Dynamics of Surfactant Adsorption and Film Stability

Transcription

1 Dynamics of Surfactant Adsorption and Film Stability Krassimir D. Danov Department of Chemical Engineering, Faculty of Chemistry Sofia University, Sofia, Bulgaria Lecture at COST P1 Student Training School Physics of droplets: Basic and Advanced Topics 1 13 July, 010, Borovets, Bulgaria Universal surface age, t u = t age /λ (ms) Surface tension, γ (mn/m) SDS 10 mm NaCl (MBPM) 7 mm 10 mm 15 mm 0 mm 30 mm 50 mm 100 mm Example: Dynamic surface tension measured by maximum bubble pressure method (MBPM) (b) Nominal surface age, t age (ms) Sofia University

2 The dynamics of surface tension correlates to the foam capacity and foamability (Examples) 0 μm The initial decrease, dσ/dt, of surface tension (Na-cas, natural ph) [5]. Foam capacity and foamability of Na-cas and WPC solutions vs. dσ/dt [5].

3 The dynamics of surface tension is related to the film lifetime and foam viscous friction (Examples) Dimensionless foam viscous stress as a function of capillary number for different surfactant solutions [6]. Film lifetime, foamability, and rate of dynamic surface tension decrease for WPC solutions as a function of ph [5].

4 Surface (total) and adsorption layer (local) deformations The surface element with an area A is extended to a new area A δa for a time interval from t to t δt. The surface dilatation, α, and the rate of surface dilatation are defined as [4]: δ A α 1 δ t α α dt A δ t 0 The adsorption changes from Γ to Γ δγ and the area per molecule in the adsorption layer from a to a δa. The adsorption layer deformation and its rate [4]: ε 1 δ a a δ t δ t ε dt ε 0 For insoluble surfactants the both deformations α and ε are equal. For soluble surfactants always the adsorption layer deformation is smaller than the surface dilatation, α > ε, because of the bulk diffusion of surfactants.

5 Surfactant mass balance equations in the bulk phase c t ( cv) j = r c bulk surfactant concentration; v fluid velocity; j diffusion flux; r source term accounting for the possible chemical reactions [1,,4]. convective flux bulk diffusion flux The Fick law of diffusion: j = D c where D is the diffusion coefficient (diffusivity). The diffusion flux is proportional to the gradient of chemical potential. Thus, for ionic surfactants [7]: j i = D i Z ec ( c i i i ψ ) kt where kt is the thermal energy, Z i is the valency of the respective ion, ψ is the electric potential. At equilibrium the diffusion flux is equal to zero and the Boltzmann distribution takes place [7]: Z e c c exp( i i = i ψ ) kt

6 Surfactant mass balance equations at surfaces Γ s ( Γvs) s js = rs < t j n > Γ adsorption; v s surface velocity; j s surface diffusion flux; r s surface source term accounting for the possible chemical reactions [1,]. surface convective flux surface diffusion flux surfactants bulk fluxes to the surface The Fick law of diffusion: j s D Γ = s where D s is the surface diffusion coefficient. Generally, the surface diffusion flux is proportional to the gradient of the surface chemical potential. For simplest surface reaction Γ1 Γ one has [8]: r1 = r = k Γ1 k Γ For surface association reaction Γ1 Γ Γ3 we have [8]: r1 = r = r3 = k Γ1Γ k Γ3 The rate constant of forward reaction is k and that of the reverse reaction is k -.

7 Solution of diffusion equation for simple surfactant For simple surfactant solution below the CMC when the surface layer is in quasiequilibrium with the contiguous bulk phase (diffusion-controlled adsorption processes): t Dt D c ( ) ( ) (0) s τ Γ t = Γ cb dτ and Γ ( t) = Γ ( cs ) π π t τ 0 (Ward and Tordai [4,9]) where c b is the input concentration, c s is the subsurface concentration, and Γ(c s ) is the adsorption isotherm. General asymptotic expression for the short times reads: Analogous asymptotic expression for the long times is: Γ t ) = Γ (0) [ c c (0)] ( b s c ( t) c e s = b Γ Γ (0) πdt Dt π where Γ e is the equilibrium value of the surfactant adsorption, that is Γ e = Γ(c b ). For both asymptotic solutions the two-dimensional equation of state, γ = γ(γ) or γ = γ(c s ), relates the measured surface tension and time.

