Surfaces impact the free energy
It takes energy to form surfaces
Sm all particles dissolve easier
There are lim its to
grinding, fine powdered sugar is about 50µm
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Liquid-gas or solid-gas interface is called a surfaceFor surfaces we define a surface tension, s , energy/area
Liquid/liquid or solid/liquid or solid/solid is just called an interfaceFor interfaces we define the interfacial energy, g, energy/area
Gibbs Surface
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Surface Excess M oles
The adsorption of “i”There could be surface excess “i” or surface depletion of “i”
G i can be positive or negative
Surface Excess Properties
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Adsorption (not Absorption) see video
Adsorption of i
Surface Excess
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-SUVH A
-pGT
V doesn’t change
If the thickness is m uch sm aller than r you can ignore curvature
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Surface Area and Curvature Energy Terms, cx = 1/rx, cy = 1/ry
Surface Tension
dl
Curved Interface (Laplace Equation psat ~ s/r)
Pressure reaches equilibrium
a
b
a
b
dl
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dAs = 0 for flat surface
Laplace EquationFor a 100 nm (1e-5 cm ) droplet of water in air (72 e-7 J/cm 2 or 7.2 Pa-cm )Pressure is 720 M Pa (7,200 Atm ospheres)
10-2
10-1
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101
102
103
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Pres
sure
, MPa
100 101 102 103 104 105 106
Size, nm
Laplace Equation
Laplace Equationfor a water dropletin air
1 µm 1 mm
1 Atm.
1,000 Atm.
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Solid interface in a 1-component system
Work to create the interface
Interfacial energy, g
Surface creation always has an energy penalty. g is always positive
Nano-particles are unstableDifferences in surface energy for different crystal surfaces leads to fibrous or lam ellar crystals
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Crystal surface energy ~ num ber of bonds * bond energyDensity of bonds decreases with M iller Indices
FCC Nearest Neighbors Num ber of bonds[111] 6 3
[110] 12 6
[100] 8 4
Liquid droplets m inim ize surface area for a given volum e
So Spheres form
At high tem peratures crystalline solids also form spheres
Because surface energy becom es less im portant
Consider a crystal w ith constant volum e w ith N facets.
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W ulff Construction
Surface excess energy
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Draw a vector from the center of a crystal to a face.Gibbs-W ulff Theorem states that the length of the vector is proportional to the surface energy
hj = l g j
M inim ization to find the lowest free energy
hj O j is proportional to the volum e of a facet so for constant volum e:
And for constant volum e:
And
So: And
Diffusion rates and twinning can alter the crystal shape for large crystals
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Pressure difference for solid crystal facets
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Tem perature dependence, entropy at interface is high, n ~ 1.2 for m etals.
For a liquid with its own vaporRem iniscent of DG = DH(1-T/T*)
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Young Dupre Equation
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Three phases and three angles
Define the phase by the angle
Take the a , q line as the vector direction then
gaq + gqbcos(q) + gbacos(a ) = 0 using the dot product of the vectors
For the q , b line as the vector direction then
gaqcos(q) + gqb + gabcos(b) = 0 using the dot product of the vectorsFor the a , b line as the vector direction then
gaqcos(a ) + gqbcos(b) + gab = 0 using the dot product of the vectors
b is a flat rigid surface, b = p
gaqcos(q) + gqb + gabcos(b) = 0gglcos(q) + g ls - ggs = 0
Spreading Param eter: S>0 wets; S<0 partially wets
For S<0
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Dihedral Angle
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Dihedral Angle in M icroscopy
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Pressure for equilibrium of a liquid droplet of size ”r”
Reversible equilibrium
At constant tem perature
Differential Laplace equation
Sm all drops evaporate, large drops grow
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In the absence of nuclei, the initial bubbles on boiling can be very sm allThese bubbles are unstable due to high pressure so boiling can be prevented leading to a superheated fluid
Equilibrium
Ideal gas
Laplace equation for pressure
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Solubility and Size, rConsider a particle of size ri in a solution of concentration xi w ith activity ai
Derivative form of the Laplace equation
Dynam ic equilibrium
For an incom pressible solid phase
Definition of activity
Solubility increases exponentially w ith reduction in size, r
(xil)r = (xil)r=∞ exp(2gsl/(rRT r)) Sm all particles dissolve to build large particles with lower solubility
-To obtain nanoparticles you need to supersaturate to a high concentration (far from equilibrium ).-Low surface energy favors nanoparticles. (Such as at high tem peratures)-High tem perature and high solid density favor nanoparticles.
