\input texp \parindent=5pt \parskip=2mm
\title{ Estimation of The Microstructure}
 \title{\&\ Properties
of  Ferritic Creep--Resistant Steels }\bigskip \singlespace
\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.}
\medskip

\singlespace


\sec{Abstract}

{\parindent=10pt \narrower \medskip 

\x Creep resistant steels must be
reliable over very long periods of time in severe environments. Their
microstructures have to be very stable, both in  the wrought and in
the welded states.    This paper reviews the  quantitative methods
for the design of steels  for elevated temperature
applications. A methodology is described for the calculation of 
complex precipitation reactions over periods extending many tens
of years.  However, microstructure alone is not enough in the design
of alloys. The estimation of  the
creep rupture stress using a neural network technique is described in
the second part of this review. The calculation of the influence of
solute--elements on the self--diffusivity of iron, which features in
many creep equations, is an emerging area in alloy design. The
methodology for such calculations is reviewed in the final section of
the paper.

\medskip}

\sec{Introduction}

\x Typical operating parameters for steels 
used in the manufacture of power plant are compared against
corresponding values for nickel alloys in aeroengines, in
\tablaa. In both cases, the service conditions are severe. But this is
especially so for steels where the service life is many decades. The
degree of reliability demanded of heat resistant steels is
therefore extraordinary, and must represent one of the highest
achievements of technology. By contrast, computers, which  are
frequently identified with advanced technology,  seldom last for
more than two years and are generally obsolete at the point of
installation! 

\bigskip
\thicksize=1pt \thinsize=0.8pt
\tablewidth=5.0truein  \begintable
Property & Aeroengine & Power Plant \cr
Temperature & $>$ 1000$\,^\circ$C & 540-750$\,^\circ$C \nr
Pressure & $\simeq 3$ bar & 160--370  bar \nr
Design life & $10^4$~h & $2.5\times 10^{5}$ h \nr
$\sigma_{100,000~h}$ & 10 MPa & 100 MPa \nr
Coating & Yes & No \nr
Forced cooling & Yes & No \nr
Single crystal & Yes & No
\endtable{\tabtit{\tablee : } {Service conditions for a
component  in the hot part of an aeroengine and one in
the hottest part of a power plant. The lower limits for the power
plant component are representative of current technology. The
stress is a 100,000~h creep rupture strength. }}

 
It should  not be surprising that
the number of variables involved in the design of creep--resistant
steels is very large -- in fact, we shall show later that there are at
least thirty parameters which need to be controlled in any
experiment or calculation of creep properties. 


The variables can ideally be taken into account using what scientists
like to call \lqq physical models", \ie theories which explain a large
class of observations, which contain few arbitrary
elements and which make verifiable predictions.  The first part of
this paper deals with such  physical models in the prediction
of microstructure.  

There is no adequate theory to deal with the second task, which is
the estimation of   creep rupture strength as a function of the steel
composition, microstructure and heat treatment. Difficult problems
like this, where the general concepts might be understood but which
are not as yet amenable to fundamental treatment, are common in
metallurgy.   To form a complete design--technology, it is 
consequently necessary to resort to careful
empiricism. The second part of this paper deals with a
semi--empirical method implemented to achieve useful results. The
combination of physical and empirical models can then be used to
attempt the design of alloys.  

The  final part of the paper deals with the estimation of the
self--diffusion coefficient of iron containing solute additions. This is
perceived to be an important parameter in creep deformation; it has
become prominent in recent work because of the availability of
commercial software capable of estimating the diffusivity.  It is
useful therefore to summarise the basis of such calculations, even
though it is too early to comment on their significance in the context
of changes in creep properties.

\sec{The Microstructure}

There is a large variety of heat--resistant steels (\tablaa).
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 known
precipitates are listed in \fagg; they determine the microstructure
and are  crucial in the development of creep strain. The task
is therefore to model the evolution of  precipitation and dissolution
reactions.

\picture{phases}{164}{218}{300}{\tabtit{\figg : }{The variety of
precipitates to be found in power plant steels. The iron--rich
carbides such as cementite form extremely rapidly, whereas
graphite forms incredibly slowly because it is difficult to nucleate.
 M$_5$C$_2$ and Z--phase are recent discoveries [2,3].}}



\midinsert { \thicksize=1pt \thinsize=0.2pt
\tablewidth=5.4truein \begintable 

Designation | C & Si & Mn & Ni & Mo & Cr & V \cr

1Cr${1\over2}$Mo \hfill | 0.15 & 0.25 & 0.50 & -- & 0.6 & 0.95 &  \nr

$1\over 4$CrMoV \hfill | 0.15 & 0.25 & 0.50 & 0.05 & 0.50 & 0.30 & 0.25
 \nr

${1\over2}$Cr${1\over2}$Mo${1\over4}$V \hfill | 0.12 & 0.25 & 0.50 & -- &
0.6 & 0.45 & 0.25\nr

