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

Analytical expressions for the melting curves of binary alloys composed of con-stituent elements with the same structure have been derived from expressions for the ratio of root mean squ

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 con-stituent 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 Cu

1−xNix agree well with the measured ones, and those for the other alloys are found to be in a reasonable agreement with experiment.

Keywords: thermodynamic lattice theory • Lindemann’s melting temperature • eutectic point • binary alloys

© Versita Sp z o.o.

The melting of materials has great scientific and

tech-nological interest The problem is understanding how to

determine the temperature at which a solid melts, i.e., its

∗

E-mail: hungnv@vnu.edu.vn

†

Present address: Dept of Metallurgy and Materials, University of

Birm-ingham, Edgbaston, BirmBirm-ingham, UK B15 2TT.

melting temperature The atomic vibrational theory has

been successfully applied by Lindemannand others [1

4] The Lindemann’s criterion [1,2] is based on the

con-cept thatthe melting occurs when the ratio of the root

mean square fluctuation (RMSF) in atomic positions on

the equilibrium lattice positions and the nearest neighbor

distance reaches a threshold value The validity of this

criterion was tested by experiment [5] This criterion

re-lates melting to a lattice vibrational instability Hence,

the thermodynamic lattice theory is one of the most

im-portant fundamentals for interpreting the thermodynamic

properties and melting of materials [1 6, 9, 18–23]

Bi-nary alloys having liquidus consisting of two branches in

their phase diagram or melting curve are called eutectics

[6] and theminimum solidification temperature is called

the eutectic temperature [6] The binary alloy phase

dia-grams have been experimentally studied [7]

Phenomeno-logical 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

con-sidered more generally as the minimum of the melting

curve X-ray absorption fine structure (XAFS) [9] in

study-ing 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

de-pendence 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

pro-cedures in computer simulations without performing free

energy calculations[14] The mechanism for the

solid-liquid 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

ele-ments 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 2is

a derivation of the analytical expressions for the atomic

mean square displacement (MSD), mean latticeenergy,

atomic mean square fluctuation (MSF), and then the ratio

of the RMSF in atomic positions on the equilibrium

lat-tice positions and the nearest neighbor distance, as well

as the melting curves of binary alloys This melting curve

contains the atomic proportion of constituent elements and

their melting temperatures in the limiting cases when the

whole elementary cell is occupied by the atoms of one of

the constituent elements Our theory is based on

Linde-mann’s idea regardingthe melting [1, 2,11, 18] so that

the derived melting curve provides information on the

Lin-demann’s melting temperatures of the binary alloys with

respect to any proportion of the constituent elements and

on their eutectic points Further we have shown that the

Gibbs energy of the eutectic binary alloy system described

by the present theory is always at minimum, satisfying the

condition of equilibrium The theory has been applied to

fcc and bcc binary alloys by carrying out several numerical

calculations (Sec 3 The results are compared to

exper-iment [7,16,17,24] and show a good correspondence

be-tween the calculated melting curves and the experimental

phase diagrams More specifically, the calculated results

for Cu1−xNixagree well with the measured ones, and those

for the other binary alloys are found to be in a reasonable

agreement with experiment The conclusions mentioning

the main results and possible applications of the derived

theory are presented in Sec 4

The binary alloy lattice is always in a state of atomic

thermal vibration so that in the lattice cell n the atomic

fluctuation function, denoted by number 1 for the 1

st

ele-ment and by number 2 for the 2

nd

element composing the

binary alloy, is given by

U 1n= 1 2 X

q

u 1q e iq.R n

+ u

∗

1q e −iq.R n
,

U 2n= 1 2 X

q

u 2q e iq.R n

+ u

∗

2q e −iq.R n
,

(1)

where

u

1q = u1e iω q t , u

2q = u2e iω q t , (2) with ω q being the lattice vibration frequency and q the

wave number

The atomic oscillating amplitude is characterized by the

MSD or Debye-Waller factor (DWF) [9,18,20–23] which

has the form [18]

W = 1 2 X

q

|K u¯q |2, (3)

where K is the scattering vector equaling a reciprocal

lat-tice vector, and ¯u qis the mean atomic vibration amplitude.

If each binary alloy lattice cell contains p atoms, where

on average s is the number of atoms of type 1 and (p − s)

is the number of atoms of type 2, then the quantity ¯u q is

given by

¯

u q=

su 1q + (p − s)u 2q

The potential energy of an oscillator is equal to its kinetic

energy, so that the mean energy of atom k vibrating with

wave number q has the form

¯

ε q = M k

u˙kq

2

Using Eqs (2), (5), the mean energy of the crystal

con-sisting of N lattice cells is given by

¯

E=X

q

¯

ε q=

X

q

NsM1ω2

q |u 1q |2

+ (p − s)M2ω2

q |u 2q |2



,

(6)

where M1, M2 are the masses of atoms of types 1 and 2,

respectively

Using the relationship between u 2q and u 1q[21], i.e

u 2q = mu 1q , m= M1

and Eqs (5), (6) we obtain the mean energy for the atomic

vibration of the qth lattice mode as

¯

ε q = N ω

2

q |u

1q |2sM

1+ M2(p − s) m

2 (8)