8 Diffusion-controlled adsorption (experimental examples) From the linear plot of the surface tension vs. t -1/ we can obtain the equilibrium surface tension (the intercept is mn/m) and the slope, which is related to the adsorption at equilibrium. Surface tension vs. time for 6 different bubbles formed in 0.05 mm Brij-58 solution. The bubbles are formed very fast and after their formation the bubble volumes are kept constant.

9 Diffusion-controlled adsorption (experimental examples) From the linear plot of the surface tension vs. t 1/ one checks the mechanism of adsorption processes. The linear plot shows that we start from a clean surface the intercept is 7.18 mn/m, which corresponds to zero initial adsorption. From the slope one can calculate the value of the diffusion coefficient. γ ( t) = γ (0) ktc b Dt π The calculated diffusion coefficient from the slope is.6x10-9 m /s, which 6 times greater than the diffusion coefficient of the individual molecule. Conclusion: For some surfactant systems the mechanism of adsorption above the CMC is diffusion-controlled but with effective diffusion coefficient larger than the diffusion coefficient of the individual molecule (see below).

10 Barrier-controlled adsorption (experimental examples) The adsorption is under barrier control when the stage of surfactant transfer from the subsurface to the surface is much slower than the diffusion stage. In this case the change of adsorption is controlled by the rate of adsorption, r ads, and the rate of desorption, r des [1,3]: dγ = dt rads( Γ, cs ) rdes( Γ ) The solution of the diffusion equation for this regime is: γ ( t) = γ e [ γ (0) γ e]exp( t / tb) The asymptotic expression for the short times is: γ (0) γ γ ( t) = γ (0) e t b t Conclusion: For barrier-controlled regime of adsorption from the initial slope of the surface tension vs. time one obtains the relaxation time of surfactant, t b.

11 Dynamics of adsorption from micellar surfactant solutions The micelles release monomers to restore the equilibrium concentrations of surfactant monomers at the surface and in the bulk. The concentration gradients give rise to diffusion of both monomers and micelles. The adsorbing component are the surfactant monomers, whereas the micelles do not adsorb. Typical distribution of micellar aggregates: m the mean aggregation number; σ the polydispersity of micelles; c 1 concentration of monomers [11,1] (Aniansson and Wall). Fast process: exchange monomers between micelles and surrounding solution. Slow process: decomposition of micelles with critical micellar sizes.

12 Maximum bubble pressure method Consecutive photographs of a bubble growing at the tip of a capillary hydrophobized by treatment with silicon oil. One measures the maximum of pressure difference inside the bubble, which is directly related to the dynamic surface tension through the Laplace equation. The experimental curve, A(t), is known [3,13].

13 Maximum bubble pressure method (universal time age) Bulk diffusion of simple surfactant... = z c D z c z t c α Mass balance at the interface... d d = z c D t α Γ Γ c bulk concentration; Γ adsorption; t time; z space coordinate; dα/dt rate of surface dilatation; A(t) area of the surface Time and space transformation [3,13,14] t t A t A z A t A y 0 ~ d (0) ) ~ ( (0) ) ( τ Bulk diffusion (classical problem)... = y c D c τ Mass balance (classical problem)... ] (0) ) ( [ d d = y c D A A Γ τ τ

14 Maximum bubble pressure method (universal time age) Apparent surface age: t age Real (universal) surface age t real < t age [13]: treal tage 1 A ( ξ ) / dξ A (0) 0

15 Maximum bubble pressure method (universal time age) Experimental curves are processed with equation [1,13]: For slow diffusion process in micellar solutions [10,11]: γ = γ eq a (t sγ age 1/ ) D D eff = σ σ D ( 1 β )(1 β m ) m m D From s γ we calculate the value of the effective diffusion coefficient. where D m is the diffusion coefficient of micelles.

