Decomposition of Alloys: the Early Stages. Proceedings of the 2nd Acta-Scripta Metallurgica Conference

243 Pages·1984·9.705 MB·English

by  P. Haasen, V. Gerold and R. Wagner (Eds.)

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Decomposition
of Alloys:
the early
stages
Proceedings of the
2nd Acta-Scripta Metallurgica Conference
Sonnenberg,
Germany,
19-23 September 1983
Edited by
P. HAASEN
Universität Göttingen, Germany
V. GEROLD
Institut für Werkstoffwissenschaften,
Stuttgart, Germany
R. WAGNER
GKSS-Forschungszentrum, Geesthacht, Germany
M. F. ASHBY
University of Cambridge, England
Sponsored by
Akademie der Wissenschaften in Göttingen
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First edition 1984
Library of Congress Cataloging in Publication Data
Acta-Scripta Metallurgica Conference (2nd : 1983 :
Sonnenberg, Germany)
Decomposition of alloys, the early stages.
(Acta-scripta metallurgica proceedings series ; 2)
Includes index.
1. Alloys — Congresses. 2. Chemistry, Metallurgica —
Congresses. I. Haasen, P. (Peter) II. Title. III. Series.
QD135.A25 1983 620.1'62 84-1078
British Library Cataloguing in Publication Data
Acta-Scripta Metallurgica Conference
(2nd : 1983 : Sonnenberg)
Decomposition of alloys. — (Acta-scripta
metallurgica proceedings series ; 2)
1. Alloys 2. Diffusion
I. Title II. Haasen, P. III. Series
669'94 TN690
ISBN 0-08-031651-4
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
P r e f a ce
The decomposition of alloys by diffusional processes is indeed a very old and well-studied
subject. It has come to life again recently as experimental methods of high resolution and
reliability have been developed, which make it possible to study the very early stages of de-
composition. Here, the theoretical descriptions can be most severely tested and compared with
experiments using different instruments. It became clear that only a highly specialized work-
shop could serve this purpose, rather than another one of the very general and large phase trans-
formation conferences that have been organized several times during the past one and a half
decade. The meeting was intended to focus on the unsolved problems of the field, rather than
the solved ones. The important task, then, was to get together a small-enough group of experts
from different backgrounds for a full discussion of the topic. Of course, not all the invited
people from different parts of the world were available during the week of September 19-23, 1983,
However, the group of 55 active contributors that met at the 'International House Sonnenberg'
in the middle of the Harz mountains near Göttingen, FR Germany, turned out to be truly as con-
genial and inspiring a team, enthusiastic and penetrating in its discussions, as I have ever
met before.
Of course, The Sonnenberg has a long tradition of such discussion-meetings, of the Arbeits-
gemeinschaft Metallphysik, and the convenient and hospitable atmosphere of the International
House, as well as the beautiful, and even partly-sunny, forest around us contributed to the
success. Dr. Fritz Wohl er and the graduate students of the Göttingen Metal Physics Institute
worked indefatigably for a smooth organization of the meeting, which was made possible by a
grant from the Stiftung Volkswagenwerk and support by Acta Metallurgica, Inc. We would like to
thank all of them for their help. The Akademie der Wissenschaften in Göttingen entrusted me
with the organization of this meeting in accordance with its charter for the promotion of inter-
disciplinary scientific cooperation. It was ample reward for my years of presidentship over the
Akademie and was greatly eased by the friendly help of the Conference committee, which also
edited these procedings: Richard Wagner, Volkmar Gerold and Michael Ashby. On behalf of the
Akademie I thank them and all participants for their great contributions.
Peter Haasen
V
RECENT DEVELOPMENTS IN THE THEORY OF THE
KINETICS OF FIRST ORDER PHASE TRANSITIONS
J. D. Gunton
Physics Department, Temple University
Philadelphia, Pa. 19122 U.S.A.
Abstract
A review is given of certain theoretical developments in the kinetics of
phase separating systems. The concept of a "coarse-grained" spinodal
curve, interface and renormalization group theories for dynamical scal-
ing, and recent extensions of the Lifshitz-Slyozov diffusion-controlled
coarsening process are discussed.
