Mobility of Point Defects: Diffusion - people.Virginia.EDUpeople.virginia.edu/~lz2n/mse6020/notes/Diffusion.pdf · University of Virginia, MSE 6020: Defects and Microstructure in

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

Mobility of Point Defects: Diffusion

Diffusion: Phenomenological description, diffusion coefficientDriving force for diffusionSpinodal decomposition vs. nucleation and growthMicroscopic picture of diffusion: Einstein relation The concept of thermal activationAtomic mechanisms of diffusion: Vacancy and interstitial diffusionDiffusion of interstitial and substitutional impuritiesDiffusion of self-interstitials and vacanciesFast diffusion paths: Diffusion along crystal defects

References:Poter &Easterling, Ch. 2, pp. 60-106Shewmon, Diffusion in solids, Ch. 2


Diffusion: Definition and Phenomenological Description“Diffusion” is transport through “random walk” - atoms, molecules, electrons, phonons, etc. are moving around randomly in a crystal. This random motion can lead to mass, heat, or charge transport. We will consider atomic diffusion that is involved in most phase transformations.

Phenomenological description of diffusion:

Fick’s first law: the diffusion flux along direction x is proportional to the concentration gradient

D - diffusion coefficient [m2s-1]dxdCDJ −= A J

C

x

dxdC

J - flux of diffusing atoms = amount of material diffusing through unit area and per unit time [mol m-2s-1] or [kg m-2s-1]

( ) ( )⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂

∂x

tx,CDt

tx,Cx

Fick’s second law for nonsteady-state diffusion: relates the rate of change of composition to the curvature of the concentration profile

C

x


Driving force for diffusionThe empirical Fick’s first law assumes proportionality between the diffusion flux and the concentration gradient. But thermodynamics tells us that any spontaneous process should go in the direction of minimization of the free energy.

BX

G

10BX

G

10

A-rich B-rich

A-rich B-rich

1α 2α

Atoms can diffuse from regions of high concentration towards the regions of low concentration – down the concentration gradient (left) as well as from the regions of low concentration towards the regions of high concentration – sometimes even up the concentration gradient (right)


Driving force for diffusion

'BX

G

10

BX

G

10

BμThe chemical potential of atoms of type A can be defined as the free energy per mole of A atoms:

2121 αA

αA

αB

αB μμ and μμ ==

Bμ

1α 2α

BBAA XμXμG +=

xμCMJ A

AAx ∂∂

−=

A and B atoms are diffusing from the regions where chemical potential is high to the regions where chemical potential is lower. The driving force for diffusion is gradient of chemical potential.

Atoms migrate so as to remove differences in chemical potential. Diffusion ceases at equilibrium, when

where MA is the atomic mobilityof A atoms.

Diffusion occur so that the free energy is minimized → is driven by the gradient of free energy

'Bμ

''Bμ

''BX

Therefore, the free energy gradient can be expressed through the chemical potential gradient and the flux can be expressed as:


Driving force for diffusion

Reconcile xμCMJ B

BBx ∂∂

−=B

BBB

B

BBBB X

μXMCμCMD

∂∂

=∂∂

=xCDJ B

Bx ∂∂

−=

- gradient of chemical potential is in the same direction as the concentration gradient

0D then 0,Xμ if B

B

B >>∂∂

- diffusion occurs against the concentration gradient! 0D then 0,Xμ if B

B

B <<∂∂

One can show that the condition for negative ∂μ/∂XB is

with

0X

G2

B

2

<∂∂

BX

G

10

1α 2α

Bμ

↑BX

0Xμ

B

B <∂∂

When the free energy curvature is negative, the diffusion is directed against the concentration gradient → spinodal decomposition


Spinodal decomposition vs. nucleation and growthLet’s consider a homogeneous solution (α phase) with composition XB

0 cooled into the miscibility gap. To decrease the total free energy of the system, the solution will decompose into α1 and α2. But what is the mechanism of this decomposition? Let’s consider small fluctuations around the average composition.