16 Effective diffusion coefficient of ionic surfactants In the case of ionic surfactants the dynamics of surface tension obeys the same rules as for nonionic surfactants but with an effective diffusion coefficients, D eff [15]: D = D eff eff ( D1, D, D3, concentrations) where D 1 is the diffusion coefficient of surface active ion, D of counterions, and D 3 of coions.

17 Typical consecutive stages of evolution of thin liquid films [4,16] a) mutual approach of slightly deformed surfaces b) the curvature at the film center inverts its sign and a "dimple" arises c) the dimple disappears and an almost plane-parallel film forms d) due to thermal fluctuations or other disturbances the film either ruptures or transforms into a thinner Newton black film e) Newton black film expands f) the final equilibrium state of the Newton black film is reached

18 DLVO Theory: Equilibrium states of free liquid films DLVO = Derjaguin, Landau, Verwey, Overbeek [1]: Π ( h) = Π el( h) Π vw ( h) Born repulsion Electrostatic component of disjoining pressure [1]: Π el( h) = Bexp( κ h) (repulsion) Van der Waals component [1]: A Π ( ) H vw h = (attraction) 3 6π h h film thickness; A H Hamaker constant; κ Debye screening parameter () Secondary film (1) Primary film Π > 0 h Π < 0 h unstable equilibrium state stable equilibrium state

19 Role of surfactants on the surface mobility [1,,16,17] Fluid motion in the film and droplet phases. The flow in the film phase is superposition of the Poiseuille flow and a flow of constant surface velocity u s. Due to the nonuniform interfacial surfactant distribution surface diffusion and convective fluxes appear. The bulk viscous stresses are balanced with the surface stresses. The gradient of surface tension (Marangoni effect) and the surface viscosity (Boussinesq-Scriven effect) suppress the interfacial mobility. Both effects characterize the interfacial rheology! (Interfacial rheology measurements) The bulk and surface diffusion suppress the gradient of surface tension and increase the mobility of interfaces! (Dynamics surface tension measurements) The mobility of film surfaces is important for a film (foam) drainage and foam rheology! As a rule the film stability is controlled by the disjoining pressure (small effect of the surface mobility is measured)!

20 Stability film thickness (example for foam films surface fluctuations) Symmetric (squeezing) modes Anti-symmetric (bending) modes The film thickness, h, is presented as a superposition of equilibrium thickness, h e, and fluctuations, h f : h = he hf and hf exp( ωt) J0( k k is the dimensionless Fourier-Bessel wave number. For positive (negative) ω the disturbances will grow (decay). From mass and momentum balance equations one derives the dispersion relationship []: 3 γ he R Π ωh Ω = [ k ](1 F ) and e s Ω = 48η DR γ h 4D for squeezing modes F s mobility function; D, D s bulk and surface diffusivity; η, η s bulk and surface viscosity; h s, E G adsorption thickness and Gibbs elasticity; R film radius: 1 ηd he Dsk he ηs 1 ds =, Bs = Ω Ω tanh Ω, F s = dsbs (1 ) he EG hs 4DR 3η he r R ) The stability film thickness, h st, is defined as a thickness at which ω = 0: Π = h γk R

21 Stability film thickness (example for foam films surface fluctuations) Ω = 3 γ he 48η DR [ R γ Π k ](1 h Fs ) General conclusions: Π < 0 h stable fluctuations. The surfactants influence the decrement of surface waves! Π > 0 Two possible cases: a) unbounded films; b) bounded films. h Π γk < stable fluctuations Unbounded films h R the stability depends on the wave number: Π < h γk unstable fluctuations R R Π Stability limit of bounded films: = j h γ where j 1 is the first root of the Bessel function, J 0. For example: AH Π A Π = Π H vw = and = 3 4 6πh h πh AH R 1/ 4 AH R 1/ 4 hst = ( ) ( ) πj γ 1 γ

