Download Kinetics and Diffusion Basic concepts in kinetics

Kinetics and Diffusion
Basic concepts in kinetics
Kinetics of phase transformations
Activation free energy barrier
Arrhenius rate equation
Diffusion in Solids - Phenomenological description
Flux, steady-state diffusion, Fick’s first law
Nonsteady-state diffusion, Fick’s second law
Atomic mechanisms of diffusion
How do atoms move through solids?
● Substitutional diffusion
● Interstitial diffusion
● High diffusivity paths, diffusion along grain
boundaries, free surfaces, dislocations
Factors that influence diffusion
● Diffusing species and host solid (size, bonding)
● Temperature
● Microstructure
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Kinetics: basic concepts
Thermodynamics can be used to predict what is the
equilibrium state for a system and to calculate the driving
force (ΔG) for a transformation from a metastable state to
a stable equilibrium state.
How fast the transformation occurs is the question
addressed by kinetics.
Let’s consider transition from a metastable to the
equilibrium state. 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
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Kinetics: basic concepts
G
Transition path
ΔG a
G1
ΔG
G2
Initial state
(metastable)
Final state
Activated state (equilibrium or
another metastable)
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.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
The concept of thermal activation
In order for 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*:
a
⎞
⎞ ~ exp⎛ − ΔH a
⎞~ exp⎛ ΔS a ⎞ exp⎛ − ΔH a
rate~ exp⎛⎜ − ΔG
⎜
⎟
⎜
⎟
⎜
⎟
k BT ⎠
kB ⎠
k BT ⎠
k BT ⎟⎠
⎝
⎝
⎝
⎝
ΔGa = ΔHa -TΔSa
Arrhenius equation can be applied to a wide range of thermally
activated processes, including diffusion that we consider next.
* 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.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
What is diffusion?
“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.
Most kinetic processes in materials involve diffusion.
Inhomogeneous materials can become homogeneous by
diffusion, compositions of phases can change by diffusion, etc.
For an active diffusion to occur, the temperature should be high
enough to overcome energy barriers to atomic motion.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Phenomenological description of diffusion: diffusion flux
The flux of diffusing atoms, J, is used to quantify how fast
diffusion occurs. The flux is defined as either number of atoms
diffusing through unit area and per unit time (e.g., atoms/m2second) or in terms of the mass flux - mass of atoms diffusing
through unit area per unit time, (e.g., kg/m2-second).
A
J
Let’s consider steady state diffusion - the diffusion flux does not
change with time. Example: diffusion of gas molecules through
a thin metal plate.
gas at
pressure P1
gas at
pressure P1 < P2
C1
C2
concentration profile
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Steady-State Diffusion: Fick’s first law
Fick’s first law: the diffusion flux along direction x is
proportional to the concentration gradient
dC
J = −D
dx
where D is the diffusion coefficient
C
dC
dx
A
J
x
Concentration gradient: dC/dx (Kg m-4) is the slope at a
particular point on concentration profile.
The minus sign in the equation means that diffusion is down the
concentration gradient.
Fick’s first law describes steady state flux in a uniform
concentration gradient.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Nonsteady-State Diffusion: Fick’s second law (I)
In most practical cases steady-state conditions are not
established, i.e. concentration gradient is not uniform and varies
with both distance and time. Let’s derive the equation that
describes nonsteady-state diffusion along the direction x.
Consider an element of material with dimensions dx, dy, and dz
dV =dx dy dz
dAx = dy dz
Jx
Jx+dx
dV
x
J x = −D
∂ C (x, t )
∂x
x+dx
J x + dx = J x +
∂J x
dx
∂x
The number of particles that diffuse into the volume dV during
time dt is JxdAxdt from the left and -Jx+dxdAxdt from the right.
