Cent. Eur. J. Phys. • 9(1) • 2011 • 222-229
DOI: 10.2478/s11534-010-0065-1

Central European Journal of Physics

A thermodynamic lattice theory on melting curve and
eutectic point of binary alloys. Application to fcc and
bcc structure
Research Article

Nguyen V. Hung1∗ , Dung T. Tran1† , Nguyen C. Toan1 , Barbara Kirchnner2
1 Department of Physics, University of Science, VNU-Hanoi,
334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
2 Wilhelm-Ostwald-Institute for Physical and Theoretical Chemistry, University of Leipzig,
Linnéstr. 2, 04103 Leipzig, Germany

Received 10 February 2010; accepted 1 July 2010

Abstract:

A thermodynamic lattice theory has been developed for determination of the melting curves and eutectic
points of binary alloys. Analytical expressions for the melting curves of binary alloys composed of constituent elements with the same structure have been derived from expressions for the ratio of root mean
square fluctuation in atomic positions on the equilibrium lattice positions and the nearest neighbor distance.
This melting curve provides information on Lindemann’s melting temperatures of binary alloys with respect
to any proportion of constituent elements, as well as on their eutectic points. The theory has been applied
to fcc and bcc structure. Numerical results for some binary alloys provide a good correspondence between
the calculated and experimental phase diagrams, where the calculated results for Cu1−x Nix agree well
with the measured ones, and those for the other alloys are found to be in a reasonable agreement with
experiment.

PACS (2008): 61.10.Ht


properties and melting of materials [1–6, 9, 18–23]. Binary alloys having liquidus consisting of two branches in
their phase diagram or melting curve are called eutectics
[6] and the minimum solidification temperature is called
the eutectic temperature [6]. The binary alloy phase diagrams have been experimentally studied [7]. Phenomenological theory of the phase diagrams of the binary eutectic
systems [8] has been developed to show qualitatively the
temperature-concentration diagrams of eutectic mixtures
using a Landau-type approach, which involves a coupling
between the liquid-solid transition order-parameters and
a specific nonlinear dependence on concentration of the
free-energy coefficients. Here the eutectic point is considered more generally as the minimum of the melting
curve. X-ray absorption fine structure (XAFS) [9] in studying melting is focused mainly on the Fourier transform
magnitudes and cumulants of XAFS. The melting curve
of materials from theory versus experiments [10] has been
studied based on quantum mechanics within the framework
of density functional theory, with use of the generalized
gradient corrections, but this is focused mainly on the dependence of the melting temperature of single elements
on pressure. Empirical rules [1, 11–13] have been used to
characterize the melting transition of solids as useful procedures in computer simulations without performing free
energy calculations [14]. The mechanism for the solidliquid phase transition based on the Lindemann’s criterion
has been studied using Monte-Carlo simulation [15], but
a complete “ab initio” theory for the melting transition is
not available [11, 15]. As such, the calculation of melting
temperature curves versus proportions of constituent elements of binary alloys and their eutectic points still can
be a useful contribution to the field.
The purpose of this work is to derive a thermodynamic
lattice theory for analytical calculation and analysis of
the melting curves or phase diagrams and eutectic points
of binary alloys composed of any constituent elements
with the same structure. Our development in Sec. 2 is

thermal vibration so that in the lattice cell n the atomic
fluctuation function, denoted by number 1 for the 1st element and by number 2 for the 2nd element composing the
binary alloy, is given by
U1n =
U2n

where

1
2

1
=
2

u1q eiq.Rn + u∗1q e−iq.Rn ,
q

u2q e

iq.Rn

+

u∗2q e−iq.Rn

(1)
,

q

given by
su1q + (p − s)u2q
¯q =
.
(4)
u
p
The potential energy of an oscillator is equal to its kinetic
energy, so that the mean energy of atom k vibrating with
223


A thermodynamic lattice theory on melting curve and eutectic point of binary alloys. Application to fcc and bcc structure

wave number q has the form
2

˙
¯ q = Mk u
¯ kq .
ε

(5)