1CrMoV  \hfill | 0.25 & 0.25 & 0.75 & 0.70 & 1.00 & 1.10 & 0.35 \nr

2$1\over 4$Cr1Mo \hfill | 0.15 & 0.25 & 0.50 & 0.10 & 1.00 & 2.30 & 0.00  \nr

 Mod. 2$1\over 4$Cr1Mo \hfill | 0.1 & 0.05 & 0.5 & 0.16 & 1.00 & 2.30 & 0.25  \hfill \nr
| & & & & &   Ti=0.03 & B=0.0024 \hfill \cr
 
3.0Cr1.5Mo \hfill | 0.1 & 0.2 & 1.0 & 0.1 & 1.5 & 3.0 & 0.1   \nr

3.5NiCrMoV \hfill | 0.24 & 0.01 & 0.20 & 3.50 & 0.45 & 1.70 & 0.10   \nr

9Cr1Mo \hfill | 0.10 & 0.60 & 0.40 & -- & 1.00 & 9.00 & --   \nr

Mod. 9Cr1Mo \hfill | 0.1 & 0.35 & 0.40 & 0.05 & 0.95 & 8.75 & 0.22 \nr
| & & & &   Nb=0.08 & N=0.05 & Al $<$0.04 \hfill \cr

9Cr$1\over2$MoWV \hfill | 0.11 & 0.04 & 0.45 & 0.05 & 0.50 & 9.00 & 0.20  \nr
  | & & & &  W=1.84 & Nb=0.07 &  N=0.05  \hfill\nr

12CrMoV \hfill | 0.20 & 0.25 & 0.50 & 0.50 & 1.00 & 11.25 & 0.30  \nr

12CrMoVW \hfill | 0.20 & 0.25 & 0.50 & 0.50 & 1.00 & 11.25 & 0.30 \hfill \nr
| & & & & &  & W=0.35 \hfill \nr

12CrMoVNb \hfill | 0.15 & 0.20 & 0.80 & 0.75 & 0.55 & 11.50 & 0.28  \hfill \nr
| & & & & &  Nb 0.30 &   N 0.06 \hfill


\endtable }\tabtit{\tablee : }{Typical compositions (wt.\% ) of
creep--resistant steels.  The range of alloys available is in fact
much larger than this, for example, there is a variety of alloys
available for bolting applications.}
\endinsert  

The results of equilibrium calculations which give the phase fractions of the
carbides as a function of the overall alloy composition and
temperature, are given in \fagg\ for the common power plant steels.
The calculations have been done using the {\it MTDATA} [1] computer
program and {\it SGTE} database, taking into account the carbide
phases and Laves phase listed, together with cementite. The chemical
elements considered are carbon, silicon, manganese, chromium, nickel,
molybdenum, vanadium, niobium and nitrogen.  ${\rm M_5C_2}$ has
recently been identified in 1Cr--0.5Mo steels [2] but along with
graphite, has not  been included in the analysis.

\picture{histogram}{160}{115}{500}{\tabtit{\figg : }{Equilibrium
fractions of carbides in some common power plant steels, as
calculated using MTDATA and the SGTE thermodynamic database for
565$\,^\circ$C (838~K). Very small fractions of vanadium and
niobium carbonitrides are are present in some steels but are not
shown. Thus, the modified 9Cr1Mo contains 0.0009 NbN and 0.003 VN,
the 9CrMoWV steel contains 0.0008 NbN and 0.0032 VN. The
calculations allowed the existence of all the carbides described in
Figure~1 with the exception of graphite, epsilon, Chi, Z--phase and
M$_5$C$_2$.}}

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

Equilibrium calculations such as those presented in \figg\ are
useful in specifying the ultimate microstructure but the results are
far   from the actual microstructures that exist during
service. It is   necessary in practice to be able to calculate
time--temperature--transformation diagrams for tempering
reactions, as a function of steel chemical composition and
tempering temperature. In order to do this, a theory capable of
handling several simultaneous precipitation reactions has been
developed [4], where the different phases influence each other, for
example by drawing the same solute from the matrix ferrite. 

