Langmuir 2002, 18, 5799-5803
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Effect of Ionic Surfactants on the Dimple Relaxation in Wetting Films R. Tsekov,* P. Letocart, E. Evstatieva, and H. J. Schulze Max Planck Group for Colloids and Surfaces, TU Bergakademie, 09599 Freiberg, Germany Received January 20, 2002. In Final Form: April 26, 2002 A procedure for simulation of the dimple relaxation in wetting films is developed based on a precise treatment of the effects of ionic surfactants. The theory is compared with experimental data for the drainage of films of 1 mM solution of sodium dodecyl sulfate. An important effect of the interfacial electrostatics to reduce substantially the Marangoni effect is discovered. It is shown that the adsorption-pressure coupling is also important for ionic surfactants, which, enhanced by the decreased Marangoni number, increases the thinning rate close to the equilibrium. The latter could be a plausible explanation of the discrepancy in the mobility determined by drainage experiments and by dynamics of artificial waves on an equilibrium film from ionic surfactant solution.
Introduction The drainage of wetting films is a complicated process passing through several stages during its evolution. Initially, the thinning at the film border is much faster than that in the film center.1 Thus, the thickness at the barrier rim reaches quickly a value close to the equilibrium thickness he; this distinctive film profile is called a dimple. Dimples are nonequilibrium structures, and the next stage in film drainage is the dimple relaxation. Under the action of capillary and disjoining forces, the dimple shrinks until the film reaches a flat shape at equilibrium.2 In a previous paper3 we have considered wetting films from a 1 mM aqueous solution of KCl on a glass surface and have discovered an important effect of the adsorption of OHions on the film rheology. The present study aims to investigate further the effect of ionic surfactants on the dimple relaxation. For this reason, wetting films are formed on a glass surface from 1 mM aqueous solution of sodium dodecyl sulfate (SDS), which is an anionic surfactant. SDS contributes to the dimple evolution either through the Marangoni effect or through the disjoining pressure; the adsorption of DS- ions generates electrostatic disjoining pressure much larger than the van der Waals component at the considered concentration.4 In the film rheology5 the dependences of the surface tension on adsorption and of the disjoining pressure on film thickness are traditionally taken into account. It is shown,6 however, that the dependence of disjoining pressure on adsorption is also important for the film behavior, especially in the case of ionic surfactants. In the present paper an original procedure is employed to simulate the dimple relaxation at the presence of SDS, which develops further our previous considerations.3 By comparison of the theory with experimental data, the surface diffusion coefficient of DSions is determined as a single fitted parameter. Thus, important information about the interfacial properties of DS- adsorption layer is obtained such as the Marangoni (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) Tsekov, R.; Letocart, P.; Schulze, H. J. Langmuir 2000, 16, 8206. (4) Letocart, P.; Radoev, B.; Schulze, H. J.; Tsekov, R. Colloids Surf., A 1999, 149, 151. (5) Ivanov, I. B., Ed. Thin Liquid Films; Dekker: New York, 1988. Valkovska, D.; Danov, K. J. Colloid Interface Sci. 2001, 241, 400. (6) Tsekov, R.; Schulze, H. J.; Radoev, B.; Letocart, P. Colloids Surf., A 1998, 142, 287.