1. A "Coarse-Grained" Spinodal Curve
In this article I will summarize some recent developments in the theory
of metastable and. unstable states. More detailed reviews have been giv-
en elsewhere recently [1-4]. Therefore I will limit my attention here
to only a few topics in this field. The first subject which I will con-
sider involves the so-called spinodal curve, which historically has
played an important conceptual role in the theory of metastable and un-
stable states. I do not want to give too much significance to this top-
ic. However, in my opinion there exists considerable confusion in the
literature concerning the concept of a (unique) spinodal curve. This
is particularly true in the experimental literature where well-inten-
tioned (and possibly correct!) scientists have attempted to determine
such a spinodal curve, in order to give a precise dynamical distinc-
tion between metastable and unstable states. In this section, I will
explain why there is probably not a unique spinodal curve for systems
with short-ranged forces. As this is a subtle subject which will ul-
timately require a mathematically rigorous solution, I will limit myself
here to a summary of my current understanding of this issue.
To begin with, it is worth pointing out that a theory of metastable and
unstable states requires a dynamical description. The concept of a
spinodal curve (if meaningful) must fundamentally emerge from a dynami-
cal theory, such as we discuss below. Indeed, if one looks at various
operational definitions which experimentalists have used in attempts to
determine a spinodal curve, one finds that they ultimately involve dy-
namical concepts (either implicitly or explicitly). We will discuss
two particular methods of determining a spinodal curve later and indi-
cate why such attempts are probably misleading for systems with finite
range forces. Before discussing these, however, it is useful to recall
the one case in which one can give a precise definition of a unique
spinodal curve. Namely, for a system which is exactly described by
mean field or van der Waals type theories, such as a model with an in-
finitely long range force. Recall that in the van der Waals picture,
there is a "van der Waals" loop which does not describe thermodynamic
equilibrium states. Rather, the outer portions of the loop (for which
the order parameter isothermal susceptibility is positive) describe
metastable states. The inner portion of the loop (for which the suscep-
tibility is negative) describes unstable states. The two points at
which the inverse susceptibility vanishes (for fixed T < T ) are known
1
2
as spinodal points and in mean field theories provide a sharp distinc-
tion between metastable and unstable states. The locus of these points
as a function of temperature defines the classical spinodal curve. Un-
fortunately, however, since mean field theory is strictly valid only
for systems with infinitely long range forces, the above definition is
not useful for most physically interesting systems. It should also be
noted that the life time of a metastable state for a mean field system
is infinite [5], so that such systems do not display the interesting dy-
namical features of experimental interest.
What can one say about the physically relevant case of a system with
finite-range forces? Here the situation is much more subtle than in
mean field theory. One can, for example, imagine giving at least two
different operational definitions of a "spinodal curve", as I indicated
above. The first would be the analogue of the van der Waals picture.
Namely, consider the problem of analytically continuing the equilibrium
free energy of a stable phase into the metastable part of the two phase
region. It is reasonably well established [1,4] that there is an es-
sential singularity in this free energy at any point on the coexistence
curve below Tc, with the analytic continuation consisting of a real and
an imaginary part. The real part f(c) is schematically indicated in
Fig. 1 by the short dashed lines.
f(C)|
Fig. 1 : Schematic graphs of the coarse grained free energy f(c) for a
particular L) and the corresponding equilibrium free energy
f(c). The dashed sections denote the real part of the analy-
tic continuation of f(c) into the metastable region.
The interpretation of this real part is that it describes the equili-
brium properties of the metastable state. That is, it would describe
such quantities as the susceptibility and specific heat in experimental
measurements of a metastable state on a time scale during which the
state has not begun to decay. If this dashed line has a unique continu-
ation the extrapolated value of the measured susceptibility would appear
to diverge at some point (for fixed T). This could provide one defini-
tion of a "spinodal" point. Indeed extrapolations of this type provide
one common means in the literature to define such a "spinodal" [6]. It
should be pointed out, however, that this definition might have little
or no physical significance. In the region where one would make such an
3
extrapolation there is no "equilibrium" metastable state: The system is
evolving on the time scale of the experiment.
