IET Science, Measurement & Technology
Research Article
Thermodynamic calculations of the Mn–Sn,
Mn–Sr and Mg–Mn–{Sn, Sr} systems
ISSN 1751-8822
Received on 8th December 2013
Accepted on 29th December 2014
doi: 10.1049/iet-smt.2013.0267
www.ietdl.org
Mohammad Aljarrah 1 ✉, Suleiman Obeidat 1, Rami Hikmat Fouad 1, Mahmoud Rababah 2,
Ahmad Almagableh 2, Awni Itradat 3
1
Industrial Engineering Department, Faculty of Engineering, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan
Mechanical Engineering Department, Faculty of Engineering, The Hashemite University, Zarqa 13115, Jordan
3
Department of Computer Engineering, Faculty of Engineering, The Hashemite University, Zarqa 13115, Jordan
✉ E-mail: maljarrah@hu.edu.jo
2
Abstract: Thermodynamic modelling of Mn–Sn and Mn–Sr binary systems is carried out using the reliable data from the
literature. Thermodynamic properties of the binary liquid solutions are described using the modified quasi-chemical
model. The calculated phase diagrams and the thermodynamic properties are found to be in good agreement with the
experimental data from the literature. A self-consistent thermodynamic database for the Mg–Mn–{Sn, Sr} systems is
constructed by combining the thermodynamic descriptions of their constituent binaries. The constructed database is
used to calculate and predict liquidus projection and invariant reactions of these ternary systems. The Mg–Mn–Sr
system has nine ternary eutectic reactions, two saddle points and eleven crystallisation fields. Mg–Mn–Sn has four
saddle points, two quasi-peritectic and six ternary eutectic reactions.
1
Introduction
Automobile manufacturers are continually searching for means to
reduce vehicle weight in order to increase fuel economy. With
their low density, high stiffness/weight ratio and high damping
capacity, magnesium alloys have the potential to make a
significant contribution to this weight reduction. Wrought
magnesium alloys hold great promise for use in structural
applications [1]. However, the possible application of wrought
magnesium alloys is strongly reduced by the low formability and
the mechanical anisotropy of this material [2]. This is mainly
caused by its hexagonal crystal structure which offers limited slip
systems at room temperature [3]. There are several attempts to
improve formability of magnesium alloys in which alloying
elements were found to play a key role [4–25]. Among them, rare
earth (RE) elements, manganese and strontium additions improve
formability of wrought magnesium alloys through grain size
refinement, activation of non-basal slip and/or weakening the
texture [16–25]. Li and RE additions to Mg are likely to bring
numerous liabilities including high cost, increased flammability
during processing and decreased corrosion resistance [1, 6, 19].
Mg–Mn-based alloys have some popularity because of their good
weldability [26]. They provide medium strength in wrought
product form such as extrusion, rolling, sheet and plate [26]. Sr
provides good grain refinement in Mg-alloys and thereby enhances
mechanical properties [23, 24, 27]. Alloying Mg with Sn is
promising for high-temperature applications. Addition of Sn to
Mg-based alloys leads to precipitations of thermally stable
compound (Mg2Sn) which provides stable microstructure and
enhances mechanical properties at 150°C [28, 29].
Thermodynamic descriptions of Mg–Mn–Sr and Mg–Mn–Sn
systems are very important for further improvement in alloy design
through providing invaluable information such as solubility of
alloying addition in Mg-matrix, heat treatment temperature,
amount of secondary phase precipitates and solidus and liquidus
temperatures.
In this paper, thermodynamic modelling of Mg–Mn–{Sn, Sr}
systems are carried out for the first time according to authors
knowledge. Among the constituent binaries, Mn–Sn and Mn–Sr
systems were re-optimised using the modified quasi-chemical model
(MQM) for the liquid solution. On the basis of calculation of phase
IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692
& The Institution of Engineering and Technology 2015
diagrams (CALPHAD) method [30], all thermodynamic information
such as experimental phase diagram data and thermodynamic
properties data are critically evaluated and optimised simultaneously.
Thermodynamic optimisation and calculations were performed in this
paper using FactSage program [31].
2
2.1
Experimental data
Mn–Sn system
Watchel and Ulrich [32] and Watchel et al. [33] investigated the
phase relationships of the Mn–Sn system for a composition range
of (0–80 at.% Mn) by metallography, magnetic susceptibility and
polarisation. Their samples were prepared from 99.999 wt.% Sn
and 99.99 wt.% Mn and melted in high-frequency induction
furnace under argon. Homogenisation was carried out by
annealing the specimen in quartz tube under argon atmosphere.
Their alloys were heat treated from 200°C for 40 days to up to
800°C for 10 h. They [32] reported that Mn2Sn phase melts
incongruently at 883°C which is comparable with 897°C, the
calculated value of Singh et al. [34] and William [35]. Watchel
and Ulrich [33] indicated that MnSn2 melts incongruently at 549°C,
compared with Singh et al. [34] and William [35] results as
548°C. Their data [32] indicates that Mn3Sn is a solid solution in
the composition range of 76.4–77.6 at.% Mn at 700°C. Singh
et al. [34] investigated Mn–Sn phase diagram below 50 at.% Sn
using optical microscopy and X-ray diffraction using high purity
Mn (99.9 at.%) and Sn (99.999 at.%) starting materials. Their
samples were cast in recrystallised alumina crucible using
high-frequency induction furnace under protective gas (argon).
