Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models
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XXIV Contents – In Detail
18.5.1 Models withSymmetricRates ........................ 778
18.5.2 Selected Results for the Coefficient of Collective
Diffusion in theRandomSite-EnergyModel ............ 780
18.6 Conclusion................................................ 783
18.7 Appendix................................................. 784
18.7.1 Derivation of the Result for the Diffusion Coefficient for
ArbitrarilyDisorderedTransitionRates................ 784
18.7.2 Derivation of the Self-Consistency Condition for the
Effective-MediumApproximation ..................... 787
18.7.3 Relation Between the Relative Displacement and the
DensityChange..................................... 789
References ..................................................... 790
19 Diffusion on Fractals
Uwe Renner, Gunter M. Sch¨utz, G¨unter Vojta ...................... 793
19.1 Introduction: WhataFractalis ............................. 793
19.2 AnomalousDiffusion:Phenomenology........................ 797
19.3 Stochastic TheoryofDiffusion on Fractals .................... 802
19.4 AnomalousDiffusion: Dynamical Dimensions ................. 803
19.5 AnomalousDiffusion and Chemical Kinetics .................. 806
19.6 Conclusion................................................ 809
References ..................................................... 810
20 Ionic Transport in Disordered Materials
Armin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer ..... 813
20.1 Introduction .............................................. 813
20.2 BasicQuantities........................................... 816
20.2.1 TracerDiffusion .................................... 816
20.2.2 Dynamic Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817
20.2.3 Probability Distribution and Incoherent Neutron
Scattering.......................................... 817
20.2.4 Spin-Lattice Relaxation.............................. 818
20.3 Ion-Conducting Glasses: Models and Numerical Technique . . . . . . 819
20.4 DispersiveTransport....................................... 822
20.5 Non-ArrheniusBehavior.................................... 832
20.6 Counterion Model and the “Nearly Constant Dielectric Loss”
Response ................................................. 835
20.7 CompositionalAnomalies................................... 839
20.8 Ion-Conducting Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843
20.8.1 LatticeModelofPolymerElectrolytes ................. 843
20.8.2 Diffusion through a Polymer Network: Dynamic
PercolationApproach................................ 846
20.8.3 DiffusioninStretched Polymers....................... 849
20.9 Conclusion................................................ 850
References ..................................................... 852
Contents – In Detail XXV
21 Concept of Mismatch and Relaxation for Self-Diffusion
and Conduction in Ionic Materials with Disordered Structure
Klaus Funke, Cornelia Cramer, Dirk Wilmer ....................... 857
21.1 Introduction .............................................. 857
21.2 Conductivity Spectra of Ion Conducting Materials . . . . . . . . . . . . . 861
21.3 Relevant Functions and Some Model Concepts for Ion Transport
inDisorderedSystems...................................... 864
21.4 CMR Equations and Model Conductivity Spectra . . . . . . . . . . . . . . 867
21.5 Scaling Properties of Model Conductivity Spectra . . . . . . . . . . . . . 871
21.6 PhysicalConceptofthe CMR............................... 874
21.7 Complete Conductivity Spectra of Solid Ion Conductors . . . . . . . . 877
21.8 Ion Dynamicsin aFragileSupercooledMelt .................. 880
21.9 Conductivities of Glassy and Crystalline Electrolytes Below
10MHz................................................... 883
21.10 LocalisedMotionatLowTemperatures....................... 887
21.11 Conclusion................................................ 891
References ..................................................... 892
22 Diffusion and Conduction in Percolation Systems
Armin Bunde, Jan W. Kantelhardt ................................ 895
22.1 Introduction .............................................. 895
22.2 The(Site-)PercolationModel ............................... 895
22.3 The Fractal Structure of Percolation Clusters near p
c
.......... 897
22.4 Further PercolationSystems ................................ 901
22.5 Diffusion onRegularLattices ............................... 903
22.6 Diffusion onPercolationClusters ............................ 904
22.7 Conductivity of Percolation Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 905
22.8 Further ElectricalProperties................................ 906
22.9 Application of the Percolation Concept: Heterogeneous Ionic
Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908
22.9.1 Interfacial Percolation and the Liang Effect. . . . . . . . . . . . . 908
22.9.2 Composite Micro- and Nanocrystalline Conductors . . . . . . 910
22.10 Conclusion................................................ 912
References ..................................................... 913
23 Statistical Theory and Molecular Dynamics of Diffusion
in Zeolites
Reinhold Haberlandt ............................................. 915
23.1 Introduction .............................................. 915
23.2 Some Notions and Relations of Statistical Physics . . . . . . . . . . . . . 916
23.2.1 StatisticalThermodynamics.......................... 916
23.2.2 StatisticalTheoryof IrreversibleProcesses ............. 919
23.3 MolecularDynamics ....................................... 922
23.3.1 GeneralRemarks ................................... 922
23.3.2 ProcedureofanMDSimulation....................... 923
XXVI Contents – In Detail
23.3.3 MethodicalHints ................................... 925
23.4 Simulation ofDiffusion inZeolites ........................... 925
23.4.1 Introduction........................................ 925
23.4.2 Simulations ........................................ 926
23.4.3 Results ............................................ 928
23.5 Conclusion................................................ 942
References ..................................................... 944
List of Contributors .......................................... 949
Index ......................................................... 955
Part I
Solids
1 Diffusion: Introduction and Case Studies in
Metals and Binary Alloys
Helmut Mehrer
1.1 Introduction
Diffusion in solids is an important topic of physical metallurgy and materials
science. Diffusion processes play a key role in the kinetics of many microstruc-
tural changes that occur during processing of metals, alloys, ceramics, semi-
conductors, glasses, and polymers. Typical examples are nucleation of new
phases, diffusive phase transformations, precipitation and dissolution of a
second phase, recrystallization, high-temperature creep, and thermal oxida-
tion. Direct technological applications concern, e.g., diffusion doping during
fabrication of microelectronic devices, solid electrolytes for battery and fuel
cells, surface hardening of steel through carburization or nitridation, diffusion
bonding, and sintering.
