Surface Forces and Liquid Films (Continued)

1. Surface Forces and Liquid Films (Continued) Krassimir D. Danov Department of Chemical Engineering, Faculty of Chemistry Sofia University, Sofia, Bulgaria Lecture at COST D43 School Fluids and Solid InterfacesSofia University, Sofia, Bulgaria 12 – 15 April,2011 Oscillatory structural forces measured by colloid probe AFM. Sofia University

2. (1) Van der Waals surface force: The Hamaker parameter, AH, depends on the film thickness, h, because of the electromagnetic retardation effect [1,4]. The expression for AH reads [4]: νe = 3.0 x 1015 Hz – main electronic absorption frequency;hP = 6.6 x 10– 34 J.s – Planck’s const; c0 = 3.0x 108 m/s – speed of light in a vacuum.

3. (2)Electrostatic (Double Layer) Surface Force (General Approach) Poisson equation in the film phase relates the electrostatic potential, y, to the bulk charge density, rb [2,5,7]: All ionic species in the bulk with concentrations, nj, follow the Boltzmann distribution (constant electro-chemical potentials): where q is the elementary charge, zj is the charge number, nj0 is the input concentration. The bulk charge density, rb is [2,5]: The first integral of the Poisson-Boltzmann equation reads: where p is the local osmotic pressure Eq. (2.1) In the case of symmetric films the electrostatic disjoining pressure (repulsion), Pel, is defined as a difference between the pressure in the film midplane, pm, and that at large film thicknesses, p0 [5]: Eq. (2.2)

4. (2)Electrostatic (Double Layer) Surface Force (General Approach) For constant surface potential, ys, ys and h are known and ym is calculated from: Eq. (2.3) The surface charge density, rs, is calculated from the charge balance at the film surface: Eq. (2.4) where ps is the osmotic pressure in the subsurface phase (at y = ys). For constant surface charge the system of equations, Eqs. (2.1), (2.3), and (2.4), is solved numerically to obtainys and ym. Charge regulation.In this case the surface charge density, rs, relates the surface potential through the condition of constant electro-chemical potentials [6] and Counterion binding Stern isotherm (KSt– Stern constant) leads to the equation For example: For (1:1) surface active ion “1” and counterion “2” with adsorptionsG1andG2

5. (3)Equilibrium Film Thicknesses, h0: Theory vs. Experiment [8] Sodium dodecyl sulfate (SDS) - NaC12H25SO4, CMC 8 mM Cetyl-trimethylammonium bromide (CTAB) - (C16H33)N(CH3)3Br, CMC 0.9 mM Cetyl-pyridinium chloride (CPC) - (C21H38NCl), CMC 1.0 mM

6. (3)Disjoining Pressure Isotherms: Theory vs. Experiment [21] Setup for measurement of disjoining pressure, (h), isotherms (Mysels-Jones porous plate cell [9]). Sodium dodecyl sulfate (SDS) Hexa-trimethylammonium bromide (HTAB)

7. (3)Disjoining Pressure Isotherms: Experiments – no Theory For small concentration of ionic surfactants the DLVO theory cannot explain experimental data.

8. (3) Colloidal – Probe AFM Measurements of Disjoining Pressure [10] Force, F, in nN for 80 mM Brij 35. Micelle volume fraction 0.257. Force/Radius, F/R, in mN.m-1 for 133 mM Brij 35. Micelle volume fraction 0.401. The aggregation number of micelles is 70. The solid lines are drawn without adjustable parameters (formulas by Trokhimchuk et al.[11]).

9. (4) Hydrodynamic Interaction in Thin Liquid Films [2,3] Two immobile surfaces of a symmetric film with thickness h(t,r) approach each others with velocity U(t). Rf is the characteristic film radius. where: t is time; r and z are the radial and vertical coordinates. Simplest version of the lubrication approximation (h<<Rf): Continuity equation: Momentum balance equation is simplified to: Simple solution: Hydrodynamic force, F: P(h) is the disjoining pressure, which accounts for the molecular interactions in the film.

10. (4) Taylor vs. Reynolds regimes [2,3] In the case of two spheres (Taylor) [12]: where hin is the initial thickness and hcr is the final critical film thickness. The life time can be defined as: where g is the gravity constant and Dr us the density difference. In the case of buoyancy force: The life time decreases with the increase of drop radii. For two disks (Reynolds) [13]: In the case of buoyancy force : The life time increases with the increase of drop radii.

11. (4) Taylor vs. Reynolds regimes Taylor regime Dickinson experiments for the life time of small drops (β-casein, κ-casein or lysozyme, 10–4 wt% protein + 100 mM NaCl, pH=7) [14]. Our experiments for the life time of small and large drops [15] (4x10-4 wt% BSA + 150 mM NaCl, pH=6.4). Strong dependence of the drops life time on the drop and film radii for tangentially immobile film surfaces.

12. (4) Lubrication Approximation and Film Profile [2,16] Two immobile surfaces of a symmetric film with thickness h(t,r) approach each others. The film profile changes with time and pm is the pressure in the meniscus. Simple solution: Continuity equation: Normal stress boundary condition: Film-profile-evolution equation (stiff nonlinear problem): The applied force is given by the expression:

13. (4) Study of Drainage and Stability of Small Foam Films Using AFM Microscopy photographs of bubbles in the AFM with schematics of the two interacting bubbles and the water film between them [17]: (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, Rs. (E) Schematic of the bubble geometry. Evolution of film profiles and rim rupture effect.