BX

G

BX

G

-BX

BX+ 0BX -

BX BX+ 0

BX

0XG2B

2

<∂∂ 0

XG2B

2

>∂∂

G decreases as a result of an arbitrary infinitesimal fluctuation in composition –the system is unstable

G increases as a result of an infinitesimal fluctuation in composition – the system is stable with respect to small fluctuations

-BX

BX+

0BX

coordinatespatial

2

B αX <<+

1

B αX >>−

0 B

B

B XXX ≈≈ −+

(fluctuations are small)


Spinodal decomposition vs. nucleation and growthNucleation and growthSpinodal decomposition

αB

1X

αB

2X 0

BX

coordinate spatial

αB

1X

αB

2X 0

BX

αB

1X

αB

2X 0

BX

atoms B

atoms B

αB

1X

αB

2X 0

BX

coordinatespatial

αB

1X

αB

2X 0

BX

αB

1X

αB

2X 0

BX

atomsB

atomsB

BX

T

10

1T

1α 2α

21 αα +

0XG2B

2

<∂∂

0XG2B

2

>∂∂ 0

XG2B

2

>∂∂

BX

G

101α 2α

0XG2B

2

=∂∂

1TT =

21 αα + 21 αα +

nucleation and growth spinodal decomposition


Spinodal decomposition vs. nucleation and growth

Although the system within the miscibility gap but outside the spinodal region is stable (metastable) with respect to small fluctuations, it is unstable to the separation into α1 and α2determined by the common tangent construction.

There is large difference in composition between α1 and α2 and large composition fluctuations are required in order to decrease the free energy. A process of formation of a large composition fluctuation is called nucleation. The phase separation is occurring in this case by nucleation and growth.

http://math.gmu.edu/~sander/movies/spinum.html

Computer simulation of spinodal decomposition in a binary alloy


Microscopic picture of diffusion

∑=

−≡≡≡N

iiii

))(r(t)r(N

(t)rsMSD1

222 01Δ rrr

Let’s first consider an intuitive (not rigorous) “derivation” of the relationship between s and D.

At the microscopic (atomic) level, diffusion is defined by random movement (“random walk”) of the diffusing species (atoms, molecules, Brownian particles). The mobility of the diffusing species can be described by their mean square displacement:

(t)rir(0)ri

rHow to relate this microscopic characteristic (the mean square distance of atomic/molecular migration) to the macroscopic transport coefficient Dused in the phenomenological Fick’s laws?

s

Suppose that in time t, the average particle moved a distance sx along the direction in which diffusion is occurring. X0

CRCL

JL

JR

sx sx

Assuming that the movement of particles is random, half of the particles moved to the left, half to the right. The total flux of particles from left to right is JL×t.

If CL is the average concentration of diffusing particles in the left zone, than the total flux per unit area is JL×t = (sxCL)/2 - half of the particles will cross the plane X0 from left to right.

The same for the flux from right to left, JR×t = (sxCR)/2


Microscopic picture of diffusion: Einstein relation Since JL×t = (sxCL)/2 and JR×t = (sxCR)/2, the net flow across X0 is J = JL – JR = sx(CL – CR)/2t

We can express (CL – CR) in terms of the concentration gradient dc/dx:(CL – CR)/sx = -(CR – CL)/sx = - dc/dx

Therefore, J = sx(CL – CR)/2t = -sx2/2t dc/dx

From the Fick’s law we also have J = -D dc/dx

Thus, D = sx2/2t or sx

2 = 2tD for 1D diffusionFor 3D diffusion s2 = sx

2 + sy2 +sz

2 = 3sx2, and D = s2/6t

X0

CRCL

JL

JR

sx sx

In general, D = s2/2dt

This expression is called Einstein relation since it was first derived by Albert Einstein in his Ph.D. thesis in 1905. It relates macroscopic transport coefficient D with microscopic information on the mean square distance of molecular migration.

