\input texp \hsize=6.0truein \vsize=9.0truein \singlespace \parskip=3mm
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\def\dec #1\par{\global\advance\secno by1\subsecno=0\bigskip  
\leftline{\bf #1}\nobreak\smallskip}  
  
\centerline{\bf MODELLING OF PHASE TRANSFORMATIONS IN STEEL
WELD METAL}\bigskip

\centerline{ H. K. D. H. Bhadeshia }
\medskip
\centerline{ University of Cambridge}
\centerline{ Department of Materials Science and Metallurgy}
\centerline{ Pembroke Street, Cambridge CB2 3QZ, U.K.}
\bigskip

{\parindent=20 pt \parskip=0.1mm \narrower \medtype \medskip
\x {\bf Abstract}

\x This paper deals with some of the latest issues in the
modelling of steel weld microstructures, building on work
already published in the open literature. The key factors reviewed
include the nature of acicular ferrite (a most desirable phase), the
formation of austenite in the heat--affected zone and finally, the
tempering reactions in welds which are post--weld heat treated for
service at elevated temperatures.  
\medskip
}

\dec{1. INTRODUCTION}

The welding process attempts to achieve perfect metallic
joints without the thermomechanical processing inherent in the
manufacture of wrought steels. This is impossible to achieve
in practice but there are many tools which can be exploited
on the path to perfection. One of these involves the
calculation of microstructure in ferritic steel welds, work which has
been documented and reviewed thoroughly [1--8]. The reviews are widely
available; their contents are not therefore repeated here. It is the
intention here to  highlight the very latest developments and
difficulties in the modelling of steel weld microstructures.

\dec{2. MICROSTRUCTURE}


Typical components of the microstructure are listed in \fagg, classified
into two essential categories: displacive and reconstructive. 

\midinsert \x{{\bf TABLE 1: }Approximate values of the shear strain $s$
and the dilatational strain $\delta$ for a variety of transformation
products in steels.}\medskip{ \thicksize=1pt \thinsize=0.8pt 
\tablewidth=5.2truein \begintable 
{Transformation} |$s$ | $\delta $ | Morphology 
\cr
 {\wid, $\alpha_w$}|  0.36 | 0.03 | Thin plates \nr
{Bainite, $\alpha_b$}|  0.22 | 0.03 |Thin plates \nr
{Martensite, $\alpha'$}|  0.24| 0.03| Thin plates  \nr
{Allotriomorphic $\alpha$} | 0 | 0.03 | irregular  \nr
{Idiomorphic $\alpha$}| 0 | 0.03 | equiaxed  \nr
{Pearlite} | 0 |0.03 | irregular 
\endtable}
\endinsert 

\picture{flow}{186}{227}{500}{{\bf FIGURE 1: } Flow chart of the
mechanisms of solid--state transformations in steel welds. The shaded
region deals with transformation during the cooling of austenite.}




In a displacive transformation, the change in crystal is achieved by a
deformation of the parent structure (Table~1). The strain energy due to
the deformation can
be minimised if the product phase adopts a thin--plate shape during
constrained transformation [9]. Consequently, martensite, bainite,
acicular ferrite and
\wid\ all occur in the form of thin plates. Furthermore, the ratio
($\Xi$) of iron to substitutional atoms is unaffected since these atoms
do not partition at any stage in the formation of the product phase.
Interstitial atoms such as carbon and nitrogen may partition without
affecting the displacive character of the change in crystal structure. A
change in which $\Xi$ remains constant but where carbon achieves equality
of chemical potential is known as {\it paraequilibrium} transformation
[10].  Thus, martensitic transformation is completely diffusionless.
Bainite, acicular ferrite and
\wid\ nucleate by a paraequilibrium mechanism. However, bainite and
acicular ferrite grow without diffusion; any excess carbon is then
partitioned into the residual austenite or precipitates as carbides. In
the case of
\wid\ the paraequilibrium state is maintained during both nucleation and
growth. 


