Chemical Industry: London, 1979;p 351-357.
the activation is carried out at higher temperatures, the process is complex with various degrees of shattering and sintering. 2. Pressure (Carbon Dioxide Chemical Potential) in the Gas Phase above the Solid CaC03/Ca0 Sample As Controlled by the Pumping System. The decomposition rate in vacuo depends on the temperature variation of the free energy change (enthalpy) of the process. 3. Purity of the Sample. Impurities tend to serve as fluxes to enhance low-temperature sintering and to close off access to the inner reaches of the activated oxide. 4. Amount of Reaction with Reaction Gases. Water, carbon dioxide, sulfur dioxides, etc., all react to modify the structure of the activation products. Studies with the vacuum microbalance system allow one to control or, in this instance, eliminate these reactions and to study the activation process alone. 5. The activation process can be defined in terms of simple, straightforward kinetic parameters. The process is first order in the amount of oxide formed, with little or no excess energy of activation, at least in vacuo a t temperatures ranging up to 650 “C. 6. The morphology of the activation can be described mathematically as a shrinking core generation of active sorption sites for pure materials, increasing capacity of virtually identical topology. Impurities tend to introduce a second-order loss term caused by sintering or channel blocking. Registry No. SO,, 12624-32-7; N2-,7727-37-9.
Stability, Critical Thickness, and the Time of Rupture of Thinning Foam and Emulsion Films A. Sharma and E. Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo,New York 14260 Received November 3, 1986. I n Final Form: March 11, 1987 A theory is developed to predict the critical thickness of rupture and the lifetimes of the tangentially immobile foam and emulsion films. The theory accounts for the stabilizing effects of film thinning on the growth of the surface corrugations and leads to an explicit expression for the critical thickness as a function of the velocity of thinning, amplitude of the surface corrugations, viscosity, surface tension, and Hamaker constant. The predictions of the theory are in both qualitative and quantitative agreement with the experimentally obtained critical thicknesses. The comparison of the theory with the data for critical thickness confirms an independent experimental observation that the velocity of film thinning depends inversely on the film radius (and not on the square of the film radius as predicted by Reynolds’ law of film thinning). The stabilization due to drainage is found to be significant in determining the lifetimes of thin films, and the importance of this effect increases as the wavelength of the perturbation increases. The present theory in conjunction with the data for critical thickness predicts the lifetimes of foam films that are in agreement with the experimental data. Further, the Hamaker constant that is obtained by fitting the experimental data to the present theory is in excellent agreement with its predictions from the Lifshitz theory.
powder coating,s paper ~ o a t i n gthin , ~ solid films used in I. Introduction microelectronics,1° and the corneal tear film.11J2 In esThe intermolecular forces induce the instability of thin films in such diverse settings as foams and emul~ions,l-~ sence, a thin film becomes unstable when the destabilizing deformation of biological membra ne^,^,^ electrostatic (1)Vrij, A. Discuss. Faraday SOC.1966,42,23. (2)Scheludko, A. Adu. Colloid Interface Sci. 1967,1, 391. (3) Ivanov, I. B.; Dimitrov, D. S. Colloid Polym. Sci. 1974,252,982. (4)Manev, E.; Scheludko, A.; Exerowa, D. Colloid Polym. Sci. 1974,
(7) Maldarelli, C.; Jain, R. K. J. Colloid Interface Sci. 1982,90, 263. (8) Cross, J. A. In Surface Contamination; Mittal, K . L., Ed.; Plenum: New York, 1979;Vol. 1! p 89. (9)Babchin, A. J.;Clish, R. L.; Warren, D. Adu. Colloid Interface Sci.
1981,14, 251. (10)Ruckenstein, E.; Dunn, C. S. Thin Solid Films 1978,51, 43. (11)Sharma, A,; Ruckenstein, E. Am. J . Optom. Physiol. Opt. 1985, 62,246. (12)Sharma, A.;Ruckenstein, E. J . Colloid Interface Sci. 1986,111, 8.