14. (5) Interfacial Dynamics and Rheology – Complex Boundary Conditions The velocities of both phases are equal at liquid/liquid interface S: The jump of bulk forces at S are compensated by the total surface forces: where Ts is the surfaceviscous stress tensor. Capillarypressure Marangonieffect Surface viscosityeffect For Newtonian interfaces (Boussinesq – Scriven law) [16]: where: Is is the surface idem factor;hdil – surface dilatational viscosity;hsh – surface shear viscosity. Rate of relative displacement of surface points

15. (5) Lubrication Approximation for Complex Fluids in the Films [18] The film phase contains one surfactant with bulk concentration, c, adsorption, G, and interfacial tension, s. Integrated-surfactant-mass-balance equation: cs – the subsurface concentration, u – the surface velocity, the mean velocity is defined as: For slow processes the deviations of concentrations and adsorptions are small and Adsorption length (known from the adsorption isotherm) The larger bulk and surface diffusivities lead to larger surface velocity (mobility)! Continuity equation for mobile surfaces:

16. (5) Lubrication Approximation for Complex Fluids in the Film [19] Tangential stress boundary condition (ms = mdil+msh – total interfacial viscosity): viscous friction(film phase) viscous friction(drop phase) Marangonieffect Boussinesqeffect For slow processes the Marangoni term has an explicit form and The Gibbs elasticity, EG, is known from the surface equation of state or from independent rheological experiments. The larger Gibbs elasticity and surface viscosity suppress the surface mobility! Normal stress boundary condition closes the problem for film evolution in time:

17. (5) Role of Surfactant on the Drainage Rate of Thin Films [19] In the case of two spheres(Taylor velocity): In the case of two plates(Reynolds velocity): Two truncated spheres In the case of surfactants for this geometry we have: characteristic surfacediffusion length bulk diffusivitynumber dimensionless film radius In the case of emulsion plane parallel films: In the case of two spherical drops:

18. (5) Inverse Systems – Surfactants in the Disperse Phase In this case the diffusion fluxes from the disperse phase are large enough to suppress the Marangoni effect and [3,20] where: rc – the density of liquid in the film phase;Fs – force arising from the disjoining pressure;d – characteristic thickness of the boundary layer in the drop phase. Surface active componentsin the disperse phase Surfactant in the continuous phase:0.1M lauryl alcohol (1); 2 mM C8H18O3S (2). Surfactantin the dispersephase (benzene films): C8H18O3S 0 mM (1); 0.1 mM (2); 2 mM (3). Film life time diagram Film life timediagram

19. Basic References 1. J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1992. 2. 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, 2004, pp. 1–38. 3. 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, 2008; pp. 197-377. Additional References 4. W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions,Cambridge Univ. Press, Cambridge, 1989. 5. B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum Press: Consultants Bureau, New York, 1987. 6. P.A. Kralchevsky, K.D. Danov, G. Broze, A. Mehreteab, Thermodynamics of ionic surfactant adsorption with account for the counterion binding: effect of salts of various valency, Langmuir 15(7) (1999) 2351–2365.

20. 7. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Butterworth-Heinemann, Oxford, 2004. 8. K.D. Danov, E.S. Basheva, P.A. Kralchevsky, K.P. Ananthapadmanabhan, A. Lips, The metastable states of foam films containing electrically charged micelles or particles: Experiment and quantitative interpretation, Adv. Colloid Interface Sci. (2011) – in press. 9. Mysels, K. J.; Jones, M. N. Direct Measurement of the Variation of Double-Layer Repulsion with Distance.Discuss. Faraday Soc. 42 (1966) 42-50. 10. N.C. Christov, K.D. Danov, Y. Zeng, P.A. Kralchevsky, R. von Klitzing, Oscillatory structural forces due to nonionic surfactant micelles: data by colloidal-probe AFM vs. theory, Langmuir 26(2) (2010) 915–923. 11. A. Trokhymchuk, D. Henderson, A. Nikolov, D.T. Wasan, A Simple Calculation of Structural and Depletion Forces for Fluids/Suspensions Confined in a Film, Langmuir 17 (2001) 4940-4947. 12. In fact, this solution does not appear in any G.I. Taylor’s publications but in the article by W. Hardy, I. Bircumshaw, Proc. R. Soc. London A 108 (1925) 1it was published.

21. 13. O. Reynolds, On the theory of lubrication, Phil. Trans. Roy. Soc. (Lond.) A177 (1886) 157234. 14. E. Dickinson, B.S. Murray, G. Stainsby, Coalescence stability of emulsion-sized droplets at a planar oil-water interface and the relationship to protein film surface rheology, J. Chem. Soc. Faraday Trans. 84 (1988) 871883. 15. T. D. Gurkov, E. S. Basheva, Hydrodynamic behavior and stability of approaching deformable drops, in: A. T. Hubbard (Ed.),Encyclopedia of Surface & Colloid Science, Marcel Dekker, New York, 2002. 16. D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology, Butterworth-Heinemann, Boston, 1991. 17. 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, PNAS 107 (2010) 11177. 18. I.B. Ivanov, D.S. Dimitrov, Thin film drainage, Chapter 7, in: I.B Ivanov (ed.), Thin Liquid Films, M. Dekker, New York, 1988.

22. 19.K.D. Danov, D.S. Valkovska, I.B. Ivanov, Effect of surfactants on the film drainage, J. Colloid Interface Sci. 211 (1999) 291–303. 20. I.B. Ivanov, Effect of surface mobility on the dynamic behavior of thin liquid film, Pure Appl. Chem. 52 (1980) 12411262. 21. K.D. Danov, E.S. Basheva, P.A. Kralchevsky, The Hydration Surface Force – an Effect Due to the Discreteness and Finite Size of Surface Ions and Bound Counterions. Curr. Opin. Colloid Interface Sci. (2011) – a manuscript in preparation.