Diffusion in Solid Metals and Alloys

809 Pages·1990·41.29 MB·English

by  H. Mehrer (auth.)

Ref. p. 301 1.1 Fick’s laws, flux of particles, isotropic and anisotropic diffusion I
1 General introduction
1.1 Fick’s laws, flux of particles, isotropic and anisotropic diffusion
The law governing diffusion processesa nd hence the redistribution of concentrations is Fick’s first law,
which for an isotropic medium or a cubic crystal can be written as
J= -D grade. (1.1)
J is the instantaneous flux of particles of a certain speciesa nd c is the concentration of the same species.T he
negative sign in (1.1) indicates the opposite direction of the flux compared to the concentration gradient. The
factor of proportionality D is denoted as diffusion coeficient or as diffusivity.
Jis expressedi n number of particles or moles per unit area and unit time and c in particles or moles per unit
volume. Consequently D has the dimension length2 time- l. In the international system of units (SIU) used in
this volume, diffusion coefficients are expressedi n m 2s - ’ . In the cgs-systemw hich is still used in the literature,
too, they are expressed in cm’ s -I . D depends on temperature, pressure and in general also on concentration.
Many metallic elements and alloys are cubic. Therefore, in many casesD is indeed a scalar quantity.
For anisotropic media and non-cubic crystals Fick’s first law generalizes to
J= - BVc, (I.3
where 9 is a symmetric second rank tensor denoted as the diffusion coefficient tensor. Equation (1.2)m eans that
the diffusion coefficient varies with direction. The diffusion flux is parallel to grade only along the three
orthogonal principal axes of diffusion. If x1, x2, xj denote these principal axes and J1, J, and J3 the pertaining
components of the diffusion flux, (1.2) may still be written as
JI=-D,;
J=-D i
2
2 ax,
J,=-D,&.
3
D,, D,, D, denote the principal diffusion coefficients. In general the diffusion flux and grad c are not parallel.
However, if yl, yZ, y3 denote the direction cosines of grad c a diffusion coefficient for the direction (rl, yZ, y3) may
be defined as
D(Y,,Yz,Y~)=Y:.D~+Y~Dz+Y~D~. (1.4)
Therefore, anisotropic diffusion is completely described by the three principal diffusion coefficients.
For crystals with orthorhombic and higher symmetry the principal axes of diffusion coincide with the axes
of crystallographic symmetry.
In uniaxial (tetragonal, hexagonal, trigonal) crystals with the unique axis parallel to the x, axis, we have
D, = D, =l=D ,. The diffusion coefficients for both directions perpendicular to the unique axis are the same and
are usually denoted as D,. The diffusion coefficient parallel to the unique axis is denoted as D,,. For uniaxial
crystals (1.4) reduces to
D(0) = D,, cos2 0 + D, sin2 8, (I.3
where ~9is the angle between diffusion direction and crystal axis. For cubic crystals we have D, = D, = D, = D
and (1.2) reduces to (1.1).
Equation (1.1) and its three dimensional generalizations provide a formal definition of the diffusion coefh-
cient as the ratio of the flux and the concentration gradient. The steady state methods for measuring diffusion
are based directly on Fick’s first law.
Land&-BBmstein
New Series III/26 Mehrer
2 1.2.1 Sandwich solution and thin layer solution [Ref. p. 30
In non-steady state situations the diffusion flux and the concentration vary with time t. In such situations
in addition to Fick’s first law ‘a balance equation is necessary.F or particles which undergo no reactions this is
the equation of continuity
Combining (1.2) and (1.6) yields
8C
ar = div (3 Vc) (1.7)
which is denoted as Fick’s second law. When the concentration varies only along a certain direction denoted
by x (1.7) becomes
If furthermore D is independent of concentration and hence of position x in the sample (1.8) reduces to
(1.9)
For most diffusion experiments either (1.1) or its generalization to the anisotropic case( 1.2),a nd in non-steady
state situations either (1.8) or (1.9), respectively provide appropriate descriptions of the diffusion process.