Moreover, their studied samples were wrapped in molybdenum
foil and sealed in evacuated fused silica capsules, then annealed
for 15 days at 500°C to 10 h at 1000°C. Singh et al. [34] reported
low solubility of Sn in αMn phase which is in agreement with
Nail’s [36] observation. Nowotny and Schubert [37] and Zwicker
[38] determined the solubility of Sn in βMn phase and reported
value of 8.7 at.% Sn at 600°C, whereas Singh et al. [34] indicated
this solubility to be about 10 at.% Sn. The experimental phase
diagram data from [32–34] are self-consistent and considered as
681
the most reliable data, therefore these data will be incorporated in the
current optimisation of Mn–Sn system.
Earlier investigations differ on the Mn3Sn phase region. According
to Nail [36], the Mn3Sn phase exists between 23 and 24.5 at.% Sn,
whereas Guillaud [39] indicated the Mn3Sn phase region between
20 and 22 at.% Sn. On the other hand, Nowotny and Schubert [37]
reported the Mn3Sn phase at 21.43 at.% Sn. The structure type of
Mn3Sn has been suggested to be Ni3Sn [34, 36, 37]. Elding-Pontén
et al. [40] investigated NiAs–Ni2In structure types in the Mn–Sn
system using X-ray diffraction and scanning electron microscopy/
energy dispersive X-ray spectroscopy (SEM/EDX). Three different
phases were detected such as high temperature phase (HTP)1
(Mn17Sn7), HTP2 (Mn8Sn5) and Mn3Sn2 phases. They reported that
there were difficulties in preparing the samples since traces of tin
were detected on the outside of the ingot. Besides, different phases
were not distinguishable because of having very similar
compositions. The chemical compositions and crystal structures of
HTP1 and HTP2 phases detected in this paper of Elding-Pontén
et al. [40] were not reported or verified in the literature. Therefore
these phases will not be incorporated in the current paper. Stange
et al. [41] studied the crystal structure of binary phases in the
Mn–Sn system using X-ray diffraction, neutron diffraction and
differential thermal analysis. They revealed the uncertainty on the
stability and composition of Mn2Sn. According to their work,
Mn2Sn consists of a high-temperature phase Mn(2−x)Sn and a
low-temperature phase Mn3Sn2 rather than Mn2Sn or HPT2.
Moreover, Mn(2−x)Sn is stable between 480 and 884°C and the
homogeneity range of this phase at 800°C is 0.18 ≤ x ≤ 0.23 and
0.28 ≤ x ≤ 0.34 at 600°C. Therefore in this paper Mn(2−x)Sn will be
treated as solid solution and Mn3Sn2 will be modelled as a
stoichiometric compound.
The phase diagram of Mn–Sn system was assessed by Massalaski
et al. [42] and Stange et al. [41]. According to these works, the
composition ranges of (αMn) and (βMn) were 0.0–1.0 and 5.5–10
at.% Sn at 400°C, respectively. Miettinen [43] calculated Mn–Sn
phase diagram using the experimental data from [32–34, 44–47].
He [43] also described the liquid phase using the random solution
model. According to Miettinen [43], the stable phases in the
Mn–Sn system were (αMn), (βMn), (γMn), (δMn), MnSn2, Mn2Sn
and Mn19Sn6 (Mn3Sn). In his paper [43], Mn–Sn phase diagram
was reported in two figures; Mn3Sn phase was treated as
stoichiometric and in the other figure Mn3Sn modelled as a solid
solution. Moreover, the calculated activity at 1000°C of Miettinen
[43] could not produce good fit with experimental data of
Eremenko et al. [47]. According to Okamoto’s review [48] on the
Mn–Sn phase diagram, the stable phases in this system are;
(αMn), (βMn), (γMn), (δMn), MnSn2, (Mn(2−x)Sn), Mn3Sn2,
(Mn3Sn) and β Sn. In this paper, the stable phases adopted by
Okamoto will be considered and re-optimised.
2.2
Mn–Sr system
The work of Obinata et al. [49] is the only experimental data that could
be found in the literature. According to their work, this system forms a
miscibility gap through the interval 0.74–96.5 at.% Sr at 1240°C and
no intermetallic compound was observed in their work. Peng et al.
[50] modelled all phases in the Mn–Sr binary system as completely
disordered solutions. In their work, Gibbs energy is described by
Redlich–Kister polynomial and model parameters were evaluated
using Thermo-Calc, whereas Janz [51] calculated liquidus projection
of Mg–Mn–Sr ternary system. However, he did not describe
thermodynamic modelling of the Mn–Sr binary system and the
optimised parameters were not reported.
In the current paper, the liquid phase of the Mn–Sr system will be
modelled using MQM to have consistent thermodynamic description
with Mg–Sn, Mg–Mn and Mg–Sr systems.
work of Ghosh et al. [52], Ghosh and Medraj [53] and Aljarrah
and Medraj [54], respectively. The calculated phase diagrams of
the Mg–Sn, Mg–Mn and Mg–Sr phase diagrams are in reasonable
agreement with all reported data in the literature. Liquid phase was
optimised using the MQM. Therefore the works of Ghosh et al.
[52], Ghosh and Medraj [53] and Aljarrah and Medraj [54] were
considered to construct a self-consistent thermodynamic database
of the Mg–Mn–{Sn, Sr} ternary systems.
