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Notes Dimple Relaxation in Wetting Films R. Tsekov,*,† P. Letocart,‡ and H. J. Schulze‡ Department of Physical Chemistry, University of Sofia, 1126 Sofia, Bulgaria, and Max Planck Research Group for Colloids and Surfaces, University of Mining and Technology, 09599 Freiberg, Germany Received March 20, 2000. In Final Form: July 20, 2000
Introduction If an air bubble in water approaches a solid surface, a wetting film is formed. There are several distinct stages of the thinning process in wetting films. Since the drainage near the film border is initially faster than the drainage in the film center, the thickness at the barrier rim reaches quickly a value close to the equilibrium one. Thus, a specific film profile known in the literature under the name of dimple is generated.1 Dimples are nonequilibrium structures, and a consequent stage in the film kinetics is the dimple relaxation. Under the co-action of capillary and surface forces the dimple shrinks2 until the film reaches the equilibrium flat shape at heq. Occasionally, the dimples grow and exhibit an oscillatory behavior due to external convection flows in the system.3 This is not, however, the case in this paper. In the present study we consider aqueous wetting films on a glass surface. The films are generated from 1 mM KCl solution in water. At this concentration of electrolyte the electrostatic disjoining pressure is much larger than the van der Waals one.4 Traditionally in film rheology5 one takes into account the dependence of the surface tension on adsorption and the dependence of the disjoining pressure on film thickness. It was shown recently6 that the dependencies of the surface tension on film thickness and of the disjoining pressure on adsorption could also be important for the film behavior. The goal of the present paper is to investigate the role of the adsorption dependence of the disjoining pressure on the dimple relaxation in wetting films. The effect of the thickness dependence of the film surface tension is neglected since the contribution of the disjoining pressure is insignificant compared to the surface tension in water. An original procedure for simulation of the process of dimple relaxation is elaborated, which develops further previous considerations.7 Thus, important information about the interfacial proper† ‡
University of Sofia. University of Mining and Technology.
(1) Platikanov, D. J. Phys. Chem. 1964, 68, 3619. (2) Hewitt, D.; Fornasiero, D.; Ralston, J.; Fisher, L. J. Chem. Soc., Faraday Trans. 1993, 89, 817. (3) Velev, O.; Gurkov, T.; Ivanov, I. B.; Borvankar, R. Phys. Rev. Lett. 1995, 75, 264. (4) Letocart, P.; Radoev, B.; Schulze, H. J.; Tsekov, R. Colloids Surf. A 1999, 149, 151. (5) Thin Liquid Films; Ivanov, I. B., Ed.; Dekker: New York, 1988. (6) Tsekov, R.; Schulze, H. J.; Radoev, B.; Letocart, P. Colloids Surf. A 1998, 142, 287. (7) Mitev, D.; Tsekov, R.; Vassilieff, C. S. Colloids Surf. B 1997, 10, 67.
ties of the water-air surface is obtained such as adsorption of ions, surface charge density, Marangoni number, etc. Theory A favorable circumstance in wetting film hydrodynamics is the huge difference between the two characteristic length scales: the film radius R and the film thickness H. It justifies the Reynolds lubrication approximation, in the frames of which the following equation is derived
12ηF∂tH ) ∂F(FH3∂Fp - 6ηFHu)
(1)
describing the evolution of the film thickness for the case of circular symmetry. Here F is the radial coordinate, t is time, η is the liquid viscosity, p is the film pressure, and u is the interfacial velocity on the liquid-air surface. The derivation of eq 1 employs nonslip boundary condition on the solid-liquid interface, which is reasonable for the present system since the glass is hydrophilic. The further applications of eq 1 require an expression for the interfacial velocity. Since the pure water dissociates to H+ and OHions, it is in fact a three-component solution. Hence, there is always adsorption of ions on the water-vapor surface. It is shown experimentally8 that this surface is negatively charged, which is natural because the structure of OHion corresponds better to a surfactant than the proton. In the present paper the OH- ion adsorption Γ is considered slightly disturbed by the flow in the film compared to its equilibrium value Γeq. Thus, in a stationary approximation the OH- mass balance on the water-air surface acquires the form
∂F(FΓequ) ) -FD∂zc
(2)
where c and D are the concentration and diffusion coefficient of the OH- ions in the bulk. The normal derivative of the concentration on the surface can be estimated from the diffusion equation in the bulk to obtain F∂zc ) -∂F(FH∂Fc). Note that this approximation is valid for small deviations from equilibrium.6 Substituting the above expression in eq 2 and integrating once the result yields
Γequ ) DH∂Fc
(3)
Another important equation for the interfacial dynamics is the tangential force balance on the liquid-vapor surface6
2ηu/H + H∂Fp ) 2∂Fσ ) 2(∂cσ)eq∂Fc
(4)
where σ is the surface tension of water. It is clear that the gradient of the adsorption of OH- ions on the surface will reflect in a gradient of the electric field, too. However, since the latter is proportional to the gradient of Γ (the OH- concentration, respectively), the whole influence on the surface tension is already taken into account by the derivative ∂cσ. The additional Maxwell tensor is neglected because it is quadratic on the electric field and therefore (8) Schulze, H. J.; Cichos, C. Z. Phys. Chem. 1972, 251, 252.