and adsorption-pressure numbers. It is shown that the interfacial electrostatics dramatically decrease the Marangoni number by suppressing the surface elasticity. This effect enhanced by the effect of the adsorption dependence of the disjoining pressure leads to an important increase of the mobility and thinning rate at small film thicknesses. General Theory of the Effect of Ionic Surfactants on the Film Drainage In the frames of the lubrication approximation the evolution of the local thickness H of an axisymmetric wetting film is governed by5
12η∂tH ) F-1∂F(FH3∂Fp - 6ηFHv)
(1)
where t is time, F is the radial coordinate, η is the liquid viscosity, p is the sucking pressure, and v is the interfacial velocity on the film free surface, while on the film-solid interface the nonslip boundary condition is involved. In the case of linear perturbation theory, the coupling between the liquid flow and ion distributions in the bulk is negligible and the local charge density q depends only on the electric potential φ. Hence, the electric force -q∂Fφ is a total differential, which is included in the pressure gradient in eq 1. Further applications require an expression for the interfacial velocity, which can be obtained by the mass balance of surfactants on the film-free surface. In water, ionic surfactants dissociate to counterions and surface-active ions. The latter adsorb on the air-water interface since they decrease its surface tension. However, due to the surface charge,7 created by the adsorption of surface-active ions, the counterions also attach to the adsorption layer to form a Stern layer.8 In the frame of a stationary approach, the mass balance of the ith kind of ions on the air-water surface is described by
F-1∂F(FΓiv) ) DisF-1∂F(FΓi∂Fµ˜ is/RT) - Dibcis∂zµ˜ ib/RT (2) where Γi is the adsorption, Dis and Dib are the surface and bulk diffusion coefficients, and cis is the subsurface concentration. Since the ions are charged species, the diffusion fluxes are generally expressed in eq 2 by the (7) Karakashev, S.; Tsekov, R.; Manev, E. Langmuir 2001, 17, 5403. (8) Kralchewsky, P.; Danov, K.; Broze, G.; Mehreteab, A. Langmuir 1999, 15, 2351.
10.1021/la025559m CCC: $22.00 © 2002 American Chemical Society Published on Web 06/19/2002
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surface µ˜ is and bulk µ˜ ib electrochemical potentials and thus all the specific interactions between the adsorbed species are taken into account. The normal gradient of the bulk electrochemical potential on the surface can be estimated from the stationary diffusion equation F-1∂F(Fci∂Fµ˜ ib) + ∂z(ci∂zµ˜ ib) ) 0 in the bulk. Since the films are thin and flat (H , R), the equation above can be approximated by F-1∂F(FHcis∂Fµ˜ ib) ) -H∂z(ci∂zµ˜ ib) ) -cis∂zµ˜ ib, where the condition of zero flux through the film-solid interface is also employed. Substituting this relation in eq 2 and integrating once the result after using the equality of the bulk and surface electrochemical potentials on the film free surface, one yields
v ) (Dis + HDibcis/Γi)∂Fµ˜ is/RT
(3)
Another important equation for the interfacial dynamics is the tangential force balance on the film-free surface5
ηv/H + H∂Fp/2 ) ∂Fσ + F
∑ziΓi∂Fφs ) -∑Γi∂Fµ˜ is
(4)
where σ is the surface tension, φs is the surface electric potential, and µis is the surface chemical potential. It is important to note that the contribution of electric forces reduces the effect of the surface tension gradient ∂Fσ ) -∑Γi∂Fµ˜ is in eq 4. Since the surface flow is compressible, the surface electric force is not a potential one in contrast to the bulk electric force. The combination of eq 3 and eq 4 allows obtaining an expression for the interfacial velocity as a function of the pressure gradient
v ) -H2∂Fp/2η(1 + Ma)
∑(RT)20κ/zi2F2η(Dis/H + Dibcis/Γi)
(6)
Introducing now eq 5 in eq 1, the latter changes to
12η∂tH ) F-1∂F[(4 + Ma)(1 + Ma)-1FH3∂Fp]
(7)
The further analysis requires expression for the film sucking pressure as a function of the local film thickness. It is supplied via the normal force balance -1
p ) pσ - σF ∂F(F∂FH) - ΠEL
∂Fp ) -σ∂F[F-1∂F(F∂FH)] - ΠEL′∂FH +
(8)
where the first term is the constant capillary pressure in the meniscus, the second term is the local capillary pressure due to the surface curvature, and the last term is the local electrostatic disjoining pressure. The latter is a function both of film thickness and of adsorption. The (9) Tian, Y.; Holt, R.; Apfel, R. J. Colloid Interface Sci. 1997, 187, 1.