The atomic mechanisms of diffusion in crystalline materials are closely
connected with defects. Point defects such as vacancies or interstitials are the
simplest defects and often mediate diffusion. Dislocations, grain boundaries,
phase boundaries, and free surfaces are other types of defects of crystalline
solids. They can act as diffusion short circuits, because the mobility of atoms
along such defects is usually much higher than in the lattice.
This chapter will concentrate on bulk diffusion in solid metals and alloys.
Most of the solid elements are metals. Furthermore, diffusion properties and
atomic mechanisms of diffusion have most thoroughly been investigated in
metallic solids. On the other hand, many of the physical concepts, which have
been developed for metals, apply to diffusion in all crystalline solids. Those
effects, which are unique to non-metallic systems such as charge effects in
ionic crystals and semiconductors, are treated in Chaps. 4 and 5.
For a comprehensive treatment of diffusion in solid matter the reader is
referred to the textbooks of Shewmon [1], Philibert [2], Heumann [3], Allnatt
and Lidiard [4], and Glicksman [5]. A critical collection of data for diffusion in
metals and alloys was edited in 1990 by Mehrer [6]. Recent developments can
be found in the proceedings of a series of international conferences on ‘Diffu-
sion in Materials’ [7–9]. The field of grain- and interphase-boundary diffusion
is described in Chap. 8 and in the book of Kaur, Mishin and Gust [10]. The
book of Crank [11] provides an excellent introduction to the mathematics of
diffusion.
4 Helmut Mehrer
1.2 Continuum Description of Diffusion
1.2.1 Fick’s Laws for Anisotropic Media
The diffusion of atoms through a solid can be described by Fick’s equations.
The first equation relates the diffusion flux j (number of atoms crossing a
unit area per second) to the gradient of the concentration c (number of atoms
per unit volume) via
j = −D∇c. (1.1)
The quantity D is denoted as diffusion coefficient tensor or as diffusivity
tensor. The dimensions of its components are length
2
time
−1
. Its SI units
are [m
2
s
−1
]. Equation (1.1) implies that D varies with direction. In general
the diffusion flux and the concentration gradient are not always antiparallel.
They are antiparallel for an isotropic medium.
For anisotropic media and non-cubic crystalline solids D is a symmetric
tensor of rank 2 [12]. Each symmetric second rank tensor can be reduced to
diagonal form. The diffusion flux is antiparallel to the concentration gradient
only for diffusion along the orthogonal principal directions. If x
1
,x
2
,x
3
denote
these directions and j
1
,j
2
,j
3
the pertaining components of the diffusion flux,
(1.1) can be written as
j
1
= −D
I
∂c
∂x
1
,j
2
= −D
II
∂c
∂x
2
,j
3
= −D
III
∂c
∂x
3
, (1.2)
where D
I
,D
II
,D
III
denote the three principal diffusivities. The diffusion
coefficient for a direction (α
1
,α
2
,α
3
) is obtained from
D(α
1
,α
2
,α
3
)=α
2
1
D
I
+ α
2
2
D
II
+ α
2
3
D
III
, (1.3)
where α
i
denote the direction cosine of the diffusion flux with axis i.Equation
(1.3) shows that anisotropic diffusion is completely described by the three
principal diffusion coefficients.