1.2 Solutions of diffusion equations for constant diffusivity
The diffusion coefficient is independent of concentration and position when diffusion occurs in chemically
homogeneous systems.S uch measurementsa re possible e.g.t hrough the use of radioactive tracer elements.S ince
these measurements require extremely small amounts of tracers, the system remains essentially homogeneous
during the diffusion. The diffusion of an interstitial solute in a metal or alloy solvent may be also described by
a constant D as long as the concentration differences are small. In section 1.2 some simple analytical solutions
of the equation (1.8) for various initial and boundary conditions are described. For more comprehensive
collections of solutions we refer to several textbooks [55H, 59C, 63S, 645,66A, 7X, 85P, 89Sl].
1.2.1 Sandwich solution and thin layer solution
A very thin layer of the diffusing specieso f total amount M per unit area is deposited at the boundary x = 0
between two identical samples. After diffusion for time t the concentration is described by
M
c(x, t) = ~ exp (1.10)
2&E
provided that the thickness of the deposited layer is much smaller than 2(D t) *‘2. (1.10) is often also called either
instantaneous sowce solution or Gaussian concentration profile. A plot of (1.10) in linear scalesi s shown in Fig. 1
for 4 different values of 2 fi. The quantity 2(0 t)‘12 is a measuref or the penetration depth and occurs in most
diffusion problems. It is often denoted as d@sion length.
Instantaneous source diffusion also occurs when a quantity M per unit area is placed as a source on the
surface of a sample and if the diffusing speciesi s consumed only by diffusion into the sample. The concentration
profile is then given by
c(xt),= - --
& ex(p 4X>d* t (1.11)
The thin layer solution is often used in radiotracer experiments for the determination of D from the concentra-
tion profile (see subsection 1.6.1.2.1).T he thin layer solution differs by a factor 2 from the sandwich solution
since in (1.11) diffusion occurs into a half-space.
Casesi n which the thin film condition is violated because of low solubility of the diffusing speciesa re not
uncommon in impurity diffusion. In such caseso ften (1.14)c an be used instead of(l.11). For a detailed discussion
of solubility-limited diffusion the reader is referred to [63 M].
Land&-B6mstein
Mehrer New Series 111126
2 1.2.1 Sandwich solution and thin layer solution [Ref. p. 30
In non-steady state situations the diffusion flux and the concentration vary with time t. In such situations
in addition to Fick’s first law ‘a balance equation is necessary.F or particles which undergo no reactions this is
the equation of continuity
Combining (1.2) and (1.6) yields
8C
ar = div (3 Vc) (1.7)
which is denoted as Fick’s second law. When the concentration varies only along a certain direction denoted
by x (1.7) becomes
If furthermore D is independent of concentration and hence of position x in the sample (1.8) reduces to
(1.9)
For most diffusion experiments either (1.1) or its generalization to the anisotropic case( 1.2),a nd in non-steady
state situations either (1.8) or (1.9), respectively provide appropriate descriptions of the diffusion process.
1.2 Solutions of diffusion equations for constant diffusivity
The diffusion coefficient is independent of concentration and position when diffusion occurs in chemically
homogeneous systems.S uch measurementsa re possible e.g.t hrough the use of radioactive tracer elements.S ince
these measurements require extremely small amounts of tracers, the system remains essentially homogeneous
during the diffusion. The diffusion of an interstitial solute in a metal or alloy solvent may be also described by
a constant D as long as the concentration differences are small. In section 1.2 some simple analytical solutions
of the equation (1.8) for various initial and boundary conditions are described. For more comprehensive
collections of solutions we refer to several textbooks [55H, 59C, 63S, 645,66A, 7X, 85P, 89Sl].