An alternative operational definition is often employed to empirically
determine a "spinodal curve", Here one extrapolates from the unstable
region of the phase diagram, using small angle scattering data obtained
in the early stages of spinodal decomposition and assuming the validity
of the Cahn linear theory. This method also involves difficulties which
we discuss later. The Cahn theory [7] is a linearization o^ a nonlinear,
continuum equation of motion for the local concentration c(r) which in-
volves a "coarse-grained" Helmholtz free energy functional H (or Ginz-
L
burg-Landau Hamiltonian), where
H = ' dr { \ K | V c| 2 + f(c)} (1.1)
L
The value of K and the shape of the free energy density f(c) (a double
well potential shown schematically in Fig. 1) depend on the (arbitrary)
choice of cell size L used in the coarse graining procedure described
below. (For simplicity we do not indicate this L dependence, but it is
crucial to remember this dependence in the discussion that follows).
The equation of motion for the average local concentration c(r) is as-
sumed to be of the Cahn-Cook form (a nonlinear Langevin equation) [7-10]:
Here M is a phenomenological (possibly L-dependent) mobility and ζ is
a noise term which represents the effects of the phonon modes of the al-
loy. This noise term is assumed to be described by a Gaussian distribu-
tion function. Again, it is important to realize that this semi-macro-
scopic equation of motion for the local concentration c("?) of one of the
alloy constituents implicitly involves the coarse-graining cell size L.
Namely, (1.2) only describes the long wavelength components of c{r), so
that only Fourier components c£ with |k| < Λ are included (with the up-
per cutoff A « L-l). Note that the dynamics of c(f) are determined
by the coarse grained |_ and hence the (L-dependent) double well poten-
tial which describes the nonequilibrium properties of experimental in-
terest, namely the evolution of metastable and unstable states, via (1.1)
and (1.2). Thus there are two quite different free energies, the coarse
grained f(c) and the analytic continuation of the equilibrium free ener-
gy f(c) mentioned above [4,11,12].
It is important to understand why K and f(c) depend on the coarse-grain-
ing size L. This results when one "derives" the continuum equation of
motion (1.1) and (1.2) from the microscopic dynamical equations which de-
scribe the atomic motion which occurs through exchange of A and B atoms,
vacancies, etc. (A microscopic model which is often assumed to provide
a simplified, but realistic description of this dynamics is a master
equation with "Kawasaki dynamics" [9,10].) To obtain equations (1.1)
and (1.2) one must "coarse grain out" certain short wavelength compon-
ents of the local concentration from the original microscopic model of
the binary alloy. For example, imagine an alloy with two interatomic
species, A and B, which can occupy the sites i of a Bravais lattice. Sup-
pose the energy for a particular configuration of these atoms is E(ci)
where (ci) denotes the configuration of atoms with ci = ± 1 depending on
whether site i is occupied by an A or B atom respectively. Now imagine
dividing the system into cells of linear size L, with the center of the
a tn cell being denoted by r*. Then the local average concentration,
a
ci^a) in cell a can be obtained by summing over the site variables in the
cell. Thus one goes from a microscopic description involving the Bravais
lattice site variables! c-j } to a semi-macroscopic description involving
the local cell concentration variables ί cfr^h The Hamiltonian H[_{c(f )}
a
is obtained from E{c·} by performing the partial trace.
-M. <c(r )} - BE {c.} (Τ.3)
a
e = Σ e
{ci}
4
with 3 = 1/kßT. The prime denotes that the sum is performed over the
local site variables subject to the constraint that the local cell con-
centration be given by ic("?)} and with the overall concentration of the
system held fixed. Thus we obtain H| of Eq. (1.1). The important point
is that the form of the free energy functional obviously depends on L.
For L = 1 it is the microscopic Hamilton!an whereas for L -* °° it approach-
es the equilibrium Helmholtz free energy, which in the two phase region
simply consists of a straight line connecting the two equilibrium (two-
phase coexistence) values of the alloy, corresponding to the A-rich and
A-poor phases respectively. Although L is arbitrary, the theorist
chooses L = b ξ, where ξ is the bulk correlation length and b is a con-
stant of order unity. The reasons for this choice are twofold [4,13]
first, the continuum model is most valid near the critical point, where
the correlation, length is large. Second, one is then naturally interest-
ed in describing the dynamics of phase separation on the length scale ξ .
(Note that the initial phase separation which occurs on a length scale of
the lattice constant is presumably not described by (1.1) and (1.2) for
this choice of L).
