1. Regarding the Young equation
Miguel A. Rodríguez Valverde
Aula C11, Aulario Mecenas
Lunes 19 Noviembre, 12:30 p.m.

2. In memory to Pierre Gilles De Gennes
prix Nobel de physique en 1991
2

3. Motivation
to avoid misdleading interpretations of the Young equation
Rusanov Platikanov Starov
3

4. The Solid Surfaces’ evil
Interfacial tension or interfacial energy?
4

5. Outline
•Historical overview
•Revised Surface Thermodynamics
•The Solid Surfaces’ evil
•Other controversial issues
•References
5

6. Outline
•Historical overview
•Revised Surface Thermodynamics
•The Solid Surfaces’ evil
•Other controversial issues
•References
6

7. Historical overview Timeline
E. A. Vogler, On the Origins of Water Wetting Terminology, in Water in Biomaterials
Science, M. Morra (ed.), John Wiley, pp. 149-182 (2001). 7

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. 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 )
L'angle de raccordement
9

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. 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. 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. 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. Outline
•Historical overview
•Revised Surface Thermodynamics
•The Solid Surfaces’ evil
•Other controversial issues
•References
14

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. 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. 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. 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. 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. 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. 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. 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 1
σ 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. 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. 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. 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. Outline
•Historical overview
•Revised Surface Thermodynamics
•The Solid Surfaces’ evil
•Other controversial issues
•References
26

27. The Solid Surfaces’ evil
Chemical equilibrium
t tα tβ
μj = μj = μj
σ αβ = γ αβ
… although
In practice… reservoirs
Air 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. The Solid Surfaces’ evil
Kinetic hysteresis (swelling)
38

39. Outline
•Historical overview
•Revised Surface Thermodynamics
•The Solid Surfaces’ evil
•Other controversial issues
•References
39

40. The Solid Surfaces’ evil
Contact angle hysteresis
(“history” of the system)
40

41. Other controversial issues
Scales of contact angle
Optically or atomically ideal surface?
extrap
41

42. Other controversial issues
Drop Size on contact angle: line tension, long-range forces…
42

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. 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. 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 solder
Polymerization shrinkage
Oxidized solder
Dental resin 45

46. Outline
•Historical overview
•Revised Surface Thermodynamics
•The Solid Surfaces’ evil
•Other controversial issues
•References
46

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 Young's
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. Coming Soon “Forever Young…”
?
Result Quantity Measure
48