1.2.1 Sandwich solution and thin layer solution
A very thin layer of the diffusing specieso f total amount M per unit area is deposited at the boundary x = 0
between two identical samples. After diffusion for time t the concentration is described by
M
c(x, t) = ~ exp (1.10)
2&E
provided that the thickness of the deposited layer is much smaller than 2(D t) *‘2. (1.10) is often also called either
instantaneous sowce solution or Gaussian concentration profile. A plot of (1.10) in linear scalesi s shown in Fig. 1
for 4 different values of 2 fi. The quantity 2(0 t)‘12 is a measuref or the penetration depth and occurs in most
diffusion problems. It is often denoted as d@sion length.
Instantaneous source diffusion also occurs when a quantity M per unit area is placed as a source on the
surface of a sample and if the diffusing speciesi s consumed only by diffusion into the sample. The concentration
profile is then given by
c(xt),= - --
& ex(p 4X>d* t (1.11)
The thin layer solution is often used in radiotracer experiments for the determination of D from the concentra-
tion profile (see subsection 1.6.1.2.1).T he thin layer solution differs by a factor 2 from the sandwich solution
since in (1.11) diffusion occurs into a half-space.
Casesi n which the thin film condition is violated because of low solubility of the diffusing speciesa re not
uncommon in impurity diffusion. In such caseso ften (1.14)c an be used instead of(l.11). For a detailed discussion
of solubility-limited diffusion the reader is referred to [63 M].
Land&-B6mstein
Mehrer New Series 111126
Ref. p. 301 1.2.2 Constant surface concentration and semi-infinite sample 3
0.6
Fig. 1. Instantaneous source (Gaussian) diffusion profiles.
The concentration normalized to the total amount Mis plot-
ted versus penetration distance x for four different values of
the diffusion length 2 @.
0 0.5 1.0 1.5 2.0 2.5 :
x-
1.2.2 Constant surface concentration and semi-infinite sample
If at t = 0 the concentration in a semi-infinite sample was c(x, 0) = c0 and if at t > 0 the surface concentra-
tion is maintained at ~(0, t) = c, the appropriate solution is
c - c,
- = erf(x/2 J&) (1.12)
co - c,
In (1.12)
erfz = -?- 5 e-“‘du (1.13)
fro
denotes the error function. A sample may be considered as semi-infinite as long as (D t)l” is very much smaller
than the sample dimension in diffusion direction.
For co = 0 (1.12) leads to
cfc, = erfc(x/2 JiYt) , (1.14)
where the complementary error function is defined by
erfcz = 1 - erfz. (1.15)
Equation (1.14) describes the in-diffusion of a certain speciesf rom a surface concentration maintained at c,. A
plot of (1.14) in linear scales is shown in Fig. 2 for 4 different values of 2(Dt)‘/2. Figure 3a and 3b show
comparisons between the instantaneous source concentration profile (1.11) and the constant surface concentra-
tion profile (1.14) in logarithmic scales either as a function of the penetration distance or as a function of the
penetration distance squared.
For c, = 0 (1.12) leads to
c/co = erf(x@ @) . (1.16)
(1.16) is the appropriate solution e.g. for the evaporation of a volatile solute element of initial concentration co
from a non-volatile solvent, or for the decarburization of a metal in an oxidizing atmosphere.
The diffusion flux per unit area which penetrates the surface is D cJ~ in the case of (1.14) and - D co/
,/&% in the case of (1.16). The total amount of diffusing substance M(t) which penetrates into the sample is
&f(t) = 2c, JzJi (1.17)
in case of (1.14) and the amount escaping from the sample in the case of (1.16) is
M(t) = 2c, JEqi . (1.18)
Equations (1.17) or (1.18) may be used in in- and out-diffusion experiments to determine D either from the total
amount of material taken up by or lost from a sample.
The solutions given in subsections 1.2.1a nd 1.2.2a re applicable as long as (D t)1/2i s very much smaller than
the sample dimensions in diffusion direction. Under such conditions the samples may be considered as infinite
or semi-infinite.