10.1021/la000425z CCC: $19.00 © 2000 American Chemical Society Published on Web 09/22/2000
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of second order of importance. Combining eq 3 and eq 4 allows to obtain expressions for the surface velocity and the gradient of OH- concentration as functions of the pressure gradient
electrostatic disjoining pressure on the film thickness, the following formula can be proposed
u ) -H2∂Fp/2η(1 + Ma)
(5)
∂Fc ) -ΓeqH∂Fp/2ηD(1 + Ma)
(6)
where F is Faraday’s constant. The charge density of the water-glass interface is taken to be 4 times larger than that of the liquid-air interface since it is the ratio of the corresponding ζ-potentials.8 As the disjoining pressure at equilibrium is equal to the pressure peq in the capillary meniscus, the adsorption length can be expressed from eq 13 as a function of peq and heq
where Ma ) -(Γ∂cσ)eq/ηD is the Marangoni number accounting for the ratio between the interfacial tension gradient and bulk viscous stress. As the concentration ofOH- ions in water is low, the adsorption obeys Henry’s law Γ ) ac with a being the adsorption length and the surface tension dependence on c is linear, σ ) σ0 - kTac. Hence, the Marangoni number acquires the form Ma ) a2kTceq/ηD. Introducing now eq 5 in eq 1, the latter changes to
12(1 + Ma)ηF∂tH ) (4 + Ma)∂F(FH3∂Fp)
(7)
The further analysis requires expression for the film pressure as a function on the film thickness. It is supplied via the normal force balance
p ) pg - σeq∂F(F∂FH)/F - Π(H,c)
(9)
where Πeq ) Π(H,ceq). Substituting here the concentration gradient from eq 6, the dependence of the pressure gradient from the film thickness can be obtained
(1 + Ma + Ap)∂Fp ) -(1 + Ma)∂F[σeq∂F(F∂FH)/F + Πeq] (10) Here, Ap ) -aH(c∂cΠ)eq/2ηD is the adsorption-pressure number, which is similar to the Marangoni number but depends substantially on the film thickness. Substituting the pressure gradient from eq 10 in eq 7, one yields an equation describing the film thickness evolution
3ηF∂tH ) -∂F{FMoH3∂F[σeq∂F(F∂FH)/F + Πeq]}
(11)
All the specific adsorption effects here are present in the so-called mobility number
Mo ) (1 + Ma/4)/(1 + Ma + Ap)
(12)
Note that in the absence of surfactant the mobility number is equal to 1. To close the problem, an expression is required for the film thickness dependence of the disjoining pressure. As was mentioned before, the van der Waals component is negligible and the only specific interaction is electrostatic. Employing the well-known exponential dependence of the
(13)
2
a2 ) 0peq exp(κheq)/8ceq F2
(14)
According to eq 13, the derivative of the electrostatic disjoining pressure on c equals (c∂cΠ)eq ) 2Πeq since the Debye length κ-1 is nearly independent of c at the relatively high KCl concentration. Hence, substituting this expression in the adsorption-pressure number, the mobility number (12) acquires the form
Mo(H) )
(8)