From the balance of incoming and outgoing particles:
(J x - J x + dx )dA x dt = dC(x, t )dV
and using expressions
for Jx and Jx+dx we have
∂C(x, t )
dx = J x − J x + dx
∂t
∂ C (x, t )
∂ ⎛ ∂ C (x, t ) ⎞
=
⎟
⎜D
∂t
∂x ⎝
∂x
⎠
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Nonsteady-State Diffusion: Fick’s second law (II)
∂ ⎛ ∂ C (x, t ) ⎞
∂ C (x, t )
=
⎜D
⎟
∂x ⎝
∂x
∂t
⎠
If dependence of D on x (and C !) is ignored,
∂ 2 C (x, t )
∂ C (x, t )
= D
∂x 2
∂t
[m-3t-1] = [m2t-1]×[m-3m-2]
Fick’s second law relates the rate of change of composition with
time to the curvature of the concentration profile:
C
C
x
C
x
x
Concentration increases with time in those parts of the system
where concentration profile has a positive curvature. And
decreases where curvature is negative.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Nonsteady-State Diffusion: Fick’s second law (III)
∂ 2 C (x, t )
∂ C (x, t )
= D
∂x 2
∂t
C
Can we apply this equation to
steady state diffusion?
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
x
Microscopic picture of diffusion (I)
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:
r
MSD ≡ s ≡ Δr (t)2
2
s
r
ri (0)
r
ri (t)
i
1
≡
N
N
r
r
2
(
r
(t)
−
r
(
0
))
∑ i i
i =1
How to relate this microscopic characteristic
(the mean square distance of atomic migration)
to the macroscopic transport coefficient D used
in the phenomenological Fick’s laws?
Let’s first consider an intuitive (not rigorous) “derivation” of the
relationship between s and D.
Suppose that in time t, the
average particle moved a
distance sx along the direction
in which diffusion is occurring.
X0
JL
CL
JR
sx
CR
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 3050,
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for Diagrams
the fluxand
from
rightLeonid
to left,
JR×t
MSE
Kinetics,
Zhigilei
= (sxCR)/2
Microscopic picture of diffusion (II)
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 concentration gradient dc/dx:
(CL – CR)/sx = -(CR – CL)/sx = - dc/dx
Therefore,
X0
J = sx(CL – CR)/2t = -sx2/2t dc/dx
CL
From the Fick’s law we
also have J = -D dc/dx
Thus, D = sx2/2t or
JL
JR
sx
sx2 = 2tD
CR
sx
for 1D diffusion
For 3D diffusion s2 = sx2 + sy2 +sz2 = 3sx2, and D = s2/6t
In general, D = s2/2dt
where d is the dimensionality of the system
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.
We will consider a more rigorous derivation of this relation after we talk about
the analytical solutions of the diffusion equation
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
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.
¾ Substitutional impurities
¾ Substitutional self-diffusion – can be studied by depositing of
a small amount of radioactive isotope of the element (tracer
diffusion)
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
MSE
3050, Phase
Diagrams and
Kinetics,
Leonid Zhigilei
bonding,
temperature,
and
microstructure.
Diffusion Mechanisms: Vacancy diffusion (I)
To 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,
ΔGmv, 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).
G
ΔGmv
Atom
Vacancy
Distance
The average thermal energy of an atom (3kBT ≈ 0.08 eV at room
temperature) is usually much smaller that the activation free
energy ΔGmv (~ 1 eV/vacancy) and a large thermal fluctuation is
needed for a jump.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Diffusion Mechanisms: Vacancy diffusion (II)
For a simple one-dimensional case, the probability of such
fluctuation or frequency of jumps, Rj, can be described by the
Arrhenius equation:
v
⎛ Δ Gm
⎞
R j = υ0 exp⎜ −
⎟
k
T
B ⎠
⎝
where ν0 is an attempt frequency related to the frequency of
atomic vibrations. The value of 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).
Rj =
frequency of atom vibrations
in the diffusion direction ν0
×
probability that a given
oscillation will move the
atom to an adjacent site
To relate this to the diffusion of atoms we have to consider the
jump frequency of a given atom in a 3D crystal.