where d is the dimensionality of the system

Historic note: Before Albert Einstein turned his attention to fundamental questions of relative velocity and acceleration (the Special and General Theories of Relativity), he published a series of papers on diffusion, viscosity, and the photoelectric effect. His papers on diffusion came from his Ph. D. thesis. Einstein's contributions were to propose: 1. that Brownian motion of particles was the basically the same process as diffusion; 2. a formula for the average distance moved in a given time during Brownian motion; 3. a formula for the diffusion coefficient in terms of the radius of the diffusing particles and other known parameters. Thus, Einstein connected the macroscopic process of diffusion with the microscopic concept of thermal motion of individual molecules. Not bad for a Ph. D. thesis!


Derivation of the Einstein relation

Diffusion equation:

Let’s multiply the diffusion equation by x2 and integrate over space:

( ) ( )2

2

xt,xCD

tt,xC

∂∂

=∂

∂

( ) ( )dxx

t,xCxDdxt,xCxt 2

222 ∫∫ ∂∂

=∂∂

( )tx 2

( ) ( ) ( ) ( ) ( ) ( )=

∂∂

−∂

∂=

∂∂

∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂

∂=

∂

∂∫∫∫∫

+∞

∞−

dxx

txCxDx

txCDxdxx

txCxxDdx

xtxCx

xDdx

xtxCxD

ttx ,2,,,, 2

22

2

22

2

( )( ) ( ) ( ) ( ) DdxtxCDtxDxCdxtxCxxDdxtxxC

xD 2,2,2,2,20 =+−=

∂∂

+∂∂

− ∫∫∫∞+

∞−

( ) ADt2tx 2 += for 1D diffusion

Let’s consider diffusion of particles that are initially concentrated at the origin of our coordination frame, - Dirac delta function ( ) ( )x0,xC δ=

Solution: ( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

π=

Dtx

DttxC

2exp

21,

2

2/1


Derivation of the Einstein relation for d dimensions

Diffusion equation:

Let’s multiply the diffusion equation by r2 and integrate over space:

Let’s consider diffusion of particles that are initially concentrated at the origin of our coordination frame, - Dirac delta function

Solution:( ) ( )t,rCDt

t,rC 2 rr

∇=∂

∂

( ) ( ) rdt,rCrDrdt,rCrt

222 rrrr∫∫ ∇=

∂∂

( )tr 2

( )( ) ( )( ) ( ) =∇⋅∇−∇⋅∇=∇=

∂

∂∫∫∫ rdtrCrDrdtrCrDrdtrCrD

ttr rrrrrrrrrr ,,, 2222

2

( ) ( ) ( )( ) ( ) ( ) =⋅∇+⋅∇−=∇⋅−∇ ∫∫∫∫ rdtrCrDrdtrCrDrdtrCrDSdtrCrD rrrrrrrrrrrrrrr,2,20,2,2

( ) ( ) dDrdtrCdDSdtrCrD 2,2,2 =+− ∫∫rrrrr ( ) AdDttr += 22

for diffusion in d-dimensional space= 1

( ) ( )r0,rC rrδ=

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

π=

Dt2rexp

Dt21t,rC

2

2/d

r

zz

yy

xx ∂

∂+

∂∂

+∂∂

≡∇r

Gradient:

Gradient of a scalar function: zz

yy

xx

)r(∂Φ∂

+∂Φ∂

+∂Φ∂

≡Φ∇rr

Divergence of a vector function:z

Fy

FxF)r(F zyx

∂∂

+∂∂

+∂∂

≡⋅∇rrr

( ) ( ) ( )( ))r(F)r()r(F)r()r(F)r( rrrrrrrrrrrr⋅Φ∇+⋅∇Φ=Φ⋅∇

The divergence theorem: ∫∫ =⋅∇SV

Sd)r(Frd)r(Frrrrrrr

2

2

2

2

2

22

zyx ∂∂

+∂∂

+∂∂

=∇⋅∇≡∇rr

Laplacian:Review of relevant calculus


The concept of thermal activationLet’s consider transition between two states in a system. The transformation between the initial and final states involves rearrangement of atoms – the system should go through a transformation (or reaction) path. Since the initial and final states are metastable or stable ones, the energy of the system increases along any transformation path between them

G1 and G2 are the Gibbs free energies of the initial and final states of the system

ΔG = G2 - G1 is the driving force for the transformation.

