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

100

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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