The difference
between acicular ferrite and bainite is that the latter generally
nucleates at austenite grain surfaces, so that sheaves of
identically oriented parallel plates
dominate the microstructure. Acicular ferrite, on the other hand,
nucleates from tiny nonmetallic inclusions;  these plates radiate from
the point nucleation sites, giving a less organised microstructure which
has a greater resistance to crack propagation. We shall return to this
remarkable microstructure in a later discussion.


Reconstructive transformations involve the diffusion of all elements,
including iron. Thus, the only strain that can be generated during the
formation of allotriomorphic ferrite, idiomorphic ferrite and pearlite
is that associated with the change of density due to transformation.
Strain plays a relatively minor role.

 Whereas there is a coordinated
transfer of at least the iron and substitutional atoms during a
displacive transformation, the flow of atoms for a reconstructive
reaction is without discipline. As a consequence, the products of
reconstructive transformations are not restricted to the austenite grain
in which they nucleate. The coordinated displacements associated with
martensite \etc\ cannot be sustained across austenite grain boundaries so
that martensite plates are confined to the grain in which they nucleate. 

The difference between these two mechanisms of transformation is far
from academic. For example, because the displacive transformation
products fail to cross austenite ($\gamma$) grain boundaries, there is  a
vestige  of the
\g\ grain boundary  which remains when transformation in completed.
This renders the {\it prior austenite grain boundaries} susceptible
to impurity segregation and intergranular embrittlement. This is not the
case with reconstructive transformations where the ferrite grows
across the austenite grain boundaries, thereby destroying them as
impurity segregation sites. This example highlights the fact that a
good understanding of  phase transformations not only
permits the calculation of microstructure but also helps in achieving
better mechanical properties. Table~2 shows the detailed information
available on each of the major phases found in steel welds; this kind of
information is essential before any calculations can be attempted.


\midinsert \x {\bf TABLE 2}: Detailed characteristics of
martensite ($\alpha'$), upper and lower bainite
($\alpha_{ub}$, $\alpha_{lb}$, acicular ferrite
$(\alpha_a$), allotriomorphic ferrite ($\alpha$),
idiomorphic ferrite $(\alpha_i$) and pearlite ($P$). An $=$
indicates consistency with comment, $\neq$ that the comment
does not apply and $\otimes$ that the comment sometimes
applies. \smallskip \thicksize=1pt
\thinsize=0.8pt
\tablewidth=5.9truein {\smalltype \begintable
Comment & $\alpha'$ & $\alpha_{lb}$ & $\alpha_{ub}$ & $\alpha_a$ &
$\alpha_w$ & $\alpha$ & $\alpha_i$ & $P$ \cr
Nucleation and growth reaction \hfill & = & = & = & = & = & = & = & = \nr
Plate shape \hfill & = & = & = & = & = & $\neq$ & $\neq$ & $\neq$ \nr
IPS shape change with large shear\hfill & = & = & = & = & = & $\neq$ &
$\neq$ & $\neq$ \nr
Diffusionless nucleation \hfill & = & $\neq$ & $\neq$ & $\neq$ & $\neq$ &
$\neq$  & $\neq$ & $\neq$ \nr
Only carbon diffuses during  nucleation \hfill & $\neq$ & = & = & = & = &
$\neq$ & $\neq$ & $\neq$ \nr
Reconstructive diffusion during nucleation \hfill & $\neq$ & $\neq$ &
$\neq$ &
$\neq$ & $\neq$ & = & = & = \nr
Often nucleates intragranularly on defects \hfill & = & $\neq$ & $\neq$ &
= &
$\neq$ & $\neq$ & = & $\neq$ \nr
Diffusionless growth \hfill & = & = & =  & = & $\neq$ & $\neq$ & $\neq$ &
$\neq$
\nr
Reconstructive diffusion during growth \hfill & $\neq$ & $\neq$ & $\neq$ &
$\neq$ &
$\neq$ & = & = & = \nr
Atomic correspondence (all atoms) during growth \hfill & = & = & = & = &
$\neq$ &
$\neq$ & $\neq$ & $\neq$ \nr
Atomic correspondence only for large atoms 
\hfill & = & = & = & = & = & $\neq$ & $\neq$ & $\neq$ \nr
Bulk redistribution of X atoms during growth \hfill & $\neq$ & $\neq$ &
$\neq$ & $\neq$ &
$\neq$ & $\otimes$ & $\otimes$ & $\otimes$ \nr

