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Regarding the Young equation

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Regarding the Young equation

  1. 1. Regarding the Young equation Miguel A. Rodríguez Valverde Aula C11, Aulario MecenasLunes 19 Noviembre, 12:30 p.m.
  2. 2. In memory to Pierre Gilles De Gennes prix Nobel de physique en 1991 2
  3. 3. Motivation to avoid misdleading interpretations of the Young equation Rusanov Platikanov Starov 3
  4. 4. The Solid Surfaces’ evil Interfacial tension or interfacial energy? 4
  5. 5. Outline •Historical overview •Revised Surface Thermodynamics •The Solid Surfaces’ evil •Other controversial issues •References 5
  6. 6. Outline •Historical overview •Revised Surface Thermodynamics •The Solid Surfaces’ evil •Other controversial issues •References 6
  7. 7. Historical overview Timeline E. A. Vogler, On the Origins of Water Wetting Terminology, in Water in BiomaterialsScience, M. Morra (ed.), John Wiley, pp. 149-182 (2001). 7
  8. 8. Historical overview Thomas Young (1773-1829) "An Essay on the Cohesion of Fluids", Philosophical Transactions of the Royal Society of London, 95, 65-87, (1805).http://gallica.bnf.fr/ark:/12148/bpt6k55900p 8
  9. 9. Historical overview Johann Carl Friedrich Gauss (1777-1855) "Principia Generalia Theoriae Figurae Fluidorum" Comment. Soc. Regiae Scient. Gottingensis Rec. 7 (1830). http://gallica.bnf.fr/ark:/12148/bpt6k99405t Gauss used the Principle of Virtual Work to unify the achievements of Young and Laplace Athanase Dupré (1808-1869) "Théorie mécanique de la chaleur”. Paris, Gauthier-Villars, 1869 wadh = γ LV (1 + cos θ ) Langle de raccordement 9
  10. 10. Historical overview Controversy about the Young equation 1. Experimental verification 2. Mechanistic derivation/interpretation 3. Restrictive concept of ideal solid surface 4. Influence of external field, contact angle hysteresis, effect of drop size… 10
  11. 11. Historical overview As result…the delirious renaming of the Young equation TOMINAGA HIROSHI, SAKATA GORO, ISHIBUCHI TOMOMI, OUMI MASAHARU, "Gamo Triangles of Wetting" (Provisional Name) and Re- verification of Dupre-Gamo Equation, Journal of the Adhesion Society of Japan (2000), 36(5) 176-178 Reply: VOLPE C D SIBONI S MANIGLIA D, About the Validity of the So-called Dupre-Gamo Equation, Journal of the Adhesion Society of Japan (2003), 39(2) 64-66 11
  12. 12. Historical overview Experimental verification of the Young equation σ SV − σ SL = γ LV cos θY No readily measurable quantities Experimentally accessible quantities Erik Johnson, “The Elusive Liquid-solid Interface”, Science 19 April 2002: Vol. 296. no. 5567, pp. 477 - 478 Equilibrium Pull-off dαβ cos θY A typical Zisman plot: -Hertz and JKR theories (Contact n- alkanes on teflon Mechanics: adhesion, friction, and 1 extrapolation to cos θ Y= 1 fracture) Aαβ σ αβ = -Adhesion (pull-off) forces2by 0.9 1 => γC = 18 ergs/cm2 SFA π 12 dαβ 0.8 γ c = σ SV − σ SL -Elastomeric solids γLV 2 20 30 12
  13. 13. Historical overview After ~ 200 years… A.I. Bailey, S.M. Kay, “A Direct Measurement of the Influence of Vapour, of Liquid and of Oriented Monolayers on the Interfacial Energy of Mica” Proc. Roy. Soc. A301 (1967), p. 47 Johnson, K. L.; Kendall, K.; Roberts, A. D. “Surface Energy and the Contact of Elastic Solid” Proc. R. Soc. London, A 1971, A324, 301. R. M. Pashley, J. N. Israelachvili, "A Comparison of Surface Forces and Interfacial Properties of Mica in Purified Surfactant Solutions" Colloids & Surfaces 2 (1981) 169-187 Manoj K. Chaudhury, George M. Whitesides, "Direct Measurement of Interfacial Interactions between Semispherical Lenses and Flat Sheets of Poly(dimethyl siloxane) and Their Chemical Derivatives," Langmuir 7 (1991):1013-1025 13