∑∆ci∂Fµ˜ is
(9)
where ΠEL′ ) (∂HΠEL){µ˜ ib} and ∆ci ) -(∂ΠEL/∂µ˜ ib)H is the difference between ionic concentration in the film and meniscus. Substituting in eq 9 the gradient of electrochemical potentials from eq 3 completed by eq 5, the dependence of the pressure gradient from the film thickness follows
(1 + Ma + Ap)∂Fp ) -(1 + Ma){σ∂F[F-1∂F(F∂FH)] + ΠEL′∂FH} (10) where Ap ) ∑HRT∆ci/2η(Dis/H + Dibcis/Γi) is the adsorption-pressure number. In the frames of the linearized Boltzmann distribution, the concentration difference corresponds to an internal potential in the film φm ) -RT∆ci/ciziF. On the other hand the potential φm is related to the electrostatic disjoining pressure, ΠEL* ≡ (ΠEL){µ˜ ib} ) 0κ2φm2/2. Hence, in the case of negative potential the concentration difference equals ∆ci ) cizi(F/RTκ)(2ΠEL*/ 0)1/2 and the adsorption-pressure number acquires the form
Ap ) (5)
Here Ma ) ∑RTΓi/Riη(Dis/H + Dibcis/Γi) is the Marangoni number and Ri ) dµ˜ is/dµis. It is experimentally observed9 that the Marangoni number in solutions of ionic surfactants is much lower than that in solutions of nonionic surfactants. A reasonable explanation of this phenomenon is the electrostatic repulsion between ions on the surface, which in the present theory is accounted for by the factor Ri. Hence, the latter is expected to be a number much larger than 1. Indeed, using the electrostatic relation 0κφs ) F∑ziΓi between the surface potential and charge density, with κ-1 being the Debye length, one yields Ri ) 1 + (ziF/RT) dφs/d ln Γi ) 1 + zi2ΓiF2/0RTκ. This result shows that the Ri factor is of the order of 50 for a 1 mM solution of monovalent ions. Hence, neglecting the unity in the relation above, the Marangoni number reduces to
Ma )
dependence of ΠEL on H has been an object of many publications,5 while the adsorption dependence of the disjoining pressure is relatively less studied. The electrostatic component depends strongly on Γi since the adsorption of ionic surfactants alters the surface charge. Hence, the gradient of the pressure from eq 8 is equal to
∑ciziFH(ΠEL*/20)1/2/κη(Dis/H + Dibcis/Γi)
(11)
Substituting the pressure gradient from eq 10 in eq 7, one yields finally an equation describing the film thickness evolution
3η∂tH ) -F-1∂F{FMoH3[σ∂F(F-1∂F(F∂FH)) + ΠEL′∂FH]} (12) where all the specific adsorption effects are represented in the mobility number
Mo ) (1 + Ma/4)/(1 + Ma + Ap)
(13)
Note that in the absence of surfactants the mobility number is equal to 1. Simulation of the Dimple Evolution It is convenient to introduce two dimensionless variables Y ) H/he and x ) F/R, where he is the equilibrium film thickness and R is the film radius. Thus, eq 12 acquires the form
ˆY ∂ tY ) L
(14)
where the evolution operator L ˆ is given by the expression
L ˆ Y ) -(he2/3ηR4)x-1∂x{xMo(Y)Y3[σhe∂x(x-1∂x(x∂xY)) + R2ΠEL′∂xY]} (15) A rigorous treatment of the dimple relaxation via eq 14 requires four boundary conditions as well as an initial profile. The exact boundary conditions can be written only far away in the meniscus, where the lubrication theory is no longer applicable. Moreover, eq 14 is valid for the latest stage of the dimple relaxation and the relevant initial profile can only be experimentally specified. Hence, it is impossible to close the mathematical problem and the
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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 film profile can be, in general, represented as a x2 power expansion y ) ∑akx2k. Since x < 1, one is able to approximate the dimple in the film region by a finite series. Previous investigations3 and the present experimental data suggest that a good approximation is the biquadratic form y ) a0 + a1x2 + a2x4. It can be additionally specified by application of the two known conditions at the barrier rim: equilibrium thickness H(R) ) he and minimum of the film profile (∂FH)R ) 0. Thus, the biquadratic polynom acquires the form
y ) 1 + (y0 - 1)(1 - x2)2
(16)
The relative thickness y0 in the film center is the only unknown function of time in eq 16. Suppose at time t the film profile is given by eq 16. The profile at time t + τ can be calculated from eq 14, which in the limit of small τ can be rewritten as
Y(x,t+τ) ) y(x,t) + τL ˆ y(x,t)
(17)
Figure 1. The experimental evolution of the relative thickness Y ) H/he in the film center x ) 0 (b) and at x ) 0.625 (O) compared with the theoretical predictions (the solid lines).