Local equilibrium at interface during growth  \hfill & $\neq$ & $\neq$ &
$\neq$ & $\neq$ &
$\neq$ & $\otimes$ & $\otimes$ & $\otimes$ \nr
Local paraequilibrium at interface during growth  \hfill & $\neq$ & $\neq$
&
$\neq$ & $\neq$ & = & $\otimes$ & $\otimes$ & $\neq$ \nr
Diffusion of carbon during transformation \hfill & $\neq$ & $\neq$ &
$\neq$ &
$\neq$ & = & = & =  & = \nr
Carbon diffusion-controlled growth \hfill & $\neq$ & $\neq$ & $\neq$ &
$\neq$ & = &
$\otimes$ & $\otimes$ & $\otimes$ \nr

Co-operative growth of ferrite and cementite \hfill & $\neq$ & $\neq$ &
$\neq$ & $\neq$ &
$\neq$ & $\neq$ & $\neq$ & = \nr
High dislocation density \hfill & = & = & = & = & $\otimes$ & $\neq$ &
$\neq$ &
$\neq$
\nr
Incomplete reaction phenomenon \hfill & $\neq$ & = & = & = & $\neq$ &
$\neq$ &
$\neq$ & $\neq$ \nr

Necessarily has a glissile interface \hfill  & = & = & = & = & = & $\neq$
&
$\neq$ &
$\neq$ \nr

Always has an orientation within the Bain region \hfill  & = & = & = & = &
= &
$\neq$ & $\neq$ & $\neq$ \nr

 Grows across austenite grain boundaries \hfill & $\neq$ & $\neq$ & $\neq$
&
$\neq$ & $\neq$ & = & = & = \nr

High interface mobility at low temperatures \hfill & = & = & = & = & = &
$\neq$ &
$\neq$ & $\neq$ \nr

Displacive transformation mechanism \hfill & = & = & = & = & = & $\neq$ &
$\neq$ &
$\neq$ \nr
Reconstructive transformation mechanism \hfill & $\neq$ & $\neq$ & $\neq$
&
$\neq$ & $\neq$ & = & = & = 
\endtable } \endinsert


\dec{3. ACICULAR FERRITE}

Of all the phases described above, acicular ferrite remains the most
controversial and the most desirable for good mechanical properties.

Acicular ferrite 
has in three--dimensions the morphology of thin, lenticular 
plates which nucleate heterogeneously on 
nonmetallic inclusions. However, some plates may  stimulate  the
nucleation of others, an effect  known as autocatalysis.

It has been argued that acicular ferrite and bainite are 
similar  in their  transformation mechanisms. Their microstructures 
differ in detail because bainite sheaves grow as a series of 
parallel platelets emanating from austenite grain {\it 
surfaces}, whereas acicular ferrite platelets nucleate 
intragranularly at {\it point} sites so that parallel 
formations of plates cannot develop. The nucleation site in 
the latter case is smaller than the ultimate thickness of the plate, 
so that the inclusion becomes engulfed by the plate of 
ferrite which it stimulates. 

The growth of both bainite and acicular ferrite causes an 
invariant--plane strain shape deformation with a large shear 
component. Consequently, plates of acicular ferrite 
cannot cross austenite grain boundaries, because the 
coordinated movement of atoms implied by the shape change 
cannot  be sustained across grains in different 
crystallographic orientations. The lattice of the acicular 
ferrite is therefore generated by a deformation of the 
austenite, so that the iron and substitutional solutes are 
unable to diffuse during the course of transformation. It is  not
surprising that the concentrations of  substitutional alloying
elements are unchanged during the  growth of acicular ferrite.  This has been
verified directly using atomic resolution chemical analysis [11].


The deformation which changes the austenite into acicular 
ferrite occurs on particular planes and directions, so that 
the ferrite structure and orientation are intimately related 
to that of the austenite. It follows that,  plates of 
acicular ferrite, like bainite, must without exception have 
an orientation relationship with the austenite. 