  14. 14. Outline •Historical overview •Revised Surface Thermodynamics •The Solid Surfaces’ evil •Other controversial issues •References 14
  15. 15. Revised Surface Thermodynamics Concept of interface β-phase α-phase One-component system (substance j) β-phase β-phase r0 α-phase α-phase Interphase Interface Josiah Willard Gibbs, “On the Equilibrium of Heterogeneous Substances”, The American Journal of Science and Arts Third Series, Vol. XVI, No. 96, December 1878, pp. 441-58 15
  16. 16. Revised Surface Thermodynamics How can we form a new interface αβ? 1 2 Stretching: Cutting-off: work against the forces of surface tension work against the cohesional forces dA dA dWγ = γ αβ dA dWσ = σ αβ dA Interfacial tension Specific excess interfacial free energy Intensive tensorial quantity Intensive scalar quantity (=surface grand thermodynamic potential) σ αβ (T , μ j ) > 0 http://www.iupac.org/reports/2001/colloid_2001/manual_of_s_and_t/node23.html
  17. 17. Revised Surface Thermodynamics Anisotropic interfaces 1 t t γ αβ ( = γ αβ : 1 2 ) t dWγ = ( γ αβ : de ) A t Surface strain tensor Interfacial stress tensor (principal directions) t ⎛ γ αβ 11 0 ⎞ ⎛ 1 0 ⎞ sh ⎛ 1 0 ⎞ γ αβ =⎜ ⎜ 0 22 ⎟ ⎟ = γ αβ ⎜ ⎟ + γ αβ ⎜ ⎟ ⎝ γ αβ ⎠ ⎝0 1⎠ ⎝ 0 −1⎠ γ αβ 22 γ αβ = + γ αβsh γ αβ 11 γ αβ γ αβ sh Isotropic stress Shear stress 17
  18. 18. Revised Surface Thermodynamics Wulff construction: the shape of things -In crystalline solids, the surface tension depends on the crystal plane and its direction -The equilibrium shape of a crystal is obtained by minimizing the integral t t Wγ = A∫∫ γ αβ : de S http://www.virginia.edu/ep/SurfaceScience/Thermodynamics.html 18
  19. 19. Revised Surface Thermodynamics β-phase Mechanical equilibrium condition (fluid interfaces) α-phase ˆ mαβ ˆ Nαβ ˆ mαβ = Nαβ × tˆ ˆ conormal unit vector χ-phase ˆ t r r t dFαβ r r t t γ αβ ≡ γ αβ ⋅ mαβ = ˆ Wγ = ∫ Fαβ ⋅ dr = A∫∫ γ αβ : de dl C S r r ˆ mαβ Eigenvector γ αβ Eigenvalue ∑ γ ij = 0 t γ αβ ⋅ mαβ = γ αβ mαβ ˆ ˆ [αβχ ] 19
  20. 20. Revised Surface Thermodynamics Neumann’s Triangle equation (no generalized Young equation!) r γ 12 m12 + γ 31m31 + γ 32 m32 = 0 ˆ ˆ ˆ 2-phase γ 12 m12 ˆ θ2 1-phase θ1 θ3 γ 31m31 ˆ γ 32 m32 ˆ 3-phase sinθ 3 sinθ 2 sinθ1 = = γ 12 γ 31 γ 32 Franz Ernst Neumann (1798 -1895) θ3 = π ⇒ θ1 = 0 or π !!! Neumann, F. ,”Vorlesungen über die Theorie der Capillarität”, 152 (Teubner, Leipzig, 1894). 20
  21. 21. Revised Surface Thermodynamics t Relation between σ αβ y γ αβ z0 ∞ Local stress tensor t t t t ∫( ) ( ) t γ αβ = P 1 − P ( z ) dz + ∫ Pβ 1 − P ( z ) dz α −∞ z0 Normal coordinate to the interface z β-phase z0 Dividing plane α-phase z0 ∞ t t ∫ ( μ ( z ) − μ ) c ( z ) dz + ∫ ( μ ( z ) − μ ) c ( z ) dz t tα t tβ σ αβ 1 = γ αβ + j j j j j j −∞ z0 t μ j ( z ) Chemical potential tensor of the substance j t t μ α = μ j ( ∞ ) “ in the bulk of α-phase j tβ t μ j = μ j ( −∞ ) “ in the bulk of β-phase c j ( z ) Local concentration of the substance j 21
  22. 22. Revised Surface Thermodynamics 3D tensors 1 t t γ αβ ( = γ αβ : 1 3 ) t t t t t γ αβ 1 ( 1 ) = ∫∫∫ P 1 − P dV + ∫∫∫ Pβ 1 − P dV A Vα α A Vβ ( ) t t 1 1σ αβ 1 = γ αβ + ∫∫∫ ( μ j − μ j )c j dV + ∫∫∫ ( μ j − μ j ) c j dV t tα t tβ A Vα A Vβ Chemical equilibrium t tα tβ μj = μj = μj σ αβ = γ αβ 22
  23. 23. Revised Surface Thermodynamics f ( x, y ) Thermodynamic equilibrium condition (solid-liquid-fluid interfaces) z Energy functional of a solid-liquid-vapor (SLV) system (ideal solid surface) E ⎡ f ( x, y ) ⎤ = ∫ σ LV dA − ∫ Δρ g f dV + ∫ ⎣ ⎦ (σ SL − σ SV ) dA Alv Vl Asl Closure conds. ∫Vl ∪Vv dV = const.; ∫ Asl ∪ Asv dA = const. Energy functional minimization local minimum condition: ⎪ΔE ⎣ f ( x, y ) ⎦ = 0 ⎧ ⎡ ⎤ ⎨ 2 ⎪ Δ E ⎡ f ( x, y ) ⎤ > 0 ⎩ ⎣ ⎦ 23