\ssec{ Overall transformation kinetics} 

 A model for a single
transformation   begins with the calculation of the nucleation and
growth rates using classical  theory, but an estimation of the volume
fraction requires impingement between particles to be taken into
account. This is generally done using the extended volume concept
of Johnson, Mehl, Avrami, and Kolmogorov [5] as illustrated in
\fagg. Suppose that two particles exist at time $t$; a small interval $\delta t$
later, new regions marked $a$, $b$, $c$ \&\ $d$ are formed assuming that
they are able to grow unrestricted in extended space whether or not the region
into which they grow is already transformed. However, only those components
of
$a$, $b$, $c$ \&\ $d$ which lie in previously untransformed matrix can
contribute to a change in the real volume of the product phase (identified by
the subscript \lq 1\rq) so that :
$$ dV_1 = (1 - {{V_1}\over {V}})dV_1^e   \numeqn $$ where  it is assumed
that the microstructure develops randomly. The superscript
$e$ refers to extended volume, $V_1$ is the volume of phase $1$ and
$V$ is the total volume. Multiplying the change in extended volume
by the probability of finding untransformed regions has the effect of
excluding regions such as $b$, which clearly cannot contribute to the
real change in volume of the product. This equation can easily be
integrated to obtain the real volume fraction, 
$${{V_1}\over{V}} = 1-\exp\bl\{ -{{V^e_1}\over{V}}  \br\} \numeqn $$
Nucleation and growth rates can readily be substituted into
$V^e_1$, leading to the
familiar  Avrami equation.

\picture{extended}{164}{95}{500}{\tabtit{\figg : }  {The concept of
extended volume. Two precipitate particles have nucleated
and grown to a finite size in the time $t$. New regions $c$
and $d$ are formed as the original particles grow, but  
$a$ \&\ $b$ are new particles, of which $b$ has formed in a region
which is already transformed.}}


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
$$dV_\beta^e = {4\over 3}\pi G_\beta^3(t-\tau)^3 I_\beta(V)~d\tau
\numeqn $$
$$ dV_\theta^e ={4\over 3}\pi G_\theta^3(t-\tau)^3I_\theta (V)~d\tau
\numeqn
$$ 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 $\beta$:

{\parindent=15pt \narrower \medskip
\zz{(i) } $\beta$ which has formed in untransformed
$\alpha$.

\zz{(ii) } $\beta$ which has formed in regions which are already
$\beta$.

\zz{(iii) } $\beta$ which has formed in regions which are already
$\theta$.

\medskip}

\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
$$dV_\beta =\left( 1 - {V_\beta + V_\theta\over V}
\right) dV_\beta ^e \numeqn $$  and similarly for $\theta$,
$$dV_\theta = \left( 1 - {V_\beta + V_\theta \over V}
\right) dV_\theta ^e
\numeqn $$

\x  $V_\beta$ is expected to be some complicated function of
$V_\theta$ so it is not generally 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.


{\bf Complex 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 [4].


{\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 [6]. 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 [1].}

\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}

The compositions of three power plant alloys used here for
illustration purposes, are shown in
\tablaa. These three  alloys, whilst of quite different chemical
compositions, show similar precipitation {\it sequences} [4,7,8]
but with vastly different rates.  For example, at 600$\,$\degg\ the
time taken before \mtwentythreecsix\ is observed is  1$\,$h in the
10CrMoV steel [4], 10$\,$h in the 3Cr1.5Mo alloy [7] and in  excess of
1000$\,$h in the
\bn steel [8]. These differences have never before been explained
prior to the simultaneous transformations model [4].

\midinsert \thicksize=1pt \thinsize=0.8pt \tablewidth=3.2 truein {
\smalltype \begintable
 {} | C & N & Mn & Cr & Mo & Ni & V & Nb  \cr 

{\bn} | 0.15 & -- & 0.50 & 2.12 & 0.9 & 0.17 & -- & --    \nr
{3Cr1.5Mo} | 0.1 & -- & 1.0 & 3.0 & 1.5 & 0.1 & 0.1 & --    \nr
{10CrMoV} | 0.11 & 0.056 & 0.50 & 10.22 & 1.42 & 0.55 & 0.20 & 0.50 
\endtable }
\tabtit {\tablee : }{ Concentration (in weight\%) of the major alloying elements in the
steels used to demonstrate the model.}
\endinsert

\sec{Microstructure Calculations}

A plot showing the predicted variation of volume fraction of each precipitate as  a
function of time at 600$\,$\degg\ is shown in \fagg . 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{all}{142}{365}{530}{\tabtit{\figg : }{The predicted evolution
of precipitate volume fractions at 600$\,^\circ$C for three power
plant materials (a) \bn  (b) 3Cr1.5Mo and (c) 10CrMoV.}}

Calculations like these can be used to design microstructures
exploiting knowledge built up over decades concerning what is good
and 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. 

We note for the moment, that this is as far as microstructure
modelling has progressed. The models are not yet capable of giving
size distributions and even if that were to be possible, there are no
physical models of creep deformation which have sufficient
precision to make use of this information. We shall not be
discouraged by this since good empirical methods are
available. The work described below originates from research by
Brun \et [9] and Cole and Bhadeshia [10].


