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University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Interfaces: Basic concepts and nucleation theory

General concepts, interfacial free energySolid-vapor interfacesLiquid-vapor interfacesThermodynamics of surface energySolid-liquid interfacesHomogeneous nucleation theoryHeterogeneous nucleation

References:Porter and Easterling, Ch. 3Allen and Thomas, Ch. 5.3Jim Howe, Interfaces in Materials


Types of interfaces in materials

Free surfaces of a crystal (solid-vapor interface)

Solid-liquid interfaces

Grain boundaries between regions of the same crystalline phase

Interphase boundaries between different solid phases

local atomic structure at interfaces is different from that in the bulk of a phase ⇒interface can be characterized by excess surface free energy per unit area γ [J/m2]

Surface free energy is closely related to surface stress (or surface tension in liquids), also measured in [J/m2]

γ+= ANGG a

for a system composed of N atoms and having surface area A:A

N atoms Ga - Gibbs free energy per atom in the bulk

NGa - Gibbs free energy of the system if all atoms would be in the bulk

Aγ - excess free energy due to the fact that some of the atoms are on the surface


Surface free energy, surface tension, and surface strain

work done to create area dA of new surface:

γ+γ=== AddAdGFdxdWFliquid film

dx

L

film) theof sides (two 2LdxdA =

force F acting on the slide wire is counteracted by surface tension f - force per unit length acting in the plane of the interface along its perimeter, i.e., F = 2fL

dA

f dAdγ

=γ⇒= 0 liquid,for

fdALdxfFdx == 2γ+γ= AddAfdA

dAdAf γ

+γ=

f dAdγ

≠γ⇒≠ 0 solids,for

for solids, surface stress f is, in general, not equal to surface energy γ

equilibrium interatomic bonds near the surface can be different from those in the bulk ⇒ one can describe this in terms of interior atoms exerting stress on the surface

increase of the surface area in response to external force involves diffusion and can be slow for solids

slide wireframe

γ+γ=== AddAdGFdxdW


Thermodynamics of surface energy

sP

s

P

STG

T−=⎟

⎠⎞

⎜⎝⎛∂∂

=⎟⎠⎞

⎜⎝⎛∂γ∂

γ+= ANGG a )( sss TSHAAGA −==γsss TSHG −==γ

Gs, Hs, Ss - specific surface free energy, enthalpy, and entropy (per unit area)

dAVdPSdTdG γ++−=

atoms on the surface have more freedom of movement and lower vibrational frequencies ⇒ higher vibrational entropyremember vibrational entropy of vacancy formation

entropy per vacancy: ( ) 0'ln >=Δz

Bv vvks

energy of vacancy formation is lower at the surface as compared to bulk of the crystal ⇒ more vacancies ⇒higher configurational entropy

TSs increasing with decreases γ 0 ⇒>

Hs is related to the broken atomic bonds and does not depend on T

from Howe

1-2- Km mJ 45.0≈⎟⎠⎞

⎜⎝⎛∂γ∂

P

SV

T

1.2 .... 1.1/ ≈γγ LVSV

)(45.02.1 TTmTLVSVm

−+γ≈γempirical relation for metals


Thermodynamics of surface energy

from Howe

3/21

m

sSV V

HC Δ=γ

similar correlations exist for other properties defined by the strength of interatomic interaction:

melting temperature, Debye temperature,elastic moduli...

heat of sublimation →


Surface energy of liquids, γLV

The decrease of surface energy with increasing Tcan be described by a semiempirical equation:

ncLVLV TT )/1(0 −γ≈γ

n ≈ 1.2 for metals

K 0 toedextrapolat of value0LVLV γ−γ

liquid becomes indistinguishable from gascLV TT →→γ when 0

gradient of chemical composition or temperature

gradient of surface tension

Marangoni effect - flow along the gradient of surface tension

tears of wine: flow from regions with higher concentration of alcohol (lower γLV) to regions where concentration of alcohol decreased due to evaporation (higher γLV)

TT Δ+

γ low

γhigh alcohol evaporation


Surface energy of liquids, γLV

surface-tension-driven convection has to be accounted for to explain experimental observations

Marangoni effect in laser synthesis of TiN

strong dependence of Tm on XN

N concentration gradientre-solidification starts at surface

Höche & Schaaf, Heat and Mass Transfer 47, 519, 2011

1-2- Km mJ 24.0≈⎟⎠⎞

⎜⎝⎛∂γ∂

P

LV

T


Surface energy of crystals, γSVdirectional dependence of the surface energy in crystals: the number of broken bonds in an fcc crystal increases from 111 to 100 to 110 faces and, in general, Hs can be expected to be higher for high hkl index of the crystal face.