Supersaturation is required for any nucleation
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Critical Nucleus and Activation Energy for Crystalline Nucleation (Gibbs)
(M /r)is molar volume
Surface increases free energy
Bulk decreases free energy
Barrier energy for nucleation at the critical nucleus size beyond which grow th is spontaneous
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Critical Nucleus and Activation Energy for Crystalline Nucleation (Gibbs)
D fusG m = D fusH m - TD fusSm Lower T leads to larger D fusG m (Driving force for crystallization)
sm aller r* and sm aller D l-sG *
Deep quench, far from equilibrium leads to nanoparticles
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Ostwald RipeningDissolution/precipitation m echanism for grain grow thConsider sm all and large grains in contact w ith a solution
Grain Grow th and Elim ination of Pores
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Heterogeneous versus Homogeneous Nucleation
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Formation of a surface nucleus versus a bulk nucleus from n monomersHom ogeneous Heterogeneous (Surface Patch)
Surface energy from the sides of the patchBulk vs n-m erSo surface excess chem ical potential
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Barrier is half the height for nucleationSize is half
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Three forms of the Gibbs-Thompson Equation
Ostwald-Freundlich Equationx = supersaturated m ole fractionx∞ = equilibrium m ole fraction
n1 = the m olar volum e
Free energy of form ation for an n-m er nanoparticle from a supersaturated solution at T
Difference in chem ical potential between a m onom er in supersaturated conditions
and equilibrium with the particle of size r
At equilibrium
For a sphere
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Three forms of the Gibbs-Thompson Equation
Ostwald-Freundlich Equation
Areas of sharp curvature nucleate and grow to fill in. Curvature k = 1/r
Second Form of GT Equation
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Three forms of the Gibbs-Thompson Equation
Third form of GT Equation/ Hoffm an-Lauritzen EquationB is a geom etric factor from 2 to 6
Crystallize from a m elt, so supersaturate by a deep quench
Free energy of a crystal form ed at supercooled tem perature T
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For fine grain particles at times a high Gibbs free energy polymorph forms
S/V ~ 1/r
135 m 2/g ~ 12 nm particles
a -Al2O 3 is the stable form but g-Al2O 3 form s for sm all particles
g-Al2O 3 has a lower surface energy
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Adsorption (Adherence to surface, can be chemical or physical)
Physical adsorption : Low enthalpy of adsorption; reversible adsorption isotherm
Chem ical adsorption : Large enthalpy of adsorption; irreversible; chem ical change to surface
AdsorbentAdsorbateSolid or Liquid
M olecules in a Liquid or Gas
Surface Excess
M oles
The adsorption
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Internal Energy of System :
Surface Excess Internal Energy:
Differential Form with respect to the area:
Subtract the total derivative from the differential form yields the
Gibbs-Duhem for Surface Excess:
-SUVH A
-pGT
Gibbs Absorption Equation
Gibbs Absorption Equation
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Gibbs Absorption Equation
Gibbs-Duhem Equation:
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Relative Adsorption , doesn’t
Va is the volum e of the a phase
Relative Adsorption
Adsorption , G , depends on the position of the “surface”
M ultiply second equation by c ratio then subtract, it doesn’t depend on the position of the surface.
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Relative Adsorption
Gibbs Surface S is located where there is no net adsorption of A
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Solutes that reduce the surface tension are adsorbed
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For an ideal gas µB = RTlnpB where pB is the partial pressure of B
Surface Activity of B
Henry’s Law for Surfaces (surface impurities change surface tension)
At infinite dilution so Henry’s Law Regim e
A sm all am ount of electronegative elem ents can have a large im pact on surface energy of m etals jA
~1000 for oxygen and sulfur
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Adsorption Isotherms
Bg-Gas species (N 2)
Bm on – Adsorbed (N 2) in an occupied surface site
Vm on – Available surface site
aBg is activity of B in the gas phase
q = GB/GBM ax Fractional Coverage
Langm uir Adsorption Isotherm
GBM ax Is the coverage for a m onolayer.
Equilibrium Constant:
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Deviation from ideal adsorption
Fowler-Gugenheim Equation
w is W from Hildebrand, z is coordination num ber
M ulti-layer adsorptionBrunauer, Em m ett and Teller
BET Equation
C is a constant p0 is the saturation pressure of the adsorbent
E1 heat of adsorption of first layerEL heat of adsorption of subsequent layers
Used to get nanoparticle size Sauter M ean Diam eter dp = 6V/S
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Surface Energy Term and Block Co-PolymersM icro-Phase Separation
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How can you predict the phase size? (M eier and Helfand Theory)Consider lam ellar m icro-phase separation.
For a sym m etric binary blend of polym ers the FH theory predicts a critical point at cN = 2.
If the sam e two polym ers are bonded they m icrophase separate at cN = 5, the bonding m akes the polym ers m ore m iscible.
Enthalpy associated with phase segregationEntropy associated with locating the junction points at the phase interface
Entropy associated with stretching the chains
Drives a positive enthalpic contribution that favors m icro-phase separation
Assum e transition from perfectly m ixed to perfectly dem ixed
An interfacial layer of thickness dt, Area per polym er chain op
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How can you predict the phase size? (M eier and Helfand Theory)Consider lam ellar m icro-phase separation.
For a sym m etric binary blend of polym ers the FH theory predicts a critical point at cN = 2.If the sam e two polym ers are bonded they m icrophase separate at cN = 5, the bonding m akes the polym ers m ore m iscible.
dA
dBdt
R02 = Nl2
R = b dAB = b(dA + dB)
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How can you predict the phase size? (M eier and Helfand Theory)Consider lam ellar m icro-phase separation.
For a sym m etric binary blend of polym ers the FH theory predicts a critical point at cN = 2.If the sam e two polym ers are bonded they m icrophase separate at cN = 5, the bonding m akes the polym ers m ore m iscible.
dA
dBdtThere is only one free param eter, for instance op,
the cross sectional area per polym er chain (Tom W itten, U Chicago)
Find the m inim um in the free energy by varying op
Ignoring “ln” term that varies slowly
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How can you predict the phase size? (M eier and Helfand Theory)Consider lam ellar m icro-phase separation.
dA
dBdt
Perfect m atch
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