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Fundamental Equations
of Colloid and Interface Science
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
γS = γSL+ γL cos θ
cos θ = (γS - γSL)/γL
Contact Angle: The Young Equation
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
• surfaces are not (almost never) absolutely smooth, i.e., they have some
roughness (perhaps the most important reason);
• equilibrium is not reached;
• surfaces can be contaminated (adsorbed liquids or even solid
particles, e.g., dust);
• surfaces may undergo some changes during contact with the test liquid;
• the spreading pressure may be significant for low values of contact angle.
Causes of contact angle hysteresis:
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
The Effect of Surface Roughness
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
The Wenzel Roughness Factor, Rf:
Takes into account the effect of roughness on contact angle.
cos θsmooth = (γS - γSL)/γL (Young Eq.)
cos θrough = Rf (γS - γSL)/γL (Wenzel)
where Rf = Areal/Ageom = cos θrough/cos θsmooth ≥ 1
If θsmooth < 90o (partial wetting) then θrough < θsmooth (better wetting)
If θsmooth > 90o (partial non-wetting) then θrough > θsmooth (poorer wetting)
http://www.lehigh.edu/~mkc4/our papers/surface roughness and contact angle.pdf
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Capillary Rise
For Water (θ = 0): h =14.8/r (in mm)
Diam. r h
1000 mm 500 mm 0.03 mm
40 mm 20 mm 0.74 mm
1 mm 0.5 mm 29 mm
0.4 mm 0.2 mm 74 mm
0.1 mm 0.05 mm 296 mm
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Capillary Rise of Water as a Function of Capillary Diameter
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
The Young–Laplace equation:
gives the pressure difference (inside–outside) across a curved surface.
The geometrical factor G is:
2 for a spherical liquid drop,
4 for a soap bubble with a two-sided film
(because it has 2 radii of curvature).
1 for a cylinder.
∆P = Pin - Pout = G γ / RC
RCPin Pout
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Water droplets:
RC ∆P
10 μm 0.15 bar
100 nm 15 bar
1 nm 1500 bar
The Young–Laplace equation:
How big can the pressure difference be?
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
dA (Helmholz free energy) = - P dV – S dT + γ dArea
At isothermal equilibrium:
0 = - P dV + γ dArea
Since P = ∆P = Pinside – Poutside
∆P = γ dArea/dV
The Young–Laplace equation:
Derivation
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
∆PdV = γ dArea
Area = 4πr2 dArea = 8πr dr
Volume = (4/3)πr3 dV = 4πr2 dr
∆P = 2 γ/r
The Young–Laplace equation:
Spherial Drop
r Pin Pout
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
∆P = γ dArea/dV
Area = 2πrL dArea = 2πL dr
Volume = πr2L dV = 2πrL dr
∆P = γ/r
The Young–Laplace equation:
Infinite Cylinder
r Pin
Pout
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
The Young–Laplace equation:
gives the pressure difference (inside–outside) across a curved surface.
The geometrical factor G is:
2 for a spherical liquid drop,
4 for a soap bubble with a two-sided film
(because it has 2 radii of curvature).
1 for a cylinder.
∆P = Pin - Pout = γ (1/ RC + 1/ RC’)RC
Pin Pout
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Water droplets:
RC ∆P
10 μm 0.15 bar
100 nm 15 bar
1 nm 1500 bar
The Young–Laplace equation:
How big can the pressure difference be?
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Wikipedia
The Young–Laplace equation:Maximum Bubble Pressure
∆P = 2 γ/r γ = Rcap∆P/2
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The Young–Laplace equation:
∆P = Pin - Pout = γ Σ(1/ RC + 1/ RC’)
∆P = 4 γ/r
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ln (Pdrop/Pflat) = (Vmliq/RT)(2γ/RC)
The Kelvin Equation:
For the vapor pressure of a liquid drop relative to a flat surface.
RC: 1 mm 100 nm 10 nm 1 nm
Pdrop/Pflat : 1.001 1.011 1.114 2.95
Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
[Valid down to at least 2.5 nm for organic liquids and 1.5 nm for water]
The fundamental difficulty of nucleation!