The consequence of f(c ) depending on the arbitrary choice of L (through
b, say) is that there is no longer a unique spinodal curve. Namely, a
spinodal curve defined by χ-ι = a2f/s c2 = 0 depends on L. Thus tnere
is no unique spinodal curve in this sense, but rather a "coarse-grained"
spinodal curve whose 1 ocation depends on L. Recently Kaski, Binder and
Gunton [12] have calcu lated such a curve, starting from a microscopic
Ising model for the bi nary alloy. (An earlier field theory calculation
has also been carried out [14]). This "coarse-grained" spinodal curve
is shown in Fig. 2 be!o w, in terms of the natural scaling variable 1/ ξ .
50.0
450
400
L = 3 *
350 L = 4 a
L = 6 *
L=8 x
300 (FB) ·
250
200
150
100
50
00 Jar*»*!*1
00 01 02 0.3 04 0.5 0.6
0-r )
Fig. 2: A scaling form of a "coarse-grained" spinodal curve deter-
mined from a calculation of the nearest neighbor cell-cell
distribution function for the three dimensional Ising model
(reference 12). Here c denotes the spinodal and c de-
s max
notes the position of the maximum in the distribution func-
tion.
Although this curve is completely different from the textbook, mean
field spinodal curve, it is in fact the quantity which occurs (implicit-
ly) in the continuum equations (1.1) and (1.2) used by Cahn [7], Cook
[8], Langer, Bar-on and Miller [13] and other authors who have attempted
5
to develop theories of spinodal decomposition. It should be clear, then,
that the second operational definition of a spinodal curve which involves
the Cahn linear theory and an extrapolation of the early time SAS data
to determine where X"'(or correspondingly a diffusion coefficient
D = MX"1) vanishes should not yield a unique spinodal. One natural in-
terpretation of this (which remains as a fundamental theoretical issue
requiring a mathematically rigorous analysis) is that in contrast to
mean field theory there is no sharp dynamical distinction between meta-
stable and unstable states. This has been suggested by a number of auth-
ors, for a variety of reasons [1,11,15-17].
We conclude this section with two speculative remarks. The first has to
do with the time of validity of the Cahn linear stability analysis.
Several authors [17-19] have suggested that for systems with finite
range forces, this time is exceedingly small and, strictly speaking,
might even be zero! (The most complete discussion to date of this is-
sue is due to Binder [19], who uses a "Ginzburg criterion" argument. )A
simple minded way in which one could see how this might be is to note
the consequences for the Cahn theory of the L-dependence discussed
above. One immediately sees that physically measurable quantities such
as the position of the peak in the scattering intensity, kZ =|(a2f/ac2)|
/2K, and the diffusion constant D would depend on the arbitrary choice
of L. Since this is not physically sensible, one could conclude that
this means that the time of validity of the linear theory is strictly
zero. A convincing argument of this type will require renormalization
(coarse-graining) of the equations of motion (1.1) and (1.2) (rather
than justH|_). This has so far only been carried out approximately by
Horner and Jungling [20] and remains a problem of considerable theore-
tical interest. It is conceivable that an analysis of the renormalized
equations of motion would show that the argument I have given above is
incorrect.
The second speculation has to do with the fact that in some systems ex-
perimentalists have been able to determine a "spinodal curve" by using
the linear theory discussed above. It should be immediately noted that
in these cases careful examination reveals that the agreement between ex-
periment and the linear theory is only semi-quantitative. The question,
however, is why should such a fit,and a corresponding estimate of a "spi-
nodal curve", be even approximately successful, if the preceding discuss-
ion is correct? I believe that the answer might lie in the fact that
the most successful examples to date of this occur for systems which are
almost mean-field like in nature, such as for certain polymer-polymer
blends [21,22]. This suggests that for such systems theorists might be
able to develop asymptotic expansions in which the mean field behavior
is the leading term. The semi-quantitative success mentioned above would
then be a consequence of the correction terms in such an expansion being
"small", but nevertheless important. At the moment no such theory exists,
so the above is simply a guess. It should be noted, however, that in-
teresting studies of systems with long range forces have been carried out
in references [19,23,24].