111 100 110

Let’s compare the surface energies of 111,100, and 110 faces by counting the nearest-neighbor broken bonds per unit are of each face

cut plane to create a 100 face

broken bonds

For each (red) atom on a new 100 surface we have to break 4 bonds to nearest neighbor atoms (green).

Surface density of atoms on a 100 plane is 2 atoms per a2 area.

Thus, the number of broken bonds per unit area is

a

22

bonds 8atoms2 atombonds4

aa=×


Surface energy of crystals, γSV

cut plane to create a 100 face

broken bonds

For each (red) atom in the top atomic layer of a new 110 surface we have to break 5 bonds to nearest neighbor atoms (green). Note that one of these bonds is to an atom in the 2nd

110 layer above the surface. This means that one bond is also broken for each atom in the 2nd layer below the surface.

Surface density on a 110 plane is 2 atoms per √2a2 area.

Thus, the number of broken bonds per unit area is a

2D unit cell of a 110 surfacea

a2

22ndst

bonds .4982

atoms2 layer 2in atom

bond 1layer 1in atom

bonds 5 aa

=×⎟⎟⎠

⎞⎜⎜⎝

⎛+

22

bonds .9264/3

atom1 atombonds3

aa=×

For each atom on a new 111 surface we have to break 3 bonds to nearest neighbor atoms.

Surface density on a 111 plane is 4 atoms per √3a2 area.

Thus, the number of broken bonds per unit area is



a face at an angle θ to a low-energy close-packed plane

energy per unit surface area:

Thus the nearest-neighbor broken-bond model predicts the following counts of the broken bonds:

θ

a

2/|)|sin(cos aESV εθ+θ=

where ε is the surface energy of close-packed plane per area of a2θ0

SVE

although contribution of Ss may smooth the energy cusps, the cusps are still present in γ(θ)

111for bonds .9262a

110for bonds .4982a

100for bonds 82a

Hence, γ111 < γ100 < γ110

This is a rather crude model, but provides a good general guidance on the relative surface energies of fcc metals.



Wulff construction - geometrical method to determine the low-energy crystal shape:

∑=

γ=γN

iii

totSV A

1

|| ii OA=γ

equilibrium shape of the crystal is inner envelope of all Wulff planes

in the presence of sharp cusps, the equilibrium shape is a polyhedron with large faces that correspond to low γSV

min1

→γ=γ ∑=

N

iii

totSV A

equilibrium shape of the crystal is defined by

construct γ-plot - surface about an origin so that the distance from the origin to the surface in a given direction is equal to the value of γSV for crystal face perpendicular to this direction

at the end of each vector draw a plane perpendicular to the vector direction (Wulff plane)

section through γ-plot of fcc crystal)011(

crystal with shape enclosed by planes A1, A2, …, AN has surface energy



The equilibrium shape defined by the Wulff construction is determined by the directional dependence of the surface energy and is the shape a crystal adopts during (equilibrium) growth

Experimental identification of equilibrium shapes of small crystals can be used to determine γSVfor different faces

STM image of a Pb crystallite equilibrated at 353 K. (221) and (112) facets are seen around the central (111) facet. Diameter of the crystal is ~550 nm.