  24. 24. Revised Surface Thermodynamics Conditioned minimization Vl = const Ω ⎡ f ( x, y ) ⎤ ≡ E ⎡ f ( x, y ) ⎤ − λVl ⎣ ⎦ ⎣ ⎦ Grand canonical potential Ω (T , Vl , μ ) Pressure difference at f = 0 λ = ΔP0 http://www.iupac.org/publications/pac/2001/pdf/7308x1349.pdf ⎧ Ω ⎫ Δρ g λ σ SL − σ SV ext ⎨ ⎬ = fext ) Alv − ext ∫ f dV − Vl + ext Asl f ( x, y ) σ σ LV f ( x , y ) σ LV σ LV ⎩ LV ⎭ ( x , y f ( x, y ) Vl Young-Laplace Eq. Young Eq. 1. The Young equation is necessary condition for the global equilibrium, but it is not sufficient condition. The Young equation is associated to whatever metastate of a real SLV system (local minima), instead of to the global equilibrium state (global minimum) 2. The Young equation is independent of the interfacial geometry and the gravitational field 24
  25. 25. Revised Surface Thermodynamics Young equation: (local) thermodynamic equilibrium condition (NOT force balance!)σ SV − σ SL = σ LV cos θY 3. Boundary condition of the Young-Laplace equation, derived by thermodynamic arguments 4. Young contact angle is a thermodynamic quantity, a conceptual, unlocated angle 25
  26. 26. Outline •Historical overview •Revised Surface Thermodynamics •The Solid Surfaces’ evil •Other controversial issues •References 26
  27. 27. The Solid Surfaces’ evil Chemical equilibrium t tα tβ μj = μj = μj σ αβ = γ αβ … although In practice… reservoirsAir Liquid vapour Liquid vapour Liquid vapour (unsaturated atmosphere) (saturated atmosphere) (oversaturated atmosphere) μ L ≠ μV j j ln PV PVplane ≈ 2γ LV vL RT H μ L = μV j j J. Phys. Chem. B, Vol. 111, No. 19, 2007 27
  28. 28. The Solid Surfaces’ evil Non-uniformity of chemical potentials near the solid surface: Absence of diffusion equilibrium -Diffusion in real solids proceeds slowly (diffusion time >> experiment time) -Ill-defined thermodynamic states σ SF ≠ γ SF The Solid Surfaces’ evil “God made solids, but surfaces were the work of the devil”------Wolfgang Pauli 28
  29. 29. The Solid Surfaces’ evil Stretching of interfaces Liquid-Fluid interface: Solid-Fluid interface:Variable number of interfacial molecules Constant number of interfacial molecules hole Plastic deformation Elastic deformation σ LF = γ LF σ SF ≠ γ SF 29
  30. 30. The Solid Surfaces’ evil But if… t tL tF μj ≈ μj ≈ μj t t 1 1 − σ SL ) 1 = γ SF − γ SL + ∫∫∫ ( μ j − μ j )c j dV − ∫∫∫ ( μ j − μ j ) c j dV t t tF t tL(σ SF A VF A VL t t t (σ SF − σ SL ) 1 ≈ γ SF − γ SL Equal principal directions σ SF − σ SL ≈ γ SF − γ SL 30
  31. 31. The Solid Surfaces’ evil Mechanical equilibrium condition (solid-liquid-fluid interfaces) m32 = − m31 ˆ ˆ V-phase γ LV L-phase ⎧γ SV − γ SL = γ LV cos θ γ SV θ ⎨ ⎩γ LV sin θ − w + r = 0 S-phase γ SL Young modulus Gravity force (active) Reaction force (passive) Elastic restoring force (passive) Ridge height R = w−r ∝ E 31
  32. 32. The Solid Surfaces’ evil The false equilibrium of a sessile drop (the Solid Surfaces’ evil cont’d) How minimize the solid stresses along the contact line? 32
  33. 33. The Solid Surfaces’ evil Deformation of the solid surface: “Everything flows, nothing stands still” Non-equilibrium (but stable) Initial configuration configuration Local equilibrium Both local and total equilibrium configuration configuration …after millions of years or minutes Hard matter versus Soft matter: timescales 33
  34. 34. The Solid Surfaces’ evil trel Dh ≡ tobs Deborah Number ~ 1: viscoelastic Silly putty Bitumen film http://www.sg.geophys.ethz.ch/geodynamics/klaus/hands-on/deborah.pdf 34
