lecture 3. microscopic dynamics and macroscopic d we can see that if we want to understand the...

1

2

3

4

5

Ian Farnan

Michaelmas Term 2004

Natural Sciences Tripos Part IB

MINERAL SCIENCES

Module B: Transport Properties

Lecture 3

Microscopic dynamics and Macroscopic D

We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required.

Consider a particle moving on a 1-D lattice

For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x)

are equally probable.

€

Δx = δ i1

n

∑ = 0






€

Δx = δ i1

n

∑ = 0

Large # hops, n






€

Δx = δ i1

n

∑ = 0

Large # hops, n

with the same individual hop distance






€

Δx = δ i1

n

∑ = 0

Large # hops, n


On average, distance moved is zero.






€

Δx = δ i1

n

∑ = 0

Large # hops, n


On average, distance moved is zero.

However, any individual particle may have moved a long way.

€

Δx = δ i1

n

∑ = 0

The mean of the squared displacements, however, will not be zero

€

Δx2 = δ i1

n

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

= δ i2

1

n

∑ = nδ 2

€

i2 = δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( ). δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( )

€

11 δ12

δ21 δ22

δ33

δnn

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

€

Δx = δ i1

n

∑ = 0


€

Δx2 = δ i1

n

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

= δ i2

1

n

∑ = nδ 2

Use this to quantify extent of diffusion/particle mobility

€


€

11 δ12

δ21 δ22

δ33

δnn

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

€

Δx = δ i1

n

∑ = 0


€

Δx2 = δ i1

n

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

= δ i2

1

n

∑ = nδ 2


€


Squared displacement

€

11 δ12

δ21 δ22

δ33

δnn

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

€

Δx = δ i1

n

∑ = 0


€

Δx2 = δ i1

n

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

= δ i2

1

n

∑ = nδ 2


€



€

11 δ12

δ21 δ22

δ33

δnn

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

n hops results in an nxn matrix

€

Δx = δ i1

n

∑ = 0


€

Δx2 = δ i1

n

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

= δ i2

1

n

∑ = nδ 2


€



€

11 δ12

δ21 δ22

δ33

δnn

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟


All diagonal elements are positive

€

Δx = δ i1

n

∑ = 0


€

Δx2 = δ i1

n

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

= δ i2

1

n

∑ = nδ 2


€



€

11 δ12

δ21 δ22

δ33

δnn

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟


All diagonal elements are positive

Off-diagonal elements can be positive or negative, on average sum to zero

This is true because when squaring i, the off-diagonal terms will sum

to zero for large n

€

iδ ki≠k

∑ = 0

We have then

€

Δx2 = δ n

If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:

€

n =t

τ


to zero for large n

€

iδ ki≠k

∑ = 0

We have then

€

Δx2 = δ n


€

n =t

τ

€

ii2

ii

∑ = nδ 2Recall,


to zero for large n

€

iδ ki≠k

∑ = 0

We have then

€

Δx2 = δ n

Average distance a particle has moved is given by:


€

n =t

τ

€

ii2

ii

∑ = nδ 2Recall,


to zero for large n

€

iδ ki≠k

∑ = 0

We have then

€

Δx2 = δ n


Elementary hop distance x square root of the number of hops


€

n =t

τ

€

ii2

ii

∑ = nδ 2Recall,


to zero for large n

€

iδ ki≠k

∑ = 0

We have then

€

Δx2 = δ n




€

n =t

τ

Total diffusion time

€

ii2

ii

∑ = nδ 2Recall,


to zero for large n

€

iδ ki≠k

∑ = 0

We have then

€

Δx2 = δ n




€

n =t

τ

Total diffusion time

Individual hop time

€

ii2

ii

∑ = nδ 2Recall,

It follows that

Δ x

2

=

t

If we nowtake the result fro m t heroot me -ansquar ed displacemen tdetermined fro m the

mean displacement i na Gaussian cur ve(solution t oFick’s 2 ndLaw for diffusi on o f a thin

source) i.e.,

Δ x

2

= 2 Dt

we can relat ethe macroscopic diffus ion constant to an elementa ry hoppi ng distance ()and an elementary hoppi ngtime () so that

D =

2

2

This is the fundamental equation that relates microscopi c atomic dynamics t othe

macroscopicall ymeasur ed diffusi .on

NB the facto r 2 in the denominator is related t othe possible directions arando mhop

might take i na 1- D lattice - this becomes 4 i n a 2 D lattice 6and in 3a D lattice.