2.4
Mg–Mn–Sn ternary system
In 1969, Kopetskii and Semenova [55] experimentally identified
phases in the Mg-rich region of the Mg–Mn–Sn system and draw
isothermal sections at 500 and 400°C. The identified phases in
their isothermal section will be compared with the current
calculations. They also reported a ternary eutectic at 554°C where
L ↔ (Mg) + αMn + Mg2Sn. The work of Kopetskii and Semenova
[55] is the only experimental data of Mg–Mn–Sn system that
could be found in the literature.
2.5
Mg–Mn–Sr ternary system
Celinkin et al. [27] investigated microstructure and creep behaviour
of eight Mg-{0.75 − 2} wt.% Mn-{3 − 5} wt.% Sr alloys in the
Mg-rich region. In their works, samples were heat treated at 225°C
for 150 h and 300°C for 96 h. They measured solid solubility of
Mn in Mg-matrix using TEM/EDS. These solubilities will be
compared with the current thermodynamic calculation. Three
phases were positively identified in the heat treated alloys, namely;
Mg17Sr2, (Mg) and α-Mn. XRD analysis in the heat treated alloys
show no change in phase constitution compared with the as-cast
alloys. In Janz’s doctoral thesis [51], liquidus projection and
isothermal sections at 500 and 400°C of the Mg–Mn–Sr ternary
system were calculated. In the current paper, a comparison
between the current liquidus projection and Janz’s work will be
discussed.
Mg–Mn–Sn and Mg–Mn–Sr systems are subsystems of
multi-components Mg-alloys. The purpose of this paper is to
provide a comprehensive thermodynamic description of the Mg–
Mn–Sn and Mg–Mn–Sr systems which is the backbone for
understanding solidification and phase equilibria in the Mg–
Mn-based alloys.
3
Thermodynamic models
3.1
Pure elements
The Gibbs free energy of a pure element with a certain structure f is
described as a function of temperature as
◦
GAf (T ) = a + bT + cT ln T + dT 2 + eT 3 + fT −1 + gT 7 + hT −9
(1)
The parameters a through h are taken from the SGTE compilation by
Dinsdale [56].
3.2
Stoichiometric compounds
Intermetallic compounds in the Mn–Sn phase diagram such as
Mn3Sn2 and MnSn2 are considered stoichiometric and the Gibbs
free energy of these compounds is described by the following
equation
Gphase,f = xi ◦ Gif + xj ◦ Gjf + DGf
2.3
Mg–Sn, Mg–Mn and Mg–Sr systems
Optimised Gibbs energy parameters of the constituent binary
systems, Mg–Sn, Mg–Mn and Mg–Sr will be adopted from the
682
(2)
where ◦ Gif and ◦ Gjf denote Gibbs free energy of elements i and j in
their standard state and ΔGf = a + bT is the Gibbs energy of
formation of the stoichiometric compound, where a and b are the
IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692
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Table 1 Crystal structure and lattice parameters of Mn3Sn phase [65]
Crystal data
Atoms
WP1
CN2
PS3
Atomic position
X
structure type
Pearson symbol
space group
space group number
lattice parameter, nm
Ni3Sn
CP2
P63/mmc
194
Mn
Sn
6h
2d
mm2
−6m2
6
6
0.16667
1/3
Y
Z
0.3333
2/3
1/4
3/4
a = b = 0.56834 and c = 0.45335, angles: α = 90, β = 90, γ = 120
WP1 Wyckoff position, CN2 coordination number and PS3 point symmetry
model parameters to be optimised based on experimental data of
phase equilibria and thermodynamic properties.
3.3
Terminal solid solutions
The Gibbs energy of a disordered solid solution phase is described
by the following equation
G = xi ◦ Gif + xj ◦ Gjf + RT [xi ln xi + xj ln xj ] + ex Gf
(3)
The configurational entropy of mixing is described by the
following equation
DS config = −R[nA ln (xA ) + nB ln (xB )]
x
xBB
xAB
+
n
+
n
ln
ln
− R nAA ln AA
BB
AB
2yA yB
y2A
y2B
(7)
where xA and xB are the overall mole fractions of the components A
and B, respectively.
Mole fraction of component A can be described as follows
where f denotes the phase in question and xi, xj denote the mole
fraction of components i and j, respectively. The excess Gibbs
energy is represented using Redlich–Kister equation
nA
nA + nB
(8)
nAA
nAA + nBB + nAB
(9)
XA =
Pair fraction
ex
Gf = xi · xj
n=m
n f
Li, j (xi
− xj )n
(4)
XAA =
n=0
with n Lfi, j = an + bn × T (n = 0, . . . , m) where n Lfi, j is the
interaction parameters and an and bn are model parameters to be
optimised using experimental phase diagram and thermodynamic
data. In Mn–Sn binary system, (αMn), (βMn), (γMn) and (δMn)
phases are treated as disordered solution model.