During isothermal transformation, the acicular ferrite 
reaction stops when the carbon concentration of the 
remaining austenite makes it impossible to decompose without 
diffusion. This implies that the  plates of acicular ferrite  
grow supersaturated with carbon, but the excess carbon is 
shortly afterwards  rejected into the remaining austenite. 
This of course, is the incomplete reaction phenomenon, where the
austenite never  reaches its equilibrium composition since the reaction
stops  at the $T_0'$ curve of the phase diagram (\fagg). The obvious 
conclusion is that acicular ferrite cannot form at 
temperatures above the bainite--start temperature, and this 
is indeed found to be the case in practice [12]. 

\picture{tzero}{137}{89}{500}{{\bf FIGURE 2: } Data from experiments in
which the austenite  is transformed isothermally to acicular ferrite,
showing that  the reaction stops when the carbon concentration of the 
austenite reaches the $T_0'$ curve (after Strangwood).} 

There are many other correlations which reveal the analogy 
between acicular ferrite and bainite. For example, the 
removal of inclusions by vacuum arc melting, without changing 
any other feature, causes an immediate change in the 
microstructure from acicular ferrite to bainite. 
The same effect can be obtained by increasing the number 
density of austenite grain nucleation sites relative to 
intragranular sites. This can be done  by refining the 
austenite grains to obtain a  transition from an acicular 
ferrite microstructure to one which is predominantly bainitic. 

 
The opposite phenomenon, in which an inclusion--containing 
steel with bainite  can be induced to transform into an 
acicular ferrite microstructure is also observed. This can be 
done by rendering the austenite grain surfaces  ineffective 
as nucleation sites, either by decorating the boundaries with 
a thin layer of inert allotriomorphic ferrite, or by 
adding a small amount of boron (30~p.p.m). The boron 
segregates to the boundaries, thereby reducing the boundary 
energy and making them less favourable sites for 
heterogeneous nucleation. In general, any method which 
increases the number density of intragranular nucleation 
sites relative to austenite grain boundary sites will favour 
the acicular ferrite microstructure. 

We have emphasized here the idea that the transformation mechanism for
acicular ferrite is identical to that for bainite. However, all phases
can nucleate on inclusions, including \wid\ [13,14]. Thewlis
\ett\ have argued that in some welds the so--called acicular ferrite may
predominantly be intragranularly nucleated \wid\ rather than bainite
[15]. They reached this conclusion by noting that the estimated
bainite--start ($B_S$) temperature was lower than that at which
coarse plates nucleated on very large inclusions (3--9 ${\rm \mu m}$ 
diameter). Although there is  uncertainty in their calculated
$B_S$ values, the  conclusion that a mixed
microstructure of intragranularly nucleated \wid\ and intragranularly
nucleated bainite (\ie acicular ferrite) was obtained seems
justified. Intragranularly nucleated
\wid\ can be distinguished readily from bainite by the scale of the
optical microstructure. 

\wid\ plates are always much coarser than bainite because what appears as
a single plate using optical microscopy is in fact an adjacent pair  of 
self accommodating plates. The shape deformation
consists of two adjacent invariant--plane strains which tend to mutually
accommodate and hence reduce the strain energy, thus allowing the plates
to be coarse [16]. A prediction made here is that   transmission electron
microscopy should reveal the two components of each of the optically
observed plates, with the adjacent variants
separated by a low--energy grain boundary [16]. (Note: there is some
confusion in [15] where the intragranularly nucleated
ferrite plates are identified with reconstructive transformation; in
steels, all ferritic phases in the form of plates grow by a displacive
mechanism with an accompanying shape deformation characterised by a large
shear [12].)

It is the present author's  opinion that the weight of evidence  
supports the conclusion that the acicular ferrite which is recognised to
be beneficial to weld metal is in fact intragranularly nucleated
bainite. And that the term acicular ferrite should be reserved for
such a fine microstructure. If coarse \wid\ forms on inclusions then it
can be called \lqq intragranularly nucleated \wid". The
names given to phases are important because they imply a mechanism of
transformation which in turn implies a methodology for the prediction of
microstructure. It is particularly important to avoid naming a
mixture of microstructures.

