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Mechanisms Diffusion in Solids
27-216 Spring 2004 A. D. Rollett
Based on Modules 12 & 13 from Glicksman, RPI
Lecture 3
2
Objectives, Topics
Objective: to explain the relationship between an atomistic understanding of atoms moving around in a crystal and the continuum description of mass flow (Fick's Laws). Topics:
Random walks Distances traveled Diffusion distances (Einstein formula, x versus t) Fick's 1st Law Mechanisms of diffusion (Lattice diffusion, Crystal Structure, Interstitials, Substitutional Diffusion, role of Vacancies, High diffusivity paths)
Lecture 3
Examinable
3
st
dC J = -D dx
Adolf Fick (1829-1901) "Monograph on Medical Physics", 1856
Examinable
Fick's 1 Law
Lecture 3
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Einstein's Formula
Einstein's famous result is or
Slide from module 12, Random Walk, Glicksman and Lupulescu
D: diffusion coefficient n: no. of steps t: time , : jump distance 3: atomic jumping frequency in 3D
D expressed on the basis of purely microscopic quantities
Diffusivity ranges in different states of matter
Range: D [ cm2/s]
Phase/State
10-6 10-4 1
10-25 10-7 10-2
Solids Liquids Gases
Examinable Lecture 3
5
Exercises
1. The diffusivity of bismuth atoms in lead at 600K is 1 10-9 cm2/sec. The radius of a lead atom, rPb, is 0.175 nm. Calculate the total distance traveled by the Bi atoms after one hour, and compare it to the expected RMS displacement for the same period. The lattice parameter a0 of the FCC Pb host crystal is:
The jump distance of a substitutional Bi atom in FCC Pb is given as:
Lecture 3
Slide from module 12, Random Walk, Glicksman and Lupulescu
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Exercises
Applying Einstein's relationship we obtain the jumping frequency
The total distance Rtot traveled by Bi atoms is equivalent to the length of a directed walk
Rtot = nl = (G t ) l = (4.9 10 6 s- 1 3600s) 0.35nm = 6.174m
The expected RMS displacement of the Bi atoms can be found
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Expanding on
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Mechanisms of diffusion in metals
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How are atoms placed in solid lattices?
Interstitial e.g. C in and Fe Substitutional e.g. Fe in and Fe
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Random Walks: Primitive (Simple) Cubic Crystals
The total jumping frequency tot is related to the specific jumping frequencies for a given structure: In primitive cubic crystals (SC) there exists one lattice site per unit cell surrounded by 6 neighbors: The individual crystallographic directions along which a diffuser can jump in SC are:
Z=6
Three of the SC jump vectors,
0 <100>
of the type a u
Examinable
are shown:
Lecture 3
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Random Walk: Body-Centered Cubic Crystals
In bodycentered cubic crystals (BCC) there exists one lattice site per unit cell surrounded by 8 nearest neighbors: The individual crystallographic directions along which a diffuser can jump in BCC are:
Z=8
Six of the BCC jump vectors, of the type are shown:
Examinable Lecture 3
12
Random Walks: Face-Centered Cubic Crystals
Z = 12
In facecentered cubic crystals (FCC) there exists one lattice site per unit cell surrounded by 12 nearest neighbors:
The individual crystallographic directions along which a diffuser can jump in FCC are:
Examinable Lecture 3
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Random Walks: Face-Centered Cubic Crystals
Six of the FCC jump vectors, of the type are shown below:
Examinable Lecture 3
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Constraints Imposed by Crystal Structure
The total displacement achieved, L, over a time interval, t , is:
The average number of jumps, <ni > over a time interval, t, is:
The expected (mean) displacement of such of diffuser after a time, t, is found as :
The mean displacement averaged over all the diffusers is:
Lecture 3
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Constraints Imposed by Crystal Structure For the cubic system all the jumping frequencies
and jump vectors are equivalent. Consequently, the equation simplifies to:
The square displacement written as:
may be
The sum of parallel and non-parallel scalar products may be written in terms of the sum of the cosines:
An expression for the meansquare displacement may be developed:
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Constraints Imposed by Crystal Structure Substituting for the definition of the jumping frequency, we obtain:
For the case of cubic crystals,
Averaging over many independent random walks,
Combining the previous equations:
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Constraints Imposed by Crystal Structure
Primitive cubic:
If the jump vectors and coordination numbers for each cubic crystal structure are inserted back, then we obtain:
Body-centered cubic:
Face-centered cubic: In terms of specific jumping frequency, i, one formula can represent the meansquare displacement for the cubic structures given above:
Examinable Lecture 3
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Interpreting Diffusivities in Cubic Crystals
= 6Dt, yields
Applying Einstein's famous result,
The diffusivity in cubic structures may also be expressed in terms of the total jumping frequency, tot,
RMS displacement
Computer simulation of ten "atoms" executing 100 Monte Carlo steps in SC, BCC, and FCC lattices.