Yu, Bonzel, and Scheffler, New J. Phys. 8, 65, 2006

Single quadrant of a polar plot of γ(θ) for unrelaxed(left) and relaxed (right) Pb crystal predicted in DFT calculations



H. Imai and Y. Oaki, MRS Bulletin 35, February issue, 138-144 (2010)


Growth instabilities, dendrites

by Harry Bhadeshia

by Nadezhda Bulgakova

SEM image by Begum Unveroglu


Growth instabilities, dendrites

http://www.lowtem.hokudai.ac.jp/ptdice/english/aletter.html

Bragard et al., Interfacial Science 10, 121, 2002

thermal dendrites

diffusional dendrites


Solid - liquid interface

Ωsm - atomic volume of the solid at Tm

Digilov, Physica B 352, 53, 2004

from Porter and Easterling

fluctuating solid–liquid interface in thermal equilibrium (MD simulation of pure Ni)

Hoyt, Asta, and Karma, Mater. Sci. Eng. R 41, 121, 2003

mTSL HC Δ=ργ − 3/2

CT = 0.32 (non-metals) - 0.45 (metals)ρ - number density of the solid phase

CT - Turnbull coefficientD. Turnbull, J. Appl. Phys. 21, 1022, 1950


Origin of the interfacial energy γConsider a solid-liquid interface as an example. Depending on the type of material and crystallographic orientation of the interface, the interface can be atomically flat (smooth, faceted) or rough (diffuse).

liquidsolid

liquidsolid

H

SvH

LvH

TS-SvmST-

G

LvmST-

spatial coordinate

SvG L

vGγSL

inte

rfac

e

Lv

Lv

Lv TS-HG =

Sv

Sv

Sv TS-HG =

free energies of liquid and solid per unit volume:


Nucleation and growth - the main mechanism of phase transformations in materials

αB

1X

αB

2X 0

BX

coordinatespatial

αB

1X

αB

2X 0

BX

αB

1X

αB

2X 0

BX

atomsB

atomsB

T

P

solid

liquid


NucleationNucleation can be

Heterogeneous – the new phase appears on the walls of the container, at impurity particles, etc.

Homogeneous – solid nuclei spontaneously appear within the undercooled phase.

solid solid

liquid liquid

homogeneous nucleation heterogeneous nucleation

supercooledliquid

Let’s consider solidification of a liquid phase undercooled below the melting temperature as a simple example of a phase transformation.


Homogeneous nucleation

solid

liquid Is the transition from undercooled liquid to a solid spherical particle in

the liquid a spontaneous one?

That is, does the Gibbs free energy decreases?

supercooledliquid

The formation of a solid nucleus leads to a Gibbs free energy change of ΔG = G2 - G1 = -VS (Gv

L – GvS) + ASLγSL

negative below Tm

always positive

1 2

VS – volume of the solid sphereASL – solid/liquid interfacial areaγSL – solid/liquid interfacial energy

ΔGv = GvL – Gv

S is the difference between free energies per unit volume of solid and liquid

at T < Tm, GvS < Gv

L – solid is the equilibrium phase


Driving force for the phase transformation (ΔGv)When a liquid is cooled below the melting temperature, there is a driving force for solidification, ΔGv = Gv

L - GvS

G

T*

ΔGv

GvS

GvL

Tm

ΔT

At any T < Tm there is a driving force for solidification ⇒ liquid solidifies at T < Tm. If energy is added/removed quickly, the system can be significantly undercooled or (supercooled). As we will see, the contribution of interfacial energy (γSL) results in a kinetic barrier for the phase transformation.

At temperature T*: Lv

*Lv

Lv ST-HG =

Sv

*Sv

Sv ST-HG =

v*

vv ST-HG ΔΔ=Δ

At temperature Tm: 0ST-HG mvm

mvv =ΔΔ=Δ

m

mvm

v THS Δ

=Δ

m

mv

m

mv*m

vv TΔTΔH

TΔHTΔHΔG =−≈

For small undercooling ΔT we can assume that ΔHv and ΔSv are independent of temperature (neglect the difference in Cp between liquid and solid)