Moreover, for an atom to jump, there must be a vacancy next to it
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Diffusion Mechanisms: Vacancy diffusion (III)
The probability for any atom in a solid to move is the product of
the probability of finding a vacancy in n
⎛ ΔG vf
⎞
eq
⎜
⎟
exp
z
=
z
−
an adjacent lattice site (fraction of atoms
⎜
k BT ⎟
N
⎝
⎠
that have a vacancy as a neighbor):
×
the rate of jumps of a vacancy (defined
⎞
by a probability of a thermal fluctuation R = υ exp⎛ − ΔGmv
⎜
⎟
0
j
k
T
B
needed to overcome the energy barrier
⎝
⎠
for vacancy motion):
The rate at which atom jumps from place to place in the crystal is
therefore
⎛ ΔG v
⎞ ⎛ ΔG v
1
⎞
atom
Rj
=
≈ υ0 z exp⎜⎜ −
τj
⎝
f
⎟exp −
k BT ⎟ ⎜⎝
⎠
m
k BT ⎟⎠
where τj is the average time between jumps for atoms.
If the distance atoms cover in each jump is a, the Einstein
r
relation Δ ri(t) 2 i = 6 Dt
can be used to estimate the diffusion
coefficient from the average time between jumps:
⎛ ΔG vf + Δ G mv ⎞
a2
a 2 υ0 z
⎟=
D=
=
exp ⎜ −
⎜
⎟
6τ j
6
k BT
⎝
⎠
⎛ Δs vf + Δs mv ⎞
⎛ Δ h vf + Δ hmv
a 2 υ0 z
⎟ exp ⎜ −
exp ⎜
⎜
⎟
⎜
6
kB
k BT
⎝
⎠
⎝
⎞
⎛
⎟ = D0 exp ⎜ − E d
⎜ k T
⎟
B
⎝
⎠
⎞
⎟⎟
⎠
where D0 is a parameter of material (both matrix and diffusing
species) and is independent of temperature, Ed is activation
energy for diffusion: E = Δh v + Δ h v
m
MSE 3050, Phase Diagrams anddKinetics,f Leonid Zhigilei
Diffusion Mechanisms: Vacancy diffusion (IV)
⎛ Δ s vf + Δ s mv
a 2 υ0 z
a2
D=
=
exp ⎜
⎜
kB
6τ j
6
⎝
⎞
⎛ Δ h vf + Δ hmv
⎟ exp ⎜ −
⎟
⎜
k BT
⎠
⎝
⎞
⎛
⎟ = D0 exp ⎜ − E d
⎜ k T
⎟
B
⎝
⎠
⎞
⎟⎟
⎠
Let’s perform an order of magnitude estimate of the average time
between jumps and the diffusion coefficient for self-diffusion in
aluminum
⎛ Δs vf + Δsmv ⎞ ⎛ Δhvf + Δhmv ⎞
1
⎟ exp⎜
⎟
τj =
exp⎜ −
⎜
⎟
⎜
⎟
υ0 z
kB
⎝
⎠ ⎝ kBT ⎠
⎛ Δs vf + Δs mv
a 2 υ0 z
D=
exp ⎜
⎜
kB
6
⎝
Δhfv = 0.72 eV
Al:
z ≈ 12
⎛ Δs vf + Δsmv ⎞
⎟ ~1
exp⎜
⎜
⎟
kB
⎝
⎠
Δhmv = 0.68 eV
⎞
⎛ Δ h vf + Δhmv
⎟ exp ⎜ −
⎟
⎜
k BT
⎠
⎝
⎞
⎟
⎟
⎠
υ0 ≈ 1013 s -1
a ≈ 3 × 10 −10 m
- e.g., Shewmon, Diffusion in solids, Ch. 2.4-2.7
T = 0ºC
T = 650ºC
τj ≈ 6 ×1011s
(less than one jump in 20000 years)
τj ≈ 4 × 10-7s
(2.5 million jumps per second)
D ≈ 3×10-32 m2/s
D ≈ 4×10-14 m2/s
17 orders of magnitude difference!
