T8: Calculating
a phase diagram in a binary system
This tutorial was created on
MatCalc version 5.23 rel 1.026
license: free
database: mc_sample_fe.tdb
This tutorial introduces the calculation of phase diagrams with
an example featuring the well-known Fe-C binary system.
Contents:
- Calculating phase boundaries
- Building up the phase diagram from the calculated boundaries
- Choosing appropriate calculation parameters
- Phase diagrams using stable and metastable equilibria
A phase diagram consists of a number of phase-boundaries separating
regions of temperature- and composition-space in which different
phases are stable at equilibrium. Published equilibrium diagrams
for the Fe-C system often show equilibria between Fe-rich phases
and both C (in the form of graphite) and cementite (Fe3C). Graphite
is the more stable of these two phases, and forms readily in
cast irons (2-4 wt.% C), but usually with great difficulty in
steels (up to 1.5 wt.% C). In the latter, it is more practically
useful to consider metastable equilibria involving Fe3C. In this
tutorial, phase diagrams for both Fe-C and Fe-Fe3C will be calculated.
1. Setting up the system
Create a new workspace file. From the 'mc_sample_fe.tdb' database,
select the elements FE and C and the phases LIQUID, BCC_A2, FCC_A1,
CEMENTITE and GRAPHITE. Enter the composition 0.05 wt.% C. Create
a new plot window of type p1 (X-Y data).
The phase diagrams will be calculated from 0 to 5 wt.% C, but
the low carbon content of 0.05 wt.% C is chosen so that the lines
bounding the high-temperature ('delta') ferrite phase, which is
only stable at relatively low carbon contents, can be found.
Firstly, the Fe-Fe3C diagram will be calculated, so it is necessary
to suspend graphite, since it is more stable than Fe3C. Open 'Global > Phase
status'. In the 'General' tab, put a tick-mark in the box beside
'suspended':

2. Fe-Fe3C phase diagram
The phase diagram is made up of a number of phase boundaries.
As was have seen in the last part of Tutorial 7, these boundaries can
be found by finding a point on the boundary, then tracing the boundary
as a function of element content. The method for tracing a boundary
for phase 'X' is as follows:
- Calculate an equilibrium at an appropriate temperature, near
to the expected boundary temperature
- Find the 'X'-phase boundary by varying the temperature with
'X' as the target phase
- Perform a stepped calculation, varying the element content
between selected limits, with 'X' as the target phase
Upper boundary of delta-ferrite, BCC_A2
Beginning at the high-temperature end of the diagram, the uppermost phase boundary at a
composition of 0.05 wt.% C is between a single-phase liquid region
and a liquid and delta-ferrite (BCC_A2) region. The first boundary
which must be found is, therefore, a BCC_A2 phase boundary.
To locate the boundary, an equilibrium must first be calculated
reasonably close to the expected boundary temperature. It can be
seen that the line lies somewhere between 1538°C (the melting
temperature of pure iron) and 1500°C. Click on the icon and enter the temperature '1530'.
(If this gives an error message, select 'Calc > Set
start values' then
repeat the equilibrium calculation.) Then click on to
open the 'Search
phase boundary' window. Select 'Temperature' in
the left-hand column and 'BCC_A2' as
the target phase. The 'Output' window
should show the following message:
Tsol 'BCC_A2': 1533.98 C (1807.14 K) iter: 4, time used: 0.00
s
This shows that a point on the boundary has been found successfully.
The next step is to trace this boundary using the stepped equilibrium
function to vary the carbon content. Click on and
choose 'Phase
boundary' from the left-hand column. Enter the range: '0' to '5', in intervals of '0.01'. Select 'C' as the varying element and
'BCC_A2' as the target phase. Leave the 'Max
T-step' value at its default value
of '20'. In the 'Options' section of
this window, check that 'Temperature
in Celsius' and 'Composition in weight
percent' are both selected.
Click on 'Go' to launch the calculation.