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Pdrop/Pflat
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Ostwald Ripening:
Small crystals or drops are more soluble than large ones.
ln [S(RC)/S(RC→∞)] = (Vmliq/RT)(2γSL/RC)
http://www.weizmann.ac.il/
complex/stavans/ostwald-
ripening-nonequilibrium-
liquid-solid-thin-layers
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Condensation
Nucleation increases the radius of
curvature and hence diminishes the
Kelvin effect on condensation.
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Plateau-Rayleigh Instability
of a liquid stream
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Plateau-Rayleigh Instability of a l iquid stream
Radii of the Stream
Radii of curvature
of the instabilities
∆P = Pin - Pout = γ Σ(1/ RC + 1/ RC’)
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Plateau-Rayleigh Instability
of a liquid stream
The two effects, in general, do not exactly cancel.
One of them will have greater magnitude than the
other, depending upon wave number and the
initial radius of the stream. When the wave
number is such that the radius of curvature of the
wave dominates that of the radius of the stream,
such components will decay over time. When the
effect of the radius of the stream dominates that
of the curvature of the wave, such components
grow exponentially with time.
The unstable components (that grow over time)are
only those where the product of the wave number
with the initial radius is less than unity (kR0 < 1).
The component that grows the fastest is the one
whose wave number satisfies the equation:
kR0 < 0.697 (Wikipedia)
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Plateau-Rayleigh Equilibrium
in the pendant drop
Water+PEG Water Glycerol
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
The Gibbs Adsorption Equation:
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
(Constant T and P):
(Euler Theorem):
BULK Phase:
SURFACE Phase:
Deriving the Gibbs Adsorption Equation:
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Deriving the Gibbs Adsorption Equation:
The Surface Excess:
For a Binary System with Solvent (Component 1) and Solute (Component 2).
= Γ1dµ
1+ Γ
2dµ
2
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Deriving the Gibbs Adsorption Equation:
The Surface Excess:
For a Binary System with Solvent (Component 1) and Solute (Component 2).
Choose an imaginary dividing surface where there is no surface excess of the solvent (Γ1=0).
= Γ1dµ
1+ Γ
2dµ
2X
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The Gibbs Excess and the localization of the Interface
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Eastoe in Colloid Science, 2nd Ed., Wiley, 2010
In the Gibbs approach to define the surface
excess concentration Γ, the Gibbs dividing
surface X’ is defined as the plane in which the
solvent excess concentration becomes zero
(the shaded area of is equal on each side of
the plane) as in (a). The (probably unknown)
thickness of the interfacial region is then τ .
The surface excess of component i will then be
the difference in the concentrations of that
component on either side of the plane within
the interfacial region (the shaded area) in (b).
The Gibbs Excess and the localization of the Interface
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The Gibbs Excess and the localization of the Interface
Tetrabutylphosphonium Bromide in Formamide
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The Gibbs Excess and the localization of the Interface
Tetrabutylphosphonium Bromide in Formamide
Eastoe in Colloid Science, 2nd Ed., Wiley, 2010
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The Gibbs Excess and the localization of the Interface
SDS & CsDS in Formamide
Difference between
the Surface Excess and
the Surface Concentration
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Deriving the Gibbs Adsorption Equation:
The Surface Excess:
For a Binary System with Solvent (Component 1) and Solute (Component 2).
Choose an imaginary dividing surface where there is no surface excess of the solvent (Γ1=0).
Then:
= Γ1dµ
1+ Γ
2dµ
2X
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Surface Tension and the Gibbs Adsorption Isotherm
Negative Adsorption
Positive Adsorption
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The Gibbs Adsorption Equation:
For a Binary System with Solvent (Component 1) and Solute (Component 2),
Choose an imaginary dividing surface where there is no surface excess of the solvent (Γ1=0).
Then:
Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
SDS
dγ/d ln [SDS]
SDS
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Surface coverage in mol/m2:
Area per Surfactant, m2/molecule:
Surface pressure at the CMC, mN/m:
Ideal Gas Area, m2/molecule:
Gibbs Excess and the Area
per Molecule at the Surface
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
The Harkins Spreading Coefficient, S:
Water/Air Oil/Air Oil/Water
S > 0 Spreading S < 0 Non-Spreading
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
The Harkins Spreading Coefficient, S:
Spreading
Non-Spreading
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Surface Tensions of a Liquid Lens on the Surface of an Immiscible Liquid
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Kontogeorgis & Kiil, Introduction to Applied Colloid and Surface Chemistry-John Wiley & Sons (2016)
Review of the Fundamental Equations