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Substitutional self-diffusion
For a given crystal structure and bonding type Ed/RTm is roughly
constant → D(T/Tm) ≈ const
Tm
K
D0
10-6 m2/s
Ed
kJ/mol
Ed
eV
Ed/RTm
Al
933
170
142
1.47
18.3
1.9
Cu
1356
31
200.3
2.08
17.8
0.59
Ni
1726
190
279.7
2.90
19.5
0.65
γ-Fe
1805
49
284.1
2.94
18.9
0.29
Cr
2130
20
308.6
3.20
17.4
0.54
bcc
V
2163
28.8
309.2
3.20
17.2
0.97
Nb
2741
1240
439.6
4.56
19.3
5.2
transition
metals
K
337
31
40.8
0.42
14.6
15
Na
371
24.2
43.8
0.45
14.2
16
Li
454
23
55.3
0.57
14.7
9.9
Ge
1211
440
324.5
3.36
32.3
4.4×10-5
Si
1683
900000
496.0
5.14
35.5
3.6×10-4
D(Tm)
10-12 m2/s
fcc
metals
bcc
alkali
metals
diamond
cubic
semicond.
from Porter and Easterling textbook
Not surprisingly, we see a correlation between Tm and Ed →
stronger interatomic bonds make it more difficult to melt material
and increase both Δh vf and Δhmv
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Diffusion of interstitial atoms
Interstitial diffusion also involve transition through the energy
barrier and can be discussed in a manner similar to the vacancy
diffusion mechanism.
The difference is that there are always sites for an interstitial
atom to jump to.
⎛ Δ G mi
a2
a 2 υ0 p
D=
=
exp ⎜⎜ −
6τ j
6
⎝ k BT
⎞
⎟⎟ =
⎠
⎛ Δ s mi
a 2 υ0 p
exp ⎜⎜
6
⎝ kB
⎛ Δ hmi
⎞
⎟⎟ exp ⎜⎜ −
⎝ k BT
⎠
⎛ E
⎞
⎟⎟ = D0 exp ⎜⎜ − d
⎝ k BT
⎠
⎞
⎟⎟
⎠
i
where p is number of neighbor interstitial sites and E d = Δhm
Small interstitial atoms of a foreign (extrinsic) type, e.g., C in Fe
or O in Si may diffuse directly through the lattice (i.e., without
the help of vacancies) and play an important role in defining
properties of materials.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Diffusion of interstitial atoms
Impurity
D0, mm2/s-1
Ed, kJ/mol
C in FCC Fe
23.4
148
C in BCC Fe
2
84.1
N in FCC Fe
91
168.6
N in BCC Fe
0.3
76.1
H in FCC Fe
0.63
43
H in BCC Fe
0.1
13.4
vacancy
mechanism →
from Porter and Easterling
textbook & Smithells Metals
Reference Book
← interstitial
impurities
⎛ E
D = D0 exp ⎜⎜ − d
⎝ k BT
⎞
⎟⎟
⎠
D0, mm2/s-1
Ed, kJ/mol
Fe in γ-Fe
49
284
Fe in α-Fe
276
250.6
Fe in δ-Fe
201
240.7
Fe in Cr
47
332
Au in Ag
85
202.1
Si in Si
146000
484.4
• 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 interstitials is typically faster as compared to the
vacancy diffusion mechanism (self-diffusion or diffusion of
substitutional atoms).
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Diffusion of self-interstitials
Intrinsic interstitials, also called “self-interstitials” are interstitials
atoms of the same kind as the atoms of the crystal.
Self-interstitials in most materials introduce strong deformations
into the lattice and have very high formation energy, Δhfi ≈ 3Δhfv
for metals. The number of equilibrium interstitials can be
estimated by an equation similar to the one derived for vacancies:
⎛ Δhi f ⎞
We can estimate that at room
i
⎟⎟
neq = N exp ⎜⎜ −
temperature in copper there is less
k BT ⎠
⎝
than one interstitial per cm3, whereas
just below the melting point there is one interstitial for every 1012
atoms – there are virtually no “equilibrium” interstitials in metals
and most other elemental crystals.
In Si, however, intrinsic interstitials play an important role in
diffusion and formation of defect structures.
Non-equilibrium self-interstitials in most materials are very
mobile, e.g., Δhmi ≈ 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 (two atoms share the
space of one), and crowdions.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Diffusion of clusters of interstitials
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.