3.4
And the coordination-equivalent fractions
yA =
Gliq = nA ◦ gAliq + nB ◦ gBliq − T DS config +
nAB liq
DgAB
2
(5)
1
1
2nAA
1
nAB
+ A
= A
ZA ZAA 2nAA + nAB
ZAB 2nAA + nAB
1
1
2nBB
1
nAB
+ B
= B
ZB ZBB
2nBB + nAB
ZBA 2nBB + nAB
◦
+
DgAB = DgAB
i≥1
io
i
gAB
XAA
+
oj
j
gAB
XBB
(6)
j≥1
jo
◦
io
, gAB
and gAB
are the model parameters to be optimised
where DgAB
◦
= a + bT .
and can be expressed as DgAB
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& The Institution of Engineering and Technology 2015
(11)
(12)
A
B
where ZAA
and ZBB
are the values of the coordination number of the
Ath atom when all nearest neighbours are A’s and B’s, respectively.
The coordination number of pure elements in the metallic solution,
Table 2 Crystal structure and lattice parameters of Mn(2−x)Sn phase [65]
Crystal data
where nA and nB are the number of moles of the components A and B,
nAB is the number of moles of (A-B) pairs, ΔS config is the
configurational entropy of mixing given for randomly distributing
the (A-B), (B-B) and (A-B) pairs.
liq
Pelton et al. [57] made modification to (5) by expanding DgAB
as a polynomial in terms of the pair fractions XAA and XBB as
shown in (6)
(10)
In addition, further modification has been made to permit
coordination number to vary with composition as follows
Liquid phase
In the current paper, the liquid phase is modelled using the MQM
where the pair approximation is utilised to describe short-range
ordering in the liquid. A detailed description of the MQM for
binary and multi-components solutions is available elsewhere
[57–59]. Only a brief description will be presented here. The
molar Gibbs energy of the liquid phase, derived from the modified
quasi-chemical theory [57], is described by the following equation
ZA nA
ZA nA + ZB nB
Atoms
WP1
CN2
PS3
Atomic
position
X
structure type
InNi2
Pearson
Hp6
symbol
space group
P63/mmc
space group
194
number
lattice
parameter,
nm
angles: α = 90, β = 90,
γ = 120
Y
Z
Mn1
Mn2
2d
2a
8
6
−3 m
−6m2
1/3
1/3
2/3
2/3
3/4
1/4
Sn
2c
8
−6m2
0
0
0
a = b = 0.4187 and c = 0.5132
WP1 Wyckoff position, CN2 coordination number and PS3 point symmetry
683
Table 3 Optimised parameters of Mn–Sn and Mn–Sr binary systems
Phase
liquid
Thermodynamic parameters (J/mole K)
Mn
Sn
= 4; ZMnSn
=5
ZMnSn
Mn–Sn liquid
0
DgMnSn
10
= −418.4 − 1.72T ; DgMnSn
= −20292.4 + 10.9T
Mn
Sr
ZMnSr
= 6; ZMnSr
=6
Mn–Sr liquid
0
DgMnSr
10
= 15 522; DgMnSr
= 5326.2
αMn
Sn in αMn
°L = − 12 552
βMn
Sn in βMn
°L = 3975 − 12.18T; 1L = 35 982; 2L = − 24 686 + 37.15T
γMn
Sn in γMn
°L = − 29 121 + 16.86T; 1L = 5188.2
δMn
Sn in δMn
◦
Mn–Sn
Mn3Sn
°L = − 14 602.2 + 1.88T
Mn3 Sn
= 3G(Mn, cbcc) + G(Sn, Diamond a4) − 17 029 − 22.84T
GMn:Sn
◦
Mn3
GMn:Sn
= 3G(Mn, cbcc) + 15 104
◦
Mn(2−x)Sn
Mn Sn
2
= 2G(Mn, cbcc) + G(Sn, Diamonda4 ) − 8410 − 23.89T
GMn:Sn
◦
◦
MnSn
GMn:Sn
= G(Mn, cbcc) + G(Sn, Diamonda4 ) − 8410 − 7.53T
Mn
3
GMn:Sn
= 3G(Mn, cbcc) + 27 196;
MnSn2
G298.15K
MnSn2
◦
Mn3Sn2
◦
Mn3 Sn2
G298.15K
◦
Mn
2
GMn:Sn
= 2G(Mn, cbcc) + 18 828
= −14 753 − 25.6T
= −7731.6 + 7.5T
Fig. 1 Re-optimised Mn–Sn phase diagram with the experimental data from the literature
Mg
Sn
Sr
Mn
ZMgMg
= ZSnSn
= ZSrSr
= ZMnMn
, was set to be 6. Since this value
gave the best possible fit for many binary systems and was also
recommended by Pelton et al. [57], Pelton and Chartrand [58] and
Mn
Sn
= 4, ZMnSn
=5
Pelton and Blander [59]. The values of ZMnSn
Sr
Mn
= ZMnSr
= 6 are chosen to permit the composition of
and ZMnSr
maximum short-range ordering in the binary system to be
consistent with the composition that corresponds to the minimum
enthalpy of mixing.
For binary systems with approximately the same number of model
parameters, random solution model and MQM can provide very
similar and good fits to binary phase diagram data. However, this
is no longer true for higher-order solutions [57–60]. Consider
Mg–Mn–Sn ternary system in which the liquid solution Mg–Sn
exhibits a strong tendency to short-range ordering [52, 60],
whereas the Mg–Mn and Mn–Sn liquid solutions are closer to
ideality. Positive deviations from ideal mixing will be observed,
centred along the Mg2Sn–Mn corner of the composition triangle
(where Mg2Sn is the binary composition of maximum short
ordering). This is a typical behaviour of ternary system which has
684
one binary liquid exhibiting large negative deviation from ideality
compared with the other binary liquids [60]. Such positive
deviations are expected because (Mg–Sn) nearest neighbour are
energetically favoured, MQM predicts a tendency for the liquid
phase to separate into clusters rich in (Mg–Sn) and clusters rich in
Mn solution. Random solution model overestimates the positive
deviations observed in such ternary system. Kang and Pelton [60]
also showed that the MQM predicts better extensions of binary
miscibility gaps into a ternary system. For the binary liquid lines
and ternary liquidus projection, the calculated miscibility gap
using MQM is flatter than that predicted by random solution model.