\sec{Creep Rupture Strength -- the Variables}

The basic principles of alloy design
for creep resistance are   well--established and well--founded on
experience. The steels must have a stable  microstructure which
contains fine alloy carbides to resist the motion of dislocations;
however, changes are inevitable over the long service time so that
there must be sufficient solid solution strengthening to ensure long
term creep resistance. There may be other requirements such as
weldability, corrosion and oxidation resistance. It is nevertheless
difficult to express the design process quantitatively given the
large number of interacting variables.  

These variables are described later in the context of calculations in
\tablaa. For the moment we note that the entire information about
microstructure and properties is in principle locked up in this set of
parameters since chemical composition and heat treatment are
comprehensively included. There may, of course, be many other
independent variables that might be considered important in creep
analysis, but these are for the  moment neglected for two reasons. 
Firstly, an empirical analysis requires experimental data; an over
ambitious list would simply reduce the dataset since publications
frequently do not report all of the necessary parameters. Secondly,
the effect of any missing variables would simply be reflected in the
uncertainties of prediction. If the predictions are noisy then they
can be improved with carefully designed experiments at a future
date. Bearing this in mind, the results to be presented are based on
some 2000 sets of experiments obtained from the published
literature. We now proceed to describe briefly the methodology.


\sec{The Neural Network Method}

Most people are familiar with  regression analysis 
where data are best--fitted to a specified relationship which is
usually linear. The result is an equation in which each of the
inputs $x_j$ is multiplied by a weight $w_j$; the sum of all such
products and a constant $\theta$ then gives an estimate of the output
$y = \sum_jw_jx_j + \theta$. It is well understood that there are
dangers in using such relationships beyond the range of fitted data.

A more general method of regression is neural network
analysis. As before, the input data  $x_j$ are multiplied by weights,
but the sum of all these products forms the argument of a hyperbolic
tangent. The output $y$ is therefore a non--linear function of
$x_j$, the function usually chosen being the hyperbolic tangent
because of its flexibility. The exact shape of the hyperbolic
tangent can be varied by altering the weights (\fagg a). Further
degrees of non--linearity can be introduced by combining several of
these hyperbolic tangents (\figg b), so that the neural network
method is able to capture almost arbitrarily non--linear
relationships. For example, it is well known that the effect of
chromium on the microstructure of steels is quite different at large
concentrations than in dilute alloys. Ordinary regression analysis
cannot cope with such changes in the form of relationships.


\picture{hytan}{135}{60}{550}{\tabtit{\figg : }{(a) Three different
hyperbolic tangent functions; the \lqq strength" of each depends on
the weights. (b) A combination of two hyperbolic tangents to produce
a more complex model.}}

A potential difficulty with the use of powerful regression methods is
the possibility of overfitting  data (\fagg). For example, one can 
produce a neural network model for a completely random set of
data. To avoid this difficulty, the experimental data can be divided
into two sets, a {\it training} dataset and a {\it test} dataset. The
model is produced using only the training data. The test data are
then used  to check that the model behaves itself when presented
with previously unseen data.

\picture{poly}{125}{90}{550}{\tabtit{\figg : }{A complicated
model may overfit the data. In this case, a linear relationship is
all that is justified by the noise in the data.}} 

Neural network models in many ways  mimic human experience and
are capable of learning or being trained to recognise the correct
science rather than nonsensical trends. Unlike human experience, these
models can be  transferred readily  between generations and steadily
developed to make design tools of lasting value. These models also
impose a discipline on the digital storage of valuable experimental
data, which may otherwise be lost with the passage of time.

The technique is extremely powerful and useful. Its application to
creep rupture strength analysis  is presented below. The details
can be found elsewhere [11] but it is important to note that the
generalisation of the model on unseen data has been tested
extensively against large quantities of information.

\sec{Calculations of  Creep Rupture Strength}

\fagg\ shows the variation in the creep rupture strength ($10^5$~h)
of a modern   \lqq 10CrMoW" creep resistant steel (\tablee) as a
function of the temperature, carbon, chromium and molybdenum
concentrations. The error bounds represent the uncertainty in
fitting the non--linear function to the training data, as 65\%\
confidence limits. There is an additional error associated with each
calculation, which is the noise in the experimental data, which is
perceived to be of the order of $\pm 2$\%.  The engineering design
of power plant is based on the ability to support a stress of 100~MPa
for $10^5$~h at the service temperature. The apparent
insensitivity of the creep rupture strength to the molybdenum or
chromium concentrations for
$10^5$~h is not surprising given that the carbides will all be
extremely coarse at that stage of life.

