Interfacial Tension, Capillarity and Surface Forces

Size: px

Start display at page:

Download "Interfacial Tension, Capillarity and Surface Forces"

Samantha Adams
6 years ago
Views:

1 Interfacial Tension, Capillarity and Surface Forces Peter A. Kralchevsky Department of Chemical Engineering, Faculty of Chemistry Sofia University, Sofia, Bulgaria Lecture at COST P1 School Physics of droplets: Basic and advanced topics Borovets, Bulgaria, 1 13 July, 1 Scenarios for drop drop interactions in view of the acting surface forces Sofia University

Molecular Origin of Surface Tension State 1 energy u AA

der Waals forces: Interaction energy: u AA The molecules

Because of the formation of surface, the energy of the

molecules per unit area The force per unit length is: L

2 Molecular Origin of Surface Tension State 1 energy u AA < State energy The molecules attract each other by van der Waals forces: Interaction energy: u AA The molecules at the surface have one neighbor less. Because of the formation of surface, the energy of the system increases with: W u AA ΓL x L y Γ number of molecules per unit area The force per unit length is: L y Lx dlx σ 1 dw uaa L dl y x Γ surface tension surface excess energy per unit area

3 σ F cosθ L Wilhelmy plate method for surface tension measurements Ludwig Wilhelmy ( )

4 Adsorption of Surfactants The surfactant adsorption lowers the surface tension hydrophilic headgroup hydrophobic tail (1) The hydrophobic tail is brought out of water () A contact of oil and water molecules is broken. Work of W w + ads CH adsorption: nw n number of CH groups in a paraffin tail w 1kT CH Traube rule T absolute temperature; k Boltzmann constant

Adsorption Isotherm and Gibbs Adsorption Equation (1) micelles in the bulk () Γ c i dσ i Γ d(ln i i i adsorption of component "i" its bulk concentration c The Gibbs eq.

5 Adsorption Isotherm and Gibbs Adsorption Equation (1) micelles in the bulk () Γ c i dσ i Γ d(ln i i i adsorption of component "i" its bulk concentration c The Gibbs eq. describes the lowering of surface tension due to surfactant adsorption. Chemical μ i μ i + potential: kt ln c i ) In Region (1) dense adsorption layer is formed, Γ const. and σ(lnc) is linear; In Region () micelles are formed in the bulk, μ i const. and σ const. CMC critical micelle concentration

Models with Localized Adsorption Frumkin Model: Adsorption isotherm c c( Γ) Γ Γ( c) Kc Γ Γ exp( Γ K adsorption constant; β accounts for the σ β Γ ) kt Surface equation of state : σ σ( Γ) σ σ

6 Models with Localized Adsorption Frumkin Model: Adsorption isotherm c c( Γ) Γ Γ( c) Kc Γ Γ exp( Γ K adsorption constant; β accounts for the σ β Γ ) kt Surface equation of state : σ σ( Γ) σ σ interaction surface tension of pure solvent Γ Γ + Γ kt ln(1 ) Γ maximal adsorption; between adsorbed molecules; + β Γ The surface is modeled as a lattice with full and empty adsorption sites. Model appropriate for adsorption on a solid surface Langmuir Model: Special case of the Frumkin model with β no interactions between the adsorbed molecules.

7 Models with Non-Localized Adsorption Van der Waals Model: Adsorption isotherm c c( Γ) Γ Γ( c) Kc Γ Γ exp( Γ Γ K adsorption constant; β accounts for the σ Γ β Γ ) Γ kt Surface equation of state : σ σ( Γ) σ σ surface tension of pure solvent Γ Γ ΓkT Γ Γ interaction + β Γ maximal adsorption; between adsorbed molecules; The surface is modeled as a lattice with full and empty adsorption sites. Model appropriate for adsorption on a liquid surface Volmer Model: Special case of the van der Waals model with β no interactions between the adsorbed molecules.

Surface tension, σ (mn/m) 8 7 6 5 4 3 SDS sodium dodecyl sulfate NaCl. mm.5 mm.8 mm 1. mm.5 mm 4. mm 5. mm 8.

8 Surface tension, σ (mn/m) SDS sodium dodecyl sulfate NaCl. mm.5 mm.8 mm 1. mm.5 mm 4. mm 5. mm 8. mm 1 mm mm 115 mm Comparison of the van der Waals and Frumkin Models Both models fit well data for σ(c) for a liquid surface (indistinguishable curves). Gibbs elasticity, E G (mn/m) E G SDS Concentration (mm) van der Waals Frumkin σ Γ Γ 115 mm SDS Concentration (mm) mm mm However, van der Waals gives the real excluded area per SDS molecule, whereas Frumkin yields a greater area: nm.4 nm Γ Γ (van der Waals) (Frumkin) The van der Waals model gives realistic values of surface elasticity, whereas the Frumkin models yields greater values.