ΔG = G2 - G1 = -VS Δ Gv + ASLγSL

For a spherical nucleus with radius r: 3

34 π rVS =

SLvr γπ rGπ r-G 23 4Δ34Δ +=

24π rASL =

rGΔ

*GΔ

*r

GΔinterfacial energy ~ r2

volume energy ~ r3

For nucleus with a radius r > r*, the Gibbs free energy will decrease if the nucleus grows

r* - critical nucleus size

ΔG* - nucleation barrier



at r = r* 08Δ4Δ 2 =+= SLv π r γGπ r-dr Gd

v

SL*

G γrΔ2

=

( )( )2

3

Δ316Δ

v

SL*

Gγπ G =

G

T*

GvS

GvL

Tm

ΔT

*rr =*SL

v r γG 2Δ =

Temperature of unstable equilibrium of a solid cluster of radius r* with undercooled liquid.



v

SL*

G γrΔ2

=( )

( )23

Δ316Δ

v

SL*

Gγπ G =

m

mv T

THG ΔΔΔ =

THT γrm

mSL*

Δ1

Δ2

⎟⎟⎠

⎞⎜⎜⎝

⎛=

( )( ) ( )22

23

Δ1

Δ316Δ

THTγπ G

m

mSL*

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Both r* and G* decrease with increasing undercooling

The difference between the Gibbs free energy of liquid and solid(“driving force” for the phase transformation) is proportional to the undercooling below the melting temperature, ΔT = Tm – T:

Therefore:

rΔG

*ΔG 1

*1r

ΔG

*2r

*ΔG 2

m12 TTT <<


Rate of homogeneous nucleation

There is an energy barrier of ΔG* for formation of a solid nucleus of critical size r*. The probability of energy fluctuation of size ΔG* is given by the Arrhenius equation and the rate of homogeneous nucleation is

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTG~νN

*

dΔexp& - nuclei per m3 per s

where νd is a pre-factor defined by the frequency with which atoms from liquid can attach to the solid nucleus. The rearrangement of atoms needed for joining the solid nucleus typically follows thesame temperature dependence as the diffusion coefficient: ⎟

⎠⎞

⎜⎝⎛ −

kTE~ν d

d exp

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −

kTG

kTE~N

*d Δexpexp&Therefore:


Rate of homogeneous nucleation

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −

kTG

kTE~N

*d Δexpexp&

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTG *Δexp~⎟

⎠⎞

⎜⎝⎛ −

kTE dexp~

N&

mTTemperature

ΔG* is too high - nucleation is suppressed

( ) ( )kTEkTG d* −<<− expΔexpd

* EG >Δ

d* EG ≤Δ ( ) ( )kTEkTG d

* −>− expΔexpΔG*~ 1/ΔT2 ⇒ decreases with T ⇒ sharp rise of nucleation (diffusion is still active)

( )kTEd−exp – too small ⇒ low atomic mobility suppresses the nucleation rate


Rate of homogeneous nucleationIn many phase transformations, it is difficult to achieve the level of undercooling that would suppress nucleation due to the drop in the atomic mobility (regime 3 in the previous slide). The nucleationtypically happens in regime 2 and is defined by the probability of energy fluctuation sufficient to overcome the activation barrier ΔG*r:

ΔTΔTcr

( )( ) ( )22

23

Δ1

Δ316Δ

THTγπ G

mv

mSL*r ⎟

⎟⎠

⎞⎜⎜⎝

⎛=using

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−= 20 Δ

expTAIN&

where A has a relatively weak dependence on temperature (as compared to ΔT2)

- very strong temperature dependence!

There is critical undercooling for homogeneous nucleation ΔTcr ⇒ there are virtually no nuclei until ΔTcr is reached, and there is an “explosive” nucleation at ΔTcr.

0

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTΔG~N

*

exp&

N&

we get


Heterogeneous nucleation

3 interfacial energies:

Balancing the interfacial tensions in the plane of the containerwall gives γLC = γSC + γLS cos(θ) ⇒ wetting angle θ is defined by cos(θ) = (γLC - γSC)/ γLS

solidnucleus

liquid

θ

γSC

γLSγLC

solid

liquidsupercooled

liquid

the new phase appears on the walls of the container, at impurity particles, grain boundaries, etc.

γLC – liquid container interface,γLS – liquid-solid interface, γSC – solid-container interface.

Example: heterogeneous nucleation of a nucleus of the shape of a spherical cap on a wall of a container.