0743-7463/87/2403-0760$01.50/0 0 1987 American Chemical Society
Langmuir, Vol. 3, No. 5, 1987 761
Thinning Foam and Emulsion Films influence of the van der Waals forces overcomes the stabilizing effects of the surface tension, Marangoni convection, surface viscosity, double-layer repulsion, and other stabilizing factors, if present (as is shown later, drainage of the fluid from the film also exerts a stabilizing influence). The velocity of film thinning due to drainage is inversely proportional to a certain power ( cm, because of the inclusion of the amplitude, E ( R ) .Thus, ~ the models employing the Reynolds’ law predict a critical thickness that coincidently is in partial agreement with h,, and not with h,. The conclusion, as was also pointed out by Radoev et al.,5is to employ the experimentally observed velocities of film thinning for predicting the critical thickness and for the purposes of model discrimination until a suitable model for the film thinning becomes available. This requirement is not very stringent, because, as is shown later, only one value of the velocity is all that is needed to predict the critical thickness for a given film radius. Having outlined the central experimental observations, the various theoretical approaches are now summarized. The earliest model of Vrij,l while conceptually sound, did not account for the influence of drainage on the growth of surface corrugations, nor did it combine the wave motion with drainage. Ivanov et a1.21and Ivanov and Dimitrov3 combined the wave motion with the drainage in the socalled quasi-static approximation. In this approximation, the details of the wave motion are inferred at each instant of time by freezing the relatively slow motion of the interface; hence, in essence, the time increments are replaced by dh/V. Here, dh is the change in the mean thickness of the film and V is the velocity of thinning. This model neglects the influence of drainage on the growth of waves and on the condition of neutral stability. That the drainage may affect profoundly the conditions for neutral stability has been shown by Gumerman and HomsyZ2for a particular wave-a wave with a wavelength twice the diameter of the thin film. The procedure of Gumerman and Homsy22was also exploited by Malhotra and WasanZ3 for a more realistic velocity distribution. These results are, however, only indicative of the possible influence of the drainage since the wave responsible for the destruction of the film is at least 1 order of magnitude smaller than the film radius and, in addition, the critical thickness is always smaller than the neutrally stable thickness. It is only the latter that is obtained by this approach22and also by the procedure used by Manev et al.4 The model of Manev et al.4appeared to fit their data because the model predictions were compared against the mean value of the critical thickness, and both the model predictions and the mean critical thickness overestimate the minimum critical thickness. The approach of Radoev et al.5 used the experimental velocities of film thinning but employed the same growth coefficient and the neutrally stable thickness as for a nonthinning film. Further, the wave motion and the drainage were not combined, and (as is shown later) their results are rather sensitive to the amplitude of the perturbations. As is discussed later, it appears that an unrealistically high value of thermal amplitude is needed to fit the experimental data when their theory is used. A (21) Ivanov, I.; Radoev, B.; Manev, E.; Scheludko, A. Trans. Faraday Soc. 1970, 66,1262. (22) Gumerman, R. J.; Homsy, G.M. Chem. Eng. Commun. 1975,2, 27. (23) Malhotra, A. K.; Wasan, D. T. Chem. Eng. Commun., in press.
consistent picture that emerges from the above considerations is that a model predicting the critical thickness should (a) incorporate the effects of drainage on the growth of surface corrugations and on the conditions for the neutral stability (this should be done for a wave (the dominant wave) that causes the fastest destruction of the film, i.e., a wave for which the corresponding critical thickness is maximum), (b) combine the drainage and the wave motion, and (c) employ the experimentally obtained velocity of film thinning. The effects of various phenomena may then be checked a posteriori. In what follows a theory is formulated, and efforts are made to keep it as simple as possible without sacrificing either the details of the various phenomena or the accuracy. The predictions of this theory are then compared with the recent experimental results of Radoev et aL5 Only theoretical estimates of the Hamaker constant and of the amplitude of the thermal perturbations are employed, and, therefore, the predictions are made without any adjustable parameter.