Thus, the consequent profile Y can be generated, which satisfies eq 14 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 fit of the profile Y(x,t+τ). Hence, to calculate the film profile, a minimization of the square of the deviation of the two functions all over the film is required
δ
∫01 [Y(x,t+τ) - y(x,t+τ)]2x dx ) 0
(18)
This criterion combined with eq 16 and eq 17 leads to the following recurrent relation
y0(t+τ) ) y0(t) + 10τ
∫01 (1 - x2)2Lˆ [y0(t)(1 - x2)2 + x2(2 - x2)]x dx (19)
It is only a matter of integration to calculate the evolution of the thickness in the film center, which introduced in eq 16 will provide the whole film profile evolution. Application To check the theory, we have performed experiments on the dimple relaxation in 1 mM SDS solution. At this concentration the van der Waals component of the disjoining pressure is negligible and the only specific interaction is due to electrostatics. Since κH . 1, one can employ the exponential model of the electrostatic disjoining pressure
ΠEL* ) pσ exp[κ(he - H)]
Figure 2. The radial distribution of the relative thickness Y ) H/he after 0, 10, 20, and 30 s.
2HRTc/κηDDSs and Ap ) -H2(RTcΠEL*)1/2/2ηDDSs, respectively. As seen, the only specific characteristic of SDS here is the surface diffusion constant DDSs. Introducing these expressions and eq 20 in eq 13, one yields an explicit expression for the mobility number
Mo(Y) ) (2ηDDSsκ + cRTheY)/(2ηDDSsκ + 4cRTheY κ(cRTpσ)1/2he2Y2 exp[κhe(1 - Y)/2]) (21)
(20)
ΠEL′ ) -κpσ exp[κ(he - H)] Here all the specific electrostatic properties are conveniently expressed by pσ and he since at equilibrium the disjoining pressure is equal to the capillary pressure in the meniscus. Looking now at the Marangoni and adsorption-pressure numbers, one can neglect in eq 6 and eq 11 the contribution of Na+ ions since their diffusion coefficients are larger than those of DS- ions. In addition, since the subsurface concentration of DS- ions is low due to the electrostatic repulsion, the effect of the bulk diffusion is also negligible, DDSs/H . DDSbcDSs/ΓDS. Hence, the Marangoni and adsorption-pressure numbers are Ma )
Finally, introducing eq 21 in eq 15, the dynamic operator acquires the form L ˆY ) -
he3
∂x × 3ηR4x x(2ηDDSsκ + cRTheY)Y3(σ∂x(x-1∂x(x∂xY)) - κR2pσ exp[κhe(1 - Y)]∂xY] 2ηDDSsκ + 4cRTheY - κ(cRTpσ)1/2he2Y2 exp[κhe(1 - Y)/2]
(22) As seen, to simulate the dimple evolution several specific characteristics are required. At temperature 25 °C and 1
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Figure 3. The radial distribution of the thinning rate -∂tH after 0, 10, 20, and 30 s.
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Figure 5. The radial distribution of the relative sucking pressure p/pσ after 0, 10, 20, and 30 s.
