Phase equilibria and phase target calculations

The target calculations ChemApp is able to perform are a very powerful tool to find one's way around a phase diagram. The following program demonstrates, for the binary system Pb-Sn, how target calculations and standard equilibrium calculations can be used in the calculation of phase constitution. Figure 1 shows the binary phase diagram Pb-Sn, as it has been calculated using the two-dimensional phase mapping module of FactSage. The items labeled will be calculated in the course of this application example.

The type of information to be obtained should include the following:

• The determination of temperatures at which a phase (e.g. the liquid) becomes stable for an alloy of a given composition (formation phase target calculation, item 1 in Figure 1)
• Retrieving the necessary mole fractions and amounts which are associated with the 'lever rule' for an alloy at a given composition and temperature (item 2 in Figure 1).
• The determination of the temperature at which a phase (e.g. a solid phase from the liquid) starts to precipitate (precipitation target calculations, item 3 in Figure 1).
• Calculations similar to the ones above, but with composition instead of temperature as the target variable (item 4 in Figure 1).

Figure 1: The Pb-Sn phase diagram (calculated using FactSage)

```C Phase equilibria and phase target calculations

C Note that this program contains target calculations,
C which cannot be performed with the 'light' version of ChemApp

PROGRAM CAF102

IMPLICIT NONE

CHARACTER NAME*24, MODEL*6

DOUBLE PRECISION VALUE, VALS(2), FSNBCT, FSNFCC,
*     ABCT, AFCC, EPS

INTEGER NOERR, ISLITE, NPHASE, NUMCON, I, NPCON,
*     IPB, ISN, I_LIQ, I_BCT, I_FCC

C Initialise ChemApp
CALL TQINI(NOERR)

C Open the thermochemical data-file pbsn.dat (system Pb-Sn)
OPEN(10, FILE='pbsn.dat', STATUS='OLD', IOSTAT=NOERR)
IF (NOERR .NE. 0) THEN
WRITE(*,*) 'ERROR: Cannot open data-file.'
STOP
ENDIF

CALL TQRFIL(NOERR)

C Close data-file
CLOSE(10)

C To find out about the phases contained in the data-file, print out the
C names of all phases and the associated models
CALL TQNOP(NPHASE, NOERR)

DO I=1, NPHASE
CALL TQGNP(I, NAME, NOERR)
CALL TQMODL(I, MODEL, NOERR)
WRITE(*,FMT='(I2,A)') I, ': ' // NAME // MODEL
ENDDO
```

 Output:  ``` 1: LIQUID RKMP 2: BCT_A5#1 RKMP 3: BCT_A5#2 RKMP 4: FCC_A1 RKMP```
```
C As can be seen from the short table printed above, all phases present
C in the data-file are mixture phases, since the model identifier
C 'PURE' does not appear anywhere. Also, the phase BCT_A5 is entered
C twice into the data-file. This is required because of the inherent
C metastable miscibility gap in the system.

C ****************************************************************
C Formation phase target calculation
C Item 1 in the Pb-Sn phase diagram
C ****************************************************************

C The first calculation will be the determination of the eutectic
C temperature. To do this, we need to define an alloy composition which,
C upon solidification, would undergo a eutectic transformation. An alloy
C consisting of 50% Sn and 50% Pb is a good candidate. We will tell
C ChemApp to use an alloy of this composition, and ask for the
C temperature (which thus is the target variable) at which the liquid
C phase first becomes stable (which thus is the target). Applying the
C 'formation phase target liquid' thus means that ChemApp will determine
C the point where the liquid phase is stable (activity is unity) and its
C equilibrium amount is the one selected by the programmer (in our case
C zero).

C We will use 'global conditions', because we are not interested in
C calculating extensive property (e.g. 'heat') balances.

C We will use the system components to input our incoming amounts. Note
C that it does not matter whether we input our incoming amounts as
C system components, or through any of the mixture phase constituents or
C stoichiometric compounds, since we are not interested in the extensive
C property balances.

C Find out the system component index numbers for Pb and Sn:
CALL TQINSC('Pb ', IPB, NOERR)
CALL TQINSC('Sn ', ISN, NOERR)

C Using the index numbers just determined, we can enter our incoming
C amounts:

C Entering 0.5 mol Pb
CALL TQSETC('IA ', 0, IPB, 0.5D0, NUMCON, NOERR)

C Entering 0.5 mol Sn
CALL TQSETC('IA ', 0, ISN, 0.5D0, NUMCON, NOERR)

C Check if we are working with the 'light' version.
C If we do, omit the following target calculation(s).
CALL TQLITE(ISLITE, NOERR)
IF (ISLITE .EQ. 1) THEN
WRITE(*,FMT='(3(1X,A,/))')
*     '*** Target calculations have been omitted here,',
*     '*** since they are not possible with the ',
*     '*** ''light'' version of ChemApp.'

ELSE

C The 'LIQUID' phase is the target phase. From the first output we see
C that it has the index number 1, which can also be determined by a call
C of TQINP:
CALL TQINP('LIQUID ', I_LIQ, NOERR)

C Using TQSETC, we define the _target_: formation phase target 'LIQUID',
C zero amount
CALL TQSETC('A ', I_LIQ, 0, 0.D0, NUMCON, NOERR)

C The _target variable_ is passed to TQCE/TQCEL: The target variable
C used is the temperature, with 300 K as the first guess. This estimate
C does not need to be very accurate.
VALS(1) = 300.0
CALL TQCE('T ', 0, 0, VALS, NOERR)

C Retrieve the calculated temperature
CALL TQGETR('T ', 0, 0, VALUE, NOERR)

WRITE(*,FMT='(1X,A,G12.5,A)') 'The eutectic temperature is ',
*     VALUE, ' K'
```