The lattice vibrations quantized as phonons obey BoseEinstein statistics. Transforming the sum over q into the
corresponding integral, and applying this to the high temperature area (T
θD ) due to the melting with θD being
the Debye temperature we obtain the DWF from Eq. (12)

Using Eqs. (2), (5), the mean energy of the crystal consisting of N lattice cells is given by


1
2

ωq ,

(8)

¯ q + 12
p n
.
NM1 ωq [s + (p − s)m]

1
[s + (p − s) m]2 |u1q |2 .
p2

¯=
E

˙ kn
Mk U

1
2p

K 2 [s + (p − s)m]2
q

(10)

atomic MSF in the form
1
N

1
N

To study the MSD Eq. (3) we use the Debye model, where
all three vibrations have the same velocity [18]. Hence, we
calculate the contribution of each polarization, taking Eq.
(11) into account, and then using Eq. (11), the MSD or
DWF Eq. (3) with all three polarizations is given by
W =

[s + (p − s)m]2

|U2n |2 = m2
n

|u1q |2 ,

(16)

q

which, by use of Eq. (14), is given by

From Eq. (4) and Eq. (7) we get the mean atomic vibration
amplitude for qth lattice mode in the form
|¯


K 2T

which is linearly proportional to the temperature T as was
mentioned already [18, 22].
From Eq. (11), and using Eq. (3) for W we obtain

q

u2q = mu1q ,

2

|U2n |2 =
n

6p2 m2 W
K 2 [s + (p − s)m]2

.

(17)

Further, using W from Eq. (13) this expression is resulted
as
1
9pm2 2 T
|U2n |2 =
.
(18)

composing the binary alloys. So that it represents the
contribution of different binary alloys consisted of different pairs of elements having the same crystal structures.
Based on the Lindemann’s criterion, the binary alloy will
be melted when this ratio R of Eq. (19) reaches a threshold value Rm , then the Lindemann’s melting temperature
Tm for a binary alloy using Eq. (19) is defined as
Tm =

[sM2 + (p − s)M1 ]
χ,
9pm

(20)

where
χ=

Rm2 kB θD2 d2
2

,

Rm2 =

1
Nd2

|U2n |2 .

(21)


s
+ (p − s)
.
p M1
M2

(24)

This equation can be solved using the successive approximation. Substituting the zero-order term with s from Eq.
(23), we obtain the 1st order term equation as
(1 − x) m
¯ 2 + x − (1 − x)

M2
M1
¯ −x
m
= 0,
M2
M1

¯ =
m

− x − (1 − x)

(25)

M1
M2


√
√ 2
s χ1 + (p − s) χ2
,
p2

(27)

containing χ1 for the 1st element and χ2 for the 2nd element,
for which we use the following limiting values
9Tm(2)
,
M2
9Tm(1)
χ1 =
,
M1
χ2 =

From this equation we obtain the mean number of atoms
of the element 1 in each binary alloy lattice cell
s=

which provides the following solution

s = 0,
(28)
s = p,


due to inter-phase interactions. Applying this to an AB
binary system, we have 4 possible phases: liquid A (Aliq),
solid A (Asol), liquid B (Bliq), and solid B (Bsol), so that
G = nAliq gAliq + nAsol gAsol + nBliq gBliq + nBsol gBsol + ∆Gmix .
(31)
According to the definition in thermodynamics, the Gibbs
energy G of a binary alloy has the following form
G = U + PV − T S,

model for a eutectic binary alloy is always at a minimum,
and the system is in a state of thermodynamic equilibrium.
Hence, we can determine the melting curves, from which
the Lindemann’s melting temperatures of the binary alloys with respect to any proportions of their constituent
elements, using Eq. (20) with Eqs. (22), (23), (26), (27),
(28) and their eutectic points using Eq. (29), can also be
determined. The eutectic isotherm is the one for which T
equals the eutectic temperature TE .

(32)

where U, P, V , T , S are internal energy, pressure, volume,
temperature, and entropy of the system, respectively. Taking the differentiation of the Gibbs energy Eq. (32) we
obtain
dG = dU + PdV + V dP − T dS − SdT .