It is worth mentioning that the random solution model does not
consider short-range ordering in the liquid phase, whereas
associate model takes short-range ordering in the liquid phase into
account with the assumption that some molecules occupy some
lattice site which is not physically sound. Moreover, random
solution model and associate model require too many parameters
in optimisation to fit the experimental data. For the associate
model and MQM, the optimisation results for binary systems are
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Table 4 Calculated invariant points of the Mn–Sn system in
comparison with the literature
Reaction
type
Reaction
Composition,
at.%Sn
T, °C
Reference
peritectic
L + (β Mn) ↔ (Mn3Sn)
22.6
22.8
35
35
66.7
66.7
66.7
66.7
36.5
36.5
40
40
40
984
984
884
884
549
549
548
548
480
480
540
540
540
this work
[33]
this work
[33]
this work
[33]
[34]
[35]
this work
[32]
this work
[32]
[43]
L + (Mn3Sn) ↔ (Mn(2
−x)Sn)
L + (Mn(2−x)Sn) ↔
MnSn2
eutectoid
peritectoid
(Mn(2−x)Sn) ↔
Mn3Sn2 + (Mn3Sn)
Mn(2−x)Sn) + MnSn2
↔ Mn3Sn2
mathematically very similar. However, Kang and Pelton [60] proved
that the associate model does not correctly predict the
thermodynamic properties of ternary and high-ordered system.
Therefore the MQM with pair approximation is used in this paper
to model the liquid phase.
3.5
Intermediate solid solutions
Gibbs energy of intermediate solid solution phase is described by the
compound energy formalism [61] which can be expressed as
Gref =
G = Gref + Gideal + Gexcess
(13)
(14)
q◦
yli ym
j , . . . , yk G (i:j:, . . . , :k)
Fig. 2 Calculated activities of Mn and Sn in the Mn–Sn liquid at
a 1244°C
b 1000°C
IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692
& The Institution of Engineering and Technology 2015
685
Fig. 3 Calculated enthalpy of mixing in Mn–Sn liquid alloys compared with the experimental data of [66] at 1397°C
Fig. 4 Calculated Mn–Sr phase diagram with the experimental data of [49]
Gideal = RT
yli ln yli
where i, j, …, k represent components or vacancy, l, m and q
represent sublattices. yli is the site fraction of component i on
sublattice l. fl is the site fraction of sublattice l relative to the total
lattice site. ◦ G(i:j:, ..., :k) represents the energy of a real or
hypothetical compound (end member). γL(i,j ):k represents the
interaction parameters between components i and j on one
sublattice when the other sublattice is occupied only by k.
crystallographic data and the solubility range of the phase during
the optimisation of the sublattice model parameters. The crystal
structure data of the Mn3Sn intermediate solid solutions is
obtained by Weitzer and Rogl [64] and listed in the Pearson
handbook [65] as shown in Table 1.
On the basis of crystallographic data of Mn3Sn phase, there are
two atoms at different sites in the unit cell with the same
coordination number and different points of symmetry as shown in
Table 1. To obtain an intermediate phase which has an ideal
stoichiometry, two sublattices are needed and each sublattice is
occupied only by one constituent species. In other words, the
direct sublattice model which is composed based on the
crystallographic data of Mn3Sn phase only is the following model
3.5.1 Mn3Sn phase: According to Hari Kumar et al. [62] and
Hari Kumar and Wollants [63], attention should be given to the
(Mn)3 : (Sn)1
fl
l
Gexcess =
686
yli ylj ym
k
g
(15)
i
L(i, j):k × (yli − ylj )g
(16)
g=0
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& The Institution of Engineering and Technology 2015
Table 5 Calculated reaction temperature of the Mn–Sr system in
Table 6 Invariant reaction in Mg–Mn–Sr ternary system
comparison with the literature
Reaction
Reaction
T, °C
Reference
gas + L1 ↔ L2
1402
1402
1236
1240
1238
1163
1165
1083
1063
772
740
772
707
707
547
547
this work
[50]
this work
[49]
[50]
this work
[50]
this work
[50]
this work
[49]
[50]
this work
[50]
this work
[50]
L1 ↔ δMn + L2
δMn + L2 ↔ γMn
γMn + L2 ↔ βMn
βMn + L2 ↔ βSr
βMn + βSr ↔ αMn
αMn + βSr ↔ αSr
Reaction
type
Mg, at.
%
Sr, at.
%
Mn, at.