The following message appears ; accept it with 'Yes'.

Now that the first phase boundary has been calculated, drag and
drop 'T$C' from the 'Variables' window (where it can be found under
'favorites') to the plot. It is useful to add gridlines to the
plot. Note that the x-axis currently only runs from zero to just
over 0.5 wt.% C. From the point identified on the phase boundary,
the calculation first proceeds in a direction of increasing carbon
content. However, the BCC_A2 phase is not stable beyond approx
0.53 wt.% C, so the calculation (which cannot reach the stop limit of 5 wt.% C) stops and the 'Continue
in other direction?' message shown above is displayed to ask whether the
user wishes to complete the rest of the calculation, in which the
carbon content is decreased from the point identified at 0.05 wt.%
C to zero.

Right-click in the plot window outside the plot area itself, and
select 'Duplicate and lock series' from the menu displayed. Select
the name of the series and click 'Duplicate' to produce a copy
of the series. In the 'options' window, rename the first series
'bcc_high_1'.
Lower boundary of liquid
The next phase boundary encountered on decreasing the temperature
is between the LIQUID + BCC_A2 region and the BCC_A2 single-phase
region, so this is a LIQUID-phase boundary. 1500°C would appear
to be a suitable temperature for the starting equilibrium to find
this line. In the 'search phase boundary' box, select 'LIQUID'.
The following line of output should appear:
Tsol 'LIQUID': 1515.43 C (1788.59 K) iter: 4, time used: 0.00
s
Then make a stepped calculation with the same start, stop and
interval values and the target phase set to 'LIQUID'.
(A start-value of zero is automatically changed by the program
to '1e-9' to facilitate the calculation.) The diagram should now
look like this:

Lower boundary of delta-ferrite (BCC_A1)
Calculate an equilibrium at 1400°C, search for BCC_A2, step
from 0 to 5 wt.% C with an interval of 0.01 wt.% C. Accept the
'Continue in other direction' message (the calculation could not reach 5 wt.% C limit, again) with 'Yes'. The boundary
temperature should be 1429.60 C (1702.76 K).
Upper boundary of austenite (gamma, FCC_A1)
Calculate an equilibrium at 1500°C, search for FCC_A1, step
from 0 to 5 wt.% C with an interval of 0.01 wt.% C. The boundary
temperature should be 1460.18 C (1733.34 K).
Upper boundary of low-temperature ('alpha') ferrite, BCC_A2
Calculate an equilibrium at 900°C, search for BCC_A2, step
from 0 to 5 wt.% C with an interval of 0.01 wt.% C. The boundary
temperature should be 888.64 C (1161.80 K). The image below shows
how the diagram should look, with appropriate labels for the lines:

The most of
the upper section is now done, with the exception of the phase
boundary between LIQUID and LIQUID + CEMENTITE for hypereutectic
compositions. This line is continuous with the cementite line between
the FCC_A1 and FCC_A1 + CEMENTITE regions and the horizontal line
between BCC_A2 + FCC_A1 and BCC_A2 + CEMENTITE regions, so it will
appear when this line is calculated.
In the lower section, there is a very narrow alpha-ferrite single-phase
region at low carbon contents. The upper part of this region is
bounded by an FCC_A1 boundary, and the lower part of the region
by a CEMENTITE boundary. Since these two boundaries are very steep,
it is necessary to modify the calculation parameters. Failure to
do this can lead to incorrectly calculated boundaries, as shown
in the diagram below (an attempt to calculate the FCC_A1 line using
the parameters used so far).

