Surface Forces and Liquid Films (Continued) Sofia University Oscillatory structural forces measured by colloid probe AFM.

(1) Van der Waals surface force: The Hamaker parameter, A H, depends on the film thickness, h, because of the electromagnetic retardation effect [1,4]. The expression for A H reads [4]:

(2) Electrostatic (Double Layer) Surface Force (General Approach) Poisson equation in the film phase relates the electrostatic potential, , to the bulk charge density,  b [2,5,7]: All ionic species in the bulk with concentrations, n j, follow the Boltzmann distribution (constant electro-chemical potentials): where q is the elementary charge, z j is the charge number, n j0 is the input concentration. The bulk charge density,  b is [2,5]: The first integral of the Poisson-Boltzmann equation reads: where p is the local osmotic pressure In the case of symmetric films the electrostatic disjoining pressure (repulsion),  el, is defined as a difference between the pressure in the film midplane, p m, and that at large film thicknesses, p 0 [5]: Eq. (2.1) Eq. (2.2)

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

(3) Equilibrium Film Thicknesses, h 0 : Theory vs. Experiment [8] Sodium dodecyl sulfate (SDS) - NaC 12 H 25 SO 4, CMC 8 mM Cetyl-trimethylammonium bromide (CTAB) - (C 16 H 33 )N(CH 3 ) 3 Br, CMC 0.9 mM Cetyl-pyridinium chloride (CPC) - (C 21 H 38 NCl), CMC 1.0 mM

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

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

(3) Colloidal – Probe AFM Measurements of Disjoining Pressure [10] Force, F, in nN for 80 mM Brij 35. Micelle volume fraction Force/Radius, F/R, in mN.m -1 for 133 mM Brij 35. Micelle volume fraction The aggregation number of micelles is 70.

(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). R f is the characteristic film radius. Simplest version of the lubrication approximation (h<<R f ): where: t is time; r and z are the radial and vertical coordinates. Continuity equation: Momentum balance equation is simplified to: Simple solution: Hydrodynamic force, F:  (h) is the disjoining pressure, which accounts for the molecular interactions in the film.

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

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

(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 p m is the pressure in the meniscus. Continuity equation: Normal stress boundary condition: Simple solution: Film-profile-evolution equation (stiff nonlinear problem): The applied force is given by the expression:

(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, R s. (E) Schematic of the bubble geometry. Evolution of film profiles and rim rupture effect.

(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 T s is the surface viscous stress tensor. Marangoni effect Capillary pressure Surface viscosity effect For Newtonian interfaces (Boussinesq – Scriven law) [16]: where: I s is the surface idem factor;  dil – surface dilatational viscosity;  sh – surface shear viscosity.

(5) Lubrication Approximation for Complex Fluids in the Films [18] The film phase contains one surfactant with bulk concentration, c, adsorption, , and interfacial tension, . The larger bulk and surface diffusivities lead to larger surface velocity (mobility)! Integrated-surfactant-mass-balance equation: Continuity equation for mobile surfaces: c s – 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)

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

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

(5) Inverse Systems – Surfactants in the Disperse Phase where:  c – the density of liquid in the film phase; F s – force arising from the disjoining pressure;  – characteristic thickness of the boundary layer in the drop phase. Surface active components 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] Surfactant in the continuous phase: 0.1M lauryl alcohol (1); 2 mM C 8 H 18 O 3 S (2). Surfactant in the disperse phase (benzene films): C 8 H 18 O 3 S 0 mM (1); 0.1 mM (2); 2 mM (3). Film life time diagram

Basic References 1. J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 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– 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 Additional References 4. W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge Univ. Press, Cambridge, B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum Press: Consultants Bureau, New York, 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.