9 Basic References 1. E.D. Shchukin, E.D. Shchukin, A.V. Pertsov, E.A. Amelina, and A.S. Zelenev, Colloid and Surface Chemistry, Elsevier, Amsterdam, 1.. P.A. Kralchevsky, K. Nagayama, Particles at Fluid Interfaces and Membranes, Elsevier, Amsterdam, 1; Chapter 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, 8; pp Additional References 4. K.D. Danov, P.A. Kralchevsky, et al. Interpretation of Surface-Tension Isotherms of n-alkanoic (Fatty) Acids by Means of the van der Waals Model, J. Colloid Interface Sci. 3() (6) V.L. Kolev, K.D. Danov, P.A. Kralchevsky, et al. Comparison of the van der Waals and Frumkin Adsorption Isotherms for SDS at Various Salt Concentrations, Langmuir 18 () P.A. Kralchevsky, K.D. Danov, V.L. Kolev, et al. Effect of Nonionic Admixtures on the Adsorption of Ionic Surfactants at Fluid Interfaces. Part 1. SDS and Dodecanol, Langmuir 19 (3) P.A. Kralchevsky, K.D. Danov, et al. Thermodynamics of Ionic Surfactant Adsorption with Account for the Counterion Binding: Effect of Salts of Various Valency, Langmuir 15 (1999)

c P p p R σ (Laplace equation) P c capillary pressure c 1 ) 1 1 ( P σ R R + + r z

10 Capillarity Laplace Equation Force-balance derivation: For general curved interfaces - two curvature radii: σ π R p π R p R π ) ( ) ( ) ( c m c d c + c m d c P p p R σ (Laplace equation) P c capillary pressure c 1 ) 1 1 ( P σ R R + + r z φ P σ r φ r φ d d tan ] sin d ) d (sin [ c φ running meniscus slope angle ) ( ) ( φ r r φ z z

Drop Shape Analysis (DSA) for Surface Tension Measurements pendant drop σ r d d r r z ( ) + z z z( r); P z σ R 1/ c 1 b dz dr Δρgz Picture by apparatus Kruss DSA1 Laplace equation; R b curvature

11 Drop Shape Analysis (DSA) for Surface Tension Measurements pendant drop σ r d d r r z ( ) + z z z( r); P z σ R 1/ c 1 b dz dr Δρgz Picture by apparatus Kruss DSA1 Laplace equation; R b curvature radius at the bottom of the drop P c capillary pressure 1) The drop profile is automatically digitized; ) Then, the data for the profile are fitted numerically by the Laplace equation using σ and R b as adjustable parameters. 3) The surface tension σ is obtained from the best fit; effect of surfactants on σ. 4) The method works with both drops and bubbles; both pendant and sessile profiles. 5) The method is accurate when the gravitational deformation (the deviation from spherical shape) is not too small.

12 Surface Force & Disjoining Pressure Surface force Force of interaction between two bodies (two phases) when the distance h between their surfaces is relatively small. Typically, h < 1 nm. Disjoining pressure, Π(h) Surface force per unit area of a plane-parallel film [1-3]. Capillary (Laplace) pressure: P c P in P l σ/r (σ surface tension) Force balance per unit area of the film surface: P l + Π P in Hence: Π P in P l P c (disjoining pressure capillary pressure) [4].

DLVO Theory: Equilibrium states of a free liquid film DLVO Derjaguin, Landau, Verwey, Overbeek [5,6]: Π( h) Πel( h) + Π vw ( h) Born repulsion Electrostatic component of disjoining pressure: Π el (

13 DLVO Theory: Equilibrium states of a free liquid film DLVO Derjaguin, Landau, Verwey, Overbeek [5,6]: Π( h) Πel( h) + Π vw ( h) Born repulsion Electrostatic component of disjoining pressure: Π el ( h) B exp( κ h) (repulsion) Van der Waals component of disjoining pressure: Π vw AH ( h) 3 6π h (attraction) () Secondary film (1) Primary film h film thickness; A H Hamaker constant; κ Debye screening parameter