Figure 6. The dependence of the mobility number on the relative thickness Y ) H/he.
Figure 4. The radial distribution of the interfacial velocity v after 0, 10, 20, and 30 s.
mM SDS concentration the surface tension σ equals10 68.5 mN/m, the viscosity η is 0.9 mPa s, and the Debye length κ-1 is 10 nm. The capillary pressure in the meniscus during the experiments is kept at pσ ) 257 Pa. At these conditions the radius R of the films is equal to 120 µm, the experimental equilibrium thickness he of the films is 56 nm, and the initial thickness in the film center h0 is 251 nm. The only unknown parameter in eq 22 is the surface diffusion coefficient DDSs. The evolution of the thickness h0 and the thickness at x ) 0.625 is monitored by the classical light interference method.4 In Figure 1 average experimental data for the drainage of five films are presented. The simulation by eq 19 is plotted with time step τ ) 1 s and DDSs ) 250 µm2/s. As seen, there is good juxtaposition between the theory and experiment. The surface diffusion coefficient DDSs is about two times lower than the bulk diffusion coefficient reported in the literature. This apparent discrepancy is due, however, to our definition of the diffusion flux through the electrochemical potentials. Thus, DDSs is the bare diffusion constant, while (10) Mysels, K. Langmuir 1986, 2, 423.
the measurable diffusion coefficient at low adsorption equals to RDSDDSs. The latter is much larger than the diffusion bulk coefficient as a consequence of the strong electrostatic repulsion of the ions on the surface. The evolution of the dimple calculated from eq 16 is plotted in Figure 2. In Figure 3 the local thinning rate -∂tH is presented at different moments. Initially, it is nearly constant in time but decreases from the film center to the rim. More interesting is the behavior of the interfacial velocity, which is presented in Figure 4. As seen, the velocity close to the barrier rim is much higher than that in the film center. This is due to the large gradient of the sucking pressure close to the rim as it is presented in Figure 5. In time the pressure goes down and the interfacial velocity tends to zero everywhere in the film. To estimate the importance of the interfacial rheology, the dependence of the mobility number on the film thickness is presented in Figure 6. For film thickness larger than 100 nm, Ap is small and Ma is large. Hence, the mobility number is close to 1/4, which corresponds to a tangentially immobile surface. If the film thickness decreases below 100 nm, the negative adsorption-pressure number decreases as well as the Marangoni number (see Figure 7). Thus, the mobility number increases and close to the equilibrium thickness Mo is as twice as larger than the initial value. Therefore, the coupling between the
Dimple Relaxation in Wetting Films
Figure 7. The dependence of the Marangoni and adsorptionpressure numbers on the relative thickness Y ) H/he.
disjoining pressure and adsorption is important and leads to substantial increase of the thinning rate at the end stage of the film drainage. It could be a plausible explanation of the discrepancy in the mobility determined by drainage experiments and by dynamics of artificial waves4 on an equilibrium film from ionic surfactant solution. Glossary Ap ci ∆ci cis Dib Dis
adsorption-pressure number bulk concentration difference between concentration in the film and meniscus subsurface concentration bulk diffusion coefficient surface diffusion coefficient
Langmuir, Vol. 18, No. 15, 2002 5803 DS0 F φ φs Γi H he h0 κ Ma µ˜ ib µ˜ is µis Mo η F p pσ ΠEL q R σ SDS t v x y
dodecyl sulfate anions dielectric constant Faraday constant electric potential surface potential adsorption local film thickness film equilibrium thickness thickness in the film center reciprocal Debye length Marangoni number bulk electrochemical potential surface electrochemical potential surface chemical potential mobility number liquid viscosity radial coordinate sucking pressure capillary pressure in the meniscus electrostatic disjoining pressure charge density film radius surface tension sodium dodecyl sulfate time interfacial velocity dimensionless radial coordinate dimensionless film thickness
Acknowledgment. R.T. and E.E. are grateful to the Alexander von Humboldt Foundation for a grant. LA025559M