 Output:  ` The eutectic temperature is 454.56 K`
```
ENDIF

C ****************************************************************
C Phase equilibrium calculation (verification of the 'lever rule')
C Item 2 in the Pb-Sn phase diagram
C ****************************************************************

C Now, for a temperature below the eutectic one, calculate the mole
C fraction of Sn in both eutectic phases (FCC and BCT_A5#1)

C The alloy composition is left unchanged (50% Sn, 50% Pb), but now the
C temperature is set to 400 K
CALL TQSETC('T ', 0, 0, 400.D0, NUMCON, NOERR)

C Calculate the equilibrium
CALL TQCE(' ', 0, 0, VALS, NOERR)

C To check the lever rule for this case, and compare it to the graphical
C representation in the Pb-Sn phase diagram, we need the mole fraction
C of Sn in the BCT_A5#1 and FCC phases (FSNBCT and FSNFCC), the amounts
C of BCC and FCC phase (ABCT and AFCC), plus the mole fraction of Sn in
C the alloy, which is 0.5.

C Get the equilibrium amount of phase BCT_A5#1 in mol
CALL TQINP('BCT_A5#1 ', I_BCT, NOERR)
CALL TQGETR('A ', I_BCT, 0, ABCT, NOERR)

C Get the mole fraction of Sn in BCT_A5#1
CALL TQGETR('XP ', I_BCT, ISN, FSNBCT, NOERR)

C Get the equilibrium amount of phase FCC in mol
CALL TQINP('FCC ', I_FCC, NOERR)
CALL TQGETR('A ', I_FCC, 0, AFCC, NOERR)

C Get the mole fraction of Sn in FCC
CALL TQGETR('XP ', I_FCC, ISN, FSNFCC, NOERR)

C For the lever rule condition to be satisfied, the following equation
C has to hold: (0.5 - FSNFCC)*AFCC - (FSNBCT - 0.5)*ABCT = EPS
C with EPS being zero within the numerical precision.
EPS = (0.5D0 - FSNFCC)*AFCC - (FSNBCT - 0.5D0) * ABCT

WRITE(*,FMT='(1X,A,G12.5)') 'The value of EPS is ', EPS
```