For uniaxial (hexagonal, tetragonal, trigonal) crystals and decagonal qua-
sicrystals with the unique axis parallel to the x
3
axis we have D
I
= D
II
=
D
III
and (1.3) reduces to
D(Θ)=D
I
sin
2
Θ + D
III
cos
2
Θ, (1.4)
where Θ denotes the angle between diffusion direction and crystal axis. For
isotropic media such as amorphous metals and inorganic glasses, cubic crys-
tals and icosahedral quasicrystals
D
I
= D
II
= D
III
≡ D. (1.5)
Then the diffusion coefficient tensor is reduced to a scalar quantity.
Steady state methods for measuring diffusion coefficients, like the perme-
ation method [3], are directly based on Fick’s first law. In non-steady state
1 Diffusion: Introduction and Case Studies in Metals and Binary Alloys 5
situations diffusion flux and concentration vary with time t and position x.
In addition to Fick’s first law a balance equation is necessary. For particles
which undergo no reactions (no chemical reaction, no reactions between dif-
ferent types of sites in a crystal, etc.) this is the continuity equation
∂c
∂t
+ ∇j =0. (1.6)
Combining (1.1) and (1.6) leads to Fick’s second equation (also called diffu-
sion equation)
∂c
∂t
= ∇(D∇c). (1.7)
1.2.2 Fick’s Second Law for Constant Diffusivity
In diffusion studies with trace elements very tiny amounts of the diffusing
species can be applied. Then the composition of the sample during the inves-
tigation does practically not change (see also Sect. 1.3) and D is independent
of the tracer concentration. Also diffusion in ideal solutions is described by a
concentration-independent diffusion coefficient. Then for diffusion in a certain
direction x (1.7) reduces to
∂c
∂t
= D
∂
2
c
∂x
2
. (1.8)
From a mathematical point of view (1.8) is a second order, linear partial
differential equation. Initial and boundary conditions are necessary for par-
ticular solutions of this equation. Solutions can be found in the book of Crank
for a variety of initial and boundary conditions [11]. These solutions permit
the determination of D from measurements of the concentration distribution
as a function of position and time. We consider two simple examples which,
however, are often relevant for the analysis of experiments.
Thin-Film Solution
If a thin layer of the diffusing species (M atoms per unit area) is concentrated
at x = 0 of a semi-infinite sample, the concentration after time t is described
by
c(x, t)=
M
√
πDt
exp
−
x
2
4Dt
. (1.9)
The quantity
√
Dt, called diffusion length, is a characteristic distance for
diffusion problems. The experimental determination of diffusion coefficients
by the tracer method discussed in Sect. 1.4 is based on (1.9). It is applicable
if
√
Dt is much larger than the initial layer thickness.
6 Helmut Mehrer
Error Function Solution
Suppose that at t = 0 the concentration of the diffusing species is c(x, 0) = 0
for x>0. Then, if for t>0 the concentration at x = 0 is maintained at
c(0,t)=c
s
, the appropriate solution of (1.8) is
c(x, t)=c
s
erfc
x
2
√
Dt
, (1.10)
where the complementary error function is defined by
erfc z ≡ 1 − erf z (1.11)
and
erf z ≡
2
√
π
z
0
exp (−u
2
)du (1.12)
denotes the error function. Equation (1.10) describes the in-diffusion of a dif-
fuser (or diffusant) into a semi-infinite sample with constant concentration
c
s
of that species at the surface. It is, e.g., applicable to the in-diffusion of
a volatile solute into a non-volatile solvent, to the carburization of a metal
in a carbon containing ambient, and to in-diffusion of a solute from an inex-
haustible diffusion source into a solvent with solubility c
s
.
1.2.3 Fick’s Second Law for Concentration-Dependent Diffusivity
Let us consider a case of great practical importance, in which the chemical
composition during diffusion varies over a certain concentration range. Dif-
fusing particles will experience different chemical environments and hence
different diffusion coefficients. This situation is denoted as interdiffusion or
as chemical diffusion. We use the symbol
˜
D to indicate that the diffusion co-
efficient is concentration dependent.
˜
D is denoted as interdiffusion coefficient
or as chemical diffusion coefficient. Fick’s second law (1.7) for diffusion in a
certain direction x then reads
∂c
∂t
=
∂
∂x
˜
D(c)
∂c
∂x
=
˜
D(c)
∂
2
c
∂x
2
+
d
˜
D(c)
dc
∂c
∂x
2
. (1.13)
Theoretical models which permit the calculation of the composition depen-
dent diffusivity from deeper principles using, e.g., statistical mechanics are
nowadays still not broadly available. Then the strategy illustrated in the
previous section of calculating the concentration field for certain initial and
boundary conditions is not applicable. Instead, it is possible to invert this
strategy and to determine the interdiffusion coefficient from a given concen-
tration field by using (1.13). The classical Boltzmann-Matano method for
extracting the concentration-dependent diffusivity
˜
D(c) from experimental
concentration-depth profiles will be considered in Sect. 1.11 of this chapter.
