ANALYTICAL REPRESENTATION OF THE LIQUIDUS
SURFACE IN TERNARY SYSTEMS
V. Filippov1, V. Il’ves2, K. Shunyaev1
1
Institute of Metallurgy, Ural Branch of Russian Academy of
Sciences, Amundsen str., 101, 620016, Ekaterinburg, Russia
2
Institute of Electrophysics, Ural Branch of Russian Academy of
Sciences, 106 Amundsen str. 620216 Ekaterinburg, Russia
Abstract
A new method for describing the phase equilibria of a
multicomponent system in the matrix form is proposed. A system of
equations which describes a higher order phase diagram consists of:
polynomial expressions correspond to the hypersurfaces representing the
phase boundaries of one-phase regions and the relationships expressing
the phase equilibria between phases. As an example, the results of
liquidus surface estimation of quasi-ternary InBi-Sn-Pb alloys are
presented. For liquidus surface representation in this system three binary
subsystems and only few experimental data on ternary alloys are used.
The method can be applied to the phase diagrams as a whole or its parts.
Introduction
Surface mount assembly processes for products that does not
experience harsh temperature environments, as well as production of
electronic devices whose operation is based on their temperature
response, such as thermal cutoff fuses, often use low-temperature solders
as substantial materials. For that purpose, various low-melting alloys are
used, since one single alloy may not be appropriate as a universal
solution. These alloys generally melt below 250 °C and are
multicomponent, usually composed of tin, lead, silver and one of low
melting element such as bismuth, indium, cadmium or gallium.
In the design, development, processing and understanding of
multicomponent alloys, their phase diagrams play an important role.
Since experimental determination of multicomponent phase diagrams
can be time-consuming, expensive and difficult, theoretical evaluation
based on a far less number of experimental data is substantial. Despite
the well knowledge of binary phase diagrams at a constant pressure,
phase equilibia for higher order systems are less studied. The phase
diagram can be calculated and investigated using both thermodynamic
89
methods and mathematic ones. The elaboration of the algorithms and
attendant software, based on a mathematical description of phase
diagrams surfaces.
In this paper a new approach in estimation of liquidus surface in
ternary phase diagrams on the basis of three binary subsystems and few
experimental data is presented. The new method is used for
determination of liquidus surfaces of low-melting InBi-Sn-Pb alloys and
results are compared with those available in the literature.
Calculation procedure
The liquidus or solidus temperature T in m-component systems
may be represented by the n-th order Scheffe polynomial expansion [1]
using atomic fractions xi:
T
n

 x    
1 i  m
i i
k 2
1 i  j  m

 n 
x x j ( xi  x j ) k  2    
 s x1s1 x2s 2 ...xms k  ,

 k  3 1 i1  i2 ...  ik  m

ij i
(1)
where s1  s2  ...sk  n ; β are constant coefficients. Any number of
coefficients β necessary to adequately represent the temperature in a
system may be included.
The liquidus or solidus surface resulted from N measurements can
be represented in matrix form:
T(1, N )  B(1, M )  X( N , M ) ,
(2)
where M is the number of coefficients β to be determined; T is column
vector of the measured values of temperature; X is concentration matrix;
B is column vector of coefficients β. The first and second arguments in
parentheses indicate the number of rows and columns, respectively. The
values of β are determined by the method of Gauss.
The phase equilibrium can be expressed in matrix form by the
relation:
T ( x1 , x2 ,..., xm )  T ( x1 , x2 ,..., xm )  ...  Tq ( x1q , x2q ,..., xmq )  Tf , (3)
90
where q is maximum number of phases in the phase equilibrium;
Tf is the phase equilibrium temperature.
For any phase diagram, matrices Xi and relations (3) are parts of a
large a matrix X. Number of sub-matrix Xi in X depends on the number
of phases in the system multiplied by the number of surfaces bounding
phase. Number of relations (3) in X is determined by the reaction
scheme.
Let's consider for example the matrix X for the liquidus surface in
the ternary system with one of the invariant point:
0( N 3 , M 3 ) 
 X1 ( N1 , M 1 ) 0( N 2 , M 2 )
 0( N , M ) X ( N , M ) 0( N , M ) 
1
1
2
2
2
3
3 ,