%
T, °C
E1
5.71
94.14
0.15
706.9
E2
82.38
17.26
0.36
621.2
E3
85.77
13.85
0.38
612.5
E4
86.61
13
0.39
610
S1
E5
94.77
94.18
5.21
5.33
0.02
0.49
603.4
602.6
E6
18.02
81.94
0.04
566
S2
E7
E8
19.55
19.66
38.24
80.42
80.3
61.77
0.03
0.04
0.03
549
547
518
E9
35.93
35.93
0.01
432
L#2 ↔ L#1 + αMn +
Sr-bcc
L#2 ↔ L#1 + αMn +
Mg23Sr6
L#2 ↔ L#1 + αMn +
Mg38Sr9
L#1 ↔ αMn + Mg17Sr2
+ Mg38Sr9
L#1 ↔ Mg17Sr2 + (Mg)
L#1 ↔ (Mg) + αMn +
Mg17Sr2
L#1 ↔ L + Sr-bcc +
αMn
L#1 ↔ L + Sr-bcc
L#1 ↔ L + Sr-fcc + αMn
L#2 ↔ L#1 + αMn +
Mg2Sr
L#2 ↔ Sr-fcc + αMn +
Mg2Sr
Fig. 5 Liquidus projection of Mg–Mn–Sr system
This model does not represent the homogeneity range of Mn3Sn
phase which was obtained by Stange et al. [41]. To achieve the
deviation from stoichiometry, it is necessary to allow mixing of
atoms in one or more sublattices. For the phases which have
relatively a narrow range of homogeneity such as Mn3Sn the
mixing is performed by ‘defects’, which may be vacancies or
anti-structure atoms (i.e. atoms at lattice sites belonging to the
other kinds of atoms in the ideal structure) [62, 63]. Since the
structure of Mn3Sn phase is not closed packed, vacancy is more
appropriate than anti-structure atom. Therefore vacancies (Va) in
Sn sublattice is the defect considered in this model. Therefore the
model takes the form
(Mn%)3 : (Sn%, Va)1
The range which is covered by this model is Mn3Sn to pure Mn.
Therefore this satisfies the homogeneity range requirement for
Mn3Sn phase which was obtained by Stange et al. [41]. Hence,
the Gibbs energy per mole of formula unit of Mn3Sn is described
by the compound energy formalism as shown in the following
equation (see (17) at the bottom of the next page)
where i is the species inside the sublattice.yIMn is the site fraction of
Mn3 Sn
Mn3 Sn
sublattice I.yIISn , yIIVa is the site fractions of lattice II.0 GMn:Sn , 0 GMn:Va
represents real or hypothetical compound (end member) energy,
Mn3 Sn
Mn3 Sn
Mn3 Sn
LMn:Sn , 0 LMn:Va , 0 LMn:Sn, Va represents the interaction parameters
which describe the interaction within the sublattice.
0
3.5.2 Thermodynamic modelling of the Mn(2−x)Sn phase:
The crystallographic data of the Mn(2−x)Sn phase were listed in
Table 2. On the basis of the crystallographic data of Mn(2−x)Sn
phase, there are three atoms at different sites in the unit cell with
different points of symmetry as shown in Table 2. To obtain an
IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692
& The Institution of Engineering and Technology 2015
687
Fig. 6 Amount of Mn in Mg solid solution calculated using equilibrium cooling compared with the experimental data [27] for
a Mg − 3 wt.% Sr–Mn
b Mg − 5 wt.% Sr–Mn
To obtain a deviation from this stoichiometry, mixing of
constituents is applied. Grouping was not allowed in this model
because the atomic position and point symmetry for each atom is
different from the other atoms as shown in Table 2. To obtain the
homogeneity range, mixing of Mn anti-structure atom in the
second lattice and vacancy (Va) in the first lattice are considered
intermediate phase which has an ideal stoichiometry, three sublattices
are required and each sublattice is occupied by only one constituent
species. In other works, the direct sublattice model which is derived
from the crystallographic data of Mn(2−x)Sn phase is as follows
(Mn1) : (Sn) : (Mn2)
GmMn3 Sn
=
Mn3 Sn
yIMn yIISn 0 GMn:Sn
+
688
+
Mn3 Sn
yIMn yIIVa 0 GMn:Va
Mn3 Sn
yIISn yIIVa yIMn 0 LMn:Sn, Va
+ RT 0.75
Mn
i=Mn
yIi
ln
yIi
+ 0.25
Va
i=Sn
yIIi
ln
yIIi
Mn3 Sn
Mn3 Sn
+ yIMn yIISn 0 LMn:Sn + yIIVa 0 LMn:Va
(17)
IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692
& The Institution of Engineering and Technology 2015
Fig. 7 Liquidus projection of Mg–Mn–Sn system
in this model. Therefore the model takes the form
Table 7 Invariant reaction in Mg–Mn–Sn ternary system
(see (18))
The optimised model parameters of different phases in the Mn–Sn
and Mn–Sr binary systems are summarised in Table 3.
4
4.1
Results and discussion
Mn–Sn binary system
The re-optimised Mn–Sn binary phase diagram in relation to the
experimental data from the literature is shown in Fig. 1. Table 4
summarises the calculated and experimental invariant points of this
system. As can be seen in the table, all the deviations between the
calculated temperature for the invariant reactions and the literature
are within ±4°C. Fig. 2 shows the calculated activities of Mn and
Sn in the Mn–Sn liquid at 1000 and 1244°C in comparison with
literature values and both show good agreement. The calculated
enthalpy of mixing for liquid at 1397°C compared with the
experimental data [66] is shown in Fig. 3. The reference states are
liquid Mn and liquid Sn. As can be seen in Fig. 3, the presently
calculated enthalpy of mixing agrees with the experimental data [66].