\picture{10Cr0.5Mo}{140}{391}{500}{\tabtit{\figg : }{Creep rupture
stress at 600$\,^\circ$C and 100,000 h for 10Cr--0.5Mo type steel}}


\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=3.3truein {\smalltype\singlespace\begintable
STEEL	& $2{1\over 4}$CrMo	 & 10CrMoW	 \cr 
Normalising temperature / K &  	1203 &1338  \nr 
Duration / h & 6 & 2  \nr
Cooling rate	& water quenched & 	air cooled  \nr
Tempering temperature / K & 	908 &	1043\nr
Duration / h & 	6 &	4  \nr
Cooling rate	& air cooled & air cooled\nr
Annealing temperature / K &	873 & 	1013 \nr
Duration / h &	2 & 4  \nr
Cooling rate	& air cooled & air cooled \nr
C wt\%	& 0.15	& 0.12	 \nr
Si	& 0.21	& 0.05	 \nr
Mn	& 0.53	& 0.64		 \nr
P	& 0.012	& 0.016	\nr
S	& 0.012	& 0.001	\nr
 	Cr & 	2.4	& 10.61 \nr
 	Mo	 & 1.01	& 0.44 \nr
  		W	& 0.01	& 1.87 \nr
 	Ni	& 0.14 & 0.32 \nr
 		Cu	& 0.16	& 0.86 \nr
 		V	& 0.01	& 0.21 \nr
 		Nb	& 0.005	& 0.01 \nr
 		N	& 0.0108	& 0.064 \nr
 	Al	& 0.018	& 0.022 \nr
 	B	& 0.0003	& 0.0022 \nr
 	Co	& 0.05	& 0.015 \nr
 Ta	& 0.0003 & 	0.0003 \nr
	O	& 0.01	& 0.01 
\endtable} \tabtit{\tablee : }{The standard set of input parameters  for two
alloys used to examine trends predicted by the neural network. The chemical
compositions are all in wt.\%} \endinsert

Similar data for the classical \bn\ steel are illustrated in \fagg. The
fitting uncertainties are smaller in this case because of the larger
quantity of available data since this alloy has been available
and studied for a much longer time.

\picture{2.25Cr1Mo}{140}{391}{500}{\tabtit{\figg : }{Creep rupture
stress  for 2.25Cr--1Mo type steel}}

Calculations like these can now be routinely carried out.
Furthermore, the models can be improved  both as
more data become available and as creep deformation becomes
better understood. The model can be used in a variety of ways.
The combined application of the physical models
presented earlier, and the neural network model has led
to predictions of novel alloys which ought to have
much better creep resistance than any comparable commercial
alloy [9]. There are long--term experiments in progress to
test these designer--alloys. Another way is to apply the models to
welding alloys, for which there are much fewer data when compared
with wrought steels.



\sec{Welding Alloys}

Weld metals and steels of matching composition seem to have
similar creep rupture properties. In fact, the chemical compositions
of weld metals and corresponding steel plates are not very
different (\tablaa). Of course, weld metal will have a higher oxygen
and nitrogen concentration but the former should not affect creep
resistance. Although differences in the nitrogen concentration are
important, they can easily be taken into account both in predicting
carbonitride formation and in the neural network model where
nitrogen is an input. 

The microstructure of an as--deposited weld metal is naturally
radically different from that of a wrought steel. However, even
this is unimportant because of the severe tempering heat
treatments used following the welding procedure,  
essentially wipe out the original microstructure and replace it
with one which is tempered and similar to that of the steel plate. It
is probably for this reason that the welding process itself is
found not to influence the creep rupture life [12]. 


 
\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=3.3truein {\medtype \begintable \singlespace
| \multispan2 \hfill 2.25Cr1Mo \hfill | \multispan2 \hfill 9Cr1Mo \hfill
\nr 
wt.\% | Plate & Weld | Plate & Weld \cr
C	|  0.110	& 0.091	| 0.110	& 0.090 \nr
Mn	| 0.390	& 0.590	| 0.040	& 0.480 \nr
Si	| 0.290	& 0.300	| 0.460	& 0.500 \nr
Cu	| 0.150	& 0.024		| & \nr
Ni	| 0.150	& 0.033	| 0.050	& 0.050 \nr
Cr	| 2.070	& 2.480	| 8.960	& 8.700 \nr
Mo	| 0.900	& 1.170	| 0.470	& 0.980 \nr
Nb	| & 		| 0.069	& 0.040 \nr
V	| 0.004	& 0.015	| 0.200	& 0.200 \nr
S	| 0.022	& 0.014		| & \nr
P	| 0.011	& 0.010 | & \nr
N	| 0.005	& 0.010	| 0.051	& 0.040 \nr
O	| 0.005	& 0.030	| 0.005	& 0.030
\endtable }\tabtit{\tablee : }{Chemical compositions, wt\%.}\endinsert

The hypothesis can be proved by examining the data on all--weld
metal tests in the published literature [12], again for a stress
rupture life of $10^5$~h. Such data are most reliable for the \bn\
type weld metals; the calculations are therefore presented for the
\bn\ weld metal listed in \tablee.  

