Recap of Lecture 19

by

John Stempien

3.14 / 3.40J / 22.71J – November 20, 2012

Diffusion in a solid results in self-generated stress,

establishing the Young-Laplace pressure at equilibrium.

2

internal external

2Young LaplaceP P P

R

• The core of the solid may spontaneously compress • Diffusion allows the solid to flow • Diffusive plating of atoms from the surface to the core occurs • Result is fewer surface atoms and lower surface energy

The chemical counterpart of the mechanical Young-

Laplace effect is called the Gibbs-Thomson effect

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• Young-Laplace – mechanical effect • Gibbs-Thomson – chemical effect • Center of material does not feel surface effects, only feels system wide stress • This manifests as a chemical potential described by Gibbs-Thomson

2

R

µ = chemical potential γ = surface energy Ω = atomic volume

The Gibbs-Thomson Effect is used to describe

Curvature Driven Flow on a Surface

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(+) Curvature R1(x’) > 0 R2(x’) > 0

Point of Low µ(x) Point of High µ(x’)

(-) Curvature R2(x) < 0

(+) Curvature R1(x) > 0

Gibbs-Thomson Now Expressed As:

1 2

1 1

( ) ( )R x R x

Chemical potential µ drives surface diffusion

• Flat surface has lower surface energy

• Material will move from peak to fill-in valley

*Here we assume γ is isotropic

The surface has a thin layer (skin) of disordered atoms

able to do surface diffusion along the surface.

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Layer Thickness 3 A

JΩ(x-Δx) JΩ(x+Δx)

• Chemical Potential at point x on the surface: 2( )x xy

• Average drift velocity of atoms in the layer: 3v M M x M xy

• Atom flux per unit surface area: J Cv

• Atom volume flux: 3J C v M xy

R R

x

y

We found the general solution for curvature-driven

surface diffusion.

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4

( ) ( 0) where surfBk t

B

DA t A t e B

k T

• k is like surface roughness • Doubling the surface roughness results in 16x increase in decay rate of the undulations • Smaller surface undulations are smeared-out or filled-in by diffusion of atoms • Longer undulation wavelengths require longer time to fill in and smooth over • Here we assumed the surface energy (γ) is isotropic

Fills in quickly by diffusion.

Fills in slowly by diffusion.

In reality, γ is not isotropic and depends on the

inclination of the surface.

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ε = energetic penalty per broken bond ao2 = exposed area from broken bond

n

sin( )

cos( )

Φ = inclination angle

2Total surface energy: ( ) cos( ) sin( )

o

na

This new expression for surface energy is visualized in

the Wulff Plot

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ε = energetic penalty per broken bond ao2 = exposed area from broken bond Φ = inclination angle

2Total surface energy: ( ) cos( ) sin( )

o

na

Cusps → atomic “discreteness”

Use the Wulff Graph to determine if an atomically flat

surface is stable or if facetting occurs instead.

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3n2n

1n

• Facetted length is longer • But γ is not isotropic so the facetted surface (meso length) could be energetically cheaper

*( )n n

3 3n 2 2n

Use the Wulff Graph to determine if an atomically flat

surface is stable or if facetting occurs instead.

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3n2n

1n

• γ* = effective surface energy if a macro facet with normal is comprised of two meso facets with normals and

n2n 3n

*( )n n

3 3n 2 2n

Tangent Circle Theorem: If γ* extends beyond circle, facetting will not happen.

Facetting will happen in the

example at left.

We can use the Wulff Construction in energy space to

determine the shape that minimizes the surface energy

of an anisotropic free-standing crystal.

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( )n n ( )n n

Wulff Plane

After tracing many Wulff planes, the center shape is the minimum surface energy shape.

From Herring, for anisotropic shapes:

2

i

i

P

h

hi

As a result of the equilibrium shape of the nanoparticle,

the melting point of the particle is suppressed with

decreasing particle radius.

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Gsolid(N,T)

γsolid

Gliq (N,T)

γliquid

Nano-particles with surface isotropy

γsolid > γliq Observed:

Tmnano α 1/R

Tm

R

G(T)

T

Gsolid

Gliquid

Tmbulk

• ΔS = Slope Gliq – Slope Gsolid = Sliq – Ssolid > 0 • Richard’s Rule for melting: ΔS ~ 1.1 kB/atom • Asolid > Aliq

• γsolid > γliq solid nano-particle less stable than liquid nano-particle

• Proof:

( , ) ( , )s s s L L LG N T A G N T A

2( ) ( ) ( ) ( )bulk bulk bulk bulk

s s L L L M S M M MA A N S T S T T T O T T

Scales as R2 Scales as R3

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End