In the current optimisation, Mn3Sn2 phase is modelled as a
stoichiometric phase. Mn3Sn2 phase was only reported by Stange
et al. [41] and adopted by Okamoto [48], whereas Miettinen [43]
excluded this phase from his calculations. According to the work
of Stange et al. [41], Mn3Sn2 appears as low-temperature
GmMn3 Sn
=
Mn(2−x)Sn
yIMn yIISn 0 GMn:Sn:Mn
+
yIMn yIVa
+
Mn(2−x)Sn
yIMn yIIMn 0 GMn:Mn:Mn
+
Type
Sn,
at.%
Mg,
at.%
Mn,
at.%
T, °C
L#2 ↔ L#1 + βMn
L#2 ↔ L#1 + L#2 +
βMn
L#2 ↔ L#1 +
Mn3Sn
L#2 ↔ L#1 + L#2 +
Mn3Sn
L#2 ↔ L#1 +
Mn3Sn + Mg2Sn
L#2 ↔ L#1 + αMn +
Mg2Sn
L#2 ↔ L#1 + Mn(2−x)
Sn + Mg2Sn
L ↔ hcp + Mg2Sn
L#2 ↔ L#1 +
Mn2Sn
S1
U1
38.83
18.69
56.6
3.6
4.57
77.71
1048.8
1047.6
S2
45.36
47.86
6.78
938.2
U2
29.95
6.32
63.73
938.2
E1
41.93
56.36
1.71
724
E2
21.51
78.19
0.3
687.8
E3
56.52
39.63
3.85
589.1
S3
S4
10.4
9–12
0.27
1–2.5
569.8
525
L#2 ↔ L#1 +
Mg2Sn + Mn3Sn2
L#2 ↔ L#1 +
Mg2Sn + MnSn2
L#2 ↔ bct +
Mg2Sn + MnSn2
E4
82.17
63.71
89.33
90–
85.5
5.19
32.43
12.64
3.86
540.5
509.3
E5
68.18
28.56
3.26
460.4
E6
91.95
8.04
0.01
202.7
Reaction
(Mn%, Va) : (Sn%, Mn) : (Mn%)
Mn(2−x)Sn
yIVa yIISn 0 GVa:Sn:Mn
[55]
stoichiometric phase in the Mn–Sn system. The latter precisely
determined the crystal structure of Mn3Sn2 using XRD and
neutron diffraction. It is worth mentioning that the synthesis of
pure Mn3Sn2 phase was difficult [41] and its crystal structure was
determined from two-phase region containing MnSn2 and Mn3Sn2.
(Mn3Sn) melts incongruently at 883°C which is in accord with the
+
Mn(2−x)Sn
yIVa yIIMn 0 GVa:Mn:Mn
+ RT 0.667
Va
yIi
i=Mn
Mn(2−x)Sn
Mn(2−x)Sn
Mn(2−x)Sn
I 0
II II
I 0
II 0
II 0 Mn(2−x)Sn
ySn LMn, Va:Sn:Mn + yMn LMn, Va:Mn:Mn + ySn yMn yMn LMn:Mn:Sn, Mn + yVa LVa:Mn:Sn, Mn
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& The Institution of Engineering and Technology 2015
Reference
ln
yIi
+ 0.333
Mn
i=Sn
yIIi
ln
yIIi
(18)
689
Fig. 8 Isothermal section of Mg–Mn–Sn in Mg-rich region compared with the experimental data of [55] at
a 500°C
b 400°C
work of [41, 48], whereas Miettinen [43] reported that Mn3Sn as
Mn19Sn6 that melts congruently.
4.2
Mn–Sr binary system
The calculated Mn–Sr phase diagram in relation to the few available
experimental data of [49] is shown in Fig. 4. Table 5 summarises the
calculated and experimental invariant points of this system. In
Sr-rich region, the monotectic reaction L1 ↔ L2 + δMn was
observed experimentally [49] and well produced in the current
calculations. The calculated monotectic reaction is 96.5 at.% Sr at
1236°C, whereas the measured one is 96.5 at.% Sr at 1240°C.
690
4.3
Mg–Mn–Sr ternary system
Thermodynamic properties of the Mg–Mn–Sr liquid were estimated
from the optimised binary parameters using Kohler extrapolation
[67]. The projection of the liquidus surface of the Mg–Mn–Sr
system is shown in Fig. 5. As can be seen in Fig. 5, the miscibility
gap covers most of the composition triangle and the primary
crystallisation field of (Mg) is very small. Since there is no
experimental data available for the entire Mg–Mn–Sr system, it is
possible that the size of the miscibility gap is over or
underestimated by the extrapolation. One important clue for
understanding Mg-alloy development is its narrow crystallisation
field of (Mg). Since changing the alloy’s composition slightly can
IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692
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lead to precipitate secondary phase(s) that enhance mechanical
properties of Mg-alloys.
Invariant reactions of Mg–Mn–Sr are listed in Table 6. The
calculated liquidus projection of Mg–Mn–Sr is divided into 11
narrow crystallisation fields: (Mg), αMn, βMn, γMn, δMn,
Mg17Sr2, Mg38Sr9, Mg23Sr6, Mg2Sr, α Sr and β Sr. The model
predicted nine ternary eutectic reactions and two saddle points. All
of these reactions are in the Mg–Sr side.