It follows that

Δ x

2

=

t



source) i.e.,

Δ x

2

= 2 Dt


D =

2

2





Substituting for n

It follows that

Δ x

2

=

t



source) i.e.,

Δ x

2

= 2 Dt


D =

2

2





Substituting for n

Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance

It follows that

Δ x

2

=

t



source) i.e.,

Δ x

2

= 2 Dt


D =

2

2





Substituting for n


‘Einstein relationship’ : relates microscopic dynamics to macroscopically measured diffusion through a fundamental hopping distance and a fundamental hopping time.

It follows that

Δ x

2

=

t



source) i.e.,

Δ x

2

= 2 Dt


D =

2

2





Substituting for n


‘Einstein relationship’ : relates microscopic dynamics to macroscopically measured diffusion through a fundamental hopping distance and a fundamental hopping time.

The factor 2 represents the probability of hops (left or right) on a 1D lattice

Dimensionality of diffusion

1D

€

12

δ 2

τ


1D

2D

€

12

δ 2

τ

€

14

δ 2

τ


1D

2D

3D

€

12

δ 2

τ

€

14

δ 2

τ

€

16

δ 2

τ


1D

2D

3D

€

12

δ 2

τ

€

14

δ 2

τ

€

16

δ 2

τ

Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.


1D

2D

3D

€

12

δ 2

τ

€

14

δ 2

τ

€

16

δ 2

τ


NB 1/ is often replaced by a

frequency of hopping,

to give:

€

D = 1g νδ

2


1D

2D

3D

€

12

δ 2

τ

€

14

δ 2

τ

€

16

δ 2

τ


NB 1/ is often replaced by a

frequency of hopping,

to give:

€

D = 1g νδ

2

2, 4 or 6

Types of vacancySchottky defect

Frenkel Defect


Frenkel Defect

Missing anion or cation in a lattice


Frenkel Defect


Frenkel Defect


Occur in pairs to maintain electrical neutrality not necessarily together.


Frenkel Defect



Vacant lattice site created by an atom moving into an interstititial position


Frenkel Defect




These are intrinsic vacancies

Other vacancies may be created by trace amounts of impurities or variable oxidation states of some constituent ions e.g. in NaCl a 2+ impurity (Ca2+, say)will require a missing anion (Cl-) as charge balance.


Frenkel Defect




These are intrinsic vacancies

Other vacancies may be created by trace amounts of impurities or variable oxidation states of some constituent ions e.g. in NaCl a 2+ impurity (Ca2+, say)will require a missing anion (Cl-) as charge balance.

These are extrinsic vacancies

Mechanisms of diffusion(1) Direct interstitial mechanisms for light gaseous elements, e.g. H2, He, N2, O2

'dissolved' in solids. Do not take up rational lattice sites.

(2) Direct vacancy mechanism - atom on rational lattice site moves to adjacent,

vacant lattice site. Flux of atoms in one direction requires a flux of vacancies

in the other direction.

(3)+(4) Exchange of lattice sites by two atoms 'squeezing past each other (3) or through

a ring mechanism (4).

(5) Intersticialcy mechanism - an interstitial atom takes up a rational lattice site as

atom occupying rational position moves off into adjacent interstitial position.

1

2

3

4

5

2

3

5

4

11










1

2

3

4

5

2

3

5

4

11

Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high.