  35. 35. The Solid Surfaces’ evil Pitch drop experiment Deborah Number >> 1: solid-like Date Event 1927 Experiment set up 1930 The stem was cut December 1938 1st drop fell February 1947 2nd drop fell April 1954 3rd drop fell May 1962 4th drop fell August 1970 5th drop fell April 1979 6th drop fell July 1988 7th drop fell November 28, 2000 8th drop fell http://www.physics.uq.edu.au/pitchdrop/pitchdrop.shtml 35
  36. 36. The Solid Surfaces’ evil Huge landslide at the Dolomites in Italy Deborah Number >>> 1: solid-like October 12, 2007 This morning an overshadowing landslide collapsed from the Cima Una peak (2.598 meters) of the Dolomites in Southern Tyrol, which is situated in Northern Italy. Teams of the Italian Civil Defense, of the Fire Department and of the Police Department are already working since a few moments after the fact, they affirmed that 60.000 cubic meters fell, obscurating the sky and complicating their work, but fortunately nothing makes think that there are injured people or victims. 36
  37. 37. The Solid Surfaces’ evil “History” of solid surfaces (the Solid Surfaces’ evil cont’d) • Formative and environmental history of the sample Cleaving, Grinding, Polishing, Etching, Sandblasting • Presence (or absence) of adsorbed species and surface contamination 37
  38. 38. The Solid Surfaces’ evil Kinetic hysteresis (swelling) 38
  39. 39. Outline •Historical overview •Revised Surface Thermodynamics •The Solid Surfaces’ evil •Other controversial issues •References 39
  40. 40. The Solid Surfaces’ evil Contact angle hysteresis (“history” of the system) 40
  41. 41. Other controversial issues Scales of contact angle Optically or atomically ideal surface? extrap 41
  42. 42. Other controversial issues Drop Size on contact angle: line tension, long-range forces… 42
  43. 43. Other controversial issues Line Tension σ slv cos θint = cos θY − γ lv rc redistribution of liquid γlv l(x) Rinteraction σsv σsl xd x Schematic drop profile. In the vicinity of the three-phase contact line there is a redistribution of liquid from the spherical cap profile observable, which is caused by the effective interface potential. 43
  44. 44. Other controversial issues Precursor film: surface pressure θapp Formation of a precursor film with an apparent contact angle θapp Young-Bangham equation σ S σ SV − σ SL = σ LV cos θY −π 44
  45. 45. Other controversial issues Drying, Soldering, Polymerization Shape of desiccated, solder, polymerized drops may not represent neither their surface tension nor their contact angle Desiccation 0s 19 min 1 s Surface oxidation 3 h 49 min 28 s 6 h 42 min 40 s Bitumen emulsion Unoxidized solderPolymerization shrinkage Oxidized solder Dental resin 45
  46. 46. Outline •Historical overview •Revised Surface Thermodynamics •The Solid Surfaces’ evil •Other controversial issues •References 46
  47. 47. References Robert Finn, The contact angle in capillarity, Physics of fluids 18, 047102, 2006 P.-G. de Gennes, F. Brochard-Wyart, and D. Quéré, Capillarity and Wetting Phenomena Springer, Berlin, 2004, p. 18 P. Roura and Joaquim Fort, Local thermodynamic derivation of Youngs equation, Journal of Colloid and Interface Science Volume 272, Issue 2, 15 April 2004, Pages 420-429 Borislav V. Toshev and Dimo Platikanov, Wetting: Gibbs’ superficial tension revisited, Colloids and Surfaces A: Physicochemical and Engineering Aspects Volume 291, Issues 1-3, 15 December 2006, Pages 177-180 A.I. Rusanov Problems of surface thermodynamics, Pure &App. Chem., Vol. 64, No. 1, pp. 111-124, 1992 C. Della Volpe, S. Siboni, and M. Morra, Comments on Some Recent Papers on Interfacial Tension and Contact Angles, Langmuir, 18 (4), 1441 -1444, 2002 47
  48. 48. Coming Soon “Forever Young…” ?Result Quantity Measure 48

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