\dec{4. AUSTENITE FORMATION}

\x The welding process inevitably heats some of the surrounding solid
metal into the austenite phase field. It is useful, therefore, to be able
to  model quantitatively the transformation of an ambient temperature
steel microstructure into austenite. This applies both to the weld metal
since a gap can be filled with many weld passes, and to the heat affected
zone of the steel plate being joined.

 The formation of austenite during heating  differs in
many ways from those transformations that occur during the
cooling of austenite. For cooling transformations, the kinetics of
decomposition  follow the classical C-curve behaviour, in which
the rate goes through a maximum as a function of the undercooling
below the equilibrium transformation temperature. This is because 
diffusion becomes sluggish with decreasing temperature, but 
the driving force for transformation increases with the
undercooling. On the other hand, both the diffusion coefficient and the
driving  force increase with the extent of superheat above the
equilibrium temperature, so that the  rate  of austenite formation
must increase indefinitely  with temperature, \fagg.

\picture{ttt}{187}{134}{300}{{\bf FIGURE 3: } The
time--temperature--transformation  curves for the
$\gamma\rightarrow\alpha$ reaction, and for the  reverse
$\alpha\rightarrow\alpha$ transformation. $\alpha$ and $\gamma$
represent ferrite and austenite, respectively. $\Delta G$ represents
the chemical driving force for transformation; $D$
is the rate--controlling diffusion coefficient.}

There is another important difference between the transformation
of austenite, and the transformation to austenite. In the former
case,  the kinetics of transformation can be
described completely in terms of the alloy composition and the
austenite grain size.  By contrast, the microstructure from which
austenite may grow can be infinitely varied. Many more variables are
therefore needed  to describe the kinetics of austenite
formation.  The  extent to which the starting  microstructure has to be
specified remains to be determined, but factors such as  particle size,
the distribution and chemistry of individual  phases, homogeneity, the
presence of  nonmetallic inclusions, \etc should all be important.

This discussion highlights the complexity of the problem.  A
fundamental attempt at modelling the formation of austenite is
therefore unlikely to be of general value, except at slow heating rates
consistent with the achievement of equilibrium. Some aspects of the
difficulties involved  have been reviewed recently for a variety of
starting microstructures [12].  Models of specific
metallurgical approaches exist for austenite formation
from a mixture of cementite and ferrite [17], from bainite [18],  and
from ferrite [19]. However, none of these are of
general applicability for the reasons described earlier.

To resolve this problem, Gavard and co--workers [20] have developed a
neural network [21] model to enable the austenite--start ($Ac_1$) and
austenite--finish ($Ac_3$) temperatures to be estimated as a function of
the steel chemical composition and the heating rate. The method
involves a non--linear regression of a vast quantity of
experimental data. The predictions can then be used to estimate the
fraction of austenite ($V_\gamma$) that forms at any temperature $T$
[21]:
$$ V_\gamma = {{1-\exp\bl\{-k_0\exp\bl\{ {-{Q}\over{kT}}\br\} \bl[
{{T-Ac_1}\over{Ac_3 - Ac_1}} \br]^n\br\}
}\over{1-\exp\{-k_0\exp(-Q/k\,Ac_3)\}}}
$$
where $Q/k=0.12\times 10^{-6}\,$K, $k_0=0.0206$ and $n=0.849$ and $k$ is
the Boltzmann constant. 

This work has not yet been applied to welding but is ready to be
incorporated into detailed models on weld metal microstructure.

\dec{5. TEMPERING REACTIONS}

There is a large range of heat--resistant steels and welding alloys,
generally rich in Cr, Mo, V, Nb and W. The ones with the lowest solute
concentrations might contain substantial quantities of allotriomorphic
ferrite and some pearlite, but the vast majority have bainitic or
martensitic microstructures in the normalised condition. After
normalising the steels are severely tempered to produce a \lqq stable"
microstructure consisting of a variety of alloy carbides in a ferritic
matrix. The task is therefore to model the evolution of  precipitation
and dissolution reactions.