1

2

3

4

5

2

3

5

4

11


2,3,4 all mechanisms proposed to move one atom onto the site of another. Elastic energy required by (3) was considered too great so a cooperative mechanism (4) was postulated.

}










1

2

3

4

5

2

3

5

4

11



(2) allows direct hopping onto adjacent vacant site, explicitly requires vacancies, (3) + (4) do not.

}










1

2

3

4

5

2

3

5

4

11



(2) allows direct hopping onto adjacent vacant site, explicitly requires vacancies, (3) + (4) do not.

}

(5) Mechanism actually observed in some fast ion conductors (see later) combination of vacancy (2) and interstitial (1) mechanisms.

Kirkendall's ExperimentFine Molybdenum

Markers

Brass

Copper

Brass block was surrounded by fine (inert) molybdenum markers and then

copper. The whole arrangement was annealed at high temperature for increasing lengths

of time. The markers moved closer together for longer times.

Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out

and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move

nward.

This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).

The markers move inward because new lattice sites are being created on the outside and

here is a net vacancy flow inward.

First direct evidence of vacancy mechanisms and their importance in solid state diffusion.

Brass - alloy of Cu/Zn


Markers

Brass

Copper






nward.






Concentration gradient exists between Cu and Cu/Zn -

Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.


Markers

Brass

Copper






nward.








What happens to the inert markers?


Markers

Brass

Copper






nward.









They move closer together the longer time goes on.

Conclusion: Zn diffuses faster than Cu.


Markers

Brass

Copper






nward.











Must be vacancy mechanism, because if direct (3) or cooperative (4) then Dcu = DZn


Markers

Brass

Copper






nward.












If markers move in then new sites are being created beyond markers with vacancy flow inwards


Markers

Brass

Copper






nward.












If markers move in then new sites are being created beyond markers with vacancy flow inwards

Direct vacancy mechanism is the predominant mechanism in solid state diffusion.

Microscopic diffusion by atoms and vacancies

Earlier, we have seen that the diffusion constant D is related to microscopic (atomistic)

processes by the elementary hop distance, r, and a time between hops, , which we now

express a s a hopping frequenc , y = 1/.

For diffusi on of a vacancy on a cubic lattice t he diffusi onconstant becomes

Dv

=

1

6

r

2

v

v

≡ vacancy hoppi ngfrequency

≡ atomic hopping frequency

Similarly for diffusi on o f interstitials

DI

=

1

6

r

2

I

However, for atomic diffusion vi a a vacancy mechanism an ato mrequires an adjacent

vacancy in orde r t omove (this is not requir edin interstitial or vacancy diffusion)

Ds

=

1

6

r

2

v

cv

cv

≡

n

N

, the concentrati on of vacancies

Ds i sknown as the se -lf diffusion constan t andrefers t othe itineran t motion of the

constitue nt atoms withi n a material which occurs wit h or withou t aconcentration (or

other potentia )l gradien .t In t heabsence of a gradient this atomic motion still occurs but

ther e isno tnet atomic movemen .t






Dv

=

1

6

r

2

v

v




DI

=

1

6

r

2

I



Ds

=

1

6

r

2

v

cv

cv

≡

n

N






V






Dv

=

1

6

r

2

v

v




DI

=

1

6

r

2

I



Ds

=

1

6

r

2

v

cv

cv

≡

n

N






V

(r = )






Dv

=

1

6

r

2

v

v




DI

=

1

6

r

2

I



Ds

=

1

6

r

2

v

cv

cv

≡

n

N






V

(r = )

V

V






Dv

=

1

6

r

2

v

v




DI

=

1

6

r

2

I



Ds

=

1

6

r

2

v

cv

cv

≡

n

N






V

(r = )

V

V

Ds can be thought of as an average mobility of indistinguishable particles.Increased by increasing the hopping freqeuncy or the concn of vacancies

Temperature dependence of D

Temperature will have a profound effect on the diffusion constant. Increasing

temperature will increase the atomic hopping frequency, . i s related t othe vibrational

frequency o f atom sin a materia l and as temperature increas esmore of the atoms will

vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop

over into t he nex tpotential wel .l

E

Δ E

A B C

A

B

C

r

I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom

t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point

energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits

equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty

that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and

hence the likelihood that it can make the jum .p This can be written

υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠

Where ΔE is t hesaddle point energy.