How about the out-of-plane component of γLV? It is expected to be balanced by the elastic response of the solid, but theoretical analysis is not straightforwardSee [PRL 106, 186103, 2011] for extracurricular reading



solidnucleus

liquid

θ

γSC

γLSγLC

the formation of the nucleus leads to a Gibbs free energy changeof ΔGr

het = -VS ΔGv + ASLγSL + ASCγSC - ASCγLC

VS = π r3 (2 + cos(θ)) (1 – cos(θ))2/3ASL = 2π r2 (1 – cos(θ)) and ASC = π r2 sin2(θ)

( ) ( )θSGθSγπ rGπ r-G rSLvhetr

hom23 Δ4Δ34Δ =

⎭⎬⎫

⎩⎨⎧ +=

( ) ( )( ) 4cos1cos2 where 2 /θθθS −+=

One can show that

at r = r* ( ) ( ) 08Δ4Δ 2 =+= θ Sπ r γGπ r-dr Gd

SLvr

v

SL*

G γrΔ2

=

( ) ( )( )

( ) *

v

SL*het GθS

Gγπ θSG hom2

3

ΔΔ3

16Δ ==

- same as for homogeneous nucleation

( ) ( )( ) 42 104cos1cos2,10 small becan )( -/θθθ Sif θS ≈−+==⇒θ o



ΔG*

ΔT

*hetGΔ

*G homΔ

active nucleation starts

crhetTΔ crThomΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTG~N

*homhom Δexp&

ΔT

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTG~N

*hethet Δexp&

homNN het && >>

hetN& homN&N&

( ) **het GθSG homΔΔ =*

hetΔG

*r

ΔG

*homΔG

heterogeneous nucleation starts at a lower undercooling


Pre-meltingfor solid/liquid/vapor interfaces, often γSolid-Vapor > γSolid-Liquid + γLiquid-Vapor

in this case, no superheating is needed for nucleation of liquid and surface melting can take place below Tm ⇒ pre-melting

Why ice is slippery?Physics Today, Dec. 2005, pp. 50-55

cross-section of an atomic cluster close to Tm(simulations by J. Sethna, Cornell University)


Observation of two-step nucleation in solid–solid phase transitions

Yi Peng et al. Nature Mater. 14, 101, 2015

single-particle resolution video microscopy of a transition from square to triangular lattice in a colloidal film

microgel colloidal spheres with σ of ~0.7 μm

packing fraction

phase transformations in a system of hard spheres (colloidal particles) confined between parallel hard plates: 1∆ → 2 → 2∆ → 3 → ... → n∆ →(n + 1) → (n + 1) ∆ → ..., where where n is the number of crystal layers, and the phases have triangular (∆) or square () symmetry.

Fortini & Dijkstra, J. Phys. Condens. Matter 18, L371, 2006


Two-step nucleation in solid–solid phase transitions

Yi Peng et al. Nature Mater. 14, 101, 2015

two-step nucleation with an intermediate liquid phase when γS-L < γS1-S2

intermediate liquid may play a role in the nucleation processes of solid–solid transitions

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid ZhigileiA.Y. Vorobyev and C. Guo, J. Appl. Phys. 117, 033103, 2015.http://www.rochester.edu/newscenter/superhydrophobic-metals-85592/

Super-hydrophobic surfaces produced by direct femtosecond laser ablationAnatoliy Y. Vorobyev and Chunlei Guo, The Institute of Optics, University of Rochester

hieratical nano- and micro-structures generated with femtosecond laser pulses ⇒ multifunctional surfaces exhibiting combined effects of enhanced broadband light absorption, superhydrophobicity & self-cleaning.

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid ZhigileiA.Y. Vorobyev and C. Guo, Optics Express 18, 6455, 2010

Superwicking surfaces produced by direct femtosecond laser ablationAnatoliy Y. Vorobyev and Chunlei Guo, The Institute of Optics, University of Rochester

Si

• Capillary spreading of liquid in smooth V-grooves• The water contact angle on laser-treated silicon and glass surfaces

is measured to be about 0º ⇒ surface is superhydrophilic


Waterfallby M.C. EscherLithograph, 1961

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