2. Self-Similar Pattern Formation
It has by now been well established that to a good first approximation
the complicated morphological changes which characterize phase separa-
tion satisfy a self-similar behavior sometimes termed dynamical scaling
[1,16]. One of the best examples of this is found in the small angle
neutron scattering studies of a single crystal Al-Zn [25,26]. In general
one finds that after a certain initial transient time, t , the nonequili-
0
brium structure factor S(k,t) can be written as
S(k,t) - R d(t) F (kR(t)), t > t , (2.1)
Q
where R(t) is a characteristic time dependent length (such as the aver-
age domain size or generalized Guinier radius [16]) and F is a so-called
scaling function. The transient time t varies from yery short times in
0
certain binary fluids to extremely long times in certain binary alloys,
reflecting the large differences in diffusion constants, etc., in such
6
systems. (We do not discuss here the generalization of the above scal-
ing form necessary near a critical point, where the bulk correlation
length becomes large.) Although a first principles understanding of
this scaling behavior is not yet available, progress is being made us-
ing dynamical interface models and/or renormalization group arguments.
A classification of the dynamic universality classes in these first or-
der phase transitions is also a fundamental problem which only now is
beginning to be addressed [2], It should be noted that one simplifying
feature of the scaling law is that the complicated dynamical evolution
of the system is described by the scaling length, whereas the overall
self-similarity in the pattern formation reflects itself in a "relative-
ly simple" scaling function, F, for the scattering intensity.
The one case in which scaling is now reasonably well understood is that
of order-disorder transitions in binary alloys or Ising antiferromagnets.
Kawasaki, Yalabik and Gunton (KYG) [27] first obtained a structure fac-
tor (using a singular perturbation theory) which exhibited a scaling
form. More recently, Ohta, Jasnow and Kawasaki (OJK) [28] obtained a
scaling function which is essentially identical to that of KYG and which
is in reasonably good agreement with existing Monte Carlo studies of the
kinetic Ising antiferromagnet [1] (which can be considered as a simple
model of a binary alloy order-disorder transition). Their work is based
on an approximate solution of the Allen-Cahn equation [29] which de-
scribes the motion of interfaces in these order-disorder systems. The
OJK theory has been generalized by Grant and Gunton [30] and by Ohta
[31] to take into account the effects of thermal noise. Finally, a re-
cent phenomenological real space renormalization group calculation for
this model with nonconserved order parameter by Mazenko and Vails [32]
yields results which are in good agreement with Monte Carlo data for the
structure function and nearest neighbor correlation function of the sys-
tem. This calculation is particularly interesting because it suggests
that a first principles renormalization group theory will eventually be
available for a wide class of problems involving the kinetics of first
order phase transitions.
Without going into technical details I will state here the phenomenolo-
gical argument put forth by Mazenko and Vails, who have also applied it
with qualitative success to a study of spinodal decomposition in a bi-
nary alloy [33]. The basic ansatz involves rather standard ideas of re-
normalization group theory as used in critical phenomena. Namely, scal-
ing suggests that the structure factor S(k,t) must be similar to
S(bk,At), where distances are rescaled by a factor b > 1 and times are
rescaled by a factor Δ"1 > 1. Their guess for the relationship between
the original and rescaled structure factors is that
S(t,t) = A(£,t) + B(t,t) S(bt,At) (2.2)
As in critical phenomena one expects that since S(k,t) depends on the in-
itial (before quench) and final (after quench) temperatures Ti and Tp,
the rescaled structure function will depend upon the renormalized tem-
peratures (couplings) Tj' and Tp'. (The dependence of these structure
factors on the original and renormalized temperatures is implicit in Eq.
(2.2)). Mazenko and Vails also assume that the functions A and B are lo-
cal quantities which equilibrate rapidly in time to their final values,
relative to the equilibration time of the structure factor. One crucial
problem which remains is to develop a self-consistent determination of the
time rescaling factor, (which is equivalent to determining the growth law
R(t)). The interested reader is referred to the literature for the de-
tails of how one implements Eq. (2.2). I simply want to stress two
points here. The first is that the recursion Eq. (2.2) has a scaling so-
lution of the form (2.1) in the long time domain, which is one attrac-
tive feature of their model. The second is that at the moment one cannot
really regard the above as a first principles renormalization group theory,
since it has not yet been derived in a convincing fashion. Nevertheless
it certainly provides a promising direction for future research in this
field.