 0( N1 , M 1 )
0( N 2 , M 2 ) X3 ( N 3 , M 3 )


X 2 (1, M 2 )
X3 (1, M 3 ) 
 X1 (1, M 1 )
(4)
where 0 is zero sub-matrix; Ni and Mi are number of experimental points
and coefficients β in i-th sub-matrix, respectively.
In the liquidus surface calculation of ternary systems usually the
coefficients of the binary systems can be treated as known, only the
coefficients of the ternary polynomial are unknown.
In general, the composition of the ternary eutectic point
( x1e , x2e , x3e ) is not known and may be found from the condition:
T1 ( x1e , x2e )  Te  0
,

e
e
T2 ( x1 , x2 )  Te  0
(5)
where Ti ( x1e , x2e ) is the temperature at the eutectic point for i-th liquidus
surface equation (1). The system of equations (5) can be solved by any
minimization method (e.g., Newton-Raphson or simplex).
Results and discussion
Experimental data for analytical representation of the liquidus
surface of binary sub-systems (Sn-Pb, InBi-Sn, InBi-Pb) are obtained
from the literature [2-4]. Measured liquidus temperatures of 9 ternary
InBi-Sn-Pb alloys are taken from ref. [5]. Results of the liquidus surface
91
calculation the quasi-ternary InBi-Sn-Pb system are presented in Fig.1.
Deviations of the experimental points from the calculated liquidus
surface not exceed 1.5 oC. The calculated and experimental [6] values of
composition and temperature of the InBi-Sn-Pb eutectic point are given
in the Table. The results of the calculation of eutectic point coordinates
obtained in [5] without the use of condition (3) (i.e., by extrapolation of
these surfaces to the point of intersection) are shown also in the Table 1
for comparison purposes.
Pb
0.00
1.00
300
0.25
0.75
x
Sn
250
0.50
200
x Pb
0.50
150
0.75
100
0.25
E
0
10
0
15
Sn 0.00
0
20
1.00
0.25
0.50
0
10
0.75
0.00
1.00 InBi
xInBi
Fig.1. Calculated projection of liquidus surface of the InBi-Sn-Pb system
Thus, the use of relation (3) in calculation makes it easier to
describe the liquidus surface in the concentration range where no
experimental points (i.e., near the eutectic point).
92
Table 1. Composition and temperature of the InBi-Sn-Pb eutectic point.
InBi
69.71
69.58
70.80
Composition, at. %
Sn
16.3
12.64
14.58
Pb
14.0
17.78
14.62
Temperature, oC
Ref.
58.0
45.2
57.9
[6]
[5]
our data
Conclusion
A new method for describing the phase equilibria of a
multicomponent system in the matrix form is proposed. Method is
applicable for any ternary system whose phase diagrams of binary
subsystems and a few appropriate experimental points are known.
Liquidus surfaces of the low-melting quasi-ternary InBi-Sn-Pb system is
constructed as illustration of the proposed method. Liquidus surfaces of
the low-melting quasi-ternary alloys are constructed as illustration of the
proposed method. Advantage of presented method is the possibility of
analytical representation and determination of liquidus temperature of
arbitrary composition of considered ternary alloy.
Acknowledgement
This work was supported by the Russian Foundation, project no.
14-03-01126-a.
1.
2.
3.
4.
5.
6.
References
Sheffe H: 'Experiments with mixtures'. J. Roy. Statist. Soc. 1958
20B (2) 344.
Karakaya I, Thompson W T: 'The Pb−Sn (Lead-Tin) system'. J.
Phase Equilibria 1988 9 (2) 144-52.
Dooley G J, Peretti E A: Phase Diagram of the System InBi-Sn.
J. Chem. Eng. Data 1964 9 (1) 90-91.
Boa D, Ansara I: 'Thermodynamic assessment of the ternary
system Bi-In-Pb'. Thermochimica Acta 1998 314 (1) 79-86
Il'ves, V G, Filippov V V, Yatsenko S P: 'Liquidus and enthalpy
of mixing of liquid alloys in quasi-ternary InBi-Cd-Pb and InBiPb-Sn systems'. Rasplavy 1994 (1) 15-22 (in Russian).
Mei Z, Holder H, Vander Plas H: 'Low-Temperature Solders'.
Hewlett-Packard Journal 1996 (8) Article 10.
93