E

Δ E

A B C

A

B

C

r







υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠


Migration energy ΔEm







E

Δ E

A B C

A

B

C

r







υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠


Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.








E

Δ E

A B C

A

B

C

r







υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠



The energy vs distance profile is a maximum at B, this is the migration

energy, ΔEm . Or the ‘saddle point energy’.








E

Δ E

A B C

A

B

C

r







υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠






Hopping frequency [v=voexp(- ΔEm / kT)] increases with temperature as the atom occupies vibrational states nearer the top of the well.







E

Δ E

A B C

A

B

C

r







υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠







Macroscopically this leads to an Arrhenian temperature dependence for D:







E

Δ E

A B C

A

B

C

r







υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠








DI,v = Do exp(- ΔEm / kT)







E

Δ E

A B C

A

B

C

r







υ = υ

0

exp

− Δ E

kT

⎛

⎝

⎞

⎠








DI,v = Do exp(- ΔEm / kT)

interstitials, vacancies

Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,

T, can be written

DI , v

= Do

exp −

Δ Em

kT

⎛

⎝

⎜

⎞

⎠

⎟

where Δ Em

is knowna st hemigrati on energy andis related t othe micoscopic energy

barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D

over a range of temperatures and making an Arrhenius plot of loge D vs 1/T

loge

D = loge

Do

−

Δ Em

R

.

1

T

T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.

For interstitial diffusion:Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,

T, can be written

DI , v

= Do

exp −

Δ Em

kT

⎛

⎝

⎜

⎞

⎠

⎟

where Δ Em




loge

D = loge

Do

−

Δ Em

R

.

1

T


For interstitial diffusion:

measure D as a function of temperature


T, can be written

DI , v

= Do

exp −

Δ Em

kT

⎛

⎝

⎜

⎞

⎠

⎟

where Δ Em




loge

D = loge

Do

−

Δ Em

R

.

1

T




Plot loge D vs 1/T


T, can be written

DI , v

= Do

exp −

Δ Em

kT

⎛

⎝

⎜

⎞

⎠

⎟

where Δ Em




loge

D = loge

Do

−

Δ Em

R

.

1

T




Plot loge D vs 1/T

gradient


T, can be written

DI , v

= Do

exp −

Δ Em

kT

⎛

⎝

⎜

⎞

⎠

⎟

where Δ Em




loge

D = loge

Do

−

Δ Em

R

.

1

T




Plot loge D vs 1/T

gradient

What about self-diffusion?


T, can be written

DI , v

= Do

exp −

Δ Em

kT

⎛

⎝

⎜

⎞

⎠

⎟

where Δ Em




loge

D = loge

Do

−

Δ Em

R

.

1

T




Plot loge D vs 1/T

gradient

What about self-diffusion?

Vacancy concentration vs temperature?


T, can be written

DI , v

= Do

exp −

Δ Em

kT

⎛

⎝

⎜

⎞

⎠

⎟

where Δ Em




loge

D = loge

Do

−

Δ Em

R

.