Land&-Bibstein
New Series III/26
4 1.2.3 Diffusion in a membrane [Ref. p. 30
0.8
0.6
I
L.7
:
0.4
0.2
0.5 1.0- 1.5 2.0 2.5 i0
x-
Fig. 2. Constant surface concentration (erfc) diffusion pro-
Iiles. The concentration normalized to the constant surface
conccntraction r/c, is plotted versus distance from the surface
for four different values of the diffusion length 2 fi.
1
10
;;I 10-2
II
; 10-s
u
lo-’
10.5
0 0.5 1.0 1.5 2.0 2.5 30 0 1.5 3.0 k.5 6.0 15 9.0
a z=x/zyz- b I’= x2/1, Lit -
Fig. 3. Instantaneous source (Gaussian) and constant surface concentration source (erfc) diffusion profiles in a semilogarith-
mic plot. The concentration normalized to the surface concentration is plotted in (a) versus the distance from the surface
normalized to the diffusion length z = x/2 ,/% and in (b) versus z*.
1.2.3 Diffusion in a membrane
In this subsection we consider two caseso f one dimensional diffusion in a membrane of thickness L bounded
by two parallel planes. If the surfaces of the membrane at x = 0 and x = L are maintained at constant con-
centrations c, and c2 as illustrated in Fig. 4a, after some delay time of the order of L2/6D (seeb elow) a stead~~
sr~te is reached which is described by
c-c, x
(1.19)
c,=c,=
According to (1.19) the concentration changes linearly from c1 to c2 through the membrane. The flux across the
membrane is given by
J = D(c, -Q/L. (1.20)
Provided that c,, c2 and L are known, D can be determined from (1.20) by measuring .I.
If the region of the membrane - L/2 < x < L/2 is initially at uniform concentration c0 and the surfacesa re
kept at constant concentration c, either desorption (c, < c,J or ahsorption (c, > cO)c an occur as illustrated in
Fig. 4 b. The ~~ort-srca~s~r~n re solution of (1.9) is described by
c - co
-,1-T! n 2= . 2Cn- 1) cos[(2n + 1) nx/L] exp[-(2n + l)2n2Dt/L2]. (1.21 a)
cs - co ”
land&BBmstcin
Mehrer New Series III,‘26
Ref. p. 301 1.2.3 Diffusion in a membrane
Solution (1.21 a) is particularly useful for large times since then only few terms in the sum contribute significantly.
The appropriate solution for small times is
c - co F (- 1y erfcW + 1) WI - x + 2 (- 1y erfcI On + 1) WI + x
(1.21 b)
cs- co n=O 2JDt n=O 2JDt
Equations (1.21 a) and (1.21 b) can be written in terms of the dimensionless parameters D t/L’ and x/L. Graphs
of (c - co)/(c, - co) versus x/(L/2) are shown in Fig. 4c for various values of 4 D tJL2.
The total amount M(t) of the diffusing species which has entered the membrane at time t with respect to the
corresponding quantity M(co) after infinite time obtained by integration of (1.21 a) is
M(t) 8 m 1
- = 1 “FO exp[- (2n + 1)2x2Dt/LZ] (1.22a)
M(a) - 2 (2n + 1)27?
and by integration of (1.21 b) is
1
I/& + 2 jJ (- I)” ierfc nL (1.22b)
n=O 2JDt
where
ierfc z = 7 erfc u du (1.22c)
I
denotes the integral of the complementary error function.
For c, = 0 the expressions (1.21) and (1.22) can be used to describe the outgassing of a gaseous or volatile
solute from a membrane. The case co = 0 describes the uptake of a gas or a solute by a thin slab of solvent
material.
in.
0 L
a x-
l
-L/2 0 L/2
b X-
Fig. 4. Concentration distributions in “plane sheet” membranes of thickness L.
(a) Steady state distribution with constant surface concentrations c1 and c2 according to (1.19).
(b) Schematic non-steady state distribution according to (1.21 a) for the caseso f absorption c, > c,, and desorption cs < cO.