111. Theory 111.1. Influence of Drainage on the Growth of Waves. The concurrent processes of f h thinning and the growth of thermal surface waves are depicted in Figures 1parts a and b. The dominant thermal wave first becomes unstable at thickness h, and grows until it destroys the primary film at a certain minimum thickness, h,. As is well-known, the wavelength of the dominant wave has a wavelength far exceeding the thickness of the neutrally stable film, h,, and, therefore, the following Navier-Stokes equations in the “thin film” approximation describe the hydrodynamics of the thin film:16J9 -a4+ - =a~o (2) az
az
a4 ap a%, -+-=I.1(3) dr ar az2 where p is the hydrodynamic pressure, u, is the radial component of the velocity, and 4 is the van der Waals interaction potential per unit volume of the liquid. The continuity equation has the form du,
dz
a + -1r -(rur) dr
=0
(4)
where u, is the normal component of the velocity. The boundary conditions at the interface, z = H(r,t)/2,are
aH
aH
at
ar u, = 0
- + u,-
= 2u,
(5) (6)
(7)
where p g is the pressure outside the film, CT is the interfacial tension, and Ar is the surface Laplacian. Equations 5, 6 , and 7 are the kinematic condition, the condition for the tangential immobility of the interface, and the condition for the pressure jump, respectively. In addition, the symmetry of the film at the z = 0 plane allows us to write u,=O atz=O (8) and aur _ -- 0 az The solution of eq 2 is
atz=O
P + 4 = p ( H / 2 ) + 4(H/2)
(10)
Langmuir, Vol. 3, No. 5, 1987 163
Thinning Foam and Emulsion Films where p(Hf2) is determined from the boundary condition 7 and @(H/2)is given by eq 11, where A is the Hamaker
$(H/2) = A/67r@
(11)
Equation 17 describes the process of film thinning, and an explicit form of the velocity, V, of film thinning may be obtained for a given pressure distribution, P d = pd(r). The choice
constant for the interactions between the molecules of the thin film. Equation 3 may now be solved with the help of boundary conditions (9) and (6) to give
q
u, = 2P
22
-
:)( p + );
(12)
The continuity equation (eq 4) in conjunction with the above expression and the boundary condition (8) yields the normal component of the velocity as '2
= - -2p i- -ir[ ar ra( g - F ) ( $ + g ) ]
(13)
Finally, the kinematic condition (5) gives the sought after equation of evolution, viz., -=---[r@($+;)] aH 1 i d at 12p r ar
(14)
The above equation describes both the drainage of the
f i i and the growthfdecay of thermal corrugations of the interface. However, these two motions may be decoupled because of the widely separated length and time scales governing the two phenomena, and the location of the interface may be decomposed into two parts, viz., H(r,t) = h(t) + 2C;,E(r,t) (15) where h(t) is the mean (averaged over the surface corrugations) thickness of the film that decreases due to the drainage, C;, is the amplitude of the thermal perturbations, and ((x,t) describes the spatial and temporal dependence of the thermal surface corrugations. It may be noted that until the very last stages of the process of rupture, viz., h = h,, the inequality 2C;,[(r,t) cm leads to an even better agreement with the theory. Of course, for R < cm, the experimental data never overestimate the critical thickness by more than 5 A; hence, the points depicted by the circles are in entire agreement with the theory. The broken curve (curve 3) in Figure 2 corresponds to the case when the effects of film thinning on the growth of perturbations are neglected, but all other parameters are the same as for curve 1. In this case, the growth coefficient is given by the first term of eq 22 only. The latter approach overestimates the critical velocity of film thinning by about a factor of 2 and also overestimates the critical thickness by about 30 A for this system. As is shown shortly, the determination of the velocity of thinning is important for calculating the lifetime of a free film. The dominant wavelength, A, = (2a/km),is about onetenth of the film radius for the system considered here. The influence of film thinning on the wave motion is, however, more important for large wavelength perturbations (see section 111.2). The estimates of the time of rupture of a foam film are needed to predict the time of collapse of foams and for the design of foam fractionat" The lifetime of a foam film with initial thickness h, is readily determined as At =
kBT -u
1 kmR
-for
large kmR (47)
where k, is the Boltzmann constant, T is the absolute temperature, km2is given by eq 32, and J1 is the Bessel function of order 1. Equation 47 appears to be more accurate inasmuch as it accounts for the dependence of amplitude on the wavelength according to Einstein's theoremz5and is also supported by other investigations.'Sz6 In any event, the results of the present theory are not very sensitive to the exact value of the thermal amplitude (eq
h"dh TI
"
Jh,
This may be evaluated to a first approximation by noting that
whereas Ivanov and Dimitrov3derive for it the expression kBT/ruk,2R2J1z(kmR)
r
V ( h )=
V,(h3
+ a3)
(hC3+ a3)
where a3 = (A/6rPA). Substituting the above expression in eq 48 gives h,3 + a3
2h, - u
2h, - u
(h,2 - ah, (h: - ah,
+ a2) (h, + + a2) (h, + a ) 2 (50)
The time of rupture can be computed from eq 50 in con-
~
(25) Einstein, A. Ann. Phys. 1910, 33, 1275. (26) Hajiloo, A.; Slattery, J. C. J. Colloid Interface Sci. 1986,112, 325.