Scheludko-Exerowa capillary cell for thin-liquid-film

Illustration: Stepwise thickness transitions in films

14 Scheludko-Exerowa capillary cell for thin-liquid-film (TLF) studies [7,8] Scheludko-Exerowa capillary cell for thin-liquid-film (TLF) studies [7,8] Measurements of: (1) Film thickness vs. time; () Contact angles of TLF; (3) Lifetime of the films. Illustration: Stepwise thickness transitions in films from.1 M solutions of the nonionic Brij 35:

interaction free energy per unit area of a plane-parallel film: ~ f ( h) Π( h ~ )dh of the surface force is much

15 Derjaguin s Approximation (1934): The energy of interaction, U, between two bodies across a film of uneven thickness, h(x,y), is [9]: U f ( h( x, y)) dxdy This approximation is valid if the range of action where f(h) is the interaction free energy per unit area of a plane-parallel film: ~ f ( h) Π( h ~ )dh of the surface force is much smaller than the surface curvature radius. h For two spheres of radii R 1 and R, this yields: U π R R + R R h 1 ( h ) f ( h)dh 1

16 Derjaguin s approximation for other geometries [1-3,1]: Sphere Plate Truncated Sphere Plate U h ( h ) πr f ( h) c dh U c h ( h ) π R f ( h) dh +πr f ( h ) Two Crossed Cylinders Two Truncated Spheres U π r r sinω h 1 ( h ) f ( h) dh U c h ( h ) π R f ( h) h +πr f ( h ) d

17 Molecular Theory of Surface Forces DLVO Forces: (1) Van der Waals force () Electrostatic (double layer) force Non-DLVO Forces: (1) Hydration repulsion () Steric interaction due to adsorbed polymer chains (3) Oscillatory structural force and Depletion attraction

Van der Waals surface forces: Π vw ( h) A H 3 6π h A H Hamaker

additive: Summation over all couples of molecules.

A A A + H 1 3 31 A 33 π ρiρ jαij ; Aij ( Aii Ajj ) 1/ Symmetric

18 Van der Waals surface forces: Π vw ( h) A H 3 6π h A H Hamaker constant Hamaker s approach [11] The interaction energy is pair-wise additive: Summation over all couples of molecules. Result [11, 1]: vdw between two molecules: u ij ( r) α r ij 6 A ij A A A A + H A 33 π ρiρ jαij ; Aij ( Aii Ajj ) 1/ Symmetric film: phase phase 1 ( 1/ 1/ ) A AH A11 A13 + A33 11 A33 > For symmetric films: always attraction!

19 Lifshitz approach to the calculation of Hamaker constant E. M. Lifshitz ( ) [13] took into account the collective effects in condensed phases (solids, liquids). (The total energy is not pair-wise additive over al pairs of molecules.) Lifshitz used the quantum field theory to derive accurate expressions in terms of [1, 14]: (i) Dielectric constants of the phases: ε 1, ε and ε 3 ; (ii) Refractive indexes of the phases: n 1, n and n 3 : A H A 13 3 ε1 ε3 kt 4 ε1 + ε3 ε ε ε3 + ε h ( )( ) Pν e n1 n3 n n3 ( ) 3/ 4( ) 3/ 4 n + n n + n Zero-frequency term: A orientation & induction interactions; kt thermal energy. ( ν ) 13 Dispersion interaction term: A ( ν > ) 13 ν e 3. x 1 15 Hz main electronic absorption frequency; h P 6.6 x 1 34 J.s Planck s const.

Electrostatic (Double Layer) Surface Force n 1m, n m n Π el excess osmotic pressure of the ions in the midplane of a symmetric film (Langmuir, 1938) [15]: Π el kt ( n + n n ) 1m m Π el > repulsion!

20 Electrostatic (Double Layer) Surface Force n 1m, n m n Π el excess osmotic pressure of the ions in the midplane of a symmetric film (Langmuir, 1938) [15]: Π el kt ( n + n n ) 1m m Π el > repulsion! n 1m, n m concentrations of (1) counterions and () coions in the midplane. n concentration of the ions in the bulk solution; ψ m potential in the midplane. For solution of a symmetric electrolyte: Z 1 Z Z; Z is the valence of the coions. Boltzmann equation; Φ m dimensionless potential in the midplane (Φ m << 1). n1 m n exp( Φ m ); nm n exp( Φ m ); Πel nkt cosh m m > ( Φ ) 1 n ktφ cosh Φ m Φ Zeψ kt m m 4 ( Φ ) 1+ + O( Φ ) m m

