Nucleation Rate

8/13/2019 Nucleation Rate

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KINETICS

MACROSCOPIC TRANSPORT

Whenever a material system is not in thermodynamic equilibrium, driving forces

arise naturally to push it toward equilibrium. The resulting atomic concentration

gradients trigger time-dependent mass transport effects that reduce free-energy

variations in the system. In solids, mass transport is accomplished by diffusion.

Fickestablished the phenomenological connection between concentration

gradients and the resultant diffusional transport through the equation

WhereD is Diffusion constant and increases in exponential fashion with

temperature according to a Maxwell-Boltzmann relation

Where Do is a constant and EDis activation energy of Diffusion

The time-varying accumulations or depletions

of atomic species is governed by equation:

Where C(x,t) is Concentration as a function of

space and time.


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Consider an initially pure thick film into which some solute diffuses from the surface. If the

film thickness is very large or effectively infinite compared to the extent of diffusion, the

situation can be physically modelled as following initial and boundary conditions

A second boundary condition that must be specified concerns the nature of the

diffusant distribution maintained at the film surface x = 0.

Substrate

Diffusantx=0Two simple cases are possible.

i) ii)

In the first, a thick layer of diffusant provides an essentially limitless external supply of atoms

maintaining a constant surface concentrationCo for all time.

In the second case, a very thin layer of diffusant provides an instantaneous source Soof

surface atoms per unit area. Here the surface concentration diminishes with time as atoms

diffuse into the underlying

substrate.

MACROSCOPIC TRANSPORT Cont...


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The two solutions are

i)

ii)

MACROSCOPIC TRANSPORT Cont...


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The fraction of vacant lattice sites was

previously given by

Where Ef is surface energy generally 1eV.

exp( )f

B

E

K T

The number of times per second that an atomsuccessfully reaches the activated state is :

Where Each atom vibrates about its equilibrium

position with a characteristic lattice frequencyv

and Emis the vacancy jump or migration energy

per atom.

exp( )m

B

EvK T

Atom fluxes that pass from plane 1 to 2 and from plane 2 to 1 are respectively given by

1 2 0

1exp( ) exp( )

6

fm

B B

EEJ v Ca

K T K T

0

2 1 0

1exp( ) exp( )( )

6

fm

B B

EE dCaJ v C a

K T K T dx

where we have substituted Ca0 for n(x) and used the factor of 1/6 to account for

bidirectional jumping in each of the three coordinate directions.

The net flux JNis the difference or 20

1exp( ) exp( )( )

6

fmN

B B

EE dCJ a v

K T K T dx


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By association with Pick's law and the expression for D

where

We get [ ]D f m

E E E 20 0

1

6D a v


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Atom Transport in a Force Field

Consider neighbouring atomic positions in a

crystalline solid where no fields are applied. The

free energy of the system has the periodicity of

the lattice and varies schematically as shown inFig a.

Imposition of an external field now biases or tilts

the system such that the free energy is lower in

site 2 relative to 1 by an amount 2G as shown in

Fig b.

The rate at which atoms move from 1 to 2 or 2 to

1 is given by


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and the net rate rNis given by the difference

By multiplying both sides of Eq a0, we obtain the atomic velocity v:

The term in brackets is essentially the diffusivity D, where GDis a diffusional activation free

energy that for all practical purposes we may equate with ED

The term 2G/a0is a measure of the molar free-energy gradient or applied force F


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When chemical equilibrium prevails, thecompeting rates are equal and rN= 0. Therefore

where G* is the molar free energy of activation

If CRis the concentration of reactants at coordinate position 1 and

CPthe concentration of products at 2, then the net rate of reaction

is proportional to


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Substrate Surfaces and Thin-Film Nucleation

Many observations of subsequent film formation have pointed to three basic growth modes:

(1) island(or Volmer-Weber), (2)layer (or Frank Van der Merwe), and (3) SK(Stranski-

Krastanov)

Island growth occurs when the smallest stable clustersnucleate on the substrate and grow in three

dimensions to form islands. This happens when atoms

or molecules in the deposit are more strongly bound

to each other than to the substrate.

the extension of the smallest stable nucleus occursoverwhelmingly in two dimensions, resulting in the

formation of planar sheets. In this growth mode the

atoms are more stronglybound to the substrate than

to each other. The first complete monolayer is then

covered with a somewhat less tightly bound second

layer. Providing the decrease in bonding energy is

continuous toward the bulk-crystal value,

the layer growth mode is sustained.

The layer plus island or Stranski-Krastanov (S-K) growth mechanism is an intermediate

combination of the preceding two modes. In this case after forming one or more

monolayers, subsequent layer growth becomes unfavorable and islands form.


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AN ATOMIC VIEW OF SUBSTRATE SURFACES


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THERMODYNAMIC ASPECTS OF NUCLEATION

SURFACE ENERGIES

Atoms at free surfaces are more energetic than

atoms within the underlying bulk because they makefewer bonds with surrounding atoms and are thus

less constrained. The difference in inter atomic

energy of atoms at these two locations is the origin

of surface energy. Alternatively, there is a

thermodynamic driving force to reduce the number

of necessarily cut, dangling bonds at the surfacethrough rebonding between atoms. We may then

equivalently view the surface energy (J/m2) in

terms of the energy reduction per unit area.

surface energies roughly span the range 0.2 to 3

J/m2 with 1 J/m2 ( = 1000 erg/cm2) being typical.


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CAPILLARITY THEORY OF HETEROGENEOUS NUCLEATION

We start by assuming that film forming

atoms or molecules in the vapor phaseimpinge on the substrate creating nuclei of

mean dimension r. The free-energy change

accompanying the formation of such an

aggregate is given by

For the spherical cap-shaped solid nucleus shown in Fig. the curved surface area (a1r2), the

projected circular area on the substrate (a2r2), and the volume (a3r3) are involved

0d G

dr

At equilibrium And the size is called critical radius r*

Correspondingly


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An aggregate smaller in size than r* disappears by shrinking, lowering G in the

process. Critical nuclei grow to supercritical dimensions by further addition of

atoms, a process which lowers G still more. In heterogeneous nucleation the

accommodating substrate catalyzes vapour condensation by lowering the energybarrier G* through a reduction of the contact angle. After substitution of the

geometric constants, it is easily shown that

If, for example, the film nucleus is elastically strained throughout because of the

bonding mismatch between film and substrate, then a term a3r3Gs, where Gsis the

strain free-energy change per unit volume, would be appropriate. In the calculation for

AG*, the denominator of Eq. 7-9 would then be altered to 27a32/ (Gv + Gs)2,

Because the sign of Gv is negative while Gs is positive, the overall energy barrier to

nucleation increases in such a case. If, however, deposition occurred on an initially

strained substrate, i.e., one with emergent cleavage steps or screw dislocations, then

stress relieve during nucleation would be manifested by a reduction of G*. Substrate

charge and impurities would similarly influence G* by affecting terms related to either

surface and volume electrostatic, chemical, etc., energies.


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The most important thing to remember about G* is its strong influence on the

density (N*) of stable nuclei that can be expected to survive.

where nsis the total nucleation site density


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Nucleation Rate

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