In the current calculations of the Mg − 3 wt.%Sr − {0 − 5}wt.%
Mn and Mg − 5 wt.%Sr − {0 − 5}wt.% Mn alloys indicate that up
to ≃ 1.2 wt.% Mn, αMn precipitates from Mg-matrix, whereas
above ≃ 1.2 wt.% Mn, α-Mn phase formation occurs from liquid
and Mg-matrix. To estimate the amount of Mn in Mg solid
solution under casting conditions, equilibrium simulations are
carried out and compared with the TEM/EDS data of Celinkin
et al. [27] as shown in Fig. 6. Fig. 6 shows that the amount of Mn
in Mg-matrix increases with Mn alloying addition up to 1.75 wt.%
Mn. According to Celikin et al. [27] works, TEM and XRD
analyses for heat treated alloys at 225°C (Mg–(3–5 wt.%)
Sr–(0.75–2 wt.%) Mn) show that the stable phases are Mg17Sr2,
Mg and αMn. These phases are in good agreement with the
current FactSage calculations. In the composition range
experimentally studied, Celikin et al. [27] observed in TEM/EDS
scan that Mn dissolves in interdendritic phase Mg17Sr2. No
experimental data could be found in the literature to prove the
amount of Mn solubility in Mg17Sr2. This demands experimental
investigation in order to verify the solid solubility of Mn in the
intermetallic phases of the Mg–Mn–Sr system.
Janz [51] calculated liquidus projection of Mg–Mn–Sr system as
extrapolation of the binary subsystems. In his work, miscibility
gap cover most of the ternary system that is in accord with the
current calculations. In Mg–Sr-rich region, Janz [51] projection
showed very narrow crystallisation field for all phases in the
ternary system compared with the current paper. In the current
calculations, liquidus projection displays smooth and gradual
curving, whereas Janz [51] calculations showed curvature of γ and
δMn phase boundaries in the ternary system. This particular
change in the curvature might be because of not modelling liquid
phase by the MQM since the MQM predicts flatter liquidus
projection compared with random solution model which is in
accord with Kang and Pelton [60] observations. It is worth
mentioning that Janz [51] and current isothermal sections of the
Mg–Mn–-Sr system at 400 and 500°C are identical.
4.4
Mg–Mn–Sn ternary system
In the present paper, the constituent binaries are extrapolated
according to the Kohler extrapolation model [67] to construct
Mg–Mn–Sn ternary system without addition of any ternary
parameters. Liquidus projection of Mg–Mn–Sn is shown in Fig. 7
and invariant reactions of this system are listed in Table 7. Ten
primary crystallisation fields are predicted in Mg–Mn–Sn system:
hcp, αMn, βMn, γMn, δMn, Mg2Sn, Mn3Sn, Mn3Sn2, Mn(2−x)Sn
and Mn2Sn.
Isothermal sections of the Mg–Mn–Sn system in the Mg-rich
region were calculated at 500 and 400°C and compared with the
work of Kopetskii and Semenova [55] as shown in Figs. 8a and b.
Experimental data covered wider region of (Mg) + Mg2Sn and
(Mg) compared with calculated isothermal section at 500°C,
whereas the predicted (Mg) + Mg2Sn + αMn region is in accord
with the work of Kopetskii and Semenova [55]. Wider
experimental regions of (Mg) + Mg2Sn and (Mg) might belong to
supersaturated solid solution of (Mg) phase. The calculated phase
regions of (Mg) + Mg2Sn + αMn, (Mg) + Mg2Sn and (Mg) are in
accord with the experimental data of [55]. It is worth mentioning
that the work of Kopetskii and Semenova [55] is the only ternary
experimental data that could be found in the literature.
The differences between Mg–Mn–Sr and Mg–Mn–Sn liquidus
projections drawn in Figs. 5 and 7, respectively, are: miscibility
gap in Mg–Mn–Sr system is wider than that of Mg–Mn–Sn,
intermetallic compounds of Mn–Sn show wide crystallisation field
IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692
& The Institution of Engineering and Technology 2015
in the ternary system and the Mg–Mn–Sr system has nine ternary
eutectic reactions, and two saddle points and 11 crystallisation
fields. Mg–Mn–Sn has four saddle points, two quasi-peritectic and
six ternary eutectic reactions.
5
Summary
A self-consistent thermodynamic database has been constructed for
the Mn–Sn, Mn–Sr, Mg–Mn–Sn and Mg–Mn–Sr systems using
CALPHAD method. The liquid phase is modelled using the MQM
to account for the short-range ordering in Mn–Sn liquid. The
model parameters of the Mn–Sn and Mn–Sr systems are evaluated
by incorporating all experimental data available in the literature.
The phase diagrams and thermodynamic properties of the two
binaries show good agreement with the experimental data. The
established database for Mg–Mn–Sr system predicted two saddle
points and nine ternary eutectics, whereas four saddle points, six
ternary eutectic and two quasi-peritectic reactions were predicted
in Mg–Mn–Sn system. This is the first attempt to construct the
ternary phase diagrams of the Mg–Mn–Sn and Mg–Mn–Sr
systems using the MQM for the liquid.
6
Acknowledgment
Financial support from the Deanship of Scientific Research and
Graduate Studies, The Hashemite University, Jordan is gratefully
acknowledged.
7
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