(c) Concentration distribution at various times in a membrane -L/2 < x < L/2 with an initial uniform concentration c0 and
surface concentration c, from (1.21) according to [75C]. The numbers on the curves are values of the dimensionless
quantity 4 D t/L'.
Land&-Bhmstein
New Series III/26 Mehrer
6 1.2.4 Diffusion in a cylinder; 1.2.5 Diffusion in a sphere [Ref. p. 30
1.2.4 Diffusion in a cylinder
We consider a long circular cylinder of radius R in which diffusion occurs everywhere radially. Concentra-
ion is then a function of distance r from the cylinder axis and of time t. If the concentration is initially uniform
and equal to c0 throughout the cylinder and if the surface concentration at r = R is maintained at c, for t 2 0,
:he solution of (.1 .9,) is
c - co m exp(- D~,2t) JO(cl,r)
-= + (1.23a)
cs - co n 1 a, Jl 6%R I .
In (1.23) J,(z) and J,(z) are the Besself unctions of the first kind with orders zero and one, respectively. The
r, are roots of
J,(cc,R) = 0
which are tabulated in tables of Besself unctions. Solutions for small times can be found in [75C]. The solution
‘or a cylinder can be written in terms of the dimensionless parameters Dt/R* and r/R. The corresponding
graphical representation is given in Fig. 5.
The quantity M(t) of the diffusing speciesw hich has entered or left the cylinder in time t with respect to the
:orresponding quantity M(co) at infinite time is obtained from (1.23) as
M(t)
-= 1 -f $ -$exp(-Daft). (1.24)
M(a) n’ n
Equations (1.23) and (1.24) can be used for cylindrical samples to describe the outgassing or the uptake of
Y solute.
I Y A //n/llrr
/I
Fig. 5. Concentration distribution at various times in a cyl-
inder of radius R with an initial uniform concentration cOa nd
constant surface concentration c, according to [7X3 The
numbers on curves are values of the dimensionless quantity
DrjR’.
0.4 0.6 0.8 1.0
r/R -
1.2.5 Diffusion in a sphere
We consider a sphere of radius R and restrict ourselves to a case where diffusion is radial. If the surface
concentration for t 2 0 is maintained at c, and if the sphere is initially loaded with a uniform concentration co
the solution is
c - co
=,+E 2 iI....? sin y exp[- n*rr* Dt/R*] (1.25)
cs - co xr “=I n
where r denotes the distance from the centre of the sphere.T he total amount of the diffusing speciesM (t) at time
t entering or leaving the sphere obtained by integration of (1.25) is given by
MO)
-= 1 - -$ “el -$ exp(- n27c2Dt/R2) (1.26)
M(a)
where M(m) denotes the total amount at infinite time. Curves showing the solution of (1.25) as a function oi
r/R for different values of the dimensionless parameter Dt/R* are reproduced in Fig. 6. Equation (1.25)a nd (1.26)
can be used to describe the outgassing or the uptake of a solute from or by a sphere.
la”ooll-Bomslel”
Mehrer New Series III/26
Ref. p. 301 1.3 Diff. eq. for cont.-dependent diffusivity; 1.4.1 Self-diff. coefficient
Fig. 6. Concentration distribution at various times in a
0.2
sphere of radius R with an initial uniform concentration cO
and constant surface concentration c, according to [75C]. The
numbers on curves are values of the dimensionless quantity
Dt/R2.
0 0.2 OA 0.6 0.8 '
r/R -
1.3 Diffusion equation for concentration-dependent diffusivity
In general the diffusion coefficient will depend on the concentration of the species,w hich also means that
the diffusion coefficient changes with position in the sample. In this case according to (1.8) Fick’s second law
must be written as
(1.27)
In (1.27) we have used d for the chemical diffusion coef$cient (see section 1.4). The solution of (1.27) in closed
form is (apart from special caseso f b(c)) usually not possible and numeric or graphic integrations of (1.27) are
necessary.T he most frequently used method of analysis is the Boltzmann-Matano method which was proposed
by Matano [33M] and is based on a transformation of (1.27) which is due to Boltzmann [1894B]. This method
is described in subsection 1.6.1.2.2.