1 Diffusion: Introduction and Case Studies in Metals and Binary Alloys 7
1.3 The Various Diffusion Coefficients
Diffusion in materials is characterized by several diffusion coefficients. In
this section we describe various experimental situations which entail differ-
ent diffusion coefficients. We will, however, concentrate on bulk diffusion in
unary and binary systems. Diffusion in ternary systems produces mathemat-
ical complexities which are beyond the scope of this chapter. We will focus
on bulk diffusion since diffusion along grain boundaries and along surfaces
is treated in Chaps. 7 and 8 of this book. In this section we will distinguish
the various diffusion coefficients by lower and upper indices. We will drop the
indices in the following sections again, whenever it is clear which diffusion
coefficient is meant.
1.3.1 Tracer Diffusion Coefficients
In diffusion studies with trace elements (tagged by their radioactivity or
by their isotopic mass) tiny amounts of the diffusing species can be used.
Although there will be a gradient in the concentration of the trace element,
its total concentration can be kept so small that the overall composition of
the sample during the investigation does practically not change
1
.Insuch
cases a constant tracer diffusion coefficient is appropriate for the analysis of
the experiments.
Self-Diffusion Coefficient
If the diffusion of A-atoms in a solid element A is studied, one speaks of
self-diffusion. Studies of self-diffusion utilize a tracer isotope A
∗
of the same
element. A typical initial configuration for a tracer self-diffusion experiment
is illustrated in Fig. 1.1a. If the applied tracer layer is very thin as compared
to the average diffusion length, the tracer self-diffusion coefficient D
A
∗
A
is
obtained from such an experiment.
The connection between the macroscopically defined tracer self-diffusion
coefficient and the atomistic picture of diffusion is the famous Einstein-
Smoluchowski relation discussed in detail in Sect. 1.6. In simple cases it reads
D
A
∗
A
= fD
E
with D
E
=
l
2
6τ
, (1.14)
where l denotes the jump length and τ the mean residence time of an atom
on a certain site of the crystal
2
.Thequantityf is the correlation factor. For
self-diffusion in cubic crystals f is a numeric factor. Its value is characteristic
1
From an atomistic viewpoint this implies that a tracer atom is not influenced by
othertraceratoms.
2
Equation (1.14) considers only the simplest case: cubic structure, all sites are
energetically equivalent, only jumps to nearest neighbours are allowed.
8 Helmut Mehrer
of the lattice geometry and the diffusion mechanism (see Sect. 1.6). In some
textbooks the quantity D
E
is denoted as the Einstein diffusion coefficient
3
.
In a homogeneous binary A
X
B
1−X
alloy or compound two tracer diffusion
coefficients for both, A
∗
and B
∗
tracer atoms, can be measured. A typical
experimental starting configuration is displayed in Fig. 1.1b. We denote the
tracer diffusion coefficients by D
A
∗
A
X
B
1−X
and D
B
∗
A
X
B
1−X
. Both tracer diffusion
coefficients will in general be different:
D
A
∗
A
X
B
1−X
= D
B
∗
A
X
B
1−X
. (1.15)
This diffusion asymmetry depends on the crystal structure of the material
and on the atomic mechanisms which mediate diffusion. Both diffusivities,
of course, also depend on temperature and composition of the alloy or com-
pound and for anisotropic media on the direction of diffusion. Self-diffusion
in metallic elements will be considered in Sect. 1.8. Self-diffusion in binary
intermetallics is the subject of Sect. 1.10.
Impurity Diffusion Coefficient
When the diffusion of a trace solute C
∗
in a monoatomic solvent A or in
a homogeneous binary solvent A
X
B
1−X
(Fig. 1.1) is measured, the tracer
diffusion coefficients
D
C
∗
A
and D
C
∗
A
X
B
1−X
are obtained. These diffusion coefficients are denoted as impurity diffusion
coefficients or sometimes also as foreign atom diffusion coefficients.
1.3.2 Chemical Diffusion (or Interdiffusion) Coefficient
So far we have considered in this section cases where the concentration gra-
dient is the only cause for the flow of matter. We have seen that such situ-
ations can be studied using tiny amounts of trace elements in an otherwise
homogeneous material. However, from a general viewpoint a diffusion flux is
proportional to the gradient of the chemical potential.
The chemical potential of a species i in a binary alloy is given by (cf.
Chap. 5)
µ
i
=
∂G
∂n
i
p,T,n
j=i
i =A, B (1.16)
In (1.16) G denotes Gibbs free energy, n
i
the number of moles of species i,
T the temperature, and p the hydrostatic pressure. The chemical potential
3
This notation is a bit misleading, since the original Einstein-Smoluchowski re-
lation relates the total macroscopic mean square displacement of atoms to the
diffusion coefficient (see Sect. 1.6), in which correlation effects are included.