The first term is the constant gas pressure, the second term is the local capillary pressure, and the last term is the disjoining pressure. The latter is a function both of film thickness and OH- ion concentration. The dependence of Π on H is an object of many publications,5 while the c dependence of the disjoining pressure is a relatively less studied effect. It is usually considered that the van der Waals component of Π does not depend substantially on c for dilute solutions. The electrostatic component, however, is strongly dependent on c since the adsorption of ionic surfactants changes the surface charge density. The gradient of the pressure from eq 8 is equal to
∂Fp ) -σeq∂F[∂F(F∂FH)/F] - ∂FΠeq - (∂cΠ)eq∂Fc
Π ) 2(4a2c2F2/0) exp(-κH)
ηD + a2kTceq/4 2
ηD + a2kTceq - 8H exp(-κH)a3ceq F2/0 (15)
It is also interesting to calculate the adsorption profile of OH- ions on the surface during the film drainage process. Combining eqs 6 and 10 and integrating the result, one yields the following expression for the deviation of the adsorption from its equilibrium value:
1 - Γ/Γeq ) 1 - c/ceq ) R H∂F{σeq∂F(F∂FH)/F + peq exp[κ(heq - H)]}
∫F 2ηD/a + 2akTc
eq
- 2peqH exp[κ(heq - H)]
dF (16)
Equation 16 allows calculating of the adsorption if the film thickness is known. According to eq 11 and eq 15, the film thickness evolution obeys the following nonlinear equation
3ηF∂tH ) (ηD + a2kTceq/4)∂F × FH3∂F{σeq∂F(F∂FH)/F + peq exp[κ(heq - H)]} ηD + a2kTceq - apeqH exp[κ(heq - H)]
(17)
Introducing the following dimensionless variables Y ) H/heq and x ) (F/R)2, eq 17 can be rewritten in the form
∂ tY ) L ˆY
(18)
where the operator L ˆ is given by the expression 3
4heq (4ηD + a2kTceq)
∂x × 3ηR2 xY3∂x[(σeqheq/R2)∂x(x∂xY) + peq exp[κheq(1 - Y)]]
L ˆY )
{
ηD + a2kTceq - apeqheqY exp[κheq(1 - Y)]
}
(19)
A rigorous treatment of the dimple relaxation via eq 18 requires four boundary conditions as well as an initial profile. Exact boundary conditions can be written on threephase contact between the bubble, solution, and capillary. However, in this region the lubrication theory and therefore eq 18 are no longer valid. Moreover, eq 18 is
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Notes
valid for the latest stage of the dimple relaxation, and the relevant initial profile can only be experimentally obtained. Hence, it is impossible to close the mathematical problem; the further treatment requires an empirical modeling. For this reason, an approximate method is developed here for calculating the evolution under the assumption that the film shape is known. The present experimental observations as well as previous investigations9 show that the shape of the dimple can be well approximated by a quadratic polynomial form of x
y(x,t) ) y0(t)(1 - x)2 + (2 - x)x
(20)
It is easy to check that this expression satisfies the boundary conditions of equilibrium thickness and minimum of the film profile at the film edge, H(R) ) heq and (∂FH)R ) 0. Since x e 1, eq 20 could be also considered as the x-power expansion of the real profile cut after the quadratic term. As seen there is only one unknown function of time in eq 20, the relative thickness y0(t) in the film center. Suppose at time t the film profile is given by eq 20. The profile at time t + τ can be calculated from eq 18, which in the limit of very small τ can be rewritten as
Y(x,t+τ) ) y(x,t) + τL ˆ y(x,t)
Figure 1. Experimental data for three films and theoretical prediction for the dimple relaxation in aqueous wetting films on glass interface.