21 Π(h)? Verwey Overbeek Formula (1948) Near single interface, the electric potential of the double layer is [6]: tanh Zeψ1 Zeψ s tanh exp κ 4kT 4kT ( z ) w Superposition approximation in the midplane: ψ m ψ 1 [6]: Φ m Π el ψ 1 ( h) Ze Zeψ s h 8 tanh exp κ kt 4kT n ktφ m 64n ktγ exp( κ h) Zeψ In the midplane 1 << 1 4kT tanh 3 ( x ) x + O( x ) Zeψ s γ tanh 4kT

22 tanh(x) 1 for x 1.5 4kT e 1mV Zeψ s γ tanh 1 4kT for Zψ 15mV s Π el ( h) 64nkT exp( κ h) Debye screening length, κ 1 : κ κ κ [NaCl].176 [CaCl.15 [MgSO ] 4 nm nm ] nm Z i κ e I εε kt ; I 1 i Z i c i (ionic strength) and c - valence and concentration of the i-th I for 1:1 el. (M) κ 1 (nm) i 1 7 pure water ion

Derjaguin Landau Verwey Overbeek (DLVO) Theory [5,6] Disjoining pressure: Π Π el + Π vw Be κ h A H 3 6π h Free energy per unit area ~ ~ B h AH of a plane-parallel film: f ( h) Π( h )dh e κ κ 1π h h

23 Derjaguin Landau Verwey Overbeek (DLVO) Theory [5,6] Disjoining pressure: Π Π el + Π vw Be κ h A H 3 6π h Free energy per unit area ~ ~ B h AH of a plane-parallel film: f ( h) Π( h )dh e κ κ 1π h h Energy of interaction between two identical spherical particles (Derjaguin approximation): U h h π R f h ~ ~ κ H ( ) ( )dh e h B π R κ A 1π h B 64n ktγ, γ tanh Zeψ s 4kT ( e electronic charge; e )

24 DLVO Theory: The electrostatic barrier U B π R κ A 1π h κ h H ( h) e The secondary minimum could cause coagulation only for big (1 μm) particles. The primary minimum is the reason for coagulation in most cases. Condition for coagulation: U max (zero height of the barrier to coagulation) du U ( hmax ) ; dh hh max

25 Hydration Repulsion At C el < 1 4 M (NaCl, KNO 3, KCl, etc.), a typical DLVO maximum is observed. At C el 1 3 M, a strong short-range repulsion is detected by the surface force apparatus the hydration repulsion [1, 16]. Empirical expression [1] for the Important: f el decreases, whereas f hydr increases with the rise of electrolyte concentration! interaction free energy per unit area: f hydr f exp( h / λ) Explanation: The hydration repulsion is due mostly to the finite size of the hydrated counterions [17]. λ f nm 3 3 mj/m

L depends on adsorption of chains, Γ [1,1].

26 Steric interaction due to adsorbed polymer chains L L 1/ ln (ideal solvent) l the length of a segment; N number of segments in a chain; In a good solvent L > L, whereas Π st ( h) ktγ 3/ L h for h < L g ; g 9 / 4 L g h L N g 3/ 4 ( 5 Γl ) 1/ 3 in a poor solvent L < L. L depends on adsorption of chains, Γ [1,1]. Alexander de Gennes theory for the case of good solvent [18,19]: The positive and the negative terms in the brackets in the above expression correspond to osmotic repulsion and elastic attraction. The validity of the Alexander de Gennes theory was experimentally confirmed; see e.g. Ref. [1].

27 Oscillatory Structural Surface Force A planar phase boundary (wall) induces ordering in the adjacent layer of a hard-sphere fluid. The overlap of the ordered zones near two walls enhances the ordering in the gap between the two walls and gives rise to the oscillatory-structural For details see the book by Israelachvili [1] force.

28 Oscillatory structural forces [1,] were observed in liquid films containing colloidal particles, e.g. latex & surfactant micelles; Nikolov et al. [1,]. The maxima of the oscillatory force could stabilize colloidal dispersions. Oscillatorystructural disjoining pressure Depletion minimum The metastable states of the film correspond to the intersection points of the oscillatory curve with the horizontal line Π P c. The stable branches of the oscillatory curve are those with Π/ h <.