ΔGa is the activation free energy barrier for the transition - the maximum energy along the transformation path relative to the energy of the initial state.

activated stateFinal state (equilibrium or another metastable)

transition pathG

GΔ1G

2G

aGΔ

initial state (metastable)


The concept of thermal activationFor a system to proceed through the transition path to the equilibrium state, it has to obtain the energy that is sufficient to overcome the activation barrier.

The energy can be obtained from thermal fluctuation (when the thermal energy is “pooled together” in a small volume). Statistical mechanics can be used to predict the probability that a system gets an energy that exceeds the activation energy. This process is called thermal activation.

The probability of such thermal fluctuation or the rate at which a transformation occurs, depends exponentially on temperature and can be described by equation that is attributed to Swedish chemist Svante Arrhenius:*

⎟⎠⎞⎜

⎝⎛−⎟

⎠⎞⎜

⎝⎛⎟

⎠⎞⎜

⎝⎛− Tk

Hk

S~TkGrate~

B

a

B

a

B

a ΔexpΔexpΔexp

* Arrhenius equation was first formulated by J. J. Hood on the basis of his studies of the variation of rate constants of some reactions with temperature. Arrhenius demonstrated that it can be applied to any thermally activated process.

Arrhenius equation can be applied to a wide range of thermally activated processes, including diffusion that we consider next.


Atomic mechanisms of diffusion

Two main mechanisms of atomic diffusion in crystals:

Atoms located at the crystal lattice sites usually diffuse by a vacancy mechanism.

Interstitial atoms diffuse by jumping from one interstitial site to another interstitial site without permanently displacing any of the matrix/solvent atoms – interstitial mechanism.

In both cases the moving atom must pass through a state of high energy – this creates energy barrier for atomic motion.

The phenomenological description in terms of 1st and 2nd Fick’s laws is valid for any atomic mechanism of diffusion. Understanding of the atomic mechanisms is important, however, for predicting the dependence of the atomic mobility (and, therefore, diffusion coefficient) on the type of interatomic bonding, temperature, and microstructure.

Substitutional impuritiesSubstitutional self-diffusion – can be studied by depositing of a small amount of radioactive isotope of the element (tracer diffusion)


Diffusion mechanisms: Vacancy diffusionTo jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. This energy necessary for the jump, ΔGm

v, is called the activation free energy for vacancy motion. It comes from the thermal energy of atomic vibrations (thermal energy of an atom in a solid <Uatom> ≈ 3kT).

AtomVacancy

Distance

The average thermal energy of an atom (3kBT ≈0.08 eV at room temperature) is usually much smaller that the activation free energy ΔGm

v (~ 1 eV/vacancy) and a large thermal fluctuation is needed for a jump.

For a simple one-dimensional case, the probability of such fluctuation or frequency of jumps, Rj, can be described by the Arrhenius equation:

where ν0 is an attempt frequency related to the frequency of atomic vibrations.

⎟⎠⎞

⎜⎝⎛−υ= Tk

GRB

vm

jΔexp0

G

ν0 is of the order of the mean vibrational frequency of an atom about its equilibrium site (usually taken to be equal to the Debye frequency)

vmGΔ


Diffusion mechanisms: Vacancy diffusion

To relate this to the diffusion of atoms we have to consider the jump frequency of a given atom in a 3D crystal. For an atom to jump, there must be a vacancy next to it.