Lower boundary of austenite (gamma, FCC_A1)
Calculate an equilibrium at 700°C then search for the FCC_A1
phase boundary, which should be at 726.62°C. Step from 0 to
5, with an increment of 0.001 instead of 0.01. This reduced increment
makes the calculation rather more time-consuming; on a slow machine,
it may be better to divide it into two parts: step from 0 to 0.02
with an increment of 0.001, duplicate and lock, step from 0.02
to 5 with an increment of 0.01.
Upper boundary of cementite
Calculate an equilibrium at 800°C. Search for the CEMENTITE
phase boundary; this also should be at 726.62°C. Using the
same settings as for the previous stepped calculation, the phase
boundary is calculated correctly, but extends in the negative temperature
direction beyond physically reasonable values. This occurs because
the calculated boundary becomes increasingly steep as the carbon
content approaches zero.
One way of removing the unwanted line portion is simply to modify
the scaling of the y-axis. (The minimum temperatures given on equilibrium
diagrams are typically 400-500°C, because it becomes difficult
to attain equilibrium over practical timescales at low temperatures.)
Another solution is to remove unwanted data points using 'Edit
series data' from the right-click menu of the plot.
Alternatively, the 'overshoot' can be avoided by modifying the
parameters of the stepped calculation. Increasing the 'max
T-step' allows greater steps in temperature, enabling the program to cope
with very steep boundaries. A suitable value is 100 or 200. Additionally,
setting the starting value of the stepped calculation to a slightly
higher value, for example 0.0005 instead of 1e-9, avoids the calculation
of the steepest parts of the phase boundary. With these modified
settings, a satisfactory appearance is obtained without the need
to change the axis scalings or delete any points.
The finished diagram
Do not forget to lock the last series to be added, so that it
is not modified when another calculation is performed. Add axis
labels and a title 'Fe-Fe3C phase diagram'. The subscript can be
obtained using HTML tags:
Fe-Fe<sub>3</sub>C phase diagram
The finished phase diagram should look like this; compare with
the sketch below to verify that all the lines have been calculated
correctly.

The image below shows an expanded version of the low-C region.

3. Fe-C phase diagram
Set up this system by removing the 'suspended' flag for GRAPHITE
under 'Global > Phase status > General'.
The calculation of the Fe-C diagram is left as an exercise to
allow the user to familiarise him- or herself with the method,
including the choice of appropriate values for the starting equilibrium
calculation temperatures and the calculation parameters. However,
suggestions are given at the foot of the page. The diagram below
shows how the finished diagram should look. (Alternatively, the
phase boundaries from the Fe-C calculations can be superposed on
the Fe-Fe3C phase diagram for comparison.)

Suggested parameters for calculation of Fe-C phase diagram
- Upper boundary of high-T bcc: Equilibrium at 1550°C (Tsol 'BCC_A2':
1533.98°C); step from 0 to 5 with interval of 0.01, max T-step
20
- Lower boundary of liquid: Equilibrium at 1500°C (Tsol 'LIQUID':
1515.43°C); step with same parameters
- Lower boundary of high-T bcc: Equilibrium at 1400°C (Tsol 'BCC_A2':
1429.60°C); same parameters
- Upper boundary of fcc: Equilibrium at 1500°C (Tsol 'FCC_A1':
1460.18°C); same parameters
- Upper boundary of low-T bcc: Equilibrium at 900°C (Tsol 'BCC_A2':
888.64°C); same parameters
- Lower boundary of fcc: Equilibrium at 700°C (Tsol
'FCC_A1': 738.02°C). For this line, the parameters must be
modified: decrease the interval to 0.001 or increase the T-step
to 100. The latter solution (an interval of 0.01 and a T-step
of 100) is acceptable because this boundary is a straight line
so is correctly defined despite there being only a few points
in this region. An error message (max no of iterations reached)
may appear, but the line is calculated successfully.
- Upper boundary of graphite:
Equilibrium at 800°C (Tsol 'GRAPHITE': 738.02°C). In this case,
the boundary is curved, so it is advisable to decrease the interval
to 0.001 as well as increasing the T-step to 100 to obtain better
resolution of the curvature. This boundary, like the cementite
boundary discussed above, tends to 'overshoot' to very low temperature
values. This can be prevented by setting the minimum carbon content
for the stepped calculation to 0.0005 instead of 1e-9.
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