29 Oscillatory Structural Surface Force Due to Colloid Particles Ordering of micelles of the nonionic surfactant Tween [4]. Methods: Mysels porous plate cell and Theoretical curve by means of the Trokhimchuk formulas [3]. The micelle aggregation number, N agg 7, is determined [4]. Scheludko-Exerowa capillary cell [7,8]

The total energy of interaction between two particles, U(h), includes contributions from all surface forces: Disjoining pressure (force per unit area of a plane parallel film): Π(h) Π vw (h) + Π el

30 The total energy of interaction between two particles, U(h), includes contributions from all surface forces: Disjoining pressure (force per unit area of a plane parallel film): Π(h) Π vw (h) + Π el (h) + Π hydr (h) + Π st (h) + Π osc (h) + ~ Energy of interaction between two particles: U ( h) πr dh dhˆ ( hˆ Π ) ~ h h U(h) U vw (h) + U el (h) + U hydr (h) + U st (h) + U osc (h) + DLVO forces Non-DLVO forces

31 Basic References 1. J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, P.A. Kralchevsky, K. Nagayama, Particles at Fluid Interfaces and Membranes, Elsevier, Amsterdam, 1; Chapter 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, 8; pp Additional References 4. B.V. Derjaguin, M.M. Kussakov, Acta Physicochim. USSR 1 (1939) B.V. Derjaguin, L.D. Landau, Theory of Stability of Highly Charged Lyophobic Sols and Adhesion of Highly Charged Particles in Solutions of Electrolytes, Acta Physicochim. USSR 14 (1941) E.J.W. Verwey, J.Th.G. Overbeek, Theory of Stability of Lyophobic Colloids, Elsevier, Amsterdam, A. Scheludko, D. Exerowa, Instrument for interferometric measuring of the thickness of microscopic foam layers. CR Acad Bulg Sci 7 (1959)

32 8. A. Scheludko, Thin Liquid Films, Adv. Colloid Interface Sci. 1 (1967) B.V. Derjaguin, Friction and Adhesion. IV. The Theory of Adhesion of Small Particles, Kolloid Zeits. 69 (1934) B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum Press: Consultants Bureau, New York, H.C. Hamaker, The London Van der Waals Attraction Between Spherical Particles Physica 4(1) (1937) B.V. Derjaguin, Theory of Stability of Colloids and Thin Liquid Films, Plenum Press: Consultants Bureau, New York, E.M. Lifshitz, The Theory of Molecular Attractive Forces between Solids, Soviet Phys. JETP (English Translation) (1956) W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge Univ. Press, Cambridge, I. Langmuir, The Role of Attractive and Repulsive Forces in the Formation of Tactoids, Thixotropic Gels, Protein Crystals and Coacervates. J. Chem. Phys. 6 (1938) R.M. Pashley, Hydration Forces between Mica Surfaces in Electrolyte Solutions, Adv. Colloid Interface Sci. 16 (198) 57-6.

33 17. V.N. Paunov, R.I. Dimova, P.A. Kralchevsky, G. Broze and A. Mehreteab. The Hydration Repulsion between Charged Surfaces as Interplay of Volume Exclusion and Dielectric Saturation Effects, J. Colloid Interface Sci. 18 (1996) S.J. Alexander, Adsorption of Chain Molecules with a Polar Head: a Scaling Description, J. Phys. (Paris) 38 (1977) P.G. de Gennes, Polymers at an Interface: a Simplified View, Adv. Colloid Interface Sci. 7 (1987) J.N. Israelachvili, R.M. Pashley, Molecular Layering of Water at Surfaces and Origin of Repulsive Hydration Forces, Nature 36 (1983) A.D. Nikolov, D.T. Wasan, P.A. Kralchevsky, I.B. Ivanov. Ordered Structures in Thinning Micellar and Latex Foam Films. In: Ordering and Organisation in Ionic Solutions (N. Ise & I. Sogami, Eds.), World Scientific, Singapore, 1988, pp A. D. Nikolov, D. T. Wasan, et. al. Ordered Micelle Structuring in Thin Films Formed from Anionic Surfactant Solutions, J. Colloid Interface Sci. 133 (1989) 1-1 & 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 (1) E.S. Basheva, P.A. Kralchevsky, K.D. Danov, K.P. Ananthapadmanabhan, A. Lips, The Colloid Structural Forces as a Tool for Particle Characterization and Control of Dispersion Stability, Phys. Chem. Chem. Phys. 9 (7)

Interfacial forces and friction on the nanometer scale: A tutorial

Interfacial forces and friction on the nanometer scale: A tutorial M. Ruths Department of Chemistry University of Massachusetts Lowell Presented at the Nanotribology Tutorial/Panel Session, STLE/ASME International