 Output:  ` The value of EPS is -0.48727E-15`
```
C As you can see, the lever rule is sufficiently satisfied.

C Check if we are working with the 'light' version.
C If we do, omit the following target calculation(s).
CALL TQLITE(ISLITE, NOERR)
IF (ISLITE .EQ. 1) THEN
WRITE(*,FMT='(3(1X,A,/))')
*     '*** Target calculations have been omitted here,',
*     '*** since they are not possible with the ',
*     '*** ''light'' version of ChemApp.'

ELSE

C ****************************************************************
C Precipitation target calculation, with temperature as target
C variable
C Item 3 in the Pb-Sn phase diagram
C ****************************************************************

C The next calculation is again a target calculation. This time, a
C 'precipitation phase target' is used to determine a point on the
C liquidus curve, the alloy composition being the same as before. Using
C a precipitation phase target tells ChemApp to search for the condition
C under which a second phase becomes stable.

C The subsequent call to TQSETC tells ChemApp to set the following
C _target_: The liquid phase is stable, and a second phase (which does
C not need to be specified) has unit activity.
CALL TQSETC('A ', I_LIQ, 0, -1.D0, NUMCON, NOERR)

C Use the temperature as _target variable_ and calculate the equilibrium
VALS(1) = 1000.0
CALL TQCE('T ', 0, 0, VALS, NOERR)

C Retrieve the calculated liquidus temperature
CALL TQGETR('T ', 0, 0, VALUE, NOERR)
WRITE(*,FMT='(1X,A,G12.5,A)') 'The liquidus temperature is ',
*     VALUE, ' K'
```

 Output:  ` The liquidus temperature is 510.45 K`
```
C ****************************************************************
C Precipitation target calculation, with composition as target
C variable
C Item 4 in the Pb-Sn phase diagram
C ****************************************************************

C A target variable which is especially useful in T-x diagrams with
C rather steep phase boundaries is the composition of an alloy. The
C following example determines the point of the FCC phase boundary at
C 500 K where LIQUID is formed (or precipitates).

C First, for convenience, remove _all_ previously set conditions. This
C is especially useful since we have to remove the conditions describing
C the fixed composition of the alloy (set toward the beginning of the
C program) anyway, as the composition is now the target variable and
C thus supposed to vary.
CALL TQREMC(-2, NOERR)

C Set the temperature to 500 K
CALL TQSETC('T ', 0, 0, 500.D0, NUMCON, NOERR)

C Set the target: The FCC phase is stable, and a second phase has unit
C activity.
CALL TQSETC('A ', I_FCC, 0, -1.D0, NUMCON, NOERR)

C If the composition is the target variable and the total input amount
C should always be 1 mol, we need to vary two compositions
C symmetrically. In the present case, we tell ChemApp to vary the input
C amount of Sn/FCC/ between 0 and 1 to find the target, and to vary
C Pb/FCC/ symmetrically between 1 and 0.

C The upper and lower limits of the composition target variables are
C passed via the array VALS to TQCE. First we pass the limits for
C Sn/FCC/.
VALS(1) = 0.0
VALS(2) = 1.0

C We need to tell ChemApp that we are not finished with our input, since
C we still need to specify the symmetrical limits for Pb/FCC/
C later. This is done by calling TQCE with the option 'IA0', which tells
C ChemApp not to perform an equilibrium calculation, but to expect more
C incoming amounts.
CALL TQCE('IA0 ', I_FCC, ISN, VALS, NOERR)

C Enter the symmetrical limits for Pb/FCC/
VALS(1) = 1.0 - VALS(1)
VALS(2) = 1.0 - VALS(2)

C With the next call to TQCE, the information ChemApp needs to perform
C a composition target is complete, so we call ChemApp with the option
C 'IA', upon which the equilibrium is calculated.
CALL TQCE('IA ', I_FCC, IPB, VALS, NOERR)

C ChemApp has now varied the composition of the alloy until the desired
C point of the phase boundary has been found. The desired value (the
C mole fraction of Sn in the FCC phase) is retrieved.
CALL TQGETR('XP ', I_FCC, ISN, FSNFCC, NOERR)
WRITE(*,FMT='(1X,A,G12.5)') 'The mole fraction of Sn is ', FSNFCC
```

 Output:  ` The mole fraction of Sn is 0.20698`
```
ENDIF
END
```

Continue with the code example "One-Dimensional Phase Mapping" or jump to the code example "Process Modelling Using Streams".

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