\fagg\ shows the very encouraging agreement between the
calculated [9,10] and measured [12] creep rupture lives of \bn\
welds. The predictions are made without any adjustment of the
models, which did not interrogate any weld metal data during their
creation. The results confirm that it is reasonable to assume that 
weld metal creep rupture life can be modelled on the basis of
wrought steels. Of course, other properties such as creep ductility
may be more sensitive to inclusion content in which case the weld
metals should exhibit a lower ductility relative to the wrought steel.

\picture{Lundin}{144}{180}{500}{\tabtit{\figg : }{Calculated (filled
circles with error bars) and measured (open circles) stress rupture
data for 2.25Cr1Mo weld metal.}}

The method can now be used to generate creep--rupture diagrams
such as that illustrated in \fagg.

\picture{3d}{144}{93}{500}{\tabtit{\figg : }{Calculated   stress rupture
data for 2.25Cr1Mo weld metal.}}


\sec{Self--Diffusion in Iron}

\x The way in which solutes affect the diffusivity of iron
atoms must be of importance in creep theory. The
estimation of these effects is now becoming possible
although the significance of the calculations in the context
of design against creep has yet to be established. We
discuss here the methodology for the calculations without
commenting on their importance, which is best determined
experimentally.  As usual, things are not simple
for iron, there is a significant effect of magnetism which must be
taken into account.

Self--diffusion in
austenitic iron  follows the Arrhenius law  (\fagg) over the
temperature range 910--1400
\deg\ for which experimental data exist. A comprehensive
analysis of the experimental data by Oikawa  [13,14] gave the
following estimates for the diffusion parameters:
$$D=(0.89 ^{+0.40}_{-0.28})\times 10^{-4}\exp\{-Q/RT\}\qquad {\rm
m^2\,s^{-1}}\qquad \hbox{with}\qquad  Q=291.3\pm 4.5{\rm
\,kJ\,mole^{-1}} \numeqn$$ 
The activation energy\footnote\dag{Strictly an enthalpy, since the
entropy terms are usually incorporated in the pre--exponential
term. The activation enthalpy $Q$ will therefore be distinguished
from the activation free energy $G^*$.} for diffusion  is larger than
that of ferrite even though the melting temperature of austenite is
slightly lower than that of ferrite. This is presumably because
austenite is more densely packed  than ferrite.

\picture{selfaust}{134}{111}{500}{\tabtit{\figg : }{Self--diffusion
in austenite (data from [15]).}}

By contrast, the self--diffusion data for ferrite show considerable
anomalies, both in the region of paramagnetic to ferromagnetic
transition and in the $\delta-$ferrite temperature range
\fagg\ [15,16,17].  The 
anomaly  has been observed also for the diffusion of nickel in ferrite
[18] and in a vast range of other
measurements including mechanical properties such as creep at
elevated temperatures.


\picture{selfdiff}{138}{111}{500}{\tabtit{\figg : }{Self--diffusion
coefficient of iron in $\alpha-$ ferrite [17]. The Curie temperature is
1043~K and the formation of austenite begins at 1184~K.}}

 
Borg and Birchenall [16] suggested that both the vacancy formation
energy and its ability to migrate are reduced in the ferromagnetic
state. A theory which accounts for the effects in terms of pairwise
magnetic interactions was proposed by Girifalco [19] and later by
Ruch \et [20]. For the fully ordered ferromagnetic state,  the
magnetic interaction energy of an atom which has $N_c$ nearest
neighbours is
$$ W = -N_c s_0 \numeqn $$
where $-s_0$ is the interaction energy per pair of atoms. This
equation is based on the reasonable assumption that only near
neighbour terms contribute significantly to $W$. When a vacancy is
created by removing an atom, $N_c$ pairs of contacts are broken,
half of which are recreated when the discarded atom is replaced at
the free surface. The enthalpy of formation of a vacancy is therefore
increased by ${1\over 2}N_cs_0$. A similar argument gives a further
contribution to the migration energy as $(N_v-N_*)s_0$ where $N_v$ is
the number of nearest neighbours to an atom paired with a vacancy,
and $N_*$ the number of nearest neighbours to an atom in its
activated state, neglecting relaxation effects. The value of $s_0$ is
proportional to the square of the saturation magnetisation $M_m\{0\}$
at absolute zero. Bearing this is mind, the activation enthalpy for
self diffusion becomes
$$ Q= Q_0\bl[1+b_{1} \bl({{M_m\{T\}}\over{M_m\{0\}}}\br)^2
\br]\numeqn $$
where $Q_0$ is the activation energy when there is no long range
magnetic order (\ie for paramagnetic ferrite). The constant $b_{1}$ is
obtained by fitting to experimental data; it can in principle be
partitioned into two components, one representing the increment (due
to magnetic spin order) in the vacancy formation energy and the other
in the migration energy. This is difficult to do in practice because
the calculations require a knowledge of the saddle point
configuration during the migration over the activation barrier. The
important point is that equation~\nnumeqn\ gives a temperature
dependence to the activation energy, making it possible to interpret
the deviations from the Arrhenius relationship.