1

T


Thermally Created Vacancies

Given that the free energy of formation of a vacancy is ΔGv then even for a 'pure'

material a tany temperature T above absolute zer ,o ther e w ill be a number of vacancies inequilibriu mwit hthe structur .e T hefracti on o f vacancies a s a functi on of temperaturedepends ont heenthal py of vac ancy formati ,on Ev.

n

N

= exp

− Ev

RT

⎛

⎝

⎜

⎞

⎠

⎟

ΔEv i s fairl y large, since it involve s brea king bonds and removing anato m fro mthe

structur ethen n/N will have a ve ry steep temperatur e dependence. If one looks a t the

equilibriu ,m thermall y generat ed vacancy concentrations at room temperatur ethey arever ysmall. I nfac t t heya re ver ymuch less than t he norma l level of impurities i n a

material .i .e, trace impurity atoms or slight off-stoichiometry, so tha tat low temperatures

the impurit y concentration is constan tbecaus e the number o f the seextrinsic defects willnot c . hange This is anadvantage as it allows ust omeasur ethe activation ener gyΔE at

lower T (fro m l n D vs 1/T plot )s and when the number of thermally generat edintrinsic

vacancies becomes important there will be a change of slope dueto the much higher

activation ener .gy T helowe r temperature slope is t heactivati on ene rgy for atomic




n

N

= exp

− Ev

RT

⎛

⎝

⎜

⎞

⎠

⎟









Temperature dependence of vacancy concn




n

N

= exp

− Ev

RT

⎛

⎝

⎜

⎞

⎠

⎟










Large entropic TΔS factor in

ΔG promotes vacancy formation even though Ev may be large.




n

N

= exp

− Ev

RT

⎛

⎝

⎜

⎞

⎠

⎟










Large entropic TΔS factor in

ΔG promotes vacancy formation even though Ev may be large.

There are very many ways to arrange a small number of vacancies over a very large number of lattice sites - See BH48

migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case

D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥


Two energetic factors controlling the temperature dependence of diffusion can often be separated.

ΔEm + Ev


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥



Example: NaCl doped with Cd


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥




Cd2+ replaces Na+ creating Na+

vacancies.


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥





vacancies.

At low temperatures this doping creates extrinsic vacancies.


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥





vacancies.


Number of thermally created vacancies is far less than extrinsic vacancies at low temperature ->

activation energy is simply ΔEm


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥





vacancies.




At high temperatures, thermally created vacancies become important


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥





vacancies.




At high temperatures, thermally created vacancies become important

Activation energy is then ΔEm + Ev

ΔEm + Ev


D = Do

exp

− Δ Em

+ Δ Ev

( )

RT

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥





vacancies.




At high temperatures, thermally created vacancies beome important

Activation energy is then ΔEm + Ev

D increases much more rapidly as new vacancies are created.

Temperature dependence of vacancy concentration


Creation of a vacancy is a highly energetic process - breaking of all bonds and removal to the surface.



In addition there is an associated volume expansion beyond that expected from the x-ray determined volume.




Nearly all atoms remain in register and there is some increase in the lattice spacing due to thermal expansion. The ‘ideal’ volume at any temperature can be determined from the lattice parameter at the same temperature




Nearly all atoms remain in register and there is some increase in the lattice spacing due to thermal expansion. The ‘ideal’ volume at any temperature can be determined from the lattice parameter at the same temperature

The real, macroscopic volume of a sample can also be measured…….

Macroscopic expansion vs ‘x-ray expansion’


Very careful x-ray diffraction and dilatation experiments showed a

difference between Δa/a and Δl/l for aluminium




Extra volume is created by vacancies in the material




Extra volume is created by vacancies in the material

The nearer the melting point the greater the number of vacancies.

Random walk, correlation factors, tracer diffusion

Random Walk

ri

R

We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.

R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j

If all hops are random and therefore do no t depend on t he precedi ng hop then the second

term in the right hand expressi on will be zero. If for som ereason the probability of a hop

is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we

need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the

substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and

bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r


Random Walk

ri

R


R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j







Random walk - > each hop is independent of the previous hop


Random Walk

ri

R


R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j








No ‘memory effect’


Random Walk

ri

R


R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j











Random Walk

ri

R


R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j










Diagonal and off-diagonal terms


Random Walk

ri

R


R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j











If motion is not random then the off-diagonal terms no longer sum to zero for a large number of hops.