Point defects in atomistic simulations [Lin et al., Phys. Rev. B 77, 214108, 2008]
175 ps
200 ps
400 ps
MSE 3050, Phase
Diagrams<110>-dumbbell
and Kinetics, Leonid Zhigilei
vacancy
interstitial
450 ps
cluster of 4
interstitials
<111>-crowdion
Diffusion – Temperature Dependence
⎛ Ed ⎞
D = D0 exp⎜ −
⎟
⎝ RT ⎠
D0 – T-independent pre-exponential (m2/s)
Ed – activation energy for diffusion (J/mol)
R – the gas constant (8.31 J/mol-K)
T – absolute temperature (K)
The above equation can be rewritten as
ln D = ln D0 −
Ed ⎛ 1 ⎞
⎜ ⎟
R ⎝T ⎠
or
Ed ⎛ 1 ⎞
log D = log D0 −
⎜ ⎟
2.3R ⎝ T ⎠
The activation energy Ed and pre-exponential D0, therefore, can
be estimated by plotting lnD vs. 1/T or logD vs. 1/T. Such plots
are called Arrhenius plots.
b = logD0
y = ax + b
Graph of log D vs 1/T
has slop of –Ed/2.3R,
intercept of ln Do
Ed
a=−
2.3R
x = 1/T
[(
) (1 T1 − 1 T2 )]
Ed = −2and
.3RKinetics,
× log D
D2
MSE 3050, Phase Diagrams
Leonid
Zhigilei
1 − log
Diffusion of interstitial and substitutional impurities
log D = log D0 −
Ed ⎛ 1 ⎞
⎜ ⎟
2.3R ⎝ T ⎠
Diffusion of interstitials is typically faster as compared to the
vacancy diffusion mechanism (self-diffusion or diffusion of
substitutional atoms).
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Fast diffusion paths (I)
More open atomic structure at defects (grain boundaries,
dislocations) can result in a significantly higher atomic mobility
along the defects.
r
MSD = Δri(t)2 = 6 Dt
i
Mean-square displacement of all
atoms in the system (B), atoms in
the grain boundary region (C), and
bulk region of the system (A).
The plots are from the computer simulation by T. Kwok, P. S.
Ho, and S. 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
MSE
Phase Diagrams
and Kinetics, Leonid Zhigilei
grain3050,
boundary
region.
Fast diffusion paths (II)
Diffusion coefficient along a defect (e.g. grain boundary) can be
also described by an Arrhenius equation, with the activation
energy for grain boundary diffusion significantly lower than the
one for the bulk.
⎛ E ddef ⎞
def
def
⎟⎟
D = D0 exp ⎜⎜ −
⎝ k BT ⎠
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.
Self-diffusion coefficients for
Ag. The diffusivity if greater
through
less
restrictive
structural regions – grain
boundaries, dislocation cores,
external surfaces.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Fast diffusion paths: Diffusion in nanocrystalline materials
image by Zhibin Lin et al.
J. Phys. Chem. C 114, 5686, 2010
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]
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Fast diffusion paths: Diffusion in nanocrystalline materials
High-resolution electron micrograph (left, [Acta Mater. 56, 5857,
2008]) and computed atomic structure (right, [Acta Mater. 45,
987, 1997]) of nanocrystalline Si.
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.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Summary on the Diffusion Mechanisms
Make sure you understand language and concepts:
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Mobility of atoms and diffusion
Activation energy
High diffusivity path
Arrhenius equation
Interstitial diffusion
Self-diffusion
Vacancy diffusion
Connection between the microscopic picture of diffusion (mean
square displacement of atoms) and the diffusion coefficient
Factors that Influence Diffusion:
¾ Temperature - diffusion rate increases very rapidly with
increasing temperature (Arrhenius dependence)
¾ Diffusion mechanism - interstitial is usually faster than
vacancy
¾ Diffusing and host species - Do, Ed is different for every
solute - solvent pair
¾ Microstructure – low-temperature diffusion is faster in
polycrystalline vs. single crystal materials because of the
accelerated diffusion along grain boundaries and dislocation
cores.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei