Multiphase diffusion in binary and ternary solid-state
systems
van Loo, F.J.J.
Published in:
Progress in Solid State Chemistry
DOI:
10.1016/0079-6786(90)90007-3
Published: 01/01/1990
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Citation for published version (APA):
Loo, van, F. J. J. (1990). Multiphase diffusion in binary and ternary solid-state systems. Progress in Solid State
Chemistry, 20(1), 47-99. DOI: 10.1016/0079-6786(90)90007-3
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MULTIPHASE DIFFUSION IN BINARY
A N D T E R N A R Y SOLID-STATE SYSTEMS
F. J. J. van Loo
Laboratory for Solid-State Chemistry and Materials,
Eindhoven University of Technology, Eindhoven, The Netherlands
1. INTRODUCTION
I f two s o l i d m a t e r i a l s
will
are in contact at a sufficiently
o c c u r . Depending on t h e n a t u r e
time,
a new c o n c e n t r a t i o n
distribution
completely miscible at that
w i t h o u t any d i s c o n t i n u i t y .
form new p h a s e s ,
related
of the starting
high temperature, interdiffusion
materials,
of the elements i s
set
t e m p e r a t u r e and a n n e a l i n g
up.
temperature, the concentration profiles
If,
however, they a r e only p a r t i a l l y
discontinuities
in
the
concentration
profiles
If
the m a t e r i a l s
are
a r e more o r l e s s smooth
m i s c i b l e o r i f t h e y r e a c t to
o c c u r which a r e
closely
to the phase diagram of the system.
/
/
C
T
To
II
II
(a)
I'!
I
I
molefraction8
'
'
IB
m
ct
y
11
I I
~__JT--
/
c
A
X
I(b)
Fig. i.i.
( a ) P h a s e d i a g r a m o f t h e b i n a r y s y s t e m A-B.
(b) A d i f f u s i o n c o u p l e A/B a n n e a l e d a t a t e m p e r a t u r e To , t o g e t h e r w i t h t h e
c o n c e n t r a t i o n - p e n e t r a t i o n curve of
B i n mole f r a c t i o n .
Hypothetical ternary isotherm A-B-C with two possible diffusion paths, 1 and 2.
component
Fig. 1.2.
47
4~
F.J. J, van Loo
Figure I.I
shows an example f o r a b i n a r y
formed i n a d d i t i o n
treatment
to the solid
solutions
f o r s u c h a n A-B d i f f u s i o n
s y s t e m A-B, i n w h i c h t h e compounds ¥ and 6 a r e
m and ~. A f t e r
the interdiffusion
coefficients
v a l u e s of t h e c o n c e n t r a t i o n s
diagram if equilibrium
The r e a s o n f o r
the creation
binary multiphase
diffusion
the phase rule
follows
it
prevail
of
that
follows
the
concentration.
therefore,
pressure
forbidden
already
since
regions
c a n b e t a k e n from t h e p h a s e
(as is discussed
thermodynamic considerations.
(i.e.
fixed
concentration
gaps
in
From
c a n b e formed i n s u c h a c o u p l e ,
to fix
t e m p e r a t u r e and p r e s s u r e and t o
precipitates
o r wavy i n t e r f a c e s )
t h e n o n l y two d e g r e e s o f freedom a r e a l l o w e d .
having been chosen,
later).
with
regions
is
However, t h e
from b a s i c
only single-phase
Two-phase
of c o m p o s i t i o n .
interfaces
b e c a u s e t h r e e d e g r e e s o f f r e e d o m would b e n e c e s s a r y
vary
profile
p h a s e s c a n n o t be p r e d i c t e d
and ~/B i n t e r f a c e s
at these interfaces
straight
couples
in the various
a r e known a s a f u n c t i o n
at the m/y, y/6
conditions
long annealing
c o u p l e a t a t e m p e r a t u r e To, a c o n c e n t r a t i o n
s e t up a s shown f o r component B. The e x a c t p r o f i l e
unless
a sufficiently
t h e r e would b e no p o s s i b i l i t y
left
are,
T e m p e r a t u r e and
across a two-phase
zone for a variation in concentration, i.e. for diffusion.
For Interdlffusion in a ternary system at a fixed temperature and pressure, the number of
degrees of freedom is three minus the number of phases. This implies that in slngle-phase
regions one might vary the concentration
(fraction) of two components, whereas in two-phase
regions the concentration of one component can be varied but then the other concentrations
are fixed. Therefore, in a ternary diffusion couple two-phase regions are allowed.
In a s y s t e m where a component i s i n v o l v e d w h i c h i s p r e s e n t
surrounding gaseous phase it
or the fraction
of this
both in the solid
is not easy to fix unambiguously parameters
component i n t h e s o l i d
phase,
since
Although nature
o n l y c h o o s e s one r e p r o d u c i b l e
couple,
a prediction
b e c a u s e of t h e i n f i n i t e
cross-section
variety
of these
of hypothetical
pressure
in the
on.
penetration
profiles
total
t h e s e may b e d i f f e r e n t
b u l k of t h e c o u p l e and a t t h e e d g e s a s w i l l b e e x p l a i n e d l a t e r
such a ternary
like
and i n t h e
profile
for
the elements
in
from t h e p h a s e d i a g r a m i s d i f f i c u l t
possibilities.
of a s y s t e m A-B-C i 8 shown, e x h i b i t i n g
In Figure 1.2 an isothermal
two s i n g l e - p h a s e
fields
y and S and a
two-phase region in which tie lines are drawn connecting coexisting equilibrium phases.
If the element C and the alloy X Interdlffuse at the temperature for this cross-section,
the resulting concentration profile mapped on this section might resemble the hypothetical
diffusion
paths
concentrations
1 or
at
completely different
In F i g s .
diagrams.
2 or
the y-6
any o t h e r ,
interface
we h a v e r e l a t e d
This is only allowed if
from e q u i l i b r i u m
contributions,
conditions
but also
at
cases,
local
longer diffusion
lines
however, these deviations
Local equilibrium
is
two s u f f i c i e n t l y
parameter.
equilibrium
mass b a l a n c e
then less
[1-3].
different
annealing
by plotting
process,
the relevant
phase
at the interfaces,
deviations
The r e l a t i o n
although
but
experiments,
from e q u i l i b r i m n
between the diffusion
i n some c a s e s m e t a s t a b l e
as is discussed
is
the diffusion
times t as a function
In c a s e of a d i f f u s i o n - c o n t r o l l e d
The
zone would b e
occur because of surface energy
times,
c a n always b e assumed i f volume d i f f u s i o n
t h e c a s e can be v e r i f i e d
exist
in understanding
are only apparent,
preserved.
times or in thin-film
certainly
obvious,
might be helpful
is
reaction
phenomena w i t h
conditions
aonealing
are reported
the
later.
annealing
diagram concentrations
for
the phase interfaces
of equilibrium
Whether t h i s
as
the diffusion
For v e r y s h o r t
phenomena and t h e p h a s e d i a g r a m i s
extensions
long
f o r p a t h s 1 and 2 a s i s d i s c u s s e d
1 . 1 and 1 . 2 ,
cannot be considered generally.
departure
as
and t h e m o r p h o l o g y of
these deviations.
In some
later.
the rate-limiting
profiles
for
step.
(at least)
of x / t ~, where x i s t h e d i s t a n c e
the plots
then have to coincide.
Multiphase Diffusion
In t h i s
s u r v e y t h e a u t h o r would l i k e
between multiphase
diffusion
to d i s c u s s
49
in a mainly qualitative
phenomena on t h e one h a n d , and k i n e t i c
way t h e r e l a t i o n
and thermodynamic ( p h a s e
d i a g r a m ) d a t a on t h e o t h e r h a n d . T h i s p a p e r t h e n f o c u s e s on:
-
-
providing experimental
or theoretical
showing how r e l a t i v e l y
evidence to settle
simple experiments
some o f t e n d i s p u t e d i s s u e s ;
can provide
information
about diffusion
and
thermodynamics;
-
predicting
and e x p e r i m e n t a l l y
verifying
multiphase
diffusion
phenomena from k i n e t i c
and
thermodynamic d a t a .
T h i s p a p e r i s b a s e d m a i n l y on t h e a u t h o r ' s
n o t meant a s a c o m p r e h e u s i y e l i t e r a t u r e
partly
-
included in this
annealing
times,
important role.
Finite
phases
a situation
o r when n u c l e a t i o n
problems exist
is referred
to specialised
or interface
and i s
not or only
energies
the appropriate
literature
on t h i s
This is expecially
formed i n
the beginning
important
in thin-film
p l a y an
places
in the
subject.
m i g h t b e consumed d u r i n g
and c o a t i n g
further
s p o k e n t h e s e phenomena c a n b e u n d e r s t o o d b y a p p l y i n g
diffusion
couples using continuously
Diffusion
in ionic
as
in
ordered
covalent
alloys.
f o r m a t i o n and m i g r a t i o n
restriction
is
multlphase diffusion.
diffusion
cersmics
In these
important
The q u a l i t a t i v e
condition
like
cases
of p o i n t d e f e c t s
mainly
are
t e c h n i q u e s where
annealing.
However,
the theory for infinite
c h a n g i n g and members f o r s u c h a c o u p l e .
systems in which an extra
diffusion
completely
this
field
o f t e n e n c o u n t e r e d a t low t e m p e r a t u r e s and
generally
well
research
couples in which the sequence of the phase formation as well as the kinetics
time-dependant.
-
couples,
However, t h e s e p r o b l e m s a r e m e n t i o n e d a t
t e x t where t h e r e a d e r
-
in this
o v e r v i e w . For i n s t a n c e ,
paper are:
Non-equilibrium diffusion
short
own e x p e r i e n c e s
survey or total
certain
especially
is different
for
of electric
the
carbides
and o f t e n
from m e t a l l i c
materials.
in this
arises,
and n i t r t d e s
the difficult
quantitative
ideas presented
neutrality
description
of
or
as
in
coupled
However,
binary
p a p e r on t e r n a r y m u l t i p h a s e
c o m p l e t e l y i n c l u d e t h e s e t y p e s of m a t e r i a l .
2. MULTIPHASE DIFFUSION IN BINARY SYST~4S
A l t h o u g h numerous s t u d i e s
on m u l t l p h a s e b i n a r y d i f f u s i o n
have bean published,
still
some
little known or even controversial issues exist. Generally, these are mostly mentioned only
as side remarks in papers in which another main subject is involved.
After a general introduction, a few typical examples of these issues are discussed in
this section in the following sequence:
-
the time invariance of the concentration versus x/t ~ plot. The reasons for deviations are
discussed, as well as the use of deviating plots to get reliable diffusion data;
-
the absence of bulk-equilibrium phases or the presence of non-equilibrlum phases in a
diffusion couple;
-
deviations
of interface
concentrations measured
compared t o b u l k - e q u i l i b r i u m
the positions
values
of the Kirkandall
in the creation
couple;
or annihilation
of v a c a n c i e s
resulting
effect;
t h e m o r p h o l o g y of t h e p h a s e i n t e r f a c e ( s )
c o m p a r i s o n of t h e r e a c t i o n
couples;
diffusion couples as
i n homogenized a l l o y s ;
and Matano p l a n e s i n a d i f f u s i o n
the role of the phase interface(s)
from t h e K i r k e n d a l l
in multlphase
layer
if grain boundary diffusion
thickness
in planar,
cylindrical
is taking place;
and s p h e r i c a l
diffusion
50
-
F . J . J . van Loo
differences
couple,
in morphology of reaction
including
-
crystallographic
-
the evaluation
occur (e.g.
-
the
differences
texture
in the reaction
of d i f f u s i o n
solid-solid
of
and a t t h e e d g e s o f a d i f f u s i o n
diameters of the half couples|
layer;
coefficients
during interstitial
evaluation
zones in the centre
d e p e n d i n g upon t h e r e l a t i v e
i n s y s t e m s where l a r g e
For a b i n a r y
the evaluation
diffusion);
diffusion
coefficients
for
vapour-solid
couples
compared w i t h
i n t o m u l t i n h a e e b i n a r y diffusion
interdiffuaion
of diffusion
process
i n w h i c h volume d i f f u s i o n
d a t a from c o n c e n t r a t i o n
plots
is
coefficient
by u s i n g t h e N a t a n o - B o l t z m a n n a n a l y s i s
a s g i v e n i n many e x c e l l e n t
e . g . Adda and P h i l i b e r t
start
the rate-lt~iting
step
i s t h e same f o r s i n 6 1 e - p h a s e end
m u l t i p h a ~ e s y ~ t a m s . The i n t e r d i f f u s i o n
We w i l l
volume c h a n g e s
couples.
General introduction
general
total
D c a n b e f o u n d from a c o n c e n t r a t i o n
reference
plot
b o o k s , s u c h as
[4]~
this
introduction
by giving
f o r m . A t t h e end we p r e s e n t
these can be used (e.g.
the necessary
these equations
and u s e f u l
in a simplified
if assumptions of a constant
total
equations
in their
form f o r c a s e s where
volume o r o f e q u a l p a r t i a l
molar
volumes a r e a l l o w e d ) .
The f i r s t
difficulty
in these plots.
interdiffusion
indeed quite
is
In fact,
the definition
process determines
a variety
System e v e r y e v a l u a t i o n
the definition
of definitions
is equally
t r a n s f o r m e d one i n t o a n o t h e r
component i b y F i c k ' s
of the dimensions for concentration
the frame of reference
first
exist.
of the evaluated
diffusion
As l o n g a s one i s w o r k i n g
good and t h e v a r i o u s
[5].
diffusion
in a consistent
It is convenient to define the interdiffusion
can b e
flux ~ for
(I)
coefficient
moles/m 3 and x i s t h e d i s t a n c e
In the case of a constant
can be
and
law i n t h e form
where D i s t h e i n t e r d i f f u s i o n
Ci*
of t h e
coefficients
coefficients
~i " - D dcidx [m°les/m2s]
concentration
and d i s t a n c e
which is chosen for the description
coordinate
total
found
i n m 2 / s ; Ci i s t h e c o n c e n t r a t i o n
i n m.
volume t h e i n t e r d i f f u s i o n
from
of component i i n
the
penetration
plot
coefficient
of
Ci
D(C i ) a t a c e r t a i n
versus
x
by
the
M a t a n o - B o l tzmann e q u a t i o n :
D(C i ) = -
(
)
~
Ci
x d Ci
(2)
The meaning o f t h e symbols f o l l o w s from F i g .
and
B between
respectively.
two s t a r t i n g
alloys
having
2.1,
initial
for the interdiffusion
B-concentrations
of
of t h e e l e m e n t s A
CB- and CB+,
5[
Multiphase Diffusion
CB
CB
CB
i
,I
I
I
X~
Fig. 2.1.
Xo,O
c u r v e f o r component B i n a d i f f u s i o n
+
c o u p l e b e t w e e n two a l l o y s w i t h s t a r t i n g c e m p o s i t i o n s CB and CB, r e s p e c t i v e l y .
Concentration-penetration
The M a t a n o - r e f e r e n c e p l a n e x o = 0 i s d e f i n e d i n such a way that t h e h o r i z o n t a l l y
hatched
a r e a s a r e e q u a l . The f l u x d e f i n e d i n t h i s way i s measured w i t h r e s p e c t t o t h i s Matano p l a n e
or, alternatively,
t o one o f t h e e n d s o f t h e c o u p l e ( s i n c e
The v a l u e o f t h e i n t e g r a l
the total
volume i s c o n s t a n t ) .
i n Eq. (2) i s g i v e n by t h e d o u b l e - h a t c h e d a r e a i n F i g . 2 . 1 ;
the
r e c i p r o c a l o f t h e c o n c e n t r a t i o n g r a d i e n t has t o be t a k e n a t t h e p o s i t i o n x* where Ci = Ci * .
If the total volume changes during the reaction (which means that the partial molar
volumes Vi a r e d e p e n d e n t on c o n c e n t r a t i o n ) ,
Eq. (2) has t o b e r e p l a c e d by [ 6 , 7 ] :
x
where t h e c o n c e n t r a t i o n u n i t Y i s d e f i n e d as (N i - N i - ) / ( N i + - N i - ) ; Ni i s t h e mole f r a c t i o n
o f component i ,
b e i n g Ni -
and Ni + a t
the undisturbed left-hand
c o u p l e and Vm i s t h e m o l a r volume. In t h i s
the left-hand
frame of r e f e r e n c e
for a system with varying total
paper
say,
point.
It
is
a constant
indeed not necessary
total
volume
volume, s i n c e i t
is split
up i n t o two
or the r i g h t - h a n d s i d e of the couple i s taken as a
t o know a p l a n e x = 0 s i n c e i n Eq. (3) t h e
distance parameter appears only in a differential
this
to,
end o f t h e c o u p l e . A Matano p l a n e x o t 0 c a n n o t be u n i q u e l y d e f i n e d i n t h i s
p l a n e s d e p e n d i n g on w h e t h e r t h e l e f t - h a n d
reference
and r i g h t - h a n d e n d s o f t h e
case the f l u x i s measured w i t h r e s p e c t
(constant
but
form. Unless s t a t e d
not necessarily
volumes) i s supposed in o r d e r to avoid n e e d l e s s c o m p l i c a t i o n s .
otherwise,
equal
partial
throughout
molar
52
F..I.J. van Loo
The e q u a t i o n s
(1-3) are related
s y s t e m by t h e f a c t
description
of
diffusivities
that
to interdiffusion
t h e same i n t e r d i f f u s i o n
the diffusion
behaviour
phenmeena, c h a r a c t e r i z e d
coefficient
of both
species.
some l a t t i c e
D can be used for the
If
o f b o t h components h a v e t o b e i n v e s t i g a t e d ,
can b e f o u n d i n s i n g l e - p h a s e
the real
-
the diffusion
p l a n e i n a p h a s e h a v e t o b e known. The o r i g i n
frame of reference
in a binary
diffusion
of this
or intrinsic
fluxes
-
relative
so-called
to
Kirkendall
c o u p l e s more o r l e s s
easily
by
using inert markers situated at the meeting interfaces at t m O. In a later paragraph the
situation
for multiphase
The u n e q u a l d i f f u s i o n
c o u p l e s i s shown t o b e more c o m p l i c a t e d .
fluxes
o f t h e components w i t h r e s p e c t
phase is compensated by a flux of vacancies
component. The r e l a t i o n
to the lattice
in the sane direction
between the interdiffusion
planes in a
as the flux of the slower
f l u x ~ i i n t h e N a t a n o and t h e i n t r i n s i c
diffusion flux Ji in the E/rkendall frames of reference is given by [8]:
~ i = J i + Ci vKIM
where v KIN e q u a l s
position
(4)
the velocity
of Ktrkendall
and t i m e w h e r e t h e c o n c e n t r a t i o n
This velocity
frame relative
v K/M c a n b e d e f i n e d f o r e a c h l a t t i c e
be measured easily
at
the lattice
t o t h e Matano f r a m e a t
that
o f component i e q u a l s Ci .
plane in the diffusion
plane corresponding
to the orisiual
z o n e , b u t can o n l y
interface
a t t - 0 by
the relation
v K/M = A x / Z t
(5)
where 5x i s t h e m a r k e r d i s p l a c e m e n t ,
i.e.
the distance
between the Kirkendall
p l a n e x K and
t h e Matano p l a n e x o = 0.
The v e l o c i t y
v K/M i s r e l a t e d
to the flux of vacancies,
w h i c h c a n b e s e e n from Eq. ( 4 ) and
the relations
~A~A + vs ~B = 0
(6)
VA JA + VB JB + Vm J v = 0
(6)
where t h e s u b s c r i p t s
A, B and V r e f e r
t o t h e components A and B and t o t h e v a c a n c i e s V. T h i s
leads to
(7)
v~/M = Vm J v
Substitution
i n t o gq. ( 4 ) , u s i n g t h e d e f i n i t i o n
v
o f t h e m o l a r volume
m
m = CA + C B
CA
CB
leads t o :
~i = Ji + Ni Jv
(8)
MuldphaseDiffusion
The i n t r i n s i c
coefficient
diffusion
coefficients
Di a r e
53
related
t o v K/M and t h e
v ~ I " = 0 a (Da - n s ) S Cal~x
(9)
= CA VA DB ÷ CB VB DA
The v a l u e s
position)
interdiffusiou
~ by the relations
for
(lO)
Di c a n b e f o u n d a t
by c o m b i n i n g Eqs.
(2,5,9)
o~,- ~ ~,,~-o:_
If the total
The r a t i o
the concentration
and ( 1 0 ) w i t h e q u a t i o n
c~ E°:~ ~-
diffusion
gq.
origin
(marker
[6]:
-
volume does n o t r e m a i n c o n s t a n t ,
between the intrinsic
in the Kirkendall
.i,
(11) h a s t o b e r e p l a c e d b y [ 6 ] :
coefficients
f o l l o w s from
Eq. ( 1 2 ) and i s e q u a l t o
[
_
(laa)
("A-"P~ "; ~c.i--.^~
÷
t_,,;
In t h e
ratio
expression
of
the
intrinsic
between brackets
fluxes,
remains.
which equals
This equation
JA/JB = -DAVB/DBVA, o n l y
considerably
simplifies
the
i f NA+ = i and
NA- = 0.
The i n t r i n s i c
diffusion
i n homogeneous a l l o y s
coefficients
are related
to the tracer
diffusion
where d l n a A / d l n N A = d l n N B / d l n a B i s c a l l e d
component i . The t e r m r A i s a c o r r e c t i o n
i s d e p e n d e n t on t h e c r y s t a l
structure
t h e thermodynamic f a c t o r
factor
originating
and t h e d i f f e r e n c e
and a i i s
D~ . 0 A . r A
DB
DT
B
T
i n Di [ 1 1 ] . From Eq. [13] i t
(I~)
and
•
~B
.
rB
the activity
free the vacancy flux effect
follows that
DA
coefficients
DiT
o f t h e same c o m p o s i t i o n b y [ 9 , 1 0 ] :
of
and
54
F . J . J . van Loo
(NA
=
T rB + NB D~ r A)
DB
"
(15)
d I n a&
d i n NA
Since the number-fixed frame of reference,
which is defined by a constant
moles a t e a c h s i d e of t h e o r i g i n x - 0 , i s o f t e n u s e d i n l i t e r a t u r e
give the following relations
(superscript
N refers
[12] i t
total
amount of
i s c o n v e n i e n t to
to the number-fixed reference
frame):
[161
N
DA
T
d In aA
=
.
d
In
"
For l a y e r s w i t h a v e r y s m a l l h o m o g e n e i t y r e g i o n AN - 0 i t
a diffusion
coefficient
approaches
Eqs. ( 3 )
from
a
penetration
z e r o and t h e a p p a r e n t
and
coefficient
(11).
A very
diffusion
useful
curve,
since
coefficient,
concept
for
this
is very difficult
the
concentration
therefore,
case
is
the
integrated
=.~
D
This quantity
coefficient
is a material
to the parabolic
case
of
a
growth constant
constant
c0ncentration-independent,
couple at least
o f t h e p h a s e , AN = N2 - N1 and Day i s
in the phase.
constant
The e q u a t i o n s g i v e n i n t h i s
In
diffusion
(17)
d N = Day " A N
the average interdiffusion
a)
(see
d e f i n e d b y Wagner [ 1 3 ] :
where N1 and N2 a r e t h e - unknown - h o m o g e n e i t y l i m i t s
related
gradient
becomes i n f i n i t e
N2
D int
to calculate
like
the normal diffusion
coefficient
and i s s i m p l y
as defined in next section.
chapter can often be simplified.
total
Eqs.
volume,
which
means
that
VA and
( 2 ) and ( 1 1 ) c a n b e u s e d f o r f i n d i n g
one o f t h e s t a r t i n g
materials
VB a r e
~ o r Di .
I f i n such a
i s a p u r e component, e . g .
C i = 0, t h e n Eq. ( 1 1 ) r e d u c e s t o :
Di(xK)" + ~ (~Cl)xK &xK ¢idx
(Note t h e s i m p l e i n t e g r a l
differential
which contains,
w h i c h means t h a t
the position
(lla)
contrary
to
Eq.
(2),
the distance
of t h e Matauo p l a n e i s n o t i n v o l v e d
p a r a m e t e r as a
[6].
Multiphase Diffusion
b)
In c a s e o f c o n s t a n t
and e q u a l p a r t i a l
e q u a t i o n s can b e s l m p l l f l e d .
fraction
In Eqs.
55
m o l a r v o l u m e s , w h i c h means VA = VB = Vm' s e v e r a l
(2),
(Ii)
end ( 1 1 a ) ,
Ci can t h e n be r e p l a c e d b y t h e mo!
Ni . F u r t h e r :
(ga)
vK/M = (DA - D B) ~NAI~¢
=
NADB
+
NBDA
(lOa)
= ~N and Di = ~
If,
(16a)
on t o p o f t h a t ,
be simplified
DB "
~e
NA = o and N:= 1 ( i . e .
diffusion
b e t w e e n two p u r e m e t a l s ) ,
Eq. ( 1 2 a ) can
to
~AdX /
(12b)
(l-NA)dX
time daoendence of the nenetratlon nlot
The a n a l y s i s
rate-limltlng
penetration
g i v e n i n t h e f o r e g o i n g p a r a g r a p h assumes t h a t
throughout
t h e whole a n n e a l i n g
curve versus x/t ~ will
layer
procedure.
be invarlant.
t h e same d i f f u s i o n
In
that
The t r e a t m e n t
zone t h e
restricts
the problem to the study of the layer growth as a function
diffusion
process
rate-llmltlng
t h i c k n e s s d and t h e a n n e a l l n g
d 2 = 2 k p t . A p a r t from p o s s l b l e
several
r e a s o n s may e x i s t
throughout
t h e whole a n n e a l i n g
time t are related
disturbing
which Invalidate
effects
this
the starting
A few o f
the
by taking as a
of t i m e .
through the parabolic
assumption.
of
materials,
time,
during the heating
process is
a plot
is simplified
diffusion
is
of one compound growing b e t w e e n
case,
which
I f t h e same
then
the
layer
growth constant
kp a s
or cooling procedure,
these are discussed
here.
a ) The m a t i n g s u r f a c e s
e.g.
an o x i d e
film
starting'materials.
may i n i t i a l l y
or a segregation
b e c o v e r e d by a t h i n
layer
originating
layer of different
from i m p u r i t i e s
composition,
present
in
the
If an oxide film at the interface acts as a barrier for diffusion, then
some incubation time will pass before an undisturbed diffusion process can begin. The plots
of the width of the diffusion layer d and d 2, respectlvely, versus time then resemble those
shown in Fig. 2.2.
56
F . J . J . van Loo
d
200
d2
T
tOG
5
Fig.
and A1 b r o k e
5
10
15
(b)
phenomenon was f o u n d i n t h e Ti-A1 s y s t e m [ 1 ~ ] . An o x i d e l a y e r b e t w e e n
down a f t e r
reaction-limited
fast
0
15
P l o t s o f l a y e r t h i c k n e s s d ( a ) and d 2 ( b ) v e r s u s t i m e i n a r b i t r a r y u n i t s i n a
multiphase diffusion couple, if an incubation time is involved before the regular
diffusion process sets in.
2.2.
An example of t h i s
Ti
I0
Pt
(a)
a
period
of
many h o u r s ,
g r o w t h o f TiA13 was f o u n d . A f t e r
and p a r a b o l i c a l l y
marked t h e t r a n s i t i o n
this
with time in a diffusion-limited
point
in the process,
indicating
during
which a
incubation
process.
slow
linear
t i m e t h e l a y e r grew v e r y
A row of p o r e s i n t h e l a y e r
the residue
of the disturbing
oxide
film.
A segregation
layer at
or preanuealing
present
the starting
layers
ternary
o n e . The d i f f u s i o n
b ) The r e a c t i o n
diffusion
present
Especially
process
as a r e s u l t
the couple
atoms s u c h as c a r b o n o r p h o s p h o r u s ,
in the bulk material,
can then be totally
amount o f t h e s e g r e g a t i n g
of heating
may g i v e r i s e
change t h e b i n a r y
different,
to enriched
system into a
but also
time-dependent
element [15,16].
b e t w e e n e l e m e n t s t o form a new compound may b e t h e r a t e - l l m l t l n g
then starts
process
Dybkov [ 1 ] .
per million
is often
of o n l y a few n a n o m e t e r s w h i c h i n f a c t
b e c a u s e of t h e l i m i t e d
Fig.
materials.
i n o n l y a few p a r t s
surface
layer
the interface
growing linearly
eventually
The p l o t s
with
time, but after
becomes r a t e - l i m i t i n g .
o f d and d 2, r e s p e c t i v e l y ,
reaching a certain
This case i8 extensively
versus
time will
step.
thickness
The
the
discussed by
then be like
shown i n
2.3.
In f a c t ,
in bulk diffusion
c o u p l e s no c l e a r l y
known t o t h e a u t h o r
for non-porous reaction
that
i n most s t u d i e s
the d2/t
and too l a r g e
plots
a spread
t i m e . Small i n c u b a t i o n
p r o v e n c a s e of r e a c t i o n - l i m i t e d
layers.
on n m l t i p h a s e
to be sure of the exact
However, i t
diffusion
relation
times or periods of non-parabollc
grovth is
must i m m e d i a t e l y b e added
show i n - e n o u g h m e a s u r e d p o i n t s
between the layer
thickness
growth are then easily
and
overlooked.
Multiphas¢ Diffusion
57
15
d2
d
T
t
10
20(
100
/
J
I
0
5
Pt
(a)
In porous
the
layers,
t
10
-----~ t
(b)
the reactlon-li~ited
substrate
dlffuslon-llslted
reacts
with
a s shown i n F i g .
'~altese
type figure
cross"
can react
layer already
the reaction
formed.
taken
15
gas
or
llquld,
In experiments
reactlon-l~ited
In the latter
from
[17].
independent of
case,
[17],
the hot-dip
In
and
by t h e m o r p h o l o g y c h a n g e o f a n o r l g l n a l l y
In the
first
case,
forms from t h e s h a r p c o r n e r s o f t h e s u b s t r a t e ,
Exsmples o f t h e c r o s s - t y p e
of i r o n
g r o w t h i s w e l l known.
the thickness
volume d i f f u s i o n
towards the gas phase is rate-limiting,
Ta and o f Co-~/C a l l o y s
galvanizing
surrounding
2.4,
with the substrate
layer
rounded e d g e s .
a
llnear
growth can be dlstlngulshed
cubic substrate
liquid
I
5
15
P l o t s o f l a y e r t h i c k n e s s d ( a ) and d 2 ( b ) v e r s u s t i m e i n a r b t t r a t y
units in a
m u l t i p h a s e d i f f u s i o n c o u p l e , i f t h e r e a c t i o n o f compound f o r m a t i o n i s t h e
rate-limiting
s t e p i n t h e b e g i n n i n g of t h e p r o c e s s .
Flg. 2.3.
which
10
attack
of
are the oxidation
alomintzing
of
titanium
typlcal
s i n c e t h e gas o r
the porous reaction
of the substrate
leading
the
atoms t h r o u g h
t o c o m p l e t e c o v e r a g e and
of m e t a l s s u c h a s Nb o r
[18] and t h e h o t - d i p
[19].
~.,~.'."
(a)
i
r:~ ¸
1
(b)
Flg. 2.4.
JP~SC 20:I-E
M o r p h o l o g i e s o c c u r r i n g d u r i n g t h e o x i d a t i o n o f an i n l t l a l l y
c a s e o f l l n e a r ( a ) and p a r a b o l i c g r o w t h ( b ) [ 1 7 ] .
cubic substrate
in
58
F.J.J. van Loo
c) In many d i f f u s i o n
c o u p l e e x p e r i m e n t s t h e l a y e r growth i s f a s t e r
p r o c e s s b e c a u s e of t h e s m a l l g r a i n s p r e s e n t a t t h e s t a r t
diffusion
process.
During t h e a n n e a l i n g p r o c e d u r e ,
s l o w e r volume d i f f u s i o n
in t h e b e g i n n i n g of the
which l e a d to a f a s t
g r a i n growth o c c u r s ,
p r o c e s s w i l l be t h e r a t e - l i m i t i n g
step.
grain-boundary
and e v e n t u a l l y
the
P l o t s of d and d 2 v e r s u s
time t h e n a p p e a r as shown in F i g . 2 . 5 .
t,00
d s" ~ j, j l
~
/ I
200
//
//
t/f
//
/
!
!
l
S
0
t. t
J
,
5
10
/
/
It
growth s t a g e .
be p r e f e r r e d
n
5
0
5
t o i n q u i r e whether a d 2 / t o r a d / t ~ p l o t can b e t t e r
Pieraggi
if
the s t e a d y - s t a t e
formed d u r i n g an i n i t i a l
to the steady-state
often arises
parabolic
[20] c o n c l u d e d i n an i n t e r e s t i n g
the layer,
does n o t c o n t r i b u t e
rate
P t
110
kp in the second
t h a t a d / t ~ p l o t has to
period of faster
i n the l a t e r
parabolic
kinetics,
stage.
This
In t h a t c a s e , t h e r e l a t i o n
(18)
i n which d and t a r e t h e t o t a l
thickness after the transient
be used in o r d e r to
constant
discussion
transient
control
in oxidation reactions.
rate
dp 2 - (d - d l ) 2 - 2 kp ( t - t l )
applies,
~
P l o t s of l a y e r t h i c k n e s s d ( a ) and d 2 (b) v e r s u s time i n a r b i t r a r y u n i t s in a
multlphaee diffusion couple, if in the beginning of the diffusion process fast
grain-boundary diffusion occurs.
the proper values for
situation
0
0~)
is interesting
describe
0
t~//I
a)
Fig. 2.5.
1
/
/
a
/
t h i c k n e s s and t i m e ,
respectively,
and d i i s the
period ending at time ti after which the steady-state
parabolic growth starts. The value dp is the thickness of the part of the layer, grown in
the p a r a b o l i c s t a g e a f t e r
time t i . This r e l a t i o n
leads to:
(18a)
d = d i + {2 kp ( t - t l ) } ~
A p l o t of d and dp v e r s u s t ~ f o r a s p e c i f i c
parallel
p l o t s f o r d and dp a f t e r
c a s e i s g i v e n in F i g . 2 . 6 a .
longer annealing times,
f o r kp from t h e s l o p e of an e x p e r i m e n t a l d / t ~ p l o t .
too l a r g e compared to t ;
related
through [20]
I t shows n e a r l y
l e a d i n g to a n e a r l y c o r r e c t v a l u e
T h i s i s t r u e f o r c a s e s where t i i s n o t
the a p p a r e n t v a l u e kp' from t h e d / t ~ p l o t and t h e r e a l v a l u e kp a r e
Muldphase Diffusion
®
59
.f
d
/ I d
/ / kp,=O05
/
//
.o3es
s'
J
f/
|
.
.
.
.
,
i
S _..,. tl/Z
2O
T
®
d2
-(-to?2
10
~
83
.../¢"
-/
/.
f*
kp:O.05
//
-(-to)l~
©
s
lo
2O
T
®
d2
d~=o,05
.~ ~ 1' t I
kp:O05
/
/
is,,'
i
!
SO
-t o
Fig.
2.6.
.
.
.
.
.i
100
--..~ t
0
to
SO
--4, t
IO0
P l o t s of d v s . t ~ ( a ) and d 2 v s . t ( b ) i n a r b i t r a r y u n i t s i n c a s e o f t h e p r e s e n c e
of an initial
l a y e r o f t h i c k n e s s d = 2 a f t e r a t r e n s i a n t p e r i o d t = 10, which
does n o t c o n t r i b u t e t o t h e r a t e - c o n t r o l
in the following parabolic stage with
= 0 . 0 5 . A v a l u e k _ ' would have b e e n f o u n d from t h e " e x p e r t m e n t a l "
curves.
~ ( c ) and ( d ) t h e J I B e p l o t s a r e shown, now f o r t h e c a s e i n w h i c h t h e i n i t i a l
l a y e r does c o n t r i b u t e t o t h e f o l l o w i n g p a r a b o l i c g r o w t h s t a g e .
kp' = kpt/(t
- t i)
(19)
which is independent of the value of d i .
If for this
case a d2/t plot
had b e a n u s e d , a wrong v a l u e f o r s t e a d y - s t a t e
bean found as can be seen by the different
a p p a r e n t v a l u e k p ' from t h e d 2 / t p l o t
slopes
of d2/t
is then related
and d 2 p / t
[20].
In F i g .
kp would have
2.6b.
The
t o t h e t e a l v a l u e o f kp t h r o u g h :
k p ' - kp [ I + d l / { 2 kp ( t - t i ) } ~ ]
as shown by P t e r a g g t
in Fig.
(20)
2.6
the sharp transition
p o i n t b e t w e e n t h e two growth
r e g i m e s i s smoothed o u t t o a c o n t i n u o u s c u r v e .
On t h e o t h e r
contribute
relation
to
hand,
if
the value
the steady-state
of the thickness
parabolic
di after
the transient
growth in the second stage,
then the following
applies:
d 2 - d i 2 ~ 2 kp ( t - t i )
p e r i o d does
(21)
60
F . J . J . van Loo
In t h i s
case,
a d2/t plot
erroneous results
leads to the correct
a s shown i n F i g .
value of kp, whereas a d/t ~ plot
2 . 6 c and d w i t h a n a p p a r e n t k p ' a c c o r d i n g t o
(22)
kp' = kpt / (t - t i + di2/2kp)
The same i s
transient
true
layer
i n c a s e s where an i n c u b a t i o n
thickness
like
give the proper kp-value,
Eq. ( 1 9 ) .
It
relative
for
is,
the parabolic
difficult
in Fig.
2.2.
diffusion
process
are
to investigate
This consideration
literature
is
on t h i n - f i l m
been reported
[21].
leads
to an apparent
role
evaluated.
of prejudice
value
is preferred
Actually,this
spread in layer
in the evaluation
plot will
kp'
of the trensient
or a d/t ~ plot
large
a substantial
d i i s z e r o and a d 2 / t
given by
l a y e r and t h e
when k i n e t i c
consideration
thickness
thoroughly morphologic items like
order to reduce the influence
P h a s e s formed i n d i f f u s i o n
involved without
case,
d e p e n d e n t on t h e k i n e t i c
d and di whether a d2/t
to use because of the possible
then necessary
time is
In that
whereas a d/t ~ plot
therefore,
thicknesses
l e a d s Co
is
data
often
measurements.
porosity
It is
and g r a i n s i z e
of the experimental
in
results!
e o u v l e s compared w i t h t h e p h l s a d l a e r ~ m
rather
closely
reaction
This,
related
diffusion
of course,
rules
to
the former one,
a sequential
nucleation
out parabolic
since
especially
of v a r i o u s
growth of all
in
phases has
layers
starting
at
t = 0. S e v e r a l c a s e s w i l l b e d i s c u s s e d :
a ) An o b v i o u s r e a s o n
layers
at
starting
for
the interface,
materials.
gives
rise
oxide films,
case,
phase is
the presence
of b a r r i e r
or the presence of impurities
segregation
of impurities,
system,
and t h e n t h e d i f f u s i o n
in the
p r e s e n t maybe o n l y i n t h e
t h e e n d members, c a n c a u s e a n e n r i c h m e n t i n t h e d i f f u s i o n
to a ternary
path might easily
zone. This
miss a binary
p h a s e , a s i s shown l a t e r .
On t h e o t h e r
stabilized
such as e.g.
In the latter
ppm-renge i n one o f
equilibrium
t h e a b s e n c e o f an e q u i l i b r i u m
hand,
t h e same e f f e c t
by i m p u r i t i e s .
can cause
the formation
Notorious examples are carbides
of a non-equilibrium
phase,
l i k e Mosgi3C [22] o r Mo6Ni6C [ 1 5 ] ,
which m i g h t b e c o n f u s e d w i t h t h e p u r e l y b i n a r y p h a s e s MosSi 3 and MoNi. The same i s t r u e f o r
o x i d e s l i k e T i 4 N i 2 0 and T i ~ F e 2 0 , w h i c h m i g h t b e c o n f u s e d w i t h t h e b i n a r y
with an,
in fact
non-existing,
when t h e s e t e r n a r y
An e a s y way t o v e r i f y
incremental couples.
apparent
the purely binary
members may b e s i n g l e - p h a s e
with
s t o p s growing a f t e r
Fig.
"compound" Ti2Fe
in series
character
[23].
This is especially
with genuine binary
o f an e q u i l i b r i u m
layer
i n s u c h a way t h a t
for
two-phase alloys.
t i m e as a r e l a t i v e l y
some t i m e s i n c e
2.7 schematically
illustrates
only this
thick
the impurity
this
A true
layer,
procedure.
confusing
thicker
is
The c o m p o s i t i o n s o f t h e end members a r e t h e n c h o s e n q u i t e
phase in question,
parabolically
binary
p h a s e s grow i n t h i n l a y e r s
compound Ti2Ni o r
t h e u s e of
close to the
p h a s e m i g h t b e formed.
equilibrium
phase
The end
t h e n grows
w h e r e a s an i m p u r i t y - s t a b i l i z e d
is totally
layers.
phase
w i t h d r a w n from t h e end members.
MultiphaseDiffusion
y
A
'NB
B
(a)
To
oY
I!
6 Sp
]1
.~.ib
-~,t
X
(¢)
(b)
v
T
:/
6
I .........
To
61
(d)
"8"
I
I
INB¥ ~
h
13
II
II
fl
i
II
I
B
(a')
Fig.
2.7.
~x
(b')
I n ( a * d) a r e shown
f o r component B i n a
y/B c o u p l e . I n ( a ' *
compound. The g r o w t h
value.
b) Especially
if
of a v e r y low d i f f u s i v i t y
subject,
treating
in the phase,
(change in free
counteracting
all;
a
threshold
e n e r g y ~G) t o b u i l d
temperature,
the
parabolically
with time.
of
formation
This situation
the
in the diffusion
in surface
may b e sO d i f f i c u l t
may a r i s e
is established
energy.
indicated
new compound
is
if
the driving
starting
At a
and
specific
growth
"A = K1 •
starts
[26] d e r i v e d e q u a t i o n s f o r t h e
d
÷ K2 • - 2t
then
higher
from t h e p u r e components A end B, of t h e
type
f
force
z o n e , e a c h l a y e r h a s t o show up from
t h e b e g i n n i n g o f t h e e x p e r i m e n t . Wagner [13] and Gurov e t a l .
d2
that it
The p h a s e t h e n does n o t form a t
[21].
favoured
in the dlfftmlon
growth o f e a c h p h a s e i n a b i n a r y c o u p l e ,
N~
zone m i g h t
[21] g i v e s a s u r v e y on t h i s
a new p h a s e i s v e r y low and m i g h t become z e r o by t h e
~G = -~00 J/cm 3 i s
of
D'Heurle
s y s t e m s where n u c l e a t i o n
of the increase
value
c) If local equilibrium
[1,21,24,25].
of p h a s e f o r m a t i o n .
influence
at the beginning of the experiment or because
t h e f o r m a t i o n of l a y e r s
of phases
a number o f s i l i c t d e
the process
(d')
t h e p h a s e d i a g r a m o f a n A-B s y s t e m , t h e p e n e t r a t i o n p l o t s
c o u p l e A/B and y / B , and t h e p l o t o f dg 2 v e r s u s t i m e i n t h e
d') the g-phase is a non-equilibrium,
impurity-stabilized
of t h e g - l a y e r i n a c o u p l e y/B t h e n r e a c h e s soon a maximum
thin films are formed, either
be g o v e r n e d b y t h e n u c l e a t i o n
dominates
(c')
(23)
62
F . J . J . van Loo
in which D(y)
is
the
interdiffusion
coefficient
in
t h e phase
y between
t h e boundary
c o n c e n t r a t i o n s NA' and NA" and d¥ i s t h e l a y e r t h i c k n e s s of phase y. K1 and K2 a r e f u n c t i o n s
of t h e c o n c e n t r a t i o n s
and d i f f u s i o n
coefficients
p a r t of Eq. (23) i s always p o s i t i v e ,
i n the o t h e r p h a s e s .
S i n c e the l e f t - h a n d
each phase has to grow to some ( p o s s i b l y v e r y s m a l l )
t h i c k n e s s d 7. Kidson [2] p u b l i s h e d an o f t e n quoted p a p e r which r e a c h e s t h e same c o n c l u s i o n ;
p u r e l y from an e q u i l i b r i u m d i f f u s i o n - l i m i t e d
albeit
growth c o n c e p t a l l
phases have to be p r e s e n t ,
in v e r y s m a l l w i d t h s ( s e e n e x t i t e m ) . The p u r e l y a l g e b r a i c n e E a t i v e growth c o n s t a n t s
f o r a phase m e n t i o n e d by Kidson may n o t be i n t e r p r e t e d
as a r e a s o n f o r t h e n o n - e x i s t e n c e of
a l a y e r of t h a t p h a s e .
If,
however, a r e a c t i o n - l i m i t e d
growth o f some phase i s i n v o l v e d , i t s
s u p p r e s s e d a c c o r d i n g to Dybkov I l l ,
phase.
l e a d i n g to an i n c u b a t i o n p e r i o d f o r t h e growth of t h i s
The r e a d e r has to be aware t h a t
previously mentioned, but it
the a u t h o r ' s
opinion,
results
this
has n o t h i n g
or microprobe analysis.
In t h a t
o b s e r v a b l e o n l y by e l e c t r o n
respect are the diffusion
should be
r e a s o n s as proposed by Dybkov. In
especially
with regard to the next item.
c o u p l e s when i n v e s t i g a t e d
case they might actually
microscopy.
be p r e s e n t
by m i c r o s c o p i c
in very thin layers,
Examples which need a t h o r o u g h a n a l y s i s
c o u p l e s Ti-A1 [14] and Cu-Si [16].
in t h i s
In b o t h systems v a r i o u s phases
p r e s e n t a c c o r d i n g to the phase d i a g r a m , b u t o n l y one phase seems to be p r e s e n t in
the diffusion
phases
problems
has y e t t o be proven e x p e r i m e n t a l l y and w i l l be one of
to be m i s s i n g i n d i f f u s i o n
seam
to do w i t h n u c l e a t i o n
from p u r e l y k i n e t i c
this prediction
the challenges for accurate experimentalists,
d) Sometimes phases
f o r m a t i o n can be
under
couples,
m i c r o m e t e r and a r e ,
diffusivity
viz.
TiA13 and Cu3Si. However, c a l c u l a t i o n s
the experimental
conditions
therefore,
in the '~issin$"
s h o u l d be p r e s e n t
microscopically
show t h a t
the m i s s i n g
in
thicknesses
f a r below 1
not observable.
The r e a s o n
is,
that
the
phases i s s e v e r a l o r d e r s of maEnitude lower than t h o s e in TiA13
and Cu~Si.
In t h e t i t a n i u m - a l u m i n i o m c a s e ,
t h e m i s s i n g phases
an i n c r e m e n t a l c o u p l e T i / T i A 1 3 . A f t e r t h e i r
c o u p l e was p l a c e d v e r s u s A1 and a n n e a l e d a g a i n .
Ti3A1 m i c r o s c o p i c a l l y
disappeared
l a y e r s remained p r e s e n t
(TiA12, TiA1 and Ti3A1) were grown i n
growth t o r e l a t i v e l y
As e x p e c t e d ,
t h i c k l a y e r s the r e m a i n i n g
t h e l a y e r s TiA12, TiA1 and
and o n l y TiA13 c o u l d be o b s e r v e d
[14].
i n v e r y t h i n l a y e r s i s now b e i n g i n v e s t i g a t e d
Whether t h e s e
i n our l a b o r a t o r y by
means of e l e c t o n m i c r o s c o p y .
Interface
concentrations in d i f f u s i o n
S i n c e the q u a n t i t a t i v e
determination,
it
a t the i n t e r f a c e s .
If the diffusion
might a p p e a r .
evaluation
is important
of d i f f u s i o n
Some of them were a l r e a d y m e n t i o n e d b e f o r e ,
-
the n e c e s s i t y
Apart from t h e s e ,
some
practical
gradients
c o u p l e . By m i c r o p r o b e a n a l y s i s
shorter
such a s :
to use i n c r e m e n t a l c o u p l e s b e c a u s e o f t h e problem of m i s s i n g p h a s e s .
a] S t e e p c o n c e n t r a t i o n
necessary
attained
c o u p l e t e c h n i q u e i s n o t used i n a p r o p e r way, a number of e r r o r s o u r c e s
the l a r g e r o l e p l a y e d by i m p u r i t i e s
always
c o u p l e s i s o f t e n used f o r phase diagram
to be s u r e t h a t b u l k - e q u i l i b r i u m v a l u e s a r e r e a l l y
-
distance
c o u o l e s comoared w i t h b u l k - e o u i l i h r i u m v a l u e s
it
difficulties
may a r i s e ,
is difficult
to measure r e l i a b l e
than about 2 pm from t h e i n t e r f a c e .
which might
cause
errors
results.
concentration values at a
This means, t h a t an e x t r a p o l a t i o n
when s t e e p
i n c r e m e n t a l c o u p l e s have to be used o r an a n a l y s i s
be made i n o r d e r to g e t r e l i a b l e
such a s :
may o c c u r in growing phases o r in t h e end members of t h e
gradients
of e q u i l i b r a t e d
are
involved.
is
Again,
two-phase a l l o y s has to
Multiphase ~iffusion
b] A c c u r a t e m i c r o p r o b e a n a l y s e s
affect
the concentration
Corrections
for this
near
interfaces
are very difficult
measurements at distances
effect
of t h e e l e m e n t s does n o t
are possible
suffer
63"
as large
but difficult
[27].
from t h e f l u o r e s c e n s e
flunrescense
and one c o u l d m e a s u r e
of that
for instance,
p u r e Cu and Co a r e clamped t o g e t h e r w i t h o u t any d i f f u s i o n
can a p p a r e n t l y
effects
In a binary couple, usually
effect
concentration
radiation
e l e m e n t by m i c r o p r o b e a n a l y s i s
if
a s 40 ~m from t h e i n t e r f a c e .
to describe
the penetration
one
the
plot.
If,
taking place,
Co-K~
b e m e a s u r e d up t o 40 pm i n t h e p u r e Cu. About ~ . 8 mole • Co a p p e a r s
to be in Cu at the interface [27]! Measuring Co-K= radiation diminishes the error, but still
an apparent
concentration
of 2 mole % Cu is found in Co at the interface, whereas
the
Cu-radlation can still be detected at 15 ~n from the interface in Co.
c] As shown f o r d i f f u s i o n
affect
couples
the high-temperature
d] Adds and P h i l i b e r t
lead
interface
of
the layer
to concentration
given by the phase diagram.
typically
concentration
(see further
growth as well as the
values
which are
results.
The e x t e r n a l
interface
identical
According to the author's
30 kg/cm 2 ( d e p e n d i n g on t h e a n n e a l i n g
and r e p r o d u c i b l e
values
slow c o o l i n g r a t e s
may e a s i l y
in this
section).
[4] g i v e a number of examples i n w h i c h t h e u s e of a n e x t e r n a l
changes the kinetics
pressures
i n t h e Cu-A1 s y s t e m [ 2 8 ] ,
with equilibrium
experience,
temperature)
pressure
pressure
concentrations.
Large
bulk phases
an external
pressure
of
is often needed to get reliable
is thought to establish
good c o n t a c t between
t h e end members ( 1 6 , 2 9 , 3 0 ] .
From a t h e o r e t i c a l
p o i n t o f view a more i n t e r e s t i n g
question
is whether a supersaturation
of one of t h e e l e m e n t s must o c c u r b e c a u s e of t h e n e e d f o r a d r i v i n g
transformation
maintain
values
[3].
a vacancy flux
are predicted,
the experimental
this
Also a slight
subject.
et
al.[3]
for
quoted
the
differences
from b i n a r y
we have n e v e r o b s e r v e d d i f f e r e n c e s
experimental
Cu-A1 s y s t e m .
found by Eifert
proof
for
large
et al.
Newly r e p o r t e d
results
probably originated
equilibrium
which exceeded
i n a s u r v e y p a p e r on
discrepancies
c o u p l e s and i n e q u i l i b r a t e d
to
alloys
[28]
between
was p u b l i s h e d by
showed t h a t
from t h e slow c o o l i n g
the
rates
in
experiments.
The o o s i t i o n
of t h e K l r k e n d a l l
Let us c o n s i d e r
schematically
and N a t a n o v l a n e s i n a d l f f t m l o n
counle
t h e f o r m a t i o n o f Ni3Sn 2 b e t w e e n t h e a d j a c e n t
which h a s b e e n s t u d i e d
b y v a n Beek e t
shown i n F i g .
a1.[33].
After
or experimentally
annealing
p h a s e s Ni3Sn 4 and Ni3Sn ,
at
792"C t h e r e s u l t
is
2.8.
The Matano p l a n e x o c a n b e f o u n d g r a p h i c a l l y
before,
force for the lattice
should be necessary
Remig [32] came t o t h e same c o n c l u s i o n
measured in diffusion
discrepancies
their
inaccuracy.
of v a c a n c i e s
Although very slight
in our laboratory
An o f t e n
concentrations
Eifert
[31].
supersaturation
e.g,
by p r e v e n t i n g
from t h e
the reaction
penetration
curve as
treated
b e t w e e n Ni3Sn 4 and NiHSn a t t h e
e d g e s o f t h e c o u p l e [34] a s shown i n F i g . 2 . 8 o r b y o t h e r i n g e n e o u s methods [ 3 5 ] .
For s i m p l i c i t y ,
line-compounds.
let
Then,
us c o n s i d e r
the position
be found b y t h e r e a c t i o n
equation
the
phases
in
the
couple
as
purely
stoichiometric
o f t h e N a t a n o p l a n e i n t h e N i 3 S n 2 - 1 a y e r can i m m e d i a t e l y
F. J. J. van Loo
Ni3Sn4
I--a--Ni3Sn2
.....
x2
d2
XO
XK
'=----b
Ni3Sn
Fig. 2.8.
S c h e m a t i c drawing of a Ni3Sn&/Ni3Sn d i f f u s i o n
formed [ 3 3 ] .
c o u p l e i n which t h e phase N i ~ a 2 i s
2 Ni3Sn + Ni3Sn ~ • 3 Ni3Sn 2
volume r a t i o
Therefore,
partial
103
:
125
d 2 / d 1 = 1251103 = 1 . 2 1 .
:
225
Because h a r d l y any change i n t o t a l
volume o c c u r s ,
the
molar volume8 VNi and VSn a r e n e a r l y c o n s t a n t , v i = . a b o u t 9 and 24.
The K i r k e n d a l l m a r k e r s ,
put i n t h e o r i g i n a l
interface
at
t = 0, were d i s p l a c e d towards
the N i 3 S n - s i d e a t t h e p o s i t i o n xK, l e a d i n g t o a r a t i o x 2 / x 1 = 1 . 8 5 .
From Table 2.1 i t
immobile ( l i n e
is
clear
a in Fig. 2.8)
that
since
this
ratio
then at
m o l e c u l e s a r e formed. I f o n l y Ni d i f f u s e s
From t h e s e d a t a i t f o l l o w s t h a t Ni d i f f u s e s
equals 0.5 only if
the Ni3Sn-side,
and Sn i s immobile t h i s
faster
Sn d i f f u s e s
and Ni i s
two t i m e s as many Ni3Sn 2
ratio x2/x 1 = 2 (line b).
than Sn.
Table 2.1
Only Sn d i f f u s e s
Only Ni d i f f u s e s
Nt3Sn4-side
Ni3Sn 4 • Ni3Sn 2 ÷ 2 8n
Ni3Sn 4 ÷ 3 Ni ~ 2 Ni3Sn 2
Ni3Sn-slde
2 Ni3Sn ÷ 2 Sn "*' 2 Ni3Sn 2
2 Ni3Sn ~ Ni3Sn 2 + 3 Ni
x21x I
0.5
2
Multiphase Diffusion
I f b o t h Ni and Sn d i f f u s e ,
to uJe if
the partial
Eq. ( 1 2 a ) a c c o u n t s
t h e n the s i m p l e c h e m i c a l e q u a t i o n s
m o l a r volumes a r e n o t e q u a l .
for
6S
this
difference
In fact,
of T a b l e 2 . 1 a r e d a n g e r o u s
the mathematically
i n a t o m i c volume a s i t
does f o r t h e p r e s e n c e of a
h o m o g e n e i t y r a n g e o f a b o u t 5 mole ~ i n Ni3Sn 2 end t h e n o n - s t o i c h i o m e t r y
The e x p e r i m e n t a l l y
1.5.
d e t e r m i n e d v a l u e o f x 2 / x 1 = 1.85 t h e n l e a d s
If only the difference
would h a v e b e e n c a l c u l a t e d
This analysis
a] The r a t i o
the
relative
DNi/Dsn of a b o u t
the ratio
of
the
reaction
side,
layer
into
is actually
reaction
equation.
Although the penetration
diffusing
component.
p l a n e d e p e n d i n g on t h e r a t i o
of t h e i n t r i n s i c
t h r o u g h Eq. ( 1 2 a ) i s n e e d e d e s p e c i a l l y
concentration
gradients
this
as
to get reliable
is larger
an
example
stoichiometric
diffusion
if
is deeper
of the Kirkendall
coefficients.
different
it
A more a c c u r a t e
partial
m o l a r volumes and
are involved.
c] The a c c u r a c y o f t h e m e a s u r e d p o s i t i o n
order
t h e end
diffusing:
Ni i s t h e f a s t e r
b] A s i m p l e mass b a l a n c e c a n p r o v i d e g l o b a l i n f o r m a t i o n a b o u t t h e p o s i t i o n
analysis
DNi/Dsn
t o show:
penetration
t o do w i t h t h e q u e s t i o n w h i c h atom s p e c i e s
is governed purely by the overall
towards the tin-rich
to a ratio
as 9.5.
i.e.
members, h a s n o t h i n g
o f t h e end members.
i n a t o m i c volume had b e e n t a k e n i n t o a c c o u n t ,
h a s b e e n g i v e n i n some d e t a i l
d2/dl,
correct
values
for
of t h e K i r k e n d a l l
the ratio
p l a n e h a s t o b e tremendous i n
of the intrinsic
diffusion
coefficients
if
t h a n a b o u t 5. T h i s h a s b e e n d e m o n s t r a t e d by B a s t i n e t a l . [34] end i s shown
for
the
above
compounds a r e
mentioned
Ni3Sn4/Ni3Sn
assumed end Eq. ( 1 2 a )
DNi/Dsn , taking VNi/Vsn = 0.38.
In fact,
is
couple
used
one must admit
in
for
that
Fig.
the
2.9,
where
calculation
the diffusion
of
couple
technique is not suited for the accurate determination of the ratio between the intrinsic
diffusion coefficients if this is larger than about 5, even if the partial molar volumes
of the components are exactly known. For a layer width of 100 ~n, the position of the
markers
for the cases DNi/Dsn = 5 and DNi/Dsn = ~ differs by 3.A ~m, which is mostly
within the accuracy limits of the experiment.
I0
DNi/Dsn
2
1
Fig. 2.9.
0.5 1~ 2
I~
I ~l!/lI
The r a t i o o f t h e d i s t a n c e s x ~ x I i n r e l a t i o n
d i f f u s i o n c o e f f i c i e n t s DNi/Dsn.
to the
ratio
of
the-intrinsic
66
F. J, J. van Loo
In t h e f o r e g o i n g
inert
it
was assumed t h a t
m a k e r s . The a n a l y s i s
accurately
coefficients.
The m a r k e r s h a v e ,
does n o t work i n a l l
proper
etching
especlally
cases,
procedure.
at
immediately,
places
the position
results
therefore,
is
debris
the
contact
of t h i s
or
of
this
row i s
of the intrinsic
The b e s t
pores,
present
the
at
the
original
row. O f t e n t h e g r a i n
which can easily
o f t h e Ni3Sn 2 l a y e r a b o v e t h e K i r k e n d a l l
i n t o and o u t o f Ni3Sn.
and Ni3Sn i s r e f l e c t e d
Kirkendall
The u s e o f t h i c k
cyllndrlcal
thick
wire
Various parts
Fig.
2.10.
structure
be understood
of the
from e . g .
p l a n e i s formed b y
g r a i n m o r p h o l o g y o f Ni3Sn ~
m o r p h o l o g y i n t h e N i 3 S n 2 - P h a a e on b o t h s i d e s of t h e
markers,
is
like
of
t u n g s t e n w i r e s o f 10 pm d i m , e r r
[38].
wire in a Ti-Ni diffusion
in
contact
with
and v a c a n c y f l u x e s
revealed
e.g.
demonstrated by Bastln
tungsten
diffusivities
particles
interface
established
p l a n e . N i c e e x a m p l e s ,my b e f o u n d i n N i - S n [33]0 Nb-Sn [ 3 6 ] , e n d NIO-A1203 [ 3 7 ] .
as was c o n v l n c i n g l y
the
by a
p l a n e i s formed b y t h e d i f f u s i o n
I n some way t h e d i f f e r e n t
in the different
diffusion
method, although it
end members was n o t
of atoms i n t o a n d o u t o f Ni3Sn 4 , w h e r e a s t h e p a r t b e l o w t h e F d r k e n d a l l
the diffusion
by using
m a r k e r s w h i c h may b e made v l s l b l e
between
different,
e.g.
plane has to be determined very
to be very thin.
can then often be seen along a straight
The p a r t
p l a n e can b e f o u n d ,
for the calculation
to use '~atural"
Small
where
phase at both sides
Fig. 2.8.
showed, t h a t
in order to find rellable
the Kirkendall
all
phases
were d i f f e r e n t ,
the wire migrated
the true position
with
Fig.
couple.
Since after
of
couple,
the
an actual
different
The d i s t o r t i o n
of an initially
circular
a f t e r 72 h r a n n e a l i n g a t 800°C [ 3 8 ] .
in
rupture
velocities.
of the Kirkendall
is very dangerous
2 . 1 0 shows t h e f a t e
plane.
o f an o r l g l n a U y
a short
diffusion
each of which
time
the
of the wire took place.
Only v e r y
small
The same e f f e c t
tungsten wire in a diffusion
thoria
could be seen
couple Ni/Ti
Multipl~ase DiffuSion
////
I I
cu
Fig.
y
the indentations
the
occurs,
intrinsic
zn
indentations
might
happen.
plane are
displaced
with respect
to the laboratory
fixed
morphology and t h i c k n e s s
Finally,
velocities
reheating
2.12,
procedures.
interface.
sometimes
from t h e o r e t i c a l
coefficients
In Ref.
to
to that
frame,
square micro-indentations
were u s e d as m a r k e r s a s shown i n F i s .
and i n f a c t
diffusion
Kirkendall
Fig.
I
2 . 1 1 . The d i s t o r t i o n
and d i s p l a c e m e n t of i n i t i a l l y
c o u p l e a f t e r ~ h r a n n e a l i n g a t 400"C [ 3 9 ] .
if mlcrohardness indentations
of
6?
the
in
[39]
it
the
is
"laboratory"
frame.
the Kirkendall
of thediffusion
If,
2.11 [ 3 9 ] . E l o n g a t i o n of
considerations,
various
also
phases,
shown t h a t
frame,
i n a Cu-Zn
d e p e n d i n g on t h e r a t i o
disrupture
if
of
the markers
the ends of
the
the
in
couple will
the
be
on t h e o t h e r h a n d , one end of t h e c o u p l e i s f i x e d
m a r k e r s move w i t h r e s p e c t
to this
f r a m e . The f i n a l
zone i s t h e same f o r b o t h e x p e r i m e n t a l
a n e a s y method can b e u s e d
to determine
the
relative
setups.
diffusion
o f t h e e l e ~ s e u t s . By g r o w i n $ a p h a s e ¥ b e t w e e n t h e p u r e end members A and B, and
t h e same c o u p l e d u r i n g
t h e n e w l y formed p a r t
It
i s now c l e a r
that
The method was u s e d e . g .
t h e same t i m e a t
of the layer
t h e same t e , , p e r a t u r e
can often
o n l y A atoms d i f f u s e
be revealed
since
a g a i n a s shown i n
by specific
the layer
grows a t
i n t h e Ni-A1 [~0] and t h e Ti-A1 [1~] s y s t e m .
etching
t h e ¥/B
6~
F . J . J . van Loo
8
B
"r~ff~"
Y
Y
"otd"
A
A
(b)
(a)
Fig.
2 . 1 2 . R e a n n e a l i n g o f a c o u p l e A/B i n w h i c h t h e y - p h a s e i s formed ( a ) l e a d s t o new
f o r m a t i o n o f ¥ o n l y a t t h e B - s i d e , showing A t o b e t h e o n l y d i f f u s i n g component.
The r o l e o f i n t e r o h a s e i n t e r f a c e
the Kirkendall
In r e c e n t p u b l i c a t i o n s
the creation
4n t h e c r e a t i o n
[8,41]
on a n n i h i l a t i o n
most s i m p l e m u l t l p h a s e
the role of Interphase
of Kirkendall
couple,
Dependent on t h e i n i t i a l
vlz.
t h e same e l e m e n t d i f f u s e s
s u c h a way t h a t
faster
the Kirkendall
2 . 1 3 c ) no d i s c o n t i n u i t y
If the ratio
fr.--
the interdiffusion
faster
coefficient
in a diffusion
InventIsated.
or a sink (Fis.
2.13 d) for vacancies
If the initial
the
was c h o s e n .
must b e a b l e
i n t h e c a s e where
concentrations
with the position
c o u p l e on
As a n e x a m p l e ,
are chosen in
of the interphase
interface
in the vacancy flux is found.
diffusivities
interesting
2.14 a) B diffuses
interfaces
wu
o f t h e end members, t h e same i n t e r f a c e
in both phases.
c h o s e n i n s u c h a way t h a t
Fig.
vacancies
plane coincides
of the intrinsic
I in the Other phase,
are
resultine
a n ct/~ c o u p l e w i t h o n l y one i n t e r f a c e ,
concentrations
t o a c t a s a s o u r c e ( F i S. 2 . 1 3 a , b , e )
(Fig.
o r mn,~JhJlat4on o f v a ~ , , e i a e
effect
i s > 1 i n one p h a s e b u t
phencqena occur.
In Fig. 2.14 the initial
t h e Matano p l a n e c o i n c i d e s
t h a n A i n t h e same r a t i o
in ~ is larger
in both the ~ and ~-phase,
than in ~; in Fig.
t h a n A i n t h e ~ - p h a s e b u t s l o w e r ~n t h e ~ - p h a s e ,
concentrations
with the phase interface.
2.14 b) B diffuses
and i n F i g . 2 . 1 4 c ) B d i f f u s e s
In
although
faster
faster
than
A in the ~-phase but slower in the ~-phase.
In case a) the interface
the ~-phase,
source,
thus acting
interface
ratio
inflection
DA/DB i s
points
for vacancies
vacancies
therefore,
at
Occurs
•
Kirkendall
at t = 0, have to split
constant
in
up and d r i f t
each phase,
in the penetration
the interface
plot
this
[~,&2].
from b o t h s i d e s
positioned
in both phases.
at the inflection
In case a)
the interface
In r e f s .
coincides
with
points
one r e a l
the
acts
is easy to imagine that
In this
as a
has been
Markers, placed at
In c a s e c ) t h e i n t e r f a c e
It
acts
8 end 41 i t
into
i n b o t h t h e ~ and t h e ~ - p h a s e .
position
might give rise
h a d a d h e r e n c e b e t w e e n t h e end members.
plots
In case b)
planes have to be defined.
coming from b o t h t h e ~ and ~ - p h a s e .
p l a n e s may b e d e f i n e d ,
penetration
for vacancies.
i n t o b o t h t h e ~ and t h e ~ - p h a s e .
i n s u c h a c a s e two r e a l
the original
If t h e
as a source
sending vacancies
shown t h a t
a b s o r b s v a c a n c i e s from t h e ~ - p h a s e and pomps more v a c a n c i e s
two
as a sink
the arrival
to the formation of pores,
case
two ' ~ i r t u a l "
of the virtual
a n d one v i r t u a l
of
and
Kirkendall
extensions
Kirkendall
of t h e
plane
Muitiphase Diffusion
69
o
I
P~
-~
xB
I
¢t
ALpha
~
XB
"- -]v
o
I
-~'0
!
I MK0+ KP "+110
Jo
J,
Jv
o
ALpha
j
-10
a)
"-'-~-:
3B
3A
•
o
Beta
•
i
1
-M KP
+10
b)
9
IMK" K]I,
I-M KP
/
E'o
Io
Io
Io
Io
Io
i:
,o°
31+
Tp
3A
3v
XB
~
JA
Jv
¢z
Beta
0
C)
,o
-10
MI-Ka,13
+10
mX.'l~~e
-10
~}a
.1'0
d)
/"
I
M.K~j tKO.
1
P
T
XB
o
o
e)
-Ib
,
o
i
o
M.Ka KP ÷1~
| M,,Ko.
Fig. 2.13.
BIta
•
Ki5
P e n e t r a t i o ~ p l o t s o f component B (mole f r a c t i o n ) i n a s y s t e m i n which DB = 10 DA
i n b o t h t h e ~ and ~ p h a s e f o r v a r i o u s
i n i t i a l end member c o n c e n t r a t i o n s and
D: - I0 D : . A l s o i s shown t h e v a c a n c y f l u x JV and i n a ) and c ) t h e g r a d i e n t
t h e v a c a n c y g l u x as a f u n c t i o n o f t h e p o s i t i o n i n t h e c o u p l e [ 8 ] .
of
70
F, J. J. van Loo
1
:+____----
£S
J8
-+A
"" ++~-+"
T
XB
0
- ~
3v
,.,.
O.
ALpha
+~0
l.M KP
-~'0
a)
IoM KP
o
;
o
0
'
0
0
0
0
i
i
i
0
0
+
0
!+
< ,,
o
:I~
o
0
0
|
J-
0
o
o
- r
o
°
I
o
-10
b)
t
I
,
"I
JA
•
-u
1
~
o
i
K= M-I KO
~
0
o°
°B
,,
I
+10
M KP
I
I
•
Atpha
c)
c'
~Vol'
[
Fig.
2.14.
~
"
K=
_,~
Beta
-.+Co
t.,
, _..._......~M.t
,
X¢=L~.._...---~
Plots of the mole fraction
o f B a n d t h e v a c a n c y f l u x 3V i n c o u p l e s w h e r e t h e
Matano i n t e r f a c e
d o e s n o t move, I n ( a ) DB/DA = l 0 i n b o t h t h e = - a n d ~ - p h a s e s , i n
<b) D ~ / D , - 10 i , the ~ - p ~ . .
and Oil" i , th, =-phase. and in ~o~ D , / D , -
t h e ~ - p h a s e a n d 10 i n t h e = - p h a s e
[
0.1 in
Multiphase Diffusion
These phenomena may b e v i s u a l i z e d
the interface
by t h e b e h a v i o u r of an i n i t i a l l y
a t t = 0 a s shown i n F i g .
The same a u t h o r s
presented
phase transformation,
'71
a model f o r
t h e dynamic a c t i o n
of t h e i n t e r f a c e
which includes
c l i m b of m i s f i t
dislocations
dislocations
in the interface
[43].
c l i m b of m i s o r i e n t a t i o n
square indentation
at
2.15.
during the
out of the interface
and
Beta
-
4b
t=O
Eb
-
H=I
?b
Fllpha
Fig.
2 . 1 5 . The c h a n g e i n m o r p h o l o g y of a s q u a r e i n d e n t a t i o n m a r k e r , o r i g i n a l l y p l a c e d a t t h e
~/~ interface at t = 0 after diffusion annealing in the situations
g i v e n by
f i g s . 2 . 1 4 a - c . The r e a l o r v i r t u a l
d i s p l a c e d o r i g i n o f t ~ e a l p h a and b e t a
K i r k e n d a l l f r a m e s o f r e f e r e n c e a r e d e s c r i b e d b y 1~ and K~ ( d e f i n e d b y t h e
constant lateral indentation width) [8].
The morvholo~v o f i n t e r o h a s e
If a slngle-crysta11ine
interfaces
if stain
effects
structures
In many d i f f u s i o n
completely
straight
formation
are given in Fig.
experiments,
in Cu-Si couples
[16,44,45]
is that
is
thin
found:
an equiaxed
needle-like
crystals
at the interfaces
immaterial
because of
graln-boundaries
the
is faster
grown i n Ni/NiA1 c o u p l e s
of
c)
are
because of the
The n e c e s s a r y
but in fact
interdiffusion
t h r o u g h t h e b u l k as e . g .
The morphology g i v e s
Cu3Si
becomes u n i m p o r t a n t .
takes place.
In c a s e
e.g.
interfaces
e.g.
o r volume d i f f u s i o n ,
the crystals.
than interdiffusion
[40].
layers
o f ¥ bounded by
formed,
occurs,
so t h a t g r a i n - b o u n d a r y d i f f u s i o n
m i g h t o c c u r by s u r f a c e
thinness
layer
are
a s shown i n c a s e b ) where s t r a i g h t
c a s e b ) , on t h e o t h e r h a n d , o n l y g r a i n - b o u n d a r y d i f f u s i o n
diffusion
In p o l y c r y e t a l l i n e
are often not completely straight.
i n c a s e a ) o n l y volume d i f f u s i o n
high temperatures
= and
2.16.
case a)
Often
substrates
p r o v i d e d t h a t no o r i e n t a t i o n
interface.
end members, t h e i n t e r f a c e s
interfaces.
f o u n d . The e x p l a n a t i o n
u s e of r e l a t i v e l y
= / 7 and y / ~ i n t e r f a c e s ,
occur which might cause a faceted
grown b e t w e e n p o l y c r y s t a l l i n e
Some t y p i c a l
is takine ,lace
l a y e r of 7 i s growing b e t w e e n s l n g l e - c r y s t a l l l n e
~, one m i g h t e x p e c t c o m p l e t e l y s t r a i g h t
relation
boundarv diffusion
no i n f o r m a t i o n
In
lateral
this
is
through
i n t h e Ni3A1 phase
about
the
relative
m o b i l i t y of t h e two c o m p o n e n t s .
In c a s e d ) ,
found e.g.
more e x a g g e r a t e d
is different
the
grains.
at
form e . g .
for various
t h e FeSn2/FeSn i n t e r f a c e
in a FeSn2/Fe couple
t h e f o r m a t i o n of Fe2A15 [46] b u l k d i f f u s i o n
crystallographic
A non-unlform nucleation
morphologies like this [44].
directions
a c c o m p a n i e d by
which leads
lateral
[33] and i n a much
is predominant, but
to a different
g r o w t h may a l s o
l e n g t h of
lead
to
72
F . J . J . van Loo
Illllltlllllllrl}tl]llilll111}llllllllllllllllll
cb'
(d)
(e)
Fig.
(f)
2.16. Various morphologies of a reaction
in the text.
Grain boundaries
low t e m p e r a t u r e ,
For i n s t a n c e ,
Bastin
especially
therefore,
interface
by g r a i n - b o u n d a r y
diffusion
Influence
of a difference
During r e s e a r c h
schematically
by (impurities
in cross-section
on i n t e r d i f f u s i n n
shown i n F i g .
If
iron stopped this
therefore,
The e x p l a n a t i o n
e.g.
type of
i s f o u n d d e p e n d s on t h e p o s i t i o n
the total
is
further
layer width increases
displaced
the formation of a solid
from t h e
solution
in
in) the grain boundary [14].
of t h e end members on t h e l a v e r morDholoev i n
of Cu and S i
the Cu3Si-layer
the Si has a larger
formed, b r o a d e r t h a n t h e o r i g i n a l
a l w a y s grow p e r p e n d i c u l a r
of
[65] a t y p i c a l
phenomenon was f o u n d , a s
2.17.
If a small piece of Si is used,
t h e Cu end member.
diffusion
The r e m o v a l of
t o t h e Matano p l a n e :
and t h e i n t e r f a c e ,
layer
2.16 e,f).
~-Ti in the grain-boundaries
Matano p l a n e . Morphology f ) c a n a l s o b e f o u n d i f
t h e end member i s f a c i l i t a t e d
(Fig.
Iron atabilise8
Whether morphology e) or f)
with respect
of i m p u r i t i e s
f o r m a t i o n of ~ - T i by g r a i n - b o u n d a r y
of iron.
enhances Ni-diffusion.
grain-boundary interdiffusion.
couple, explained
the morphology of the reaction
under the influence
[34] f o u n d e x t e n s i v e
Ni i n t o m-Ti i n t h e p r e s e n c e o f t r a c e s
of t h e r e l e v a n t
in a binary diffusion
i n t h e end members may a l s o a f f e c t
at relatively
o f m-Ti a n d ,
layer
grows i n e x a c t l y
cross-section
t h e same c r o s s - s e c t i o n
t h a n Cu, a l e n s - s h a p e d
layer
Cu end member. I n b o t h c a s e s t h e c o l u m n a r g r a i n s
to the original
into
is
of Cu3Si
interface.
has to be found in the fact,
converting
S i i n t o a Cu3Si mass o f t h e same s h a p e ( a l t h o u g h w i t h a volume o f
a b o u t 7/3 t£mes t h e volume o f t h e r e a c t e d
diffuse
by s u r f a c e
cross-section
Conversely,
of
diffusion
Cu,
the faster
Si).
straight
component i n
figure,
the original
Cu atoms d i f f u s e
t h a t Cu i s t h e o n l y d i f f u s i n g
Cu3Si. In t h e l e f t - h a n d
t h r o u g h Cu3Si t o t h e a v a i l a b l e
In the right-hand
figure,
a l o n g t h e Cu3Si t o r e a c t w i t h Si a v a i l a b l e
leading
diffusing
to
the
lens-shaped
morphology
component can b e i d e n t i f i e d
of
the
Si,
h o w e v e r , Cu can
outside
reaction
the original
layer.
from t h i s m o r p h o l o g y .
Multiphas¢ Diffusion
73
1rll flail
Cu
Illliliiiillll
IIIIIiiii
$i
5i
F i g . 2.17. D i f f e r e n t u o r p h o l o g i e s of the C u ~ i r e a c t i o n l a y e r in a Cu/Si d i f f u s i o n couple
dependent on the r e l a t i v e d i ~ e t e r s of the end members. The columnar g r a i n s in
Cu3Si a r e i n d i c a t e d .
G e n e r a l l y , the edges and f r e e s u r f a c e s of the d i f f u s i o n couple g i v e e x t r a information on
the l a y e r growth mechanian and c o n c e n t r a t i o n l i m i t s of the v a r i o u s phases. Often s u r f a c e
d i f f u s i o n or e v a p o r a t i o n / c o n d e n s a t i o n p r o c e s s e s g i v e r i s e
~ t
to the formation of a l a r g e r
og phases r i c h in s u b s t r a t e m a t e r i a l compared to the r e ~ l a r
of an impeded supply of d i f f u s i n g m a t e r i a l
d i f f u s i o n zene because
(see c - p h a s e in F i g . 2.18) In f a c t ,
a kind of
inc r em en t al couple i s p r e s e n t a t th e s e edges. In t e r n a r y d i f f u s i o n c o u p l e s , t h i s phenomenon
can be used to even more advantage (see c h a p t e r 3).
8
A
F i g . 2.18. Development of d i f f u s i o n l a y e r s a t the edges of a d i f f u s i o n couple.
~ : 1 4r
74
F.J.J.
Comvarison of
the reaction
laver
thickness
van
Loo
in olanar,
cylindrical
and a o h e r i c a l
difftmion
CQuDles
It
makes a d i f f e r e n c e
when l a y e r
d i s c u s s e d up t o now o r i n e . g .
coated wires,
fibrous
thickness
diameter.
time t,
2 r o,
studied
or spherical
composites or in sintering
l a y e r &By b e t w e e n a n i n f i n i t e
with
growth is
cylindrical
or
in
planar
couples,
diffusion
couples
problems. Let us look to the growth of a
m a t r i x B and p u r e A w h i c h h a s t h e d i m e n s i o n s o f e i t h e r
a cylinder
with
diameter
as
which are important in e.g.
2 r o,
or
else
a sphere
of
the
a plate
same
The v o l u m e o f one s o l e &By i s z t i m e s t h e v o l u D e o f one m o l e o f p u r e A. A f t e r
the Situation
(for the plate)
shown i n F i g .
a equals
half
I
ro
I
2.19 arises,
the unreacted
where a e q u a l s
thickness
the unreacted
radius,
some
or
o f A.
-I
I
v'-
t
B
A
I
A
3y
A By
I
I
I
I
I
(a)
Fig.
2.19.
The r e a c t i o n b e t w e e n a p l a t e o f p u r e A w i t h t h i c k n e s s 2 r o ( a ) a n d o f a s p h e r e o r
c y l i n d e r o f p u r e A w i t h d i a m e t e r 2 r o ( b ) w i t h a m a t r i x B, f o r m i n g t h e compound
ABy o f l a y e r t h i c k n e s s d .
The c o n v e r s i o n f a c t o r
==
r - a
o
r
r
~=
r
~=
r
3
0
3
o
(b)
a
;
r
o
2
2
-a
g
2
r
o
-
a
m, i . e .
the fraction
of A which
reacted,
is:
(]Plate)
=I-~
o
2
L_ = 1 - ~
2
r
o
;
3
(Cylinder)
3
;
has
~L.
3
r
o
=
1 -
~
(Sphere)
(2~)
Multiphase Diffusion
From g e o m e t r i c a l c o n s i d e r a t i o n s
75
the f o l l o w i n g r e l a t i o n s
exist
between = and t h e l a y e r
t h i c k n e s s d:
dp/r o - z
(Plate)
1
d c l r o = {i + (s - I)~} I12 - ( I - z ) l 1 2
(Cylinder)
d s / r o = {I + (z -
(Sphere)
We a r e
I)~}
I13
interested
rate-limiting
-
(I
-
a ) 113
i n = and d as a f u n c t i o n
of
(25)
time if
volume d i f f u s i o n
is
the
s t e p . As an example, c o n s i d e r t h e f o r m a t i o n of an A B y - l a y e r , which grows in a
p l a n a r c o u p l e a c c o r d i n g to t h e p a r a b o l i c
For s h o r t a n n e a l i n g
longer annealing,
times,
law d 2 = 2 kpt
(~
= parabolic
growth c o n s t a n t ) .
t h e l a y e r w i d t h s a r e e q u a l f o r the t h r e e m o r p h o l o g i e s , b u t f o r
differences
arise
from the f a c t
that
t h e s u r f a c e a r e a of the i n n e r and
o u t e r s h e l l of the l a y e r w i l l change, c o n t r a r y to the c o n s t a n t s u r f a c e in p l a n a r c o u p l e s .
A number of e q u a t i o n s a r e g i v e n in the l i t e r a t u r e
v a l u e s of z [ 4 7 - 5 2 ] .
i n v a r i o u s d e g r e e s o f a p p r o x i m a t i o n and
The a u t h o r has nowhere found a c l e a r c o m p a r i s o n , e x p r e s s e d in the same
quantles and, therefore, a number of useful relations is given here between the values of =,
ro,
t and t h e p l a n a r
parabolic
growth c o n s t a n t
kp.
The d e r i v a t i o n
of
these equations
p r o c e e d s a l o n g t h e same l i n e s as used by C a r t e r [ 4 7 ] , b u t s p e c i a l c a r e i s taken i n d e f i n i n g
the v a r i o u s q u a n t i t i e s
which l e a d s to e x p r e s s i o n s s l i g h t l y
different
from t h o s e g i v e n in the
literature:
(z~) 2 = 2 k p t l r ~
"2 [
2
Plate
--
(26)
Cylinder
, In
z~l(l+(z-1)~}213-s(1-~)213=2kp t i t - ~o
Sphere
--
(z~)2=d21r2=2k
t/r 2
p o
p
o
Plate
~-/
(z~)2=4d21r2=Sk t l r 2
c o
p
o
Cylinder
~
(z~)2=9d2/r2=18k
t/r 2
s
o
p
o
Sphere
z-1
z
-
- 1:
F o r low v a l u e s
By p l o t t i n g
(1-~13)-(Z-~)213-2kptlr2 o
of a the Zqs.
( 2 5 ) and ( 2 6 ) r e d u c e t o :
(27)
the c o n v e r s i o n f a c t o r ~ v e r s u s t ~ a c c o r d i n g to Eqs. (26) f o r z = 2 as shown
in Fig. 2.20, straight lines are found with slopes of I, 2 and 3 times (2kp)~/Zro for values
of = lower than about 0.35. Therefore, from each slope the true value for the parabolic rate
constant of a planar couple can be found.
76
F . J . J . van Loo
0.5
0 '/
0
5
10
15
- tV2(a. .)
20
F i g . 2 . 2 0 . A p l o t o f t h e c o n v e r s i o n f a c t o r ~ v e r s u s t ~ f o r a p l a t e o f t h i c k n e s s 200 ( p ) , a
c y l i n d e r ( c ) and a s p h e r e ( s ) b o t h o f d i a m e t e r 200 o f m a t e r i a l A r e a c t i n g w i t h a
matrix B according to A + B ~ AB. Volume diffusion is the rate limiting step. The
volume r a t i o AB/A = 2, t h e p a r a b o l i c g r o w t h c o n s t a n t f o r t h e p l a t e kp = 50, a l l
in arbitrary units.
Through Eq.
(25) t h e a / t ~ p l o t
can b e c o n v e r t e d i n t o a d / t ~ p l o t a s shown i n F i g . 2.21
for s - 2. Consistent with the foregoing result, up to a thickness of about 0.35 ro the
thicknesses are nearly
about
equal for
1.1 r o (= = 0 . 9 5 )
for
the three couple types.
a cylinder
It
t a k e s t h i c k n e s s v a l u e s up t o
and 0 . 7 r O (= m 0 . 9 0 )
f o r a s p h e r e t o f i n d an
i n c r e a s e o f 10% i n l a y e r w i d t h compared w i t h a p l a n a r c o u p l e . For l o w e r v a l u e s o f s ,
the
deviations are larger.
However, in practice, the deviations are probably much larger than predicted from these
geometrical relations.
One reason has bean pointed out by Beke
[53]: diffusion into a
cylinder or sphere may build up stresses which cannot be released as easily as in a planar
c o u p l e s . These s t r e s s e s
vol,--e changes are
especially
may i n f l u e n c e
involved.
when t h e
cylinder
surrounding material.
In t h i s
Also
or
t h e t h i c k n e s s o f t h e growing l a y e r ( s ) ,
the
Kirkendall
sphere material
effect
is
can c a u s e
diffusing
case, a large increase of the total
the formation of hollow s p h e r e s [5~].
large
much f a s t e r
especially
when
deviations,
than
the
volume may o c c u r due t o
Multiphase Diffusion
77
P
200
150
C
d (..u.)
S
1
IO0
5O
0
5
0
10
20
15
tl~(a.u.)
F i g . 2 . 2 1 . A p l o t of t h e t h i c k n e s s d of t h e AB r e a c t i o n l a y e r v e r s u s t ~ f o r t h e s i t u a t i o n
described in Fig. 2.20.
Texture i n r e a c t i o n l a y e r s
Very o f t e n t h e c r y s t a l s
orientation
i n our
l a b o r a t o r y and t h e
crystallosraphic
parallel
of t h e p h a s e s In d l f f u s l o n - s r c r ~ n r e a c t i o n l a y e r s have a p r e f e r r e d
in t h e srowth ( d i f f u s i o n )
T h i s t e x t u r e has b e e n s t u d i e d e x t e n s i v e l y
result
found:
is
the direction
of
the
lonsest
a x l a i s i n v a r i a b l y o r i e n t e d p e r p e n d i c u l a r to t h e d i f f u s i o n d i r e c t i o n ,
to t h e i n t e r f a c e
repeat-dlstance
followins
direction.
i.e.
[ 5 5 , 5 6 ] . Very o f t e n a n o t h e r r u l e of thumb 4s found: t h e s h o r t e s t
in the c r y s t a l
lattlce
is aligned parallel
of d l f f u a l o n
texture).
layer;
i n t h e " o l d e s t " p a r t of t h e l a y e r sometimes changes i n t e x t u r e a r e found due to
recrystallization.
These t e x t u r e s a r e most pronounced a t
to the d i r e c t i o n
(fiber
t h e s r o w t h f r o n t of the r e a c t i o n
78
F.J.J. van Loo
The p r e s e n c e o f a t e x t u r e
quite
for
is important since often chemical and mechanical properties
d e p e n d e n t on t h e c r y s t a l l o g r a p h i c
instance,
diffusion
the
S-FeZnl0 layer
direction
[57,58].
which c a u s e s r e l a t i v e l y
is
direction.
In that
direction
thick reaction
despite
the various
tricks
hexagonal axis
the diffusion
layers.
process,
perpendicular
to the
o f Zn t h r o u s h S i s v e r y f a s t
For economic r e a s o n s i t would b e v e r y h e l p f u l
if the hexagonal axis could be forced to orient
Unfortunately,
In the well-known galvanizing
formed w i t h i t s
are
itself
parallel
to the diffusion
direction.
we u s e d , n a t u r e d i d n o t a l l o w us t o i n t e r f e r e
with
its above-mentioned rules.
The e v a l u a t i o n
Equations
coefficients
of diffusion
coefficients
i n c o u p l e s i n w h i c h l a r e s volume e h a n ~ a s o c c u r
( 3 ) and ( 1 2 ) c a n b e u s e d w i t h o u t a n y p r o b l e m f o r f i n d i n g
the (inter)diffusion
from a p e n e t r a t i o n
changes occur.
A special
curve
example i s
t i t a n i u m and c a r b o n [ 1 0 ] . The p a r t i a l
in multiphase
couples,
even if
very
large
t h e g r o w t h of a t i t a n i u m
carbide
layer
between pure
m o l a r volume of c a r b o n i n t i t a n i u m c a r b i d e i s
virtually
zero, which leads to equal values
intrinsic
diffusion
coefficient
~he e v a l t t a t i o n
of diffusion
for the interdiffusion
coefficient
D and t h e
o f c a r b o n DC i n t i t a n i u m c a r b i d e :
=
= CTi VTi DC + CC 'Vc DTi
volume
(lOa)
DC
coefficients
in vaoour-solid
c o u n l e s compared w i t h s o l i d - s o l i d
~ouples
Let u s c o m p a r e , f o r i n s t a n c e ,
hand,
equilibrium
first
t h e g r o w t h o f a n FeO l a y e r b e t w e e n Fe end Fe304 on t h e one ~
and b e t w e e n Fe and oxygen g a s w i t h
case
reaction
on t h e o t h e r
hand [ 1 3 ] .
than in the latter
The t h i c k n e s s
of iron,
t h e h o m o g e n e i t y r a n g e i n FeO i s
important for this
According
to
that
the
Fe0/Fe304
be larger
in the
FeO grows from two s i d e s :
o f oxygen from Fe304.
are schematically
taken as zero,
to
The p e n e t r a t i o n
shown i n F i g .
2.22.
the
curves,
For simplicity,
which i s o n l y a r o u g h a p p r o x i m a t i o n b u t n o t
discussion.
Wagner's
FeO-phase c a n b e w r i t t e n
analysis
[13]
the
integrated
diffusion
coefficient
for
the
for the Fe/Fe30 A couple as:
( l - O . S ) CO.S-~/7)
Int=
equal
of the FeO-layer will
because of the fact,
o f oxygen w i t h i r o n and t h e l o s s
e x p r e s s e d i n mole f r a c t i o n s
a n oxygen a c t i v i t y
d~/2t
4/7
=
I-~6"d~/2t
For the Fe/oxygen couple one finds:
Bint=
(I-0.5)(0.5-0), d~/2t = ~. d2/2t
(I-0)
Since
the
integrate~
diffusion
d 1 = 2d 2. In t h e s e s i m p l e c a s e s
m o l a r volumes a r e i r r e l e v a n t
coefficient
is
a material
constant,
(only line-compounds are involved)
and a p l o t w i t h m o l a r f r a c t i o n
it
follows
the variations
can b e u s e d .
that
in partial
Multiphase Diffusion
I f t h e mole f r a c t i o n
o f oxygen had b e e n u s e d f o r
79
the plots,
t h e same r e s u l t s
must be
obtained. This means that for the oxygen gas the mole fraction must be taken as N O = I,
irrespective
of
its
pressure
or activity.
Differences
i n oxygen a c t i v i t y
only reflect
themselves through differences in the defect structure in changes in the interdiffuslon
coefficient
for the FeO-phase.
Fe
1
Fe
0.8
0.6
FeO
FeO
Fe3Oz
----4W
dI
0.~.
0.2
02
Ca)
(b)
F i g . 2 . 2 2 . The f o r m a t i o n o f a F e O - l a y e r i n a d i f f u s i o n c o u p l e F e / F e ~ 6 ( a ) and i n an
o x i d a t i o n e x p e r i m e n t o f Fe i n an oxygen a t m o s p h e r e e q u a l t o t h e FeO/Fe30 A
e q u i l i b r i u m ( b ) , a n n e a l e d f o r t h e same p e r i o d o f t i m e .
The same p r i n c i p l e
can be u s e d i f
AyBl_y, h e a t e d i n a c l o s e d v e s s e l
t h r o u g h t h e gas p h a s e .
diffusion
is s t u d i e d between the a l l o y s
w i t h no d i r e c t
In t h e i d e a l
case,
contact.
AqBI_ q and
D i f f u s i o n t h e n has t o p r o c e e d
t h e same c o n s t a n t s u r f a c e c o n c e n t r a t i o n NA = p
(q<p<y) has to o c c u r i n t h e e q u i l i b r i u m c o n d i t i o n on t h e o u t e r s u r f a c e o f b o t h a l l o y s .
practice,
sometimes a d i f f e r e n t
but
constant value
is
found f o r b o t h s u r f a c e s
In
[59-61],
showing an activity gap in the gas phase.
In F i g . 2.23
it
is
assumed t h a t
only element A evaporates,
and
that
the
surface
concentration on both alloys equals N A = p. The alloy AyBl_y has lost material (thickness
d e c r e a s e x 0 - x 1) and AqBI_ q h a s g a i n e d m a t e r i a l
(thickness increase x 2 - Xo').
volumes o f A and B a r e
surface
equal,
x0 - x 1 = x 2 - x o ' The t o t a l amount o f A, l o s t
the o t h e r hand,
original
it
the
total
of both a l l o y s
is
I f t h e molar
equal,
then
0
from t h e a l l o y AyBl_y, e q u a l s t h e sum o f t h e a r e a C and D. On
a l s o e q u a l s t h e sum o f t h e a r e a D and E. T h e r e f o r e , one may f i n d t h e
s u r f a c e x o by e q u a t i n g
Area C = Area E.
and i f
80
F . J . J . van Loo
In t h e same way t h e &mount o f component A g a i n e d b y t h e AqBI_ q a l l o y e q u a l s t h e a t e o f
t h e a r e a C + K + H, b u t a l s o e q u a l s K + G ÷ F, from w h i c h i t
suface xo
Area F
w
follows that
the orig~
i s £ound by e q u a t i n g
,,
Area H
t
Under t h e a b o v e m e n t i o n e d c o n d i t i o n s
i n w h i c h x 0 - x I = x 2 - x 0 , one m i s h t s h i f t
the
t
couple halves in
s u c h a way t h a t x 0 = x 0
and x 1 = x 2. One t h a n s e t s F i g . 2 . 2 ~ , s h o w i n | t h ~
Area C + F - g = Area H
!
and, t h e r e f o r e ,
the plane x0 = x0
"'1
I
~NA t
E:
s
i
I
-4
Ay Bl.y
is identical
plF
t o t h e Hatano p l a n e .
t"A
i
i
I
0
I
J
I
x:~i ..j
x
(a)
q K'~
x0
x2 x6
Xl:X2 x0:x ~
(b]
F i g . 2 . 2 3 . The a n n e a l i n g o f t h e a l l o y s A ~ l _ y and AqBI_a t o g e t h e r i n a c l o s e d v e s s e l . A f t e r
e q u i l i b r i u m i s r e a c h e d , t h e same s u r f a c e c o n g e n t r a t i o n Na - p i s p r e s e n t oqz b o t h
a l l o y s . The p o s i t £ o n s o f t h e Mateno p l a n e s x o and x o ' a r e i n d i c a t e d f o r b o t h
alloys,
they coincide with the original surfaces. ~e
new s u r f a c e s a f t e r
d i g £ u s i o n a n n e a l i n g a r e g i v e n by x 1 and x 2, r e s p e c t i v e l y .
F i g . 2 . 2 4 . The s i t u a t i o n i n a p u r e s o l i d - s t a t e
d i f f u s i o n c o u p l e A y B l _ ~ A ~ l _ o ~ o b t a i n e d by
o v e r l a p p i n g o f F i g s . 2.23 a ) and b ) u n t i l t h e coamon c o ~ c e n c r a t ~ o n p .
Multiphase Diffusion
From t h i s a n a l y s i s
t h e f o l l o w i n g c o n c l u s i o n s can be drawn:
1) S i n c e t h e d i f f u s i o n
coefficient,
calculated
t o b e t h e same f o r e a c h c o n c e n t r a t i o n
follows that
81
from t h e p r o f i l e s
independent of the total
amounts o f t h e a l l o y s
it
f
w i t h o u t any d e p e n d e n c e on t h e a c t u a l
a l l o y AyBl_y by e q u a t i n g t h e a r e a s F and H. I f
equal, corrections
Matano p l a n e
profile
composition of the master
t h e m o l a r volumes o f A and B a r e n o t
shown i n F i g . 2.23 a ,
(now s i t u a t e d
outside
the
alloy)
i.e.
the material-losing
can b e
found
Area C = Area E. I t would l e a d t o c o m p l e t e l y wrong r e s u l t s
plane as the plane x3, d i v i d i n g the c o n c e n t r a t i o n p r o f i l e
in area (x3-xl-P-r
in
alloy,
i f one would d e f i n e t h e Natano
i n t h e a l l o y i n two p a r t s e q u a l
the total
s u r f a c e o f one a l l o y i s much l a r g e r
a p l a t e o f AqBI_ q embedded i n a l a r g e e x c e s s o f AyBl_y p a r t i c l e s )
kind of a n a l y s i s
the
s u c h a way t h a t
= r-s-t).
3) I f b o t h A and B a r e e v a p o r a t i n g , o r i f
(e.g.
can
have t o be made as d i s c u s s e d p r e v i o u s l y .
2) From t h e d i f f u s i o n
In t h e f i r s t
2.23 and 2.2~ has
i n F i g . 2.23 b t h e Matano p l a n e h a s t o be t h e p l a n e x 0 . T h i s p o s i t i o n
be found g e o m e t r i c a l l y
the other
in Figs.
applies
case,
provided that
the net difference
This means t h a t an e q u a l t r a n p o r t
infinite
diffusion
couples still
than
a same
can he assumed.
(A-B) has t o b e e q u a l t o Area D+E = Area K+G+F.
o f A and B l e a d s t o a Matano p l a n e c o i n c i d i n g w i t h t h e
outer surfaces of both ~ltoy~.
In t h e s e c o n d c a s e ,
t h e c o n c e n t r a t i o n p a t t h e o u t e r s u r f a c e would be much c l o s e r
to the
t
value y, making x0-x I < x2-x o . The two penetration curves can then not be joined to one
continuous curve llke in Fig. 2.2~: two different Matano planes can be defined, one for
each alloy.
A discontinuity
in the slope will arise at the outer surfaces
in such a way, that the
same diffusion coefficient is found whether the one or the other alloy is evaluated at
their outer surface concentration.
In f a c t ,
the message i s simple:
penetration
curve,
p o s i t i o n of the o r i g i n a l
extension
of
if
diffusion
the mathematical position
these
surface,
coefficients
which may be s i t u a t e d
considerations
have t o be c a l c u l a t e d
from a
o f t h e Matano p l a n e h a s t o c o i n c i d e w i t h t h e
interesting
inside or outside the alloy.
conclusions
position of, and the composition at the Kirkendall
plane
can be drawn
[42].
From an
concerning
In the specific
the
example
chosen here, this plane coincides with x I at N A = p in Fig. 2.2A.
3. MULTIPHASE DIFFUSION IN TERNARY SYSTEMS
In t h e
introduction
the
important difference
a l r e a d y m e n t i o n e d . The e x t r a
degree of
b e t w e e n b i n a r y and t e r n a r y
freedom i n a t e r n a r y
system gives
rise
to
in d i f f u s i o n
the composltion of the d i f f u s i o n
zone and t h e p h a s e d i a g r a m becomes much more c o m p l i c a t e d .
the
diffusion
average
diffusion
path:
this
compositions in
zone.
between
p a p e r , K i r k a l d y and Brown [62] f o r m u l a t e d a number o r r u l e s which r e l a t e
the composition of the diffusion
so-called
the relation
the
o c c u r r e n c e of two-phased l a y e r s
In a c l a s s l c a l
c o u p l e s , end e s p e e l a l l y
s y s t e m s was
zone w i t h t h e p h a s e d i a g r a m . They u s e d t h e c o n c e p t o f t h e
i s a l i n e on t h e t e r n a r y i s o t h e r m , r e p r e s e n t i n g
planes
parallel
to
the
original
interface
the locus of
throughout
the
82
F, J. J. van Loo
C
}
A /--I-.
~ [-:-X
%,--X B
X
T
C
T
(a)
C
(b)
C
A
XlT
B
X
A
0
C
Fig.
3.1.
o
°
o
o
°T°
q.,o
c
oz-porec,o
o
(c)
B
X
0
o
(d)
Examples o f p o s s i b l e m o r p h o l o g i e s f o r t h e r e a c t i o n l a y e r i n a d i f f u s i o n c o u p l e
X/C. The c o r r e s p o n d i n g d i f f u s i o n p a t h s a r e p l o t t e d on t h e i s o t h e r m s [ 6 5 ] .
Multiphase Diffusion
In F i g .
viz.
T.
3.1 a hypothetical
p h a s e d i a g r a m shows t h e e x i s t e n c e
t h e t h r e e e l e m e n t s A, B and C, two b i n a r y
In
Figure 3.1
morphologies
four
diffusion
of t h e d i f f u s i o n
t h e most i m p o r t a n t
created.
paths
are
given,
l i k e T and Z i n F i g .
3.lb.
together
for the diffusion
viz.
the conservation
the path to cross the straight
o n c e , and g i v e s c o n s t r a i n t s
about the relative
of s i x s i n g l e - p h a s e
compounds X and Z, and t h e t e r n a r y
zone,
constraint,
This forces
83
with
the
re$ions,
ccQpound
corresponding
c o u p l e C v e r s u s X. They a l l
o f m a s s : no m a t e r i a l
fulfill
can b e l o s t
or
c o n n e c t i o n l i n e b e t w e e n C and X a t l e a s t
thicknesses
of the various
O b v i o u s l y , t h e Z - p h a s e h a s t o be much t h i n n e r
diffusion
layers
t h a n t h e T - p h a s e in
o r d e r t o comply w i t h t h e mass b a l a n c e .
Not o n l y t h e r e l a t i v e
related
T-phase fast,
barrier,
path.
then obviously
i n t h e c a s e of F i g .
faster
thicknesses,
to the diffusion
3.1.d.
but also the total
If e.g.
the total
layer width for Figs.
In the first
examples,
w h e r e a s i n t h e c a s e of F i g . 3 . 1 . d ,
diffusion
Z-particles.
thickness
the total
3.1.b or c will be smaller than
layer
Model f o r t h e n r e d i c t i o n
phenomenon a r e g i v e n i n R e f s .
thickness
a c t s as a k i n d of
i s g o v e r n e d by t h e
hampered by t h e d i s c o n t i n u o u s
63 and 64.
of the laver seuuence
For a number o f r e l a t i v e l y
simple systems it
from t h e p h a s e d i a g r a m and k i n e t i c
65-67 f o r s o - c a l l e d
layer is
i s v e r y slow and i n t h e
the continuous Z-layer
in the T-phase, which is probably only little
Examples o f t h i s
of the diffusion
in the Z-phase the diffusion
displacement
is possible
to predict
the diffusion
d a t a a c c o r d i n g t o a model e x t e n s i v e l y
reactions
described
path
in Refs.
of the type
A+BX~AX+B
a l t h o u g h more i n t r i c a t e
Referring
using three
reactions
for details
real
c a n a l s o be u n d e r s t o o d by u s i n g t h i s model.
especially
to Ref.
systems as examples.
[65] we w i l l
J u s t summarise h e r e t h e main i d e a s ,
The most i m p o r t a n t s t a r t i n g
point
is,
t h a t no e l e m e n t
~ntrlnsically diffuses, i.e. jumps from one lattice site to another, up its own activity
gradient.
interdiffuse
The word
up i t s
"intrinsically"
own a c t i v i t y
e l e m e n t by t h e i n t r i n s i c
one l a t t i c e
its
site
own a c t i v i t y
diffusion
to another
gradient.
is very
gradient,
important here.
but
that
behaviour of the other
the third
element drifts
gradient
does often
t h a n imposed upon t h i s
two: b y t h e i r
a t o m i c J,-,ps from
w h i c h i n d e e d may be up
T h i s h a s m a t h e m a t i c a l l y b e e n shown i n [ 4 2 ] .
for an element in a diffusion
always have t h e same d i r e c t i o n .
In a t e r n a r y
Darken e x p e r i m e n t
instance,
[68,69],
is
in a direction
The r e a d e r must b e v e r y w e l l aware of t h e d i f f e r e n c e
the concentration
An element
transport
for
system, this
up-hill
between the activity
couple.
In a b i n a r y
is not the case.
diffusion
of
carbon
is
gradient
system,
and
these
In t h e well-known
f o u n d from a
concentration point of view, but from an activity point of view carbon diffuses down its own
activity gradlent throughout the whole couple.
Let us take as a first example the reaction
Co + Cu20 • CoO + 2Cu
at 10OO'C. The relevant part of the phase diagram is shown in Fig. 3.2a, whereas the oxygen
activity from the Cu/Cu20 side towards the Co/CoO side is shown in Fig. 3.2.b. From the
phase stability diagram, the reaction proceeds because CoO is much more stable than Cu20.
FA
F.J.J. van Loo
Diagram 3 . 2 . b i s c o n s t r u c t e d f r c a a knowledge of the thermodynamic &O'-values for Co0 and
Cu20 , but may i n a q u a l i t a t i v e way a l s o be c o n s t r u c t e d from the phase diagram by the slope
of the t i e - l i n e s
[65].
~o0
~
Cu2 CoO 3
CuO
1.2
,8"
=..
.
=o
i
8
\\\
.4
0
-6-
\ \\,
\
,
CoO * Cu20
{a)
nNi/nNi~nFe
-10"
|!
~ NiO
CoO ~" Cu
w
l
f•"Fel-xO
/
.14 ¸
-12
•]
7 -(Fe,N0
Co * Cu
Co
I
!
.2
.4
"
I
|
.6
.8
F,
Cu
.I
!
.4
I
.;
.6
Ni
Ibl
(b)
"E -10
•0.
8
Co
CoO
~-10
"12
100 IJm I
(C)
Fig. 3 . 2 .
Fig. 3 . 3 .
.J
Fe Fel.xO ~
~
NiO
(c)
a) Phase r e l a t i o n s and d i f f u s i o n path i n Co-Cu-O system a t IO00"C.
b) The oxygen p r e s s u r e as a f u n c t i o n og the metal mole r a t i o . ~ e dots r e p r e s e n t
the oxygen p r e s s u r e i n e q u i l i b r i u m with the s o l i d s o l u t i o n s Cu(Co) and Co(Cu).
¢) A schematic r e p r e s e n t a t i o n of the l a y e r morpholo$y and e q u i l i b r i u m oxygen
p r e s s u r e i n the couple Cu/Cu20 a f t e r a heat treatment of 100 h a t IO00"C [65].
Phase r e l a t i o n s and d i f f u s i o n path (a) as well as the e q u i l i b r i u m oxygen pressure
(b) i n the Fe-Ni-O system a t 1000"C; ¢) r e p r e s e n t s the l a y e r morphology and
oxygen p r e s s u r e i n an Fe/NiO d i f f u s i o n couple a f t e r 100 h a n n e a l i n g a t 1000"C
[65].
Multiphasc Diffusion
In F i g . 3 . 2 . c
the actual
oxygen i s
continuously
holds
the activities
for
85
layer morphology in a diffusion
r i s i n s frme one s i d e
the other
to
of Co and Cu, w h i c h a r e n o t
thermodyuamic p o i n t o f v i e w t h i s m o r p h o l o g y i s a l l o w e d ,
in Fig. 3.2.a.
From a k i n e t i c
through the resulting
the diffusion
layer
diffuse
path,
i.e.
Co-Cu-CoO-Cu20,
grounds.
through the Cu-layer
It
is not forbidden
The r a t e
limiting
predictions
step is
about the total
for
thermody~mmic r e a s o n s ,
but
oxygen i 8 immobile and Co h a s t o
t o form CoO and t h e s o l i d
steps
a
[65,70].
would mean t h a t
i o n s have t o move t h r o u g h Co0. T h e s e k i n e t i c
frol
path as given
s i n c e t h e d i f f ~ m i o n o f oxyKe~
[70,71].
t h r o u g h CoO, and i n d e e d q u a n t i t a t i v e
on k i n e t i c
of
The same
that
leadin 8 to a difftmion
p o i n t o f view no p r o b l e m e x i s t s ,
turn out to agree with the experiments
The r e v e r s e
very unlikely
without any extrm~m.
shown. T h i s m e a n s ,
C u - l a y e r t o form Co0 i s v e r y f a s t
of Co-ions
thickness
c o u p l e i s s h o ~ u . The a c t i v i t y
solution
of Co i n Cu20, w h e r e a s Cu
a r e much s l o w e r t h a n t h o s e i n t h e l a y e r
s e q u e n c e shown i n F i g . 3 . 2 . c .
In t h e s y s t e m F e - N i - O ,
the slopes of all
oxygen a c t i v i t y
tie
the phase diagram is a type related
lines
h a v e t h e same s i g n ,
as a function
sequence Fe/Fe-oxide/Ni/NiO is
however, i s d i f f e r e n t
but penetrate
in the sense,
confirmed by (65].
t h e Fe-Ni s o l i d
grows
that
t h e formed F e - N i s o l i d
t w o - p h a s e d l a y e r was p r e d i c t e d
Fe-ions
of the metal ratio,
then predicted.
Fe/Fe-oxides/Ni/NiO
i n a more o r l e s s
the
morphology is
can,
Then e a c h a c c i d e n t a l
straight
however,
interface
still
and b . A l a y e r
that
the diffusion
layer,
not
be recognieed
stable.
at
at
layer
way. T h i s
[70,72]
and
o f oxygen t h r o u g h
compared t o t h e f a s t
perturbation
is
finger-shaped
shown b y Rapp and c o w o r k e r s
i s now s l o w e r and r a t e - l i m i t i n g ,
through the oxides.
preferentially:
3.3.a
m o r p h o l o g y of t h e r e a c t i o n
t h e i r o n o x i d e s a r e n o t formed i n a s t r a i g h t
solution
this
a s shown i n F i g .
The a c t u a l
and e x p e r i m e n t a l l y
The r e a s o n f o r
solution
t o Co-Cu-O i n t h e s e n s e t h a t
w h i c h l e a d s t o a same t y p e o f p l o t f o r t h e
diffusion
of
the Fe-oxide/Ni boundary
The
Original
the boundaries
sequence
Fe/Fe-oxide
and
Ni/NiO.
-12Cu2S * Ni3S 2
'-
(Cu,Ni)
(b)
Cu
.2
(a)
.4
.6
.8
~
Z
Ni
c
~ nNi/nNi ÷ nc u
!
u,Ni
Cu2S
___....4~ nNil nNi ÷ nc u
l "
"ls-ir~P
;
I
I
"
I
(c)
Fig. 3 . 4 .
Phase relation8 and diffusion path (a) as well as the equilibrium sulfur pressure
(b) i n t h e Cu-Ni-S s y s t e m a t 500"C; ( c ) r e p r e s e n t s t h e l a y e r m o r p h o l o g y and
s u l f u r p r e s s u r e i n a Cu/Ni3S 2 d i f f u s i o n c o u p l e a f t e r 100 h a n n e a l i n g a t 500°C
[65].
~6
F.J.J.
van L o o
An e s s e n t i a l l y d i f f e r e n t phase diagram i s shown f o r the Cu-Ni-S system in F i g . 3 . ~ . a and
b. Because of the changing s i g n of the s l o p e s of the t i e l i n e s l e a v i n g from the m e t a l l i c
solid solution,
a maximum i n
the s u l f u r a c t i v i t y
is
found in the
three-phase
region
Cu2S-Ni3S2-Cu,Ni. Then s u l f u r i s thermodynamically n o t allowed to d i f f u s e i n t r i n s i c a l l y
Ni3S 2 to the C u - s i d e , because then i t would d i f f u s e up i t s
least at
the s t a r t
of the p r o c e s s s u l f u r
is
own a c t i v i t y
gradient.
immobile, which leads to an i n i t i a l
from
So a t
layer
sequence Cu/Cu,Ni/Cu2S/Ni3S 2 as indeed found [65].
The p r e d i c t i o n s from t h i s kind of c o n s i d e r a t i o n s f o r r e a s o n a b l y uncomplicated systems
show a p e r f e c t agreement w i t h the e x p e r i m e n t a l r e s u l t s ,
[65,66].
16 of which a r e presented in
In more complicated systems, a d a pt i o n s have to be made such as shown f o r the
Fe-Nt-B and Fe-Co-B systems [73]. In s t e m a r i s i n g , the morphology of the d i f f u s i o n zone in a
t e r n a r y couple can be p r e d i c t e d by a schame such as shown in F i g . 3 . 5 . Because complicated
systems have n o t y e t been s t u d i e d in t h i s r e s p e c t (with an e x c e p t i o n f o r the Ti-A1203 and
Ti-SiO 2 systems i n v e s t i g a t e d by B i l l e l
[7&]), i t might be s a f e r to conclude t h a t the model
can d e f i n i t e l y p r e d i c t which l a y e r morphology i s ~
,
leavin& open in some cases more
then one p o s s i b i l i t y which the model cannot decide [73]. F u r t h e r r e s e a r c h i s t h i s f i e l d to
find additional c r i t e r i a
i s p r o g r e s s i n g . The expart~.-nts r e c e n t l y published by the group of
Chang [ e . g :
75] on the i n t e r r a c i a l
interesting
in t h i s r e s p e c t , a l t h o u g h in t h e s e r e a c t i o n s n o n - e q u i l i b r i u m s i t u a t i o n s a r i s e
after
a n n e a l i n g times or low t e m p e r a t u r e s .
short
r e a c t i o n s between metals and Calliom Arsenide are
The phenomena r e p o r t e d f o r a Pt-CsAs
d i f f u s i o n couple a t 600"C [75] a r e c o m p l e t e l y in l i n e with the p r e d i c t i o n s by our model.
~X
sxl
ax
mI
".8-
.8-
g
+
+
=<
-x
I .4,
¢: .4
o
^
,~.
I
.4
.6
I
.8
0
B
A
a)
c)
Fig. 3,5.
.
.4
.6
.8
b)
o~.,,.~,
d)
Basic s o r p h o l o g i e s of the r e a c t i o n zone f o r the dlsplacement r e a c t i o n
A + BX~ B + AX in r e l a t i o n to two p o s s i b l e types of phase diagrams. In (a) the
t i e l i n e s have the same s i g n of s l o p e s , l e a d i n g to morphologies shown in (c)
depending on the k i n e t i c s i n the system. In (b) the d i f f e r e n t sign of the slopes
of the t i e l i n e s p e r m it s only morphology ( d ) .
Multiphas¢ Diffusion
ExDerie~ntal determination
The r a t e - l i s i t i n g
elegant
that
step
way b y a p p l y i n g
materials
in the diffusion
one o f
the
the application
o f an e x t r a
proving that
however,
diffusion
a NiO-layer
drastically
influence
decreases
step
process
reaction
b e f o r e making t h e d i f f u s i o n
c o u p l e w i t h Ni d i d n o t
layers,
of t h e r a t e - l i m i t i n e
87
c a n sometJJes b e f o u n d i n a r a t h e r
products
already
c o u p l e [ 7 1 ] . For i n s t a n c e ,
layer
to
one o f
the
on t h e
Ni-surface
rate,
shown
o f p u r e Cu on t h e Cu20 end member i n a d i f f u s i o n
the thickness
o f t h e n e w l y formed Cu and N i 0 - r e a c t i o n
o f oxygen t h r o u g h Cu i s n o t t h e r a t e - l i m i t i n g
the reaction
starting
i t was e x p e r i m e n t a l l y
before
making a
proving that
diffusion
couple
the diffusion
step.
Applying,
with
Cu20
of N i - i o n s
t h r o u g h NiO
is the rate-limltlng step.
A n o t h e r way o f
starting
layers
finding
materials
are often
the rate-limiting
which are not in direct
step
is
contact
by i n v e s t i g a t i n g
with
formed on t h e s e s i d e s b y e v a p o r a t i o n
the other
as s c h e m a t i c a l l y
Cu-layer,
shown i n F i g .
3.6.
capsule
solid-state
formed a t
diffusion
the diffusion
the
couple,
outer
the ratio
which i s g o v e r n e d b y t h e slow d i f f u s i o n
follows
that
the NiO-layer).
The t h i c k n e s s
t h e much f a s t e r
diffusion
i s a b o u t 90 [ 7 0 ] ,
Cu20,
b/a actually
couple,
of Ni-ions.
the thickness
diffusion
was much t h i c k e r
(From t h e r e a c t i o n
of the Cu-layer is
the diffusion
The
in the
in
e q u a t i o n and t h e
1.5 times the thickness
of t h e C u - l a y e r on t h e f r e e s u r f a c e s
for both the free
couple.
than
t h a n a l l o w e d b y t h e f o r m a t i o n of NiO,
calculated
In a Cu20-Fe d i f f u s i o n
surfaces
of
o f Cu20 i s d e p e n d e n t on
o f oxygen atoms t h r o u g h Cu. The t h e o r e t i c a l l y
since
s i d e s of t h e
forming a NiO-layer
b e i n g a b o u t 90. The r e a s o n i s c l e a r :
in agreement with the observed value.
does n o t a p p e a r ,
diffusion
of
c o u p l e t h e C u - l a y e r c a n n o t grow f a s t e r
m o l a r volumes i t
effect
sides
[65].
in which extra
From t h e f r e e
t h i c k a s t h e one formed i n t h e g e n u i n e s o l i d - s t a t e
however,
the
end member. D i f f u s i o n
Cu20 end member oxygen e v a p o r a t e d and r e a c t e d w i t h t h e e x c e s s of n i c k e l ,
w h i c h was e q u a l l y
of
o f one o r more of t h e e l e m e n t s
T h i s was f o u n d i n a Ni-Cu20 c o u p l e a n n e a l e d i n an e v a c u a t e d s i l i c a
p i e c e s of Ni were p r e s e n t ,
the sides
ratio
b/a
couple this
o f Cu20 a s w e l l a s t h a t
in the
o f oxygen t h r o u g h Cu i s r a t e - l i m i t i n g .
A third possibility is the use of a diffusion couple in which an alloy of the metals is
used instead of a pure metal. This was demonstrated for the Ni-Cu-O system [63] by making
couples of the type NixCUl_x/CU20. For a value of x lower than about 0.~, Ni cannot be
supplied fast enough by diffusion in the end member alloy to form a continuous layer of NiO.
Then, if fact, internal oxidation occurs, and the diffusion of oxygen through the Cu-Ni
alloy
is rate-limiting.
The t o t a l
layer width (alloy
+ NiO-precipitates)
t h a n i n t h e c a s e where a c o n t i n u o u s N i O - l a y e r i s f o r m e d , p r o v i n g t h a t
NiO was t h e r a t e - l i m i t i n g
i s now much l a r g e r
the diffusion
through
s t e p i n a Ni-Cu20 c o u p l e .
~/ ..;-':t::i.......~.:
~:~li!i:411',,'--N i 0
m m
Fig. 3.6.
m mm
m m /
Schematic illustratlon
o f t h e l a y e r m o r p h o l o g y i n a d i f f u s i o n c o u p l e Ni/Cu20 a t
1000*C, a n n e a l e d i n a c l o s e d s i l i c a c a p s u l e i n t h e p r e s e n c e of e x c e s s N i .
88
F.J.J. van Loo
In t h e same s y s t e m , a c o m p a r a b l e e f f e c t
present
in
the
NiO-particles
faster
Cu20-half
of
a Ni/Cu20
to a continuous
total
layer,
was f o u n d i f s m a l l amounts o f c h l o r i d e
couple.
leading
These
prevented
the
to randomly dispersed
layer growth, again id.ontifying the rate-limiting
sintering
i o n s were
of
NiO p a r t i c l e s
the
and much
step in the diffusion
process
[64].
Influence of kinetic
p a r a m e t e r s on t h e d i f f u s i o n
Up t o now we were d i s c u s s i n g
oath
the morphology of the reaction
products
phase stabillty
d i a g r a m s m a i n l y from a thermodynamic p o i n t o f v i e w .
and e s p e c i a l l y
in solid
solutions
formed from t h e s t a r t i n g
in relation
In slngle-phase
materials
kinetic
to the
regions,
factors
are
o f t h e end members, t h e d i f f u s i o n
path
a l s o v e r y i m p o r t a n t a s shown i n [ 4 2 , 7 6 ] .
As a g e n e r a l
starts
trend,
in the single-phase
in the direction
of the faster
component, and away from t h e f a s t e r
this
component. I f t h e r a t i o
this
rule,
together
that
the diffusion
from t h i s
kinetic
For i n s t a n c e ,
diffusing
prediction
BC, t h e n t h e d i f f u s i o n
information
For t h e a u t h o r ,
to undertake,
path.
work on t h i s
purposes.
(Fe-Si-C/Fe-C)
Note,
however,
c a n n o t b e found
of decreasing
carbon activity
d e t e r m i n e d by t h e t h e r m o d y n a m i c s .
path, needs detailed
task
Darken e x p e r i m e n t
path follows the direction
in a phase diagram such as Fig. 3.7,
of t h e e x a c t p o s i t i o n
dependence.
permits a rough idea about the
w h i c h i s enough f o r many p r a c t i c a l
path in the classical
this
I f we now c o n s i d e r
path can be estimated
C is
about the diffusivities
calculation
seems,
determination
[77,78],
of a l l
generally
is
the fastest
a diffusion
by these
c o u p l e of
considerations.
o f p o i n t p, as w e l l a s t h e e x a c t c u r v a t u r e
and an e x p e r i m e n t a l
Backhaus-Ricoult
region.
where t h e e l e m e n t
e l e m e n t s and t h e i r
speaking,
mutual
a too formidable
t h e o n l y way t o o b t a i n
who p e r f o r m e d b o t h e x p e r i m e n t a l
The
of the diffusion
an exact
and t h e o r e t i c a l
p r o b l e m came t o t h e same c o n c l u s i o n .
C
BC
B
Fig.
3.7.
in
r e m a i n s t h e same t h r o u g h o u t t h e whole s y s t e m ,
of t h e mass b a l a n c e ,
component i n t h e s i n g l e - p h a s e
pure A versus
diffltsion
path,
component i n t h e end member p o o r i n t h i s
component i n t h e end member w h i c h i s r i c h
of the mobilittes
model, since this
and i s e x c l u s i v e l y
diffusing
diffusing
with the constraints
c o u r s e of t h e d i f f u s i o n
regions
E s t i m a t e d d i f f u s i o n p a t h ( : : : . ) f o r a d i f f u s i o n c o u p l e AIBC w i t h o n l y t h e
knowledge t h a t t h e e l e m e n t C i s t h e most m o b i l e component. The t i e l i n e s a r e
indicated as thin dotted lines.
Multiphas¢ Diffusion
I n f l u e n c e o f t h e elm n a r t i a l
89
o r e s s u r e on t h e d i f ~ u s i o n o a t h
As soon a s E a s e s s u c h a s oxyEen and n i t r o E e n , b u t u n d e r some c i r c u m s t a n c e s a l s o s u l f u r or
phosphorus, take
part
in the reaction,
t h e gas at~eosphere d e c i d e s w h i c h r e a c t i o n s
o c c u r . Take a s an e . ~ m p l e t h e r e a c t i o n b e t w e e n Ni and A1203 ( s e e a l s o
[78,66]).
~,ight
The p h a s e
d i a g r a m i n t h e N i - r i c h c o r n e r i s s c h e m a t i c a l l y shown i n F i E. 3 . 8 .
I f t h e a n n e a l i n g o f a Ni/AI203 d i f f u s i o n
e x t r e m e l y low i n oxygen p a r t i a l
pressure
c o u p l e i s p e r f o r m e d i n an a t m o s p h e r e which i s
the diffumion path will
start
from v i r t u a l l y
the
p u r e N i - c o r n e r a l o n E p a t h 1. No new r e a c t i o n p r o d u c t i s formed and o n l y a s o l i d s o l u t i o n of
A1 and O i n Ni i s f o u n d . OxyEen p e n e t r a t e s much d e e p e r i n t o n i c k e l
their
difference
in diffusivity,
obey t h e mass b a l a n c e ,
If,
i.e.
however, a n n e a l i n g
solution
have t o o c c u r i n t h e r a t i o
takes
o f oxygen i n p u r e N i ,
corner position,
s u c h a way t h a t
i.e.
place at
3:2.
an oxygen l e v e l
the diffusion
path starts
which a l r e a d y c r e a t e s
at
a point different
p a t h 2. Then, a c c o r d i n g t o t h e mass b a l a n c e ,
spinel
is
t h a n aluminum b e c a u s e o f
b u t t h e a b s o l u t e v a l u e s o f t h e d i s s o l v e d amounts have t o
f o r m e d . T h i s i s what was p r e d i c t e d
a solid
from the
the path might develop in
and e x p e r i m e n t a l l y found by
Backhaus-Rtcoult [78].
This effect
can p r o b a b l y i n f l u e n c e t h e morphology o f t h e r e a c t i o n
diameter of the couple halves.
In Fi E. 3.9 t h e c o u p l e p u r e Ni/A1203 i s a n n e a l e d i n an oxyEen
a t m o s p h e r e e q u a l t o t h e e q u i l i b r i u m p r e s s u r e Ni/NiO. However, i t
diffuse
far
saturated
enough t o
reach point a in the couple.
solid solution
experimentally
spinel
o f oxygen i n Ni i s r e a d i l y
formation at
positions
of the i n i t i a l
could easily
reactions
diffuse
is
alone the interface
and form s p i n e l
Ni and Si3N 6 [ 7 9 ] .
e x p e r i m e n t a l i n f o r m a t i o n on t h i s
0
JPSSC 20:1-1;
on t h e
oxygen
we a r e
tryin s to set
more
behaviour.
C)
-phase
D i f f u s i o n p a t h s f o r Ni/A1203 c o u p l e s a t d i f f e r e n t
[66,78].
o f oxygen and
e v e r y w h e r e . The same h o l d s f o r
Ni
FiE. 3.8.
solution
t h e r e i s n o t good c o n t a c t ,
In o u r l a b o r a t o r y
triple-point
on t h e o t h e r h a n d , a
found p r o b a b l y d e p e n d s l a r g e l y
c o n t a c t b e t w e e n Ni and A1203. I f
between e.E.
2
At p o i n t s b ,
t a k e s t i m e f o r oxygen t o
f o r m e d . T h e r e f o r e , one e x p e c t s t o f i n d
b and o n l y a s o l i d
aluminum i n Ni a t p o i n t a . Whether o r n o t t h i s
quality
l a y e r d e p e n d e n t on t h e
;
At
oxygen p a r t i a l
pressures
90
F.J.J. vanLoo
o2 (NqNO)
a
Ni
Fi&.
3.9.
P r e d i c t e d d i f f e r e n t r e a c t i o n schemes i n a Ni/A120 3 couple in the middle (a) and
the t r i p l e p o i n t s (b) during a n n e a l i n g in an oxygen atmosphere equal to the
p r e s s u r e o£ the Ni/Ni0 e q u i l i b r i u m .
A r e l a t e d experiment f o r the Cu-Si-P system has a l r e a d y been performed [80]. The reacticm
between Cu3P emd Si appears to be l a r g e l y i n f l u e n c e d by the phosphorus p a r t i a l p r e s s u r e , as
was shown by performing the r e a c t i o n a t 800 K i n two ways, v i e . in a c l o s e d s i l i c a v e s s e l or
in a dynemlc vacuum of 10 -9 a t = .
In the f i r s t
c a s e , the phases Cu3Sl end SiP were formed in the l a y e r sequence p r e d i c t e d
by our model ac c o r d i n g to the phase dlaKrem and the d l f f u s i v l t l e s
as shown in Fig. 3.10 a
and b. I n s i d e the c a p s u l e the p a r t i a l P 4 - p r e s s u r e of SiP (about 0.03 atm.) was p r e s e n t .
In the dynamic vacuum clrc~nnstances, the SiP phase i s n o t s t a b l e because of i t s high
p a r t i a l phosphorus p r e s s u r e and the phase diagrem then looks l i k e F i g . 3.10 c. Accordin& to
the same model, the l a y e r sequence as shown i n Fig. 3.10 d has to be formed, whereby
phosporus e v a p o r a t e s a t the i n t e r f a c e Cu3P/Cu3SI. This indeed was found e x p e r l m e n t a l l y [80].
/
p
/~
~
P~gas
~
// ~
dosed
system
•
/ /
\
openvac.
system
10" atm.