Random Walk

ri

R


R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j












They are correlated by a factor, f


Random Walk

ri

R


R = r

i

i = 1

n

∑

Squari ng gives

R

2

= r

i

i , j

n

∑ . r

j

separatingR

2

= ri

2

i = 1

n

∑ + r

i

i ≠ j

n

∑ . r

j












They are correlated by a factor, f

€

f = r ii≠ j

n

∑ .r j

A tracer atom has a higher probability of moving back to the vacancy it has just left.

Jumps of a tracer are thus correlated because they depend on the direction of the previous

jump.

The correlation factor takes account of the fact that the total displacement acquired by a

tracer atom is less than that acquired in a true random walk because of these ‘wasted’

jumps back and forth on the same two sites. In general, a full matrix calculation of the

correlation term is required. However, a simple approximation can produce some

reasonable values of f.

f = 1 −

2

z

where z is the coordination number.

i.e., 2 jumps a re wast edif the tracer jumps back int othe adjacen t vacancy and theprobability of this happening i s1/z, the number of neighbouring sites.



jump.






f = 1 −

2

z



Tracer diffusion is correlated (non-random) - why?



jump.






f = 1 −

2

z




Origin of the problem is distinguishable and indistinguishable particles



jump.






f = 1 −

2

z





tracer atom has a higher probability of hopping back into a site it has just left because it is distinguishable.



jump.






f = 1 −

2

z






We call this a ‘correlation’ or a ‘memory effect’



jump.






f = 1 −

2

z






We call this a ‘correlation’ or a ‘memory effect’

Random walk of a tracer will be less than that of a self–diffusing atom by a factor, f.

Approximate and actual values of f for different lattices

lattice

2D square

2D hexagonal

diamond

simple cubic

BCC

FCC

z_

4

6

4

6

8

12

approx.f (1-2/z)

1/2

2/3

1/2

2/3

3/4

5/6 (0.833)

calculated f

0.467

0.560

0.5

0.655

0.72

0.78


lattice

2D square

2D hexagonal

diamond

simple cubic

BCC

FCC

z_

4

6

4

6

8

12

approx.f (1-2/z)

1/2

2/3

1/2

2/3

3/4

5/6 (0.833)

calculated f

0.467

0.560

0.5

0.655

0.72

0.78

f = 1 - 2/z


lattice

2D square

2D hexagonal

diamond

simple cubic

BCC

FCC

z_

4

6

4

6

8

12

approx.f (1-2/z)

1/2

2/3

1/2

2/3

3/4

5/6 (0.833)

calculated f

0.467

0.560

0.5

0.655

0.72

0.78

f = 1 - 2/z

Total displacement for n jumps (recall, √n) for a tracer is less than for a true random walk because jumps are wasted back and forth on a site.


lattice

2D square

2D hexagonal

diamond

simple cubic

BCC

FCC

z_

4

6

4

6

8

12

approx.f (1-2/z)

1/2

2/3

1/2

2/3

3/4

5/6 (0.833)

calculated f

0.467

0.560

0.5

0.655

0.72

0.78

f = 1 - 2/z


These hops do not contribute to the total displacement.


lattice

2D square

2D hexagonal

diamond

simple cubic

BCC

FCC

z_

4

6

4

6

8

12

approx.f (1-2/z)

1/2

2/3

1/2

2/3

3/4

5/6 (0.833)

calculated f

0.467

0.560

0.5

0.655

0.72

0.78

f = 1 - 2/z



Self–diffusion constant, Ds = DT / f


lattice

2D square

2D hexagonal

diamond

simple cubic

BCC

FCC

z_

4

6

4

6

8

12

approx.f (1-2/z)

1/2

2/3

1/2

2/3

3/4

5/6 (0.833)

calculated f

0.467

0.560

0.5

0.655

0.72

0.78

f = 1 - 2/z



Self–diffusion constant, Ds = DT / f

Tracer diffusion

lecture 3. microscopic dynamics and macroscopic d we can see that if we want to understand the...

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