(49)
(27) Narsimhan, G.; Ruckenstein, E. Langmuir 1986, 2, 230.
Thinning Foam and Emulsion Films
Langmuir, Vol. 3, No. 5, 1987 767
Table 11. Comparison of Theoretical and Experimental Values of the Time of Rupture (Drainage Time A t ( 8 ) ) Using Various Approaches exptl values from ref 14 film radius predictions of present calculated by neglecting calculated by Reynold's 102R, cm a b theory (ea 41 and 50)' stabilization due to drainaged law of film thinninge 0.5 19 f 1 31 f 1 21 11 >40 1.0 29 1 45 1 35 18 >120 2.0 43 f 1 63 2 46 24 >380 5.0 67 f 3 105 f 3 73 38 >1500
*
*
-
mol/L SDS, +0.1 mol/L NaCl, Pi= 490 dyn/cm. *3.5 X 04.3 x mol/L SDS, +0.25 mol/L NaCl, PA= 380 dyn/cm. mol/L mol/L SDS, +0.1 mol/L NaCl, P, 400 dyn/cm. e4.3 X lo-* mol/L SDS, +0.1 mol/L SDS, +0.1 mol/L NaCl, P,' -400 dyn/cm. NaCl, Pi= 490 dyn/cm. -
I
junction with eq 42 if the critical thickness h, is experimentally determined. Note that in the absence of the theoretical result, eq 42, both the velocity and the critical thickness are to be measured experimentally for obtaining the time of rupture. The entries in the fourth column of Table I1 are the times of rupture as computed from eq 50 and 41 with the data of Radoev et al.5 for the critical thickness and for an initial thickness of 2000 A. The critical velocity is then obtained from eq 41 or 42 and the time of rupture from eq 50. Manev et al.14 have determined the times of rupture of a foam film for a similar system, and their experimental results are also reported in Table I1 (these are also for an initial thickness h, = 2000 A). The theoretical predictions are in good agreement with the experimental values, especially in view of the fact that none of the parameters is adjusted, and the conditions for the systems of Manev et al.14 and Radoev et ale5,although similar, are not entirely identical. The critical thicknesses used are taken from the latter source because Manev et al.14 report only the mean critical thickness. In contrast, neglecting the stabilization effect due to the drainage decreases the time of rupture by about a factor of 2, and the comparison with the experimental values becomes poor (Table 11). Finally, as is apparent from entries in the last column of Table 11, if the Reynolds' law of film thinning is used for calculating At, the results are highly unrealistic. In conclusion, the process of film thinning has an important stabilizing influence on the process of rupture, and its neglect leads to a decrease in the time of rupture by about a factor of 2. The use of Reynolds' law of film thinning certainly cannot be justified, as eq 42 gives the following dependence of the critical thickness on the film radius: 1 a hc4.164 a -
v,
R" PA(h2
+ a3)
where it is assumed that the velocity of thinning is inversely proportional to the mth power of the film radius. According to the Reynolds' law of thinning, m = 2 and, therefore, the slope of h, vs. R on a log-log plot varies between 0.28 and 0.48. The experimentally observed slope is between 0.11 and 0.14 (Figure 5 of Radoev et al.5). However, the experimentally observed value of the exponent m is much smaller than 2 and is about O.8.l3J4 I t is in view of this that the present theory predicts the slope of h, vs. R on a log-log plot to be 0.11 when PA>> A/61~h,3 and about 0.19 when PA