⎟⎠⎞

⎜⎝⎛ Δ−υ= Tk

GRB

vm

j exp0frequency of atom vibrations in the diffusion direction ν0

× probability that a given oscillation will move the atom to an adjacent site

The probability for any atom in a solid to move is the product of

The rate at which atom jumps from place to place in the crystal is therefore

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−= Tk

Gzz Nn

B

vfeq exp

⎟⎠⎞

⎜⎝⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−υ≈= Tk

GTk

GzB

vm

B

vf expexp

τ1R 0

j

atomj

the probability of finding a vacancy in an adjacent lattice site (fraction of atoms that have a vacancy as a neighbor):

and the rate of jumps of a vacancy (defined by a probability of a thermal fluctuation needed to overcome the energy barrier for vacancy motion)

where τj is the average time between jumps for atoms.

⎟⎠⎞

⎜⎝⎛ Δ−υ= Tk

GRB

vm

j exp0


Diffusion mechanisms: Vacancy diffusionIf the distance atoms cover in each jump is a, we can use the Einstein relation to estimate the diffusion coefficient from the average time between jumps:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δ−⎟

⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δυ=⎟

⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δ−

υ==

TkED

Tkhh

kss

zaTk

GG za

τaD

B

d

B

vm

vf

B

vm

vf

B

vm

vf

j

expexpexp6

exp66 0

02

022

where D0 is a parameter of material (both matrix and diffusing species) and is independent of temperature, Ed is activation energy for diffusion.

Dt(t)rΔii 62 =

r

Order of magnitude estimate of the average time between jumps and the diffusion coefficient for self-diffusion in aluminum

eV 0.72vf =Δh

eV 0.68vm =Δh

-1130 s10≈υ

21z ≈

m103a 10−×≈

τj ≈ 6 ×1011s at T = 0ºC(less than one jump in 20000 years)

τj ≈ 4 × 10-7s at T = 650ºC(2.5 million jumps per second)

D ≈ 3×10-32 m2/s at T = 0ºCD ≈ 4×10-14 m2/s at T = 650ºC

17 orders of magnitude difference!

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δ⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δ−

υ=

Tkhh

kss

z

τB

vm

vf

B

vm

vf

j expexp1

0

1~exp ⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δ

B

vm

vf

kss

- see Shewmon, Diffusion in solids, Ch. 2.4-2.7

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δ−⎟

⎟⎠

⎞⎜⎜⎝

⎛ Δ+Δυ=

Tkhh

kss

zaDB

vm

vf

B

vm

vf expexp

60

2

vm

vfd hhE Δ+Δ=

/sm 102.1 260

−×≈D


Substitutional self-diffusionFor a given crystal structure and bonding type Ed/RTm is roughly constant → D(T/Tm) ≈ const

496.0

324.555.3

43.8

40.8

439.6309.2

308.6284.1

279.7200.3

142

Ed, kJ/mol

5.14

3.360.57

0.45

0.42

4.563.20

3.202.94

2.902.08

1.47

Ed, eV

0.2918.9491805γ-Fe0.5417.4202130Cr

0.9717.228.82163V5.219.312402741Nb

1514.631337K

1614.224.2371Na

9.914.723454Li

4.4×10-532.34401211Ge

3.6×10-435.59000001683Si

0.6519.51901726Ni0.5917.8311356Cu

1.918.3170933Al

D(Tm), 10-12 m2/sEd/RTmD0, 10-6 m2/sTm, K

from Porter and Easterling textbook

fcc metals

bcc transition metals

bcc alkali metals

diamond cubicsemiconductors


Diffusion mechanisms: Interstitials

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δυ=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−

υ==

TkED

Tkh

ks pa

TkG pa

τaD

B

d

B

im

B

im

B

im

j

expexpexp6

exp66 0

02

022

Interstitial diffusion also involve transition through the energy barrier and can be analyzed in a manner similar to the vacancy diffusion mechanism.