(33)

Since dU = T dS − PdV the differential of Eq. (33) is
changed into
dG = V dP − SdT .

Eq. (13).
Hence, comparing Eq. (36) to Eq. (35) we obtain
dT
= 0,
dt

(37)

which shows that in our model if the system is energetically isolated, the temperature does not change with
time. This is consistent with the meaning of temperature
being that the mean atomic vibration amplitude is timeindependent.
Therefore, in Eq. (34) we have dP = 0 and dT = 0, so
that dG = 0. This means that the Gibbs energy in our
226

Figure 1.

Possible typical phase diagrams of a binary alloy formed
by components A and B.


Nguyen V. Hung, Dung T. Tran, Nguyen C. Toan, Barbara Kirchnner

3. Application to fcc and bcc binary
alloys and discussions of numerical
results
Now we apply the derived theory to the fcc and bcc binary
alloys. It is apparent that 81 atom on the vertex and 12 atom
on the surface of the fcc are localized in the elementary
cell. Hence, the total number of atoms in a fcc elementary


Cu1−x Agx Cu1−x Alx Cu1−x Nix Cr1−x Rbx Cs1−x Rbx Cr1−x Mox

xE , Present
xE , Expt.
TE , Present
TE , Expt.

0.7107
0.719 [24]
1170.0
1123.5 [17]

0.7089
0.672 [16]
887.0
870.0 [7]

0.0
1.0
0.0 [7]
1358.0
312.6
1356.0 [7]

0.3212
0.357 [7]
288.0
285.8 [7]


the host atoms, and the pure dopant elements, where the
whole elementary cell is occupied by the dopant atoms.
Figure 2 also shows the rate at which the atoms become
more weakly bonded after Cu and Cs were mixed by the
doping elements Ag and Rb, respectively, because the
227


A thermodynamic lattice theory on melting curve and eutectic point of binary alloys. Application to fcc and bcc structure

Figure 2.

Calculated melting curves providing information on Lindemann’s melting temperature and eutectic points of binary
alloys Cu1−x Agx (fcc) and Cs1−x Rbx (bcc), where the results for Cs1−x Rbx are compared to experiment [7].

Figure 3.

Calculated melting curves of binary alloys Cu1−x Nix (fcc)
and Cr1−x Rbx (bcc), where the results for Cu1−x Nix are
compared to experiment [7].

x

melting temperature decreases up to the eutectic point,
and more tightly bonded after the eutectic point because
the melting temperature increases. Figure 3 for Cu1−x Nix
shows the rate that the atoms become more tightly bonded
after the host element Cu was doped by Ni because the
melting temperature increases. But for Cr1−x Rbx it shows
the rate that the atoms become more weakly bonded after


0.90

1396
1388
292.6
291.4

1468
1461
287.5
286.0

1538
1531
290.0
287.4

1611
1605
295.0
293.5

1687
1684
305.0
304.0

Calculated Lindemann’s melting temperatures Tm (K ) of
Cu1−x Nix (fcc) and Cs1−x Rbx (bcc) with respect to different

of binary alloys correspond to their experimental phase
diagrams, where the results for Cu1−x Nix agree well with
the measured ones and those for the other fcc and bcc
binary alloys are found to be in a reasonable agreement
with experiment.
The calculated melting curve also shows the rate that the
atoms of binary alloys become either more tightly or more
weakly bonded (the host element becomes either harder
or softer) after the host element was mixed by the doping
element to be a binary alloy. This behavior may be useful
for technological applications.
The present numerical calculations have been carried out
for fcc and bcc binary alloys, but it also can be applied
to those composed by the constituent elements with the
same other structure by calculation of the atomic number
in their elementary cells. For this reason, the derived theory may prove to be a simple and effective method for prediction of the Lindemann’s melting temperatures, eutectic
points and eutectic isotherms of binary alloys composed
by any proportions of constituent elements with the same
structure.

ical Chemistry, University of Leipzig for the supports and
hospitality during his stay here. This work ist supported
by the research project No. 103.01.09.09 of NAFOSTED.

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