SiP
Cu
Si
Y Cu3Si
Cu('/([ II~\V/
Y Cu3Si
\ Si
P~ .gas
!
(b)
Cu3P ~
Si
(O)
Cu3P
Cu3Si
|
J
Fig. 3.10. D i f f e r e n t phase diagrams f o r the Cu-Si-P system and r e a c t i o n l a y e r s in S i / C u ~
d i f f u s i o n couples when annealed in a c l o s e d system (a and b) or in a dynamic
vacuem of I 0 - a ~ (c and d) [80].
Multiphase Diffusion
The p h o s p h o r u e gas c o u l d e s c a p e d e s p i t e
The c o u p l e d i d n o t
temperatures
fall
apart.
tamperatures,
no p l a s t i c
applying an external
P o r e s were v i s i b l e
where Cu3Si was r a t h e r
ductile
at
and,
Related
the last
to
change a d i f f u ~ i o n
the
d e f o r m a t i o n of Cu3Si o c c u r r e d ,
of 20 kg/cm 2 (2 NPa).
interface
Cu3P/Cu3Si a t
plastically
deformed.
so t h a t p o r e s were p r e s e n t
bish
At low
in such
columns b e t w e e n Cu3P and S i [ 8 0 ] .
o f t h e end members on t h e d i f f m a l o n Da~h
subject
path.
pressure
therefore,
l a r g e d i m m s i o n e t h a t Cu3Si seemed t o form i n i s o l a t e d
I n f l m m e e of t h e s t o i e h i o m a t r v
91
is
the question whether small deviations
The a n s w e r depends on t h e s y s t a m a c t u a l l y
in stoichicmetry
at
hand,
as s e e n by
t a k i n g two e x a m p l e s .
In t h e c a s e o f Ni/Cu20 t r e a t e d
I f Cu20 would b e s l i g h t l y
before,
supersaturated
the stoichicaetry
i n o x y g e n , CuO would b e p r e s e n t
b u t i t would b e r e d u c e d t o Cu20 b y t h e f o r ~ a t i o n
further
reduction
form. A large
o f Cu20 w i l l
excess
I f Cu20 i s u n d e r s a t u r a t e d
reaction
o f NiO. I f t h i s
o c c u r and e v e n t u a l l y
o f oxygen would l e a d
r e d u c t i o n of Cu20 s i n c e t h e n e c e s s a r y
of Cu20 i s n o t v e r y i m p o r t a n t .
transport
in oxygen,
the layer
to a thick
i.e.
from t h e b e g i n n i n g ,
is only a small ~sount,
the
s e q u e n c e Cu20/Cu/NiO/Ni w i l l
NiO-layer,
delaying
the
further
o f Ni i o n s t h r o u g h NiO i s t o o s l o w .
if
some f r e e Cu i s p r e s e n t ,
i s s e e n b e c a u s e t h e p r e s e n c e of a Cu l a y e r does n o t i n f l u e n c e
no d i f f e r e n c e
the reaction
in
rate or
layer sequence.
In o t h e r
types of systems,
system Ti-8i-C
the reaction
with excess Si or C [81].
In the first
t o SiC, w h e r e a s i n t h e l a s t
for the large differences
our laboratory,
the influence
of t h e s t o i c h i o m e t r y
might be larger.
In t h e
b e t w e e n Ti and SiC m i g h t be c o m p l e t e l y changed by t a k i n g SiC
case this
case,
given by various
we a r e c u r r e n t l y
it
is not possible
is quite well possible.
research
investigating
t o form t i t a n i u m c a r b i d e
This effect
groups studying
the system especially
next
probably accounts
this
system [82-8~].
in this
respect
In
[81].
The o c c u r r e n c e o f a D e r l o d l c l a v e r f o r m a t i o n
In their
periodic
recent
book,
precipitation
K i r k a l d y and Young [85]
g i v e a number o f e x a m p l e s
was f o u n d i n t e r n a r y
and h i g h e r o r d e r
systmns.
the well-known Liesegang bands observed e.g.
i n some o x i d a t i o n
experiments.
In o u r l a b o r a t o r y
Oslnakl et al.
Zn-Co2Si. Nehan and J a c k s o n
structure
Fig.
layer
those
these
3.11 a - c
in
the relevant
sequence for
with,
al.
Pt-SiC
reported
couples
different
specific
respectively,
diffusion
couples,
species
It
Kirkaldy's
of
periodic
structure.
structures
is
closely
interesting
approach.
to
In
p a t h and
from o u r thermodynamic model.
( a s shown b y m a r k e r e x p e r i m e n t s ) ,
Fe, Co and S i t o form t h e v a r i o u s
intermetallics
FexZny, CoxZny
a r e t h e n F e S i , CoSi and C, r e s p e c t i v e l y .
was i n d e e d f o u n d e x p e r i m e n t a l l y
FexZry,
Z n - F e 3 S i and
with the diffusion
after
short
After
annealing
CoxZny and N i x S i y a t
regular
distances
times,
longer annealing
z o n e s showed t h e F e S i , CoSi and C p h a s e s as p o r o u s l a y e r s ,
matrices
to
t h e same t y p e of
periodic
[90].
from
as p r e d i c t e d
the only diffusing
p h a s e s F e S i , CoSi and C were formed a s p o r o u s l a y e r s .
the diffusion
[89]
Recently,
p h a s e d i a g r a m s a r e shown t o g e t h e r
and N i x S i y . The p r o d u c t s l e f t
This situation
for
a way c o m p l e t e l y
The components Zn and Ni a r e
reacting
et
couple Ni-SiC.
i n N i - S i C were r e p o r t e d
systems
87] f o u n d e x a m p l e s i n t h e c o u p l e s
and S c h i e p e r s
in the case of the diffusion
resembling
consider
[88]
[86,
in which a
They a r e r e l a t e d
whereby t h e
t i m e s , however,
precipitated
in the
and f o r m i n g a n i n t e r e s t i n g
92
F.J.J.
van Loo
Fhs~ ~ I
-3
j~
N
3
395"C
=u
-2
Zn
Y', m~--'~-
Si
zn
I
¥2Zn
Si
I°[°
,1 l 3s,
Si C 2Si
(a)
(b)
NiSi;
l-0
÷
Z
u~
=t
NiSi
Si C
6
Y
P_
(Ni)
Ni
C
N, j,rY
N,,s,2r,jc[s,c
(c)
Fig. 3.11. Schematic i s o t h e r m a l phase diagrams and the p r e d i c t e d l a y e r sequences in s p e c i f i c
d i f f u s i o n couples in the systems Fe-Si-Zn ( a ) , Co-SI-Zn (b) and Sl-C-Ni ( c ) .
Multiphase Diffusion
Our q u a l i t a t i v e
studied cases
interfaces
a certain
explanation
is
the following,
FeHSi-Zn and S i C - N i .
Because o f
93
t a k i n g a s an example t h e most i n t e n s i v e l y
the mechanical stresses
b e t w e e n FeHSi and FeSi o r SiC and C, r e s p e c t i v e l y ,
thickness before
up a t
the
t h e s e p h a s e s can o n l y grow t o
t h e y l o s e c o n t a c t w i t h t h e end member. I m m e d i a t e l y t h e r e a f t e r ,
t h e p h a s e s a r e formed a g a i n a t t h i s
periodically
built
end meaber t o t h e same l a y e r t h i c k n e s s ,
etc.
Then t h e
o c c u r r i n g p h a s e s a l w a y s s t a y i n c o n t a c t w i t h one end member and a r e n e v e r i n
c o n t a c t w i t h t h e o t h e r o n e , w h i c h i s n o t t h e c a s e i f a L i e s e g a n g mechanism i s o p e r a t i v e .
structure
r e p o r t e d f o r P t - S i C [90] can b e e x p l a i n e d i n t h e same way.
This i d e a was c o r r o b o r a t e d b y e x p e r i m e n t s i n
starting
stresses.
material
was v e r y
The i n i t i a l
thin
and,
t h e F e - S i - Z n s y s t e m i n w h i c h t h e Fe3Si
therefore,
capable
I f l e s s Si i s p r e s e n t
in the starting
material,
enough t o form a l a y e r o f FeSi a d j a c e n t
loses its
periodic
of
d e f o r m a t i o n by m e c h a n i c a l
l a y e r s e q u e n c e was s u s t a i n e d f o r much l o n g e r t i m e s , and t h e F e S i - l a y e r
grew t o an a p p r e c i a b l e t h i c k n e s s b e f o r e t h e l a y e r l o s t
fast
The
structure,
and a t
contact.
the interdiffusion
o f Fe and S i i s n o t
t o t h e Fe-Zn compounds. The r e a c t i o n
low s i l i c o n
contents,
random p r e c i p i t a t e s
layer
o f FeSi a r e
found i n t h e Fe-Zn i n t e r m e t a l l i c s .
For t h e SiC-Ni s y s t e m s , t h e e x p e r i m e n t s w i t h t h i n SiC end members have n o t y e t b e e n made
but are in p r o g r e s s .
of
graphite
precipitates
layer
in
nicely
[67,91].
sintering
In t h e
The c l o s e l y
Fe3Si
[89],
It
in
SiC-Fe d i f f u s i o n
S i C - F e x N i l _x
coincide with a larger
This e f f e c t
might be r e l a t e d
c o u p l e s show random p r e c i p i t a t e s
couples,
the
o r s m a l l e r atom r a t i o
to the different
bands
and
random
of Ni/Fe in the r e a c t i o n
i n f l u e n c e o f Fe and Ni on t h e
behaviour of graphite.
c o u p l e Co2Si/Zn
[87],
p e r i o d i c bands not p a r a l l e l
direction
related
whereas
i n t h e Co2Si c r y s t a l
would be
quantitative
worthwhile
way, r a t h e r
a strange
effect
to the interface,
to
was f o u n d ;
but parallel
from w h i c h t h e r e a c t i o n
understand
these
l a r g e number o f d i f f u s i o n
a more
t h a n i n t h e somewhat vague and q u a l i t a t i v e
each diffusion
crystallographic
products originated.
phenomena i n
Determination of a ternary isothermal cross-section
Since for
t h e p h a s e CoSi formed i n
to a certain
f u n d s m e n t a l and
terms used here.
uslne dlffu~ion counles
couple a unique diffusion
path is
followed,
in principle,
c o u p l e s must b e s t u d i e d e x p e r i m e n t a l l y f o r a t e r n a r y
a
isotherm.
C
Z
P
T
R
A
B
X
(a)
Cl
P
T
Q
(b)
F i g . 3 . 1 2 . D e t e r m i n a t i o n o f an i s o t h e r m a l s e c t i o n o f a p h a s e diagram l i k e
multiphase diffusion couples like (b).
( a ) from s e l e c t e d
9.~
F . J . J . van Leo
However,
this
materials.
larger
number can b e d e c r e a s e d
The c h a n c e t o h i t
figure,
triangle
have to be exactly
t h e n f r a m one c o u p l e
two t h r e e - p h a s e
the
three-phase
triangles
triangle
I n t h i s way, a few c o u p l e s c a n g i v e a l l
Extenslon to mmternarv
The g r a p h i c a l
in plotting
a c l e a r way a r e s o m e t i m e s f r u s t r a t i n g .
[67,91] a clear
equilibrium
consequence of
diffusion
All phases at
with
SiC,
projections
of
of
the
1,
2 the
P/Q/R e v e n more i n f o r m a t i o n
can
the
the Si-side
other
the
type
is possible.
phase diagram in
axis,
By t a k i n g t h e F e - N i - S i t r i a n g l e
t h e s y s t e m c a n b e p r e s e n t e d a s shown
of the broken line
compositions
are
in
only these
SiC/FexNil_ x.
experimental
path in a quaternary
I n some s i m p l e s y s t e m s s u c h a s t h e F e - N i - S i - C s y s t e m
representation
t h e thermodynamic m o d e l ,
couples
In area
from a r e a
the necessary information.
a diffusion
a s a b a s e a n d u s i n g NC/NFe+NNi+NSi a s a v e r t i c a l
in Fig. 3.13 a.
can be found.
T-Z-C, w h e r e a s
in
Ivet~uns
difficulties
mentioned before
i s much
known i n o r d e r t o t m d e r s t a n d
P end Q s h o u l d show a m o r p h o l o g y i n d i c a t e d
T-X-B c a n be f o u n d . By making s a n d w i c h c o u p l e s l i k e
be g a i n e d .
as starting
p r o c e d u r e i s shown s c h e m a t i c a l l y .
couple between the alloys
microprobe measurements reveal
by using polyphase alloys
at which three phases are in equilibrium
triangles
In Fig. 3.12 this
If a diffusion
this
interfaces
then, but these three-phase
the phase diagram.
appreciably
For
diffusion
paths
in the base triangle
equilibriom
latter
with
compositions
a number o f
these
on
triangle
the base
are in
graphite.
As a
c a n b e formed i n
compositions,
are
the
shown i n
Fig. 3.13 b [91].
If quaternary
presenting
phases are formed,
diffusion
the visualization
paths is better
i s more d i f f i c u l t .
Yet,
this
way of
than using the usual pyramid-type of a quaternary
phase
diagram.
4 . CONCLUSIONS
The r e a d e r will h a v e n o t i c e d
quite
qualitative
of m u l t l p h a a e
that
in character.
diffusion,
for
It
this
is
paper, expeclally
indeed intended
the possibilities
and f o r t h e e x p e r i m e n t a l means t o s e t u s e f u l
The a u t h o r
subject,
after
like
is fully
e.g.
a number of y e a r s
diffusion
processes
happens in a ternary
concentration
this
aware o f t h e f a c t
[~, 8 5 ] .
giant
qualitative
that
It
is,
diffusion
to gather
feeling
that
data,
about multiphase
correct
might occur,
with students
of,
not
if
on t h i s
and s c i e n t i s t s
especially
ternary,
I f one w a n t s t o know e x a c t l y
coefficients
what
f o r e a c h p h a s e , and t h e i r
h a v e t o b e known. I t i s t h e a u t h o r ' s
diffusion
and t h e a u t h o r h o p e s t o have c o n t r i b u t e d
which reactions
descriptions
used.
four diffusion
will
s y s t e m s , was
for the process
good b o o k s h a v e b e e n w r i t t e n
experience
to be really
couple,
these
several
his
mathe~tically
and t e m p e r a t u r e - d e p e n d e n c e ,
task
of predicting
a feeling
information.
however,
are too intricate
t h e p a r t on t e r n a r y
to present
ever be completed soon.
feeling
Therefore,
that
a
r e m a i n s v e r y i m p o r t a n t f o r t h e coming y e a r s ,
t o t h a t aim by t h i s work.
Multiphase Diffusion
c.
C
95
C
Ni
Fe
Ni
b)
Fig. 3.13. a) The isothermal q u a t e r n a r y phase diagram Fe-Ni-Si-C, e x p e r i m e n t a l l y determined
a t 850"c [91]
b) d i f f u s i o n patha projected on the Fe-Ni-Si section of the couples SiC/FesoNi20,
SiC/Fe6oNi~o , SiC/Fe40Nt60 and SiC/Fe20Ni80 [67,91].
F.J.J. van Loo
96
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LIST OF SYMBOLS
a
= unreacted radius or half the unreacted plate
ai
= activity of component i=YiN i
Ci
= concentration of component i in moles/m 3
Ci
= initial
c o n c e n t r a t i o n of component i on t h e l e f t - h a n d
diffusion
C~
1
av
s i d e of t h e
couple
= i n i t i a l c o n c e n t r a t i o n of component i on t h e r i g h t - h a n d s i d e o f t h e
diffusion couple
= interdiffusion
Di
T
Di
t h i c k n e s s i n m, Eq. 24
= intrinsic
= tracer
coefficient
diffusion
diffusion
coefficient
coefficient
= average interdiffusion
Din," integrated
(also called chemical diffusion
i n m2/s, Eq. 2
of component i i n m2/s, Eq. 11
of component i i n m 2 / s , Eq. 13
coefficient
interdiffusion
coefficient)
i n m 2 / s , Eq. 17
coefficient
i n m 2 / s , Eq. 17
d
= total
di
= layer thickness after
layer thickness in m
dp
= l a y e r t h i c k n e s s on a p l a t e d u r i n g t h e p a r a b o l i c growth p e r i o d , Eqs. 18 and 25
a transient
dc, dp and d s = l a y e r t h i c k n e s s e s
period ti,
in cylindric,
Eq. 18
p l a n a r and s p h e r i c a l
diffusion couples during diffusion-llmlted growth, Eq. 25
Ji
= interdiffusion
f l u x of component i i n t h e Matano frame of r e f e r e n c e i n m o l e l s / m 2 s ,
Eq. l
Ji
= intrinsic
diffusion
moles/m2s, Eq. 4
f l u x of component i i n t h e K i r k e n d a l l
frame of r e f e r e n c e
in
Multiphase Diffusion
kp
= p a r a b o l i c g r o w t h c o n s t a n t i n m 2 / s , Eq. 18
k'
i
P
99
apparent value of k
from a d / t ~ p l o t , Eq. 19
P
o f component i . For s u p e r s c r i p t - o r +, s e e Ci
Ni
= mole f r a c t i o n
R
= gas c o n s t a n t
ri
r0
= d i m e n s i o n l e s s f a c t o r o r i g i n a t i n g from t h e v a c a n c y f l u x e f f e c t , Eq. 13
r a d i u s o f c y l i n d e r o r s p h e r e , o r h a l f t h e p l a t e t h i c k n e s s i n m, Eq. 24
T
= temperature in Kelvin unless stated otherwise
homogeneity width of a phase
= 8,314 J/mole K
t
= diffusion time in seconds
ti
= duration of the transient period before the parabolic growth sets in, in s, Eq. 18
Vm
= molar volume in m3/mole of atoms
partial molar volume of component i in m3/mole of atoms
vKIM= velocity of local inert markers relative to the Matano frame of reference in
m/s,
Eq. 5
X
~e
coordinate in the diffusion
direction
in m
c o o r d i n a t e o f t h e o r i g i n o f t h e Natano frame o f r e f e r e n c e
X O
xK
= coordinate of the Kirkendall plane
Y
= concentration unit for component i = (Ni-Ni-)/(Ni+-Ni-), Eq. 3
Z
=
x K - x 0 i n m, Eq. 5
CL
7i
ratio between the volume of one mole ABy and one mole A, Eq. 25
conversion factor, Eq. 24
= activity coefficient of component i
An a s t e r i s k
The s u p e r s c r i p t
(as i n Ci* o r Y*) r e f e r s t o a d e f i n i t e v a l u e o f t h e q u a n t i t y i n q u e s t i o n .
N a s i n Eq. 16 d e n o t e s q u a n t i t i e s
in the number-fixed frame of r e f e r e n c e .