Interfacial dynamics Interfacial dynamics = dynamic processes at fluid interfaces upon their deformation Interfacial rheological properties: elasticity, viscosity, yield stress, Relation between macroscopic and molecular levels of description?! Aims of presentation: Basic concepts in interfacial dynamics. Illustrative examples. A. Role of interfacial dynamics. B. Interfacial rheology. C. Foam-wall viscous friction.
A. Role of interfacial dynamics in foams - two selected examples 1. Surface perturbations in foam rheology. Surface stress in the process of foam drainage Longitudinal section of the Plateau channel Viscous bulk stress dvz τ V =µ dx Matching bulk and surface flows VS = VZ x = 0 τ =τ τ =? S V ( ) S
B. Interfacial rheology. 1. Phenomenological approach - shear deformation α = S δx y Surface shear elasticity τ S = ESαS Surface shear viscosity dαs τ =η =η α& dt S S S S The shear rheological properties are sensitive to the molecular interactions in the adsorption layer. More complex models (constitutive equations) Visco-elastic (Kelvin) Visco-elastic (Maxwell) τ= Eα+ηα& Visco-plastic (Bingham) τ& τ α= & + E η Shear thinning (power law) τ=τ 0 +ηα& τ = kα& n
. Phenomenological approach - dilatational deformation. Surface dilatational elasticity τ = E α α = d d d d δa A For insoluble monolayers: dσ σ = EGαd Ed = EG = d ln A 0 σ EQ σ EQ +τd Surface dilatational viscosity & dα dt d τ d = ηdαd α d = = & d ( ln A) dt The dilatational rheological properties are sensitive to both, molecular interactions and adsorption kinetics Apparent viscosity Characteristic times: Experimental time, t EXP Adsorption time, t ADS dσ dσδam dσ τd δσ = δγ = = δln A dγ dγ A dlnγ tads δln A E δln A ( E tads ) =µ α& texp texp G G APP d M M Similarly for the molecule rearrangement in the monolayer, t R
Coupling between bulk and surface flows 1. Balance of bulk and surface stress Equal velocities ( ) V = V z = z S B S Stress balance τ B = τ S Bulk viscous stress Vx V0 τ B =µ µ z h Surface stress dσ d u τ S = +µ S { dx 1443 dx surface elasticity surface viscosity. Surface mass balance for the surfactant ( V Γ) Γ Γ S = DS + + J t x x BULK J BULK C = DB z z= z S Equations for determining Γ(x,t) and σ(x,t) ( ) σ=σ Γ Coupled set of equations for C(x,z), V x (x,z), Γ(x), V S Very complex hydrodynamic problem!
Experimental methods for studying the interfacial rheological properties Shear mode Dilatational mode ( ) ( E ) σ t = σ +τ = σ + α +η α& () EQ D EQ d d d d Expanding or oscillating drop Continuous shear or oscillations P C = σ R d 1. Continuous type of measurement (steady-state flow) Shear Dilatational τ S = τs( α& S) ( ln ) = d A dt const τ = τ + kα& S 0 n For diffusion control, steady-state is difficult to achieve
. Stress-relaxation experiments Strain Maxwell model Stress τ& τ α= & + E η t τ () t = αeexp t R t =η E E τ/ α R / The relaxation is not exponential for diffusion controlled adsorption σ(t) 1/ t 3. Strain-relaxation experiments (creep flow) Kelvin model τ α = E () t 1 exp ( t/ t ) R
4. Oscillatory measurements (shear, dilatation) Kelvin model dα τ=τ EL +τ V = Eα+η dt α ( ) & cos( ) α= A sin ωt α= A ω ωt ( ) cos( ) ( t ) τ = Aα Esin ω t +ηω ω t = = Aτ sin ω +ϕ α ( ) ( ) A τ, ϕ E ω, η ω ( A / A ) E ( ) τ α = + ηω sinϕ= surface modulus E ηω ( ) + ηω Example: Diffusion controlled adsorption- dilatation (Lucassen, van den Tempel) A 0, Γ 0, σ 0 A 0 +δa 0 Γ < Γ 0 A 0 Γ > Γ 0 σ > σ 0 Elasticity σ < σ 0 D D / E( ω ) = E ω + ω G ω + ω + 1 D D A 0 -δa 0 Γ > Γ 0 σ < σ 0 ω dr ω D = D dc σ 0 Apparent viscosity E / G ω D ηω ( ) = ω ω D + ω D + 1 Insoluble monolayers or fast oscillations: ω D E = E G ; η = 0 ω / ; ωη ω 0 ω D Slow oscillations: ω D 0 E(ω) E G ω D 0; η ( ) E G
Expected rheological response in various systems System Schematics Note Low molecular mass surfactants Adsorption Synthetic polymers Visco-elastic Shear thinning Globular proteins Solid particles + Yield stress Time dependence Plastic Slow adsorption C. Friction between foam and solid wall Wall slip friction V W V 0 Wall slip is usually significant (incl. rheo-experiments) Role of surface mobility of the bubbles.