\def\bn{2${1 \over 4}$Cr1Mo\ }

 In order to calculate time--temperature--transformation diagrams for
carbide reactions, a theory capable of handling several simultaneous
precipitation reactions has been developed [23], where the different
phases influence each other, for example by drawing the same solute from
the matrix ferrite. 

In practice, there are many cases where several transformations occur
together. The different reactions interfere with each
other in a way which is seminal to the development of power plant
microstructures. The principles involved are first illustrated with an
example in which $\beta$ and $\theta$ precipitate at the same time
from the parent phase which is designated $\alpha$. For the sake of
discussion it is assumed that the nucleation and growth rates do not
change with time and that the particles grow isotropically.

The increase in the extended volume due to particles nucleated in a time
interval $t=\tau$\ to $t=\tau + d\tau$ is, therefore, given by
$$dV_\beta^e = {4\over 3}\pi G_\beta^3(t-\tau)^3 I_\beta(V)~d\tau
\qquad\hbox{and}\qquad dV_\theta^e ={4\over 3}\pi
G_\theta^3(t-\tau)^3I_\theta (V)~d\tau
\numeqn
$$ where $G_\beta {\rm ,\ }G_\theta {\rm ,\ }I_\beta {\rm
\ and\ }I_\theta
$ are the growth and nucleation rates of $\beta$ and
$\theta$ respectively, all of which are assumed here to be independent of time.
$V$ is the total volume of the system. For each phase, the increase in extended
volume will consist of three separate parts. Thus, for $\beta$: {(i) }
$\beta$ which has formed in untransformed
$\alpha$. {(ii) } $\beta$ which are formed in regions which are
already $\beta$. {(iii) }
$\beta$ which has formed in regions which are already
$\theta$.


\x Only $\beta$ formed in untransformed $\alpha$ will contribute to the real
volume of $\beta$. On average a fraction $\left( 1 - {V_\beta + V_\theta
\over V} \right) $ of the extended volume will be in previously untransformed
material. It follows that the increase in real volume of $\beta$ is given by
$$dV_\beta =\left( 1 - {V_\beta + V_\theta\over V}
\right) dV_\beta ^e \qquad\hbox{and}\qquad dV_\theta = \left( 1 -
{V_\beta + V_\theta \over V}
\right) dV_\theta ^e
\numeqn $$

\x Generally $V_\beta$ will be some complicated function of $V_\theta$ and it is
not possible to integrate these expressions analytically to  find the
relationship between the real and extended volumes.  Numerical
integration is straightforward and offers the opportunity to change
the boundary conditions for nucleation and growth as
transformation proceeds, to account for the change in the matrix
composition during the course of reaction. The method can in
principle be applied to any number of simultaneous reactions.


The multiple reactions found in power plant
steels have important complications which can all be dealt with in
the scheme of simultaneous transformations as presented above.
The phases interfere with each other not only by reducing the
volume available for transformation, but also by removing solute
from the matrix and
 thereby changing its composition. This change in matrix
composition affects the growth and nucleation rates of the phases.
 The main features of the application of the theory to power
plant steels are summarised below; a full description is given
in references [23,24].


{\parindent=15pt \narrower \medskip
\zz{$\bullet$ }{The model allows for the simultaneous precipitation of
\mtwox, \mtwentythreecsix, \msevencthree, \msixc\ and Laves
phase.
\mthreec\ is assumed to nucleate instantaneously with the
paraequilibrium composition. Subsequent enrichment of
\mthreec\ as it approaches its equilibrium composition is accounted
for.}

\zz{$\bullet$ }{All the phases, except \mthreec, form close to their
equilibrium composition. The driving forces and compositions of the
precipitating phases are calculated using MTDATA [25].}

\zz{$\bullet$ }{The interaction between the precipitating phases is accounted
for by considering the change in the average solute level in the matrix as each
phase forms.}

\zz{$\bullet$ }{The model does not require prior knowledge of the precipitation
sequence.}