t n e m e c a l p s i d S M R
1 2 1 0 8 6 4 2 0 0
F C C S C B C C
FCC SC BCC
Examinable Lecture 3
2 4 16 8 1 0 /2 n
n1/2 n1/2
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Interpreting Diffusivities in Cubic Crystals
11, 520 "atoms" executing 100 Monte Carlo steps in SC, BCC, and FCC lattices:
12 10
RMS displacement
8 6 4 2 0 0 2
FCC FCC SC SC
BCC BCC
RMS displacement
4
n
6 1/2 n 1/2
8
10
12
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Diffusion Mechanisms
Interstitial diffusion Ring diffusion Vacancy-assisted diffusion Interstitialcy diffusion
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How can atoms move: Interstitial?
Any barriers?
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Interstitial Diffusion: Schematic of Atom Movements
h
a0
a
0
h
d
i
saddle-point plane
saddlepoint plane
diffusive jump
Examinable
saddle-point plane
saddlepoint plane
Lecture 3
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Diffusion Mechanisms
The inplane local strain, , may be approximated as:
The unstrained interatomic spacing <100> along directions in BCC crystals may be found using the standard formula:
The Boltzmann probability to acquire an energy, E* >> kBT, is:
Lecture 3
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Comparison of calculated saddlepoint strain energies with experimentally determined interstitial diffusion activation energies for some BCC materials
Solvent
Solvent - iron
Tantalum Vanadium
- Impurity Strain, Saddle-point energy, (kcal/mol)
Impurity
Energy Activation (kcal/mol) ( 3.0 /)
20.1 38.5 33 34 29 34.2 26.9
Strain () 0.077
0.406 0.338 0.298 0.334 0.302 0.298 0.267
( ) 1.3 /
25 42 35 22 19 20 17
Niobium
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Ring Diffusion
A
B
A: direct exchange B: cyclic exchange
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How can atoms move: Substitutional?
Any barriers?
A vacancy has to exist!!
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VacancyAssisted Diffusion
The FCC lattice geometry means that the size of the `window', W, through which an atom must pass is given by:
2
Da
Da
aa0 0
1
d
Vacant Vacant 4 5
1
2
a0 a
=a 0 [110]
3
0
W
3 4
(a) Examinable
= a0 [110]
plan view
Lecture 3
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Interstitialcy Diffusion
Dilated configuration of a diffusion jump "window" on the {110} plane in FCC Interstitialcy diffusion showing the movement of a crowdion effect
cr owdion crowdion
W
Lecture 3
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Diffusion in FCC Crystals
Random walk theory shows that in 3-D:
For FCC, the specific jump vector i = (a0 /2) u<110>, so its projection is:
If the thermal vacancy is Nv , the probability of having a vacant nearest-neighbor site adjacent to any lattice atom in FCC is:
Examinable Lecture 3
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Diffusion in FCC Crystals
The total jumping frequency of the atom in FCC crystals is
Substituting for total jumping frequency
The expression for the diffusivity, DFCC , for vacancy-assisted diffusion in FCC crystals is
Examinable Lecture 3
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Lattice Vacancies
The Gibbs free energy of a crystal containing monovacancies is:
Combinatoric theory shows that the number of distinguishable ways of distributing nv vacancies on Nl lattice sites is:
Stirling's approximation for large arguments of the factorial is:
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Lattice Vacancies
The previous equation may be approximated as: The equilibrium concentration of lattice vacancies forming at a particular temperature is defined as:
Differentiating with respect to nv, gives after a few steps of algebra:
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Lattice Vacancies
The minimum Gibbs function occurs when:
The last term may be approximated as:
Substituting gives:
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Lattice Vacancies
Solving for the equilibrium lattice monovacancy concentration yields the result:
or
The fractional concentration of monovacancies, at any temperature, may be estimated as:
Examinable Lecture 3
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Lattice Vacancies
Vacancy Formation Enthalpies in Solids
FCC
()
0.68 1.12 0.89 1.29 1.4 1.78 1.32 1.85 0.57
()
3.0 2.65 2.8 2.1 4.0 1.6
( )
2.02 2.68 2.12 1.80 1.34 2.3 1.4 1.1
Lecture 3
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Divacancies
The probability, p2 , for an FCC material, that one of the nearest neighbors to this vacant site is a second vacancy The total number of divacancies, nv2 , present in the crystal
Dividing both sides by Nl and multiplying by 1/2
The temperature dependence of noninteracting divacancies
Lecture 3
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Divacancies
One must consider the additional mass-action equilibrium
Suggesting
Applying the mass-action law to the divacancy-pair orientation in FCC
The divacancy concentration arising from interacting lattice monovacancies
Lecture 3
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Divacancie sbe The...