Aims of the presentation: Approaches for studying foam-wall friction. Illustration of role of interfacial properties. CONTENTS 1. Role of wall-slip in foam rheology.. Theoretical approach for description. 3. Illustrative experimental results. Foam rheological properties Constitutive rheological relation τ = τ + k & γ 0 n F τ 0 - yield stress (elastic origin) γ& F - shear rate [1/s] Herschel-Bulkley fluid (HB fluid) Princen, 1985
Role of wall-slip in foam rheology d γ& F V W & γ APP = V0 d V 0 τ = τ + k & γ 0 n F γ& F = γ& APP V W d The true shear rate in the foam, γ F, is lower in presence of wall-slip Example: Effect of wall-slip on shear stress 100 Foam stabilised by 3 wt % Betaine, Φ = 90 % Shear Stress (Pa) 10 τ 0 no wall slip with wall slip 1 0.1 Perfect slip (plug foam flow) 0.01 0.1 1 10 100 1000 Shear rate (s -1 )
Theoretical approach to describe wall-slip friction Foam Bubble V x (z) V W V W Friction Force Bubble-Wall F FR Vx = µ z Contact zone z= 0 da Shape of the dynamic wetting film formed between bubble and solid wall 4.0 D, Soap solution, 40 % Glycerin 3.5 3.0.5 V 0 = 0.8 mm/s h 1 h 1 h, µm.0 1.5 h 1.0 V 0 = 0. mm/s 0.5 V 0 = 0.1 mm/s 0.0 350 400 450 500 550 600 650 x, µm V 0 = 0. mm/s The film thickness increases with the increase of V 0
Lubrication equation for bubble-wall friction dp dx V = µ z x Bubble Surface stress balance z V x (z) V µ z dσ = + dx x µ S z= h d u dx x V W Surface elasticity Surface viscosity Important factors: Relative velocity, V W Bubble size, R B, and volume fraction, Φ Liquid viscosity, µ Surface mobility (surface elasticity and viscosity) Two limiting cases are theoretically predicted Tangentially immobile surface (high surface modulus) Tangentially mobile surface (low surface modulus) σ τw R3 Ca 1/ Ca µ V σ W = σ Ca τw R3 /3
Experimental results effect of surface mobility on foam-wall friction 10-1 τ = W kv n 0 Soap τ W R 3 /σ 10 - n = 1/ Synthetic surfactants 10-3 n = /3 10-6 10-5 10-4 10-3 Capillary number, Ca Scaling of the data with solution viscosity R 3 τw Ca σ / 3 Ca µ V σ W = Dimensionless Stress 10-1 1 wt % Na Laurate wt % K Cocoylglycinate 3 wt % Betaine 0 to 60 % Glycerol 10 - m = /3 10-3 10-6 10-5 10-4 10-3 Dimensionless shear rate, Ca
Sheared bulk foam - effect of surface mobility Dimensionless Viscous Stress 10 0 10-1 10-10 -3 1 wt % Na Laurate 1 wt % K Soap n = 0.5 n = 0.40 3 wt % Betaine 0 to 70 % Glycerol 10-7 10-6 10-5 10-4 10-3 10-10 -1 Dimensionless shear rate, Ca F τ = τ + k & γ F 0 n F Ca F µγ& R = σ Conclusions Interfacial dynamic properties play a very important role in foam dynamics (rheology, drainage, foam generation, ) Interfacial properties can be characterized by phenomenological parameters (e.g., surface viscosity, elasticity, and yield stress) These phenomenological parameters are governed by the adsorbed surfactant species (incl. exchange with the solution) Two limiting cases are often recognized: Tangentially immobile (alkylcarboxylates, proteins, synthetic polymers, particles) Tangentially mobile (most synthetic surfactants)
SINCERE THANKS Dr. S. Tcholakova - Help in preparation of the presentation. Miss M. Paraskova - Preparation of some of the figures. Other colleagues from: Laboratory of Chemical Physics & Engineering Faculty of Chemistry, University of Sofia, Sofia, Bulgaria