\zz{$\bullet$ }{Dissolution of non--equilibrium phases is incorporated
as a natural event.}

\zz{$\bullet$ }{A single set of fitting parameters for the nucleation equations
(site densities and surface energies) has been found which is applicable to a
wide range of power plant steels.}

\medskip}



A plot showing the predicted variation of volume fraction of each precipitate as  a
function of time at 600$\,$\degg\ is shown in \fagg . These
results have been shown to be consistent with experiments; the
precipitation kinetics of
\mtwentythreecsix\ are  predicted to be much slower in the \bn\ steel
compared to the 10CrMoV and  3Cr1.5Mo alloys. One contributing factor is
that in the
\bn\ steel a relatively  large volume fraction of \mtwox\ and
\msevencthree\ form prior to \mtwentythreecsix. These deplete
the matrix and therefore suppress 
\mtwentythreecsix\ precipitation. The volume fraction of \mtwox\
which forms in the  10CrMoV steel is relatively small, and there
remains a considerable excess of solute  in the matrix, allowing
\mtwentythreecsix\  to precipitate rapidly. Similarly, in the 3Cr1.5Mo steel
the volume fractions of \mtwox\   and \msevencthree\   are insufficient to
suppress
\mtwentythreecsix\  precipitation to the same extent as in the \bn\ steel.

\mtwentythreecsix\  is frequently observed  in the form of coarse
particles which are less effective in hindering creep deformation.
Delaying its precipitation would have the effect of stabilising the
finer dispersions of \mtwox\   and MX to longer times with a
possible enhancement of creep strength. 

\picture{carbide}{142}{123}{400}{{\bf FIGURE 4:}  The predicted evolution
of precipitate volume fractions at 600$\,^\circ$C for  \bn\ steel [23].}

Calculations like these can be used to design microstructures
exploiting knowledge built up over decades concerning what is good
or bad for creep strength. It is often argued that Laves phase
formation is bad for creep resistance -- it leads to a reduction in the
concentration of solid solution strengthening elements; since the
Laves precipitates are few and coarse, they do not themselves
contribute significantly to strength. The model presented here can
be used to design against Laves phase formation. 

\dec{6. CONCLUSIONS}

It is impossible in a short paper such as this to do justice to the
progress that has been made on the modelling of weld metal
microstructures. The references listed in this paper are much better
reviews in this respect. The latest issues, which have been the focus of
this paper, show that there is continued progress with aspects of basic
science grappling with some of the most complex technology. What could
possibly be a more exciting research topic for metallurgists? 

\vfill\eject

\dec{REFERENCES}

\parskip=0.5mm

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pp. 419--427.

\zz{[2]} Ashby, M. F. and Easterling, K. E., Acta Metallurgica 30,
(1982) 1969--1978.


\zz{[3]} Bhadeshia, H. K. D. H., Materials Science and Technology 8,
pp. 123--133 (1992).

\zz{[4]} Bhadeshia, H. K. D. H. and Svensson, L--E., Mathematical
Modelling of Weld Phenomena, eds H. Cerjak and K. E. Easterling, The
Institute of Materials, London, 1993, pp. 109--182.

\zz{[5]} Svensson, L.--E., Control of Microstructure and Properties in
Steel Arc Welds, London, CRC Press, 1993.
 
\zz{[6]}  Zacharia, T., Vitek, J. M., Goldak, J. A.,  Debroy, T. A., 
Rappaz M. and Bhadeshia, H. K. D. H., Modelling and Simulation in
Materials Science and Engineering,  3, pp. 265--288 (1995).

\zz{[7]} Bhadeshia, H. K. D. H.,  Mathematical
Modelling of Weld Phenomena II, eds H. Cerjak and H. K. D. H. Bhadeshia,
The Institute of Materials, London, 1995, pp. 71--118.

\zz{[8]} Grong, O., Metallurgical Modelling of Welding, 2nd edition,
London, Institute of Materials, 1997.

\zz{[9]} Christian, J. W., Acta Metallurgica 6, pp. 377-379 (1958).

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