22 Ω Stability, transitional and critical film thicknesses [,18] 3 γ he R Π = [ k ](1 Fs ) Stability thickness the first unstable mode appears. 48η DR γ h Generally, with the decrease of the film thickness, dπ/dh increases and the region of unstable modes becomes wider. Finally, the film breaks at a given critical film thickness, h cr, with the most dangerous wave number, k cr. The film thickness at which the most dangerous mode appears, Ω(k cr ) = 0, is called transitional film thickness, h tr. Critical thickness of foam films stabilized with 0.4 mm SDS in the presence of NaCl. 50 mm NaCl 100 mm NaCl

23 Other possibilities for bubble (drop) coalescence (numerical results) Red curves: In the presence of VdW component. The film ruptures in its central zone pimple instability. Blue curves: In the presence of VdW and electrostatic components. The film ruptures at its periphery rim instability. Evolution of film profile in the presence of different components of disjoining pressure [19]. Example: For raising bubble under the action of buoyancy force, the thickness at which the pimple appears, h p, is calculated from [,0]: hp AH R 1/ 4 3 = ( ) where Fb = πr gδρ 1F 3 b

24 Bubbles in the AFM [1] Microscopy photographs of bubbles in the AFM with schematics of the two interacting bubbles and the water film between them: (A) Side view of the bubble anchored on the tip of the cantilever. (B) Plan view of the custom-made cantilever with the hydrophobized circular anchor. (C) Side perspective of the bubble on the substrate. (D) Bottom view of the bubble showing the dark circular contact zone of radius, a (in focus) on the substrate and the bubble of radius, R s. (E) Schematic of the bubble geometry. Evolution of film profiles and rim rupture effect.

25 Waves at a single interface [,] k is the wave number; ω is the frequency; η s is the sum of dilatational and shear surface viscosity; g is the gravity acceleration; ρ is the bulk density; n is the vertical complex wave number and E * is the complex surface elasticity. Dispersion relationship for simple surfactant solution: 0 )] ( )[ ( )] ( ) ( )[ ( s * 3 3 s * 3 = ωη ω η ρ γ ωη ωη ω η ρ γ ω η ρω i E ik k gk k kn i i E nk k n i gk i k i k i (Lucassen and van den Tempel[3,4]), 1 1, / s G * h D q q q q i q E E i k n ω η ωρ = = = Capillary waves: for damped waves 3 gk k k i ρ γ ω η ρω = for non - damped waves 3 γk ρgk ρω = Pure longitudinal waves (longitudinal wave techniques): ) ( ) ( s * ωη ω η i E nk n k i = Non-damped waves at the surface between two liquids: gk k B A B A ) ( ) ( 3 ρ ρ γ ω ρ ρ =

26 Systems with linear and nonlinear response (interfacial rheology experiments interface with adsorbed surfactants) Input signal α ( t) = α0 sin( ω t) (surface dilatation) Output signal (interfacial tension) γ ( t) γ e = Er ( ω ) α0 sin( ω t) Ei ( ω ) α0 cos( ω t) For linear systems: E r (ω) elastic module and E i (ω) loss module and for non-linear systems: E r (ω,α 0 ) and E i (ω,α 0 )! α Linear system (arbitrary input) Input signal (surface dilatation) ( t) = α0( ω)sin( ω t) dω 0 Output signal (interfacial tension) γ ( t) γ = G ( ω)sin( ω t)dω G ( ω)cos( ω t) dω e 0 r Complex elasticity E * = Er iei 0 i For linear systems: E r (ω) = G r (ω)/α 0 (ω) and E i (ω) = G i (ω)/α 0 (ω)