(21)
Thus, the consequent profile Y can be generated, which satisfies eq 18, but in contrast to y does not obey the necessary boundary conditions. To apply the boundary conditions, we postulate that the real profile y(x,t+τ) is the best interpolation of the profile Y(x,t+τ). So, to calculate the film profile, a minimization of the square of the deviation of the two functions all over the film is required
∫01[y(x,t+τ) - Y(x,t+τ)]2 dx ) min
(22)
This criterion combined with eq 20 and eq 21 leads to the following recurrent relation
y0(t+τ) ) y0(t) + 5τ
∫01(1 - x)2Lˆ [y0(t)(1 - x)2 +
(2 - x)x] dx (23)
It is a problem only of integration to calculate the evolution of the thickness in the film center, which introduced in eq 20 will provide the whole film profile evolution. Moreover, the adsorption profile can be also calculated from eq 16, which in the new notations reads
1 - Γ/Γeq ) 2 1heqy∂x{(4σeqheq/R )∂x(x∂xy)
∫x
+ peq exp[κheq(1 - y)]}
2ηD/a + 2akTceq - 2peqheqy exp{κheq(1 - y)]
dx
(24) Application To check the theory, we have compared it with experimental results of dimple relaxation. In Figure 1 the experimental data for the drainage in the center of three films are presented. The evolution of the thickness h0 is monitored by the classical light interference method. The equilibrium thickness of the films is heq ) 42.4 nm, while the capillary pressure in the meniscus during the experiments is kept at peq ) 292 Pa. Since at 1 mM concentration of KCl the Debye thickness is κ-1 ) 10 nm and ceq ) 0.1 (9) Tsekov, R.; Ruckenstein, E. Colloids Surf. A 1994, 82, 255.
Figure 2. Calculated adsorption distribution on the waterair surface at various times: 1 s (solid), 30 s (long dash), 40 s (short dash), 50 s (dash and point), 60 s (point).
µM at pH ) 7, one can estimate the adsorption length from eq 14 as a ) 0.14 mm. This is a reasonable value since the concentration of the OH- ions is very low, and it is well-known that the slope of the adsorption isotherms is large at c approaching zero. The value of a corresponds to adsorption and surface charge density on the waterair surface Γeq ) 14 nmol/m2 and qeq ) 1.35 mC/m2, respectively. The radius of the films is R ) 120 µm, while the other parameters at the considered temperature of 25 °C possess the following values taken from the CRC Handbook: D ) 5270 µm2/s, σeq ) 72 mN/m, and η ) 0.89 mPa s. Thus, all the parameters needed for the dimple relaxation process simulation are present. In Figure 1 the simulation is plotted with time step τ ) 0.1 s. As seen there is remarkable juxtaposition between the theory and experiment. The distribution of the OH- ions on the water-air surface is also simulated by using eq 24, and the results are presented in Figure 2. Initially, the adsorption in the film center is about 30% less than Γeq due to film drainage. With advancing time the thinning rate of the film is going down, and the adsorption increases to reach its equilibrium value at the end of the process. To estimate the importance of the coupling between the disjoining pressure and adsorption, the dependence of the mobility number on the film thickness is presented in Figure 3. For film thickness larger than 80 nm, Ap is small and the mobility number is 1/4, which corresponds
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to substantial increase of the thinning rate at the end stages of the film drainage compared to the case without coupling. Conclusion
Figure 3. Calculated dependence of the mobility number on the film thickness.
to a tangentially immobile surface. If the film thickness decreases below 80 nm, the adsorption-pressure number becomes negative. This increases the mobility number, and close to the equilibrium thickness Mo is twice larger than the initial value. Hence, the coupling between the disjoining pressure and adsorption is important and leads
In the present paper a simulation procedure for the dimple relaxation in wetting films is developed. The juxtaposition between the theory and experiment is remarkable. It confirms the assumption that OH- ions adsorb on the water-air surface. The calculated adsorption and surface charge density are reasonable and commensurable with previous results. It is important to note here that the Marangoni number of this original surfactant in water is about 1000. Hence, the water-air surface is tangentially immobile even in the case of pure water. This is important novelty since in hydrodynamics the water-air surface is usually considered as a free one with zero stress. Acknowledgment. R. Tsekov is grateful to the Alexander von Humboldt Foundation for a granted fellowship. LA000425Z