The difference is that there are always sites for an interstitial atom to jump to.

where p is number of neighbor interstitial sites and imd hE Δ=

76.10.3N in BCC Fe

168.691N in FCC Fe

430.63H in FCC Fe

13.40.1H in BCC Fe

84.12C in BCC Fe14823.4C in FCC Fe

Ed, kJ/molD0, mm2/s-1Impurity

240.7201Fe in δ-Fe

202.185Au in Ag33247Fe in Cr

250.6276Fe in α-Fe28449Fe in γ-Fe

484.4146000Si in Si

Ed, kJ/molD0, mm2/s-1

from Porter and Easterling textbook & Smithells Metals Reference Book

vacancymechanism →

← interstitial impurities

• Smaller atoms cause less distortion of the lattice during migration and diffuse more readily than big ones (the atomic diameters decrease from C to N to H).

• Diffusion is faster in more open lattices


Diffusion of interstitial and substitutional impurities

Diffusion of interstitials is typically faster as compared to the vacancy diffusion mechanism (self-diffusion or diffusion of substitutional atoms).

⎟⎠⎞

⎜⎝⎛−=

T1

2.3RlogDlogD 0

dE


Diffusion of self-interstitialsAs discussed earlier, intrinsic interstitials (self-interstitials) have very high formation energy, Δhf

i

≈ 3Δhfv for metals and the number of equilibrium self-interstitials is negligible. In Si, however,

intrinsic interstitials may play an important role in diffusion and formation of defect structures.Non-equilibrium self-interstitials in most materials are very mobile, e.g., Δhm

i ≈ 0.5Δhmv for

metals and they quickly diffuse out of the bulk of the crystal after being formed.

Self-interstitials can move through formation of intermediate low-energy configurations, e.g. dumbbells and crowdions.

Self-interstitials can form clusters, e.g., bundles of crowdions oriented along a close-packed direction.Clusters of interstitials can form prismatic dislocation loops that can undergo thermally-induced 1D diffusion.

One-dimensional motion of an almost isolated ½[111] loop at 575 K. A loop continuously moves in a direction parallel to its Burgers vector

Arakawa et al., Science 318, 956, 2007

Schematic view of the observation of the 1D glide motion of a interstitial-type prismatic perfect dislocation loop by TEM.


Diffusion of vacanciesIf the diffusion of vacancies is of interest (in contract to the diffusion of atoms through the vacancy mechanism), a vacancy always has z sites to jump to and the diffusion coefficient is

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δυ=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−

υ=

TkED

Tkh

ks za

TkG zaD

B

d

B

vm

B

vm

B

vm expexpexp

6exp

6 00

20

2

Thus, the diffusion coefficient for vacancies, Dv, is much larger than the diffusion coefficient for substitutional self-diffusion, DSD:

vmd hE Δ=

veqSDv XDD /= NnX v

eqveq /=where

Diffusion in substitutional alloysIn a binary AB system, the values of Δhm

v and Δhfv can be

different for the two components and the diffusion is characterized by two diffusion coefficients, DA and DB.

x

AX

J

1

0

AJ

BJvJ

dxdCDJ A

AA −=

dxdCDJ A

BB =

dxdCDDJJJ A

BABAv )( −=−−= tCv

∂∂

0

0

xJ

tC vv

∂∂

−=∂∂

vacanciesdestroyed

vacanciescreated

Kirkendall effect: movement of the lattice as a result of inter-diffusion of A and B


Diffusion with correlated jumpsExtracting information on the vacancy diffusion coefficient from tracer diffusion experiments is complicated by correlated jumps.

Diffusion of vacancies (and interstitials) = random walk, i.e., after each jump the vacancy can jump to any of its nearest neighbor sites with equal probability.