27 Basic References 1. P.A. Kralchevsky, K.D. Danov, N.D. Denkov. Chemical physics of colloid systems and Interfaces, Chapter 7 in Handbook of Surface and Colloid Chemistry", (Third Edition; K. S. Birdi, Ed.). CRC Press, Boca Raton, 008; pp K.D. Danov, P.A. Kralchevsky, I.B. Ivanov, Equilibrium and dynamics of surfactant adsorption monolayers and thin liquid films, in: G. Broze (Ed.), Handbook of Detergents. Part. A: Properties, Marcel Dekker, 1999, pp S.S. Dukhin, G. Kretzschmar, R. Miller, Dynamics of adsorption at liquid interfaces. Theory, experiment, application, Elsevier, Amsterdam, K.D. Danov, Effect of surfactants on drop stability and thin film drainage, in: V. Starov, I.B. Ivanov (Eds.), Fluid Mechanics of Surfactant and Polymer Solutions, Springer, New York, 004, pp

28 Additional References 5. K.G. Marinova, E.S. Basheva, B. Nenova, M. Temelska, A. Y. Mirarefi, B. Campbell, I.B. Ivanov, Physico-chemical factors controlling the foamability and foam stability of milk proteins: Sodium caseinate and whey protein concentrates, Food Hydrocolloids 3 (009) N.D. Denkov, S. Tcholakova, K. Golemanov, K.P. Ananthpadmanabhan, A. Lips, The role of surfactant type and bubble surface mobility in foam rheology, Soft Matter 5 (009) W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge Univ. Press, Cambridge, I.B. Ivanov, K.D. Danov, K.P. Ananthapadmanabhan, A. Lips, Interfacial rheology of adsorbed layers with surface reaction: on the origin of the dilatational surface viscosity, Adv. Colloid Interface Sci (005) A.F.H. Ward, L. Tordai, J. Chem. Phys. 14 (1946) K.D. Danov, D.S. Valkovska, P.A. Kralchevsky, Adsorption relaxation for nonionic surfactants under mixed barrier diffusion and micellization diffusion control, J. Colloid Interface Sci. 51 (00) 18 5.

29 11. K.D. Danov, P.A. Kralchevsky, N.D. Denkov, K.P. Ananthapadmanabhan, A. Lips, Mass transport in micellar surfactant solutions. 1. Relaxation of micelle concentration, aggregation number and polydispersity, Adv. Colloid Interface Sci. 119 (006) A.E.G. Aniansson, S.N. Wall, J. Phys. Chem. 78 (1974) N.C. Christov, K.D. Danov, P.A. Kralchevsky, K.P. Ananthapadmanabhan, A. Lips, Maximum bubble pressure method: universal surface age and transport mechanism in surfactant solutions, Langmuir (006) P. Joos, Dynamic Surface Phenomena, VSP BV, AH Zeist, The Netherlands, K.D. Danov, P.A. Kralchevsky, K.P. Ananthapadmanabhan, A. Lips, Influence of electrolytes on the dynamic surface tension of ionic surfactant solutions: expanding and immobile interfaces, J. Colloid Interface Sci. 303 (006) Thin Liquid Films (I.B. Ivanov, Ed.) Marcel Dekker, I.B. Ivanov, Pure Appl. Chem. 5 (1980) D.S. Valkovska, K.D. Danov, I.B. Ivanov, Stability of draining plane parallel films containing surfactants, Adv. Colloid Interface Sci. 96 (00) S.S. Tabakova, K.D. Danov, Effect of disjoining pressure on the drainage and relaxation dynamics of liquid films, J. Colloid Interface Sci. 336 (009)

30 0. D.S. Valkovska, K.D. Danov, I.B. Ivanov, Surfactants role on the deformation of colliding small bubbles, Colloids Surfaces A 156 (1999) I.U. Vakarelski, R. Manica, X. Tang, S.Y. O Shea, G.W. Stevens, F. Grieser, R.R. Dagastine, D.Y.C. Chan, Dynamic interactions between microbubbles in water, 107 (010) M.G. Hedge, J.C. Slattery, J. Colloid Interface Sci., 35 (1971) J. Lucassen, M. van den Tempel, Chem. Eng. Sci. 7 (197) J. Lucassen, M. van den Tempel, J. Colloid Interface Sci. 41 (197) 491.