Surfaces impact the free energy - University of Cincinnati ...€¦ · Surfaces impact the free...

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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