1.4 The various diffusion coefficients
In this section various experimental situations and the various diffusion coefficients which they entail are
described. In order to permit a clear distinction between the various diffusion coefficients in the present chapter
the symbol "D" is used for the diffusion coefficient in combination with lower and upper indices. However, the
indices are dropped again in the following sections of chapter 1 and in the data chapters of the whole volume
whenever it is clear which diffusion coefficient is considered.
1.4.1 Self-diffusion coefficient
1.4.1.1 Pure elements
If in a solid of element A the diffusion of A atoms is studied, one speaks about self-diffusion. Studies of
self-diffusion usually utilize tracer atoms A* of the same element. In most experiments tracers are marked by
their radioactivity. A typical situation for a radiotracer experiment is shown in Fig. 7a. The isotopic mass or
the nuclear spin is sometimes used as tag for tracer atoms as well. The tracer self-&&ion coefficient DF is in
a microscopic picture according to
DA’ =fI”
A 62
related to the jump length 1o f atomic jumps and to the mean residence time of atoms r on a certain site in a
,c rystalline solid. The correlation factor f is in the caseo f self-diffusion often only a numeric factor which depends
on the crystal structure and on the diffusion mechanism [7OLl]. Tracer self-diffusion data in pure metallic
elements are listed in chapter 2.
Land&Biimstein
Mehrer
New Series III/26
Ref. p. 301 1.3 Diff. eq. for cont.-dependent diffusivity; 1.4.1 Self-diff. coefficient
Fig. 6. Concentration distribution at various times in a
0.2
sphere of radius R with an initial uniform concentration cO
and constant surface concentration c, according to [75C]. The
numbers on curves are values of the dimensionless quantity
Dt/R2.
0 0.2 OA 0.6 0.8 '
r/R -
1.3 Diffusion equation for concentration-dependent diffusivity
In general the diffusion coefficient will depend on the concentration of the species,w hich also means that
the diffusion coefficient changes with position in the sample. In this case according to (1.8) Fick’s second law
must be written as
(1.27)
In (1.27) we have used d for the chemical diffusion coef$cient (see section 1.4). The solution of (1.27) in closed
form is (apart from special caseso f b(c)) usually not possible and numeric or graphic integrations of (1.27) are
necessary.T he most frequently used method of analysis is the Boltzmann-Matano method which was proposed
by Matano [33M] and is based on a transformation of (1.27) which is due to Boltzmann [1894B]. This method
is described in subsection 1.6.1.2.2.
1.4 The various diffusion coefficients
In this section various experimental situations and the various diffusion coefficients which they entail are
described. In order to permit a clear distinction between the various diffusion coefficients in the present chapter
the symbol "D" is used for the diffusion coefficient in combination with lower and upper indices. However, the
indices are dropped again in the following sections of chapter 1 and in the data chapters of the whole volume
whenever it is clear which diffusion coefficient is considered.
1.4.1 Self-diffusion coefficient
1.4.1.1 Pure elements
If in a solid of element A the diffusion of A atoms is studied, one speaks about self-diffusion. Studies of
self-diffusion usually utilize tracer atoms A* of the same element. In most experiments tracers are marked by
their radioactivity. A typical situation for a radiotracer experiment is shown in Fig. 7a. The isotopic mass or
the nuclear spin is sometimes used as tag for tracer atoms as well. The tracer self-&&ion coefficient DF is in
a microscopic picture according to
DA’ =fI”
A 62
related to the jump length 1o f atomic jumps and to the mean residence time of atoms r on a certain site in a
,c rystalline solid. The correlation factor f is in the caseo f self-diffusion often only a numeric factor which depends
on the crystal structure and on the diffusion mechanism [7OLl]. Tracer self-diffusion data in pure metallic
elements are listed in chapter 2.