Very precise measurements by
Iijima \et\ [17] confirm the relation over a wide temperature range
(766--1148~K), with
$Q_0 = 250.6 \pm 3.8\,{\rm kJ\,mol^{-1}}$, $b_{1} = 0.156 \pm 0.003$
and the pre--exponential factor in the Arrhenius equation given by
$D_0 = (2.76\,^{+1.42}_{-1.04})\times 10^{-4}\,{\rm m^2\,s^{-1}}$ 
(\figg). The temperature dependence of the self--diffusion
coefficient apparently shows the normal Arrhenius relationship above
the Curie temperature. Magnetisation data are necessary to use this
diffusion model; these have been reported, for example, by Potter
[21] and Crangle and Goodman [22].



There are two further issues which need to be considered.
Firstly, early research had indicated that the diffusion anomaly
extends to temperatures beyond the Curie temperature, which would be
inconsistent with a theory which relies on just the  the long--range
magnetic--order parameter since the magnetisation  terms then tends to
zero towards $T_c$. Modern high--precision measurements on well
prepared samples do not reveal anomalous effects beyond $T_c$.

Secondly, measurements for $\delta$--ferrite at very
high temperatures do not fit the extrapolated data for paramagnetic
$\alpha$--ferrite (\fagg). Instead, for $\delta$--ferrite it is found
that $D_0 = 9.21\times 10^{-3} \,{\rm m^2\,s^{-1}}$ with a higher
activation energy of $Q=296\,{\rm kJ\,mol^{-1}}$. It has been
postulated that at high temperatures diffusion by a divacancy
mechanism makes a contribution which is additional to that from the
usual monovacancy mechanism [17], though an
alternative explanation is that the neglect of short--range order
effects near $T_c$ gives an error in the slope of the Arrhenius plot
for much higher temperatures. Nevertheless, the  precision of
the measurements made by Iijima and co--workers casts some
doubt on whether a diffusion anomaly exists for $T>T_c$. 
Similar observations have been reported for the self--diffusion in
cubic close--packed cobalt, where the diffusion anomaly is very small
indeed when compared with iron. In spite of this, the Arrhenius plot
curves upwards as the melting temperature is approached, supporting the
hypothesis of an increasing contribution from the divacancy mechanism
[23]. The estimated values of the diffusion parameters
for self--diffusion by the divacancy mechanism in ferrite are:
 $$D_{2v}  =5.2 \exp\bl\{ - {{406,000}\over{RT}}\br\} \qquad {\rm
m^2\,s^{-1}}\numeqn $$
where the units of the activation energy are in ${\rm J\,mol^{-1}}$.
This makes $D_{2v}$ about half $D_v$ at the melting temperature.

\picture{deltaself}{128}{169}{500}{\tabtit{\figg : }{Self--diffusion
coefficient of iron in $\delta-$ ferrite [17]. The $\gamma\rightarrow
\delta$ transformation occurs at 1665~K and melting at 1811~K.}}

Short--range magnetic order persists to temperatures well in excess
of the Curie temperature. A number of models have been proposed which
account for short--range order effects and at the same time allow
$\delta$--ferrite and paramagnetic
$\alpha$--ferrite to be treated identically. The models do involve a
larger number of fitting parameters. In addition they do not seem to
explain why the diffusion anomaly vanishes at $T/T_c\simeq
1.16,$ whereas the short range order parameter remains significant to
even higher temperatures (\fagg). 