The jump sequence of a tracer atom, however, is not a random walk since the vacancy that allowed the tracer atom to jump last time may still be nearby.

the tracer atom has a higher probability to jump back → the jumps are correlated

the mean square displacement will be lower then for the random walk

SDtheorytracer DfD ×=

where f is the correlation factorDtheory is the theoretical value for self-diffusion

without account for the correlationDtracer is the diffusion coefficient for tracer atoms

MSD for vacancies increases with time, but the tracer atom moves back and forth around its original position (on average, there is the same number of vacancies coming from the left and right sides) → f = 0

diffusion in 1D through vacancy mechanism:


Diffusion with correlated jumps

Correlation coefficient f can be calculated if diffusion mechanism and lattice structure are known. For example, an approximate expression for vacancy diffusion mechanism in a crystal with coordination number z is

11

+−

≈zzf it considers only the correlation between jumps while

the vacancy is still in the nearest neighbor position

85.0112112=

+−

≈ffor fcc: while more accurate value is 0.78

For substitutional impurities, correlation effects may be more complicated since the vacancy migration may not be completely random

For interstitial impurities, correlation effects may appear due to the interaction with vacancies, kick-out mechanism (interstitial impurity goes into substitutional position and kicks out a self-interstitial), etc.

For more detailed discussion, see Balluffi, Allen, Carter, Kinetics of Materials, Chs. 7.2.3 and 8.2


Fast diffusion paths

Computer simulation by Kwok, Ho, and Yip. Initial atomic positions are shown by the circles, trajectories of atoms are shown by lines. We can see the difference between atomic mobility in the bulk crystal and in the grain boundary region.

Mean-square displacement of all atoms in the system (B), atoms in the grain boundary region (C), and bulk region of the system (A).

More open atomic structure at defects (grain boundaries, dislocations) can result in a significantly higher atomic mobility along the defects.

Dt(t)rΔMSDii 62 ==

r


Fast diffusion pathsDiffusion coefficient along a defect (e.g. grain boundary) can be also described by an Arrhenius equation, with an effective activation energy for diffusion along the defect significantly lower than the one for the bulk.

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TkEDD

B

defddefdef exp0

Self-diffusion coefficients for Ag. The diffusivity if greater through less restrictive structural regions – grain boundaries, dislocation cores, external surfaces.

However, the effective cross-sectional area of the defects is only a small fraction of the total area of the bulk (e.g., an effective thickness of a grain boundary is ~0.5 nm). The diffusion along defects is less sensitive to the temperature change → becomes important at low T.


Fast diffusion paths: Diffusion in nanocrystalline materials

Arrhenius plots for 59Fe diffusivities in nanocrystalline Fe and other alloys compared to the crystalline Fe (ferrite)[Wurschum et al. Adv. Eng. Mat. 5, 365, 2003]

image by Lin et al.J. Phys. Chem. C 114, 5686, 2010


nanocrystalline Si:

high-resolution TEM (left, Acta Mater. 56, 5857, 2008)

computed atomic structure(right, Acta Mater. 45, 987, 1997)

Wurschum et al., Defect Diffus. Forum 143-147, 1463, 1997

volume fraction of grain boundary regions: ~50%

Diffusivity is enhanced by ~30 orders of magnitude

Fast diffusion paths: Diffusion in nanocrystalline materials


Effect of grain boundary melting

Experiments: [Herth et al., Phys. Rev. Lett. 92, 095901, 2004] 59Fe tracer diffusion in nanocrystalline Nd2Fe14B at temperatures close to the intergranular melting transition: the diffusion coefficient in the intergranular liquid layers is lower than in bulk melts indicating a hampered atomic mobility due to confinement.

Schematic diagram for self-diffusion in high-angle GBs in fcc metals obtained in MD simulations [Phil. Mag. A 79, 2735, 1999]

Mobility of Point Defects: Diffusion - people.Virginia.EDUpeople.virginia.edu/~lz2n/mse6020/notes/Diffusion.pdf · University of Virginia, MSE 6020: Defects and Microstructure in

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