Land&Biimstein
Mehrer
New Series III/26
8 1.4.1 Self-diffusion coefficient [Ref. p. 30
a b C d
Fig. 7. Various situations for diffusion experiments which entail different diffu-
sion cocflicients:
(a) thin layer of A* on A: tracer self-diffusion in pure elements
(b) thin layer of B* on A: impurity diGsion in pure elements
(c) thin layer of A* or B* on AB alloy: tracer self-diffusion in homogeneous alloys
(d) diffusion couple of metals A and B: interdiffusion of two metals A and B.
1.4.1.2 Homogeneous alloys
In a homogeneous binary AB alloy two tracer self-diffusion coefficients for both A* and B* tracer atoms can
be measured. They are denoted as Dii and DAB;),r espectively. A typical experimental situation is illustrated in
Fig. 7c. Since in a radiotracer experiment the concentration of A* or B* is usually negligible the alloy compo-
sition is not modified by the diffusing species.I n general the tracer self-diffusion coeflicients depend on the alloy
composition.
Results on self-diffusion in &/tlte binary alloys containing small atomic fractions X, are frequently represent-
ed in terms of
0;; = D,^;(X,) = D,^‘[l + b,X, + b2X; . ..I. (1.29a)
Then D:rf is denoted as the sol~~t seljrdtj%oa coeffjcient and DIi as the solute diJiusion coeflcienf. Experimen-
tal measurements of Dii(X,) are usually well represented by (1.29a) and b,, b, etc. are denoted as solvent
enhnrtcet~lcnfric tors. D,A’(O) is the tracer self-diffusion coefficient in the pure solvent. b, is largely determined by
perturbations due to isolated solute atoms, b, by pairs of solute atoms, and so on.
For similar reasons,t he soltrt~ dijirsion coej’kient DrR at low concentrations, can be representedb y a power
series dependence
D:f = D,“;(X,) = Dr[l + B, X, + B,X; . ..I. (1.29b)
Dy is then also denoted as impurity diffusion coefficient of species B in solvent A. B,, B, etc. are denoted as
solute enhnncement factors.
Depending on the specific alloy system, the one component is more or less soluble in the other component,
i.e. the primary and terminal phases extend over wider or smaller composition ranges. A primary phase of an
alloy AB is the solution of element B in A and thus has the same crystal structure as element A, whereas the
terminal phase crystallizes in the crystal structure of element B. For higher concentrations these alloys exhibit
usually short-range or even long-range atomic order, which may cause substantial deviations from the be-
haviour represented by the equations (1.29a, b). Attempts to describe the diffusion coefficients in these concen-
trared alloys as a function of composition theoretically or even empirically are less successful than for dilute
alloys [84Bl]. For a limited number of alloy systems the primary/terminal phase extends over the whole
composition range, sometimes with a tendency of atomic long-range order at higher concentrations and lower
temperatures. This ordering has a profound influence on the diffusion coefficients of both components. An
example is the Fe-Co system.
In contrast, many alloy systemse xhibit intermediate phases. In the phase diagram these phasesa re separated
from the primary or terminal phases or from each other by two-phase regions. They usually crystallize in
ordered structures. These may be completely different from the crystal structures of the pure components.
Therefore the self-diffusion coeilicients in these materials can not be related to those in the pure constituents at
all. A scarce number of these intermediate phasess how an order-disorder transition at higher temperatures with
a considerable influence on the diffusion characteristics. Ordered intermediate phasesa re also called intermetal-
lit compods. The number of measurementso f self-diffusion coefficients in intermetallic compounds is relatively
small. but it is clear already that the detailed atomic defect structure of these materials is essential to their
self-diffusion behaviour. Tracer self-diffusion data in binary alloys and in intermediate phases are listed in
chapter 4.
Land&BBmstein
New Series III,/26