To cope with this latter difficulty, Ku\v cera  [24] defined a 
fictitious paramagnetic temperature ($T_p>T_c$) beyond which there is
no measurable effect on diffusion of (short--range) magnetisation
with
$$Q  = Q_0(1 + q\{T\}) \numeqn $$ with the complicated function $
q\{T\}$ being such that contributions from long--range order go to zero
at $T_c$ and those from short--range order become zero at $T_p$.
Zener [25] argued for a temperature dependent activation free energy
because   an atom during its course of migration strains the
surrounding lattice, and the resistance of the lattice to such
deformation changes with temperature. He further suggested that the
temperature coefficient of the ratio $G^*/G^*_0$ should therefore be
the same as that of the shear moduli $\mu_e/\mu_e^o$, where in each
case the subscripts indicate the value at $T=0$. Any temperature
dependence of the activation free energy can be incorporated into the
pre--exponential term in the diffusion coefficient via the activation
entropy, the change in which becomes: 
$$ \Delta S^* \simeq {b_{2}\over
T_m}{{d(\mu_e/\mu_e^o)}\over{d(T/T_m)}}
\numeqn $$ where $T_m$ is the melting temperature and $b_{2}$ is a
fitted constant. Since the shear modulus also depends on the magnetic
state, Ku\v cera incorporated the function $q\{T\}$ in the
pre--exponential factor as well:
$$ D = D_0 \exp\br\{{{0.35q\{T\}Q_0}\over{RT_m}}\bl\}\exp\bl\{
-{{Q}\over{RT}}\br\} \numeqn $$

\picture{magorder}{132}{112}{500}{\tabtit{\figg : }{The
calculated long--range and short--range magnetic order parameters for
bcc iron [24]. The reduced temperature is the ratio of
the absolute temperature to the Curie temperature.}}

Other models have involved empirical modifications of the Ku\v
cera approach, for example by substitution of the excess enthalpy
attributed to the magnetic effect for the function $q\{T\}$ whilst
retaining the same form as equation~\nnumeqn\ [26], or by explicitly
including the shear moduli in the calculations [27,28]. Some models
include two additional terms in the activation energy function, one
containing the variation in the long range order parameter and the
other the variation in the elastic modulus ([28,29]. J\"onsson [30]
pointed out that these models are difficult to distinguish as empirical
representations because the  excess enthalpy and elastic modulus
vary in a similar manner with the temperature. Neither the excess
enthalpy nor the difference in shear moduli between the
ferromagnetic and paramagnetic states reach zero at the
temperature where the diffusion anomalies vanish.

Bearing in mind that enthalpy data are more frequent than modulus
data for alloys, J\"onsson adapted Braun and Feller--Kniepmeier's
method where the diffusion anomaly is expressed purely as a function
of the excess enthalpy, so that in equation~\nnumeqn\ the function $q$
becomes
$$ q\{T\} = b_{3}~ ^{mg}\Delta H\{T\} $$
where $b_{3}$ is a fitting constant and $^{mg}\Delta H\{T\}$ is the
excess magnetic enthalpy at the temperature $T$. Equation~\nnumeqn\
contains the melting temperature whose meaning in the context of
diffusion is not clear for an alloy where melting occurs over a range
of temperatures. However, there is a well known empirical
relationship between the activation energy for diffusion and the
melting temperature of a pure metal,
$$ Q_0 = 140 T_m \qquad {\rm J\,mol^{-1}}$$ so that the melting
temperature can be eliminated from equation~\nnumeqn. By fitting the
equation to ferritic iron, J\"onsson was able to demonstrate that
with appropriate thermodynamic data for $^{mg}\Delta
H$\footnote\dag{a function of the Curie temperature and the
magnetic moment per atom. },
he was able to estimate fairly well the tracer diffusion of elements
such as cobalt, nickel in ferrite both in the ferromagnetic and
paramagnetic regions. The activation energy and
$D_o$ for the paramagnetic state derived by fitting in each case but
$b_{3}$ was kept fixed,  assumed to depend only on the lattice
geometry. 

To summarise, it is possible,  when the thermodynamic data
necessary for the estimation of $^{mg}\Delta H\{T\}$ are available, to
calculate the effect of solute elements on the tracer diffusion
coefficient of iron. It will be interesting to see how such information
can be exploited in the design of creep--resistant steels.

\sec{Acknowledgments}

I am grateful to the organisers Andrew Strang and Malcolm Mclean for
the excellent organisation and for financial support. It is a pleasure
also to acknowledge very useful discussions with Dr Joe Robson and
Mr Dave Cole.

\sec{Conclusions}

It is now possible to attempt a quantitative design of heat
resistant steels and welding alloys. This is true both with respect
to the kinetics of microstructural evolution and in the
estimation of creep rupture strength. The combined models
provide for the first time an ability to predict new alloys. It would
now be interesting for industry to set some challenges, which would
stimulate theoretical predictions and finally experimental
verification. The whole process from the conception of an alloy to
its verification should take much less time than has previously
been the case.

In the longer term it is necessary for the microstructure models to
predict particle size and spatial distributions, and the effect of
stress and strain on transformation kinetics. Such information can
then be  used in a more sophisticated mechanical model, perhaps
based on dislocation and recovery theory. 


\sec{References}

{\parindent=10pt \narrower \parskip=0mm \medskip
\def\ref#1#2#3#4#5#6{\item{#1. } #2: {\it #3} {\bf #4}  
{#6}{(#5)}} 


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\medskip}


\bye