The mean energy for this qth lattice mode with p atoms

in a lattice cell calculated using the phonon energy with

¯

n q as the mean number of phonons is given by

¯

ε q = p



¯

n q+1

2



so that, using Eq (8) and Eq.(9) we obtain

|u 1q |2

=

p~ n¯q+1

2

NM1ω q [s + (p − s)m]

From Eq (4) and Eq (7) we get the mean atomic vibration

amplitude for qth lattice mode in the form

| u¯q |2

= 1

p2[s + (p − s) m]

2|u 1q |2. (11)

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 = 1

2p

X

q

K2

[s + (p − s)m]

2 n¯q+ 1

~

NM1ω q [s + (p − s) m]

.

(12)

The lattice vibrations quantized as phonons obey

Bose-Einstein statistics Transforming the sum over q into the

corresponding integral, and applying this to the high

tem-perature area (T  θ D ) due to the melting with θ Dbeing

the Debye temperature we obtain the DWF from Eq (12)

W = 3

2p [M2s + (p − s)M1] ~2K2T

M1M2k B θ2

D

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

X

q

|u 1q |2

=

6p2W

K2

[s + (p − s)m]

2. (14)

The mean crystal lattice energy has been calculated as

¯

E=X k,n

M k

U˙kn

2

= X

k,n

X

q

M k ω2

q |U knq |2. (15)

Using this expression and Eqs (6 (7) we derive the

atomic MSF in the form

1

N

X

n

|U 2n |2

= m

2

X

q

|u 1q |2, (16)

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

1

N

X

n

|U 2n |2

=

6p

2m2W

K2

[s + (p − s)m]

2

Further, using W from Eq (13) this expression is resulted

as

1

N

X

n

|U

2n |2

=

9pm2~2T

M1[s + (p − s)m] k B θ2

D (18)

Hence, at T  θ Dthe MSF in atomic positions about the

equilibrium lattice positions is determined by Eq (18)

which is linearly proportional to the temperature T

Therefore, at a given temperature T the quantity R

de-fined by the ratio of the RMSF in atomic positions on the

equilibrium lattice positions and the nearest neighbour

distance d is given by

R= 1

d

s

9pm2~2T

M1[s + (p − s)m] k B θ2

D

This expression for R contains the parameters p and s

which are different for different binary alloy structures

such as fcc and bcc, which will be determined below

(Sec.3), as well as the parameter m concerning the atomic

mass M1 of element 1 and atomic mass M2 of element 2

composing the binary alloys So that it represents the

contribution of different binary alloys consisted of

differ-ent pairs of elemdiffer-ents 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

thresh-old value R m, then the Lindemann’s melting temperature

T m for a binary alloy using Eq.(19) is defined as

T m=

[sM2+ (p − s)M1]

9pm

where

χ= R

2

m k B θ2

D d2

~2

, R2

m= 1

Nd2

X

n

|U 2n |2. (21)

This expression for the Lindemann’s melting temperature

can be applied to different binary alloys composed of

dif-ferent pairs of elements, with the atomic masses M1 and

M2 having the same crystal structures defined by the

pa-rameters p and s.

If we denote x as proportion of the mass of the element 1

in the binary alloy, then we have

sM1+ (p − s) M2. (22)

From this equation we obtain the mean number of atoms

of the element 1 in each binary alloy lattice cell

m (1 − x ) + x . (23)

We consider one element to be the host and another the

dopant Since the tendency to be the host is equal for

both constituent elements we take the average of the

pa-rameter m with respect to the atomic mass proportion of

the constituent elements in the binary alloy as

¯

m=

1

p



s M2

M1 + (p − s)

M

1

M2



This equation can be solved using the successive

approx-imation Substituting the zero-order term with s from Eq.

(23), we obtain the 1storder term equation as

(1 − x ) ¯ m2

+



x − (1 − x )

M1

M2



¯

m − x M2

M1 = 0, (25)

which provides the following solution

¯

m=

−hx − (1 − x )

M

1

M

2

i +

√

∆

2 (1 − x )

,

∆ =



x − (1 − x ) M1

M2



+ 4x (1 − x )

M2

M1,

(26)

replacing m in Eq (20) for the calculation of Lindemann’s

melting temperature

The threshold value R mof the ratio of RMSF in atomic

po-sitions on the equilibrium lattice popo-sitions and the nearest

neighbor distance at melting is contained in χ which will

be obtained by an averaging procedure The average of χ

can not be directly based on χ1 and χ2because it has the form of Eq (21) containing R

2

m, i.e., the second order of

R m, while the other averages have been realized based on

the first order of the displacement That is why we have

to perform the average for χ

1

and then obtain

χ = s √ χ1+ (p − s) √ χ22

p2

containing χ1for the 1

st

element and χ2for the 2

nd

element,

for which we use the following limiting values

χ2= 9T m(2)

M2 , s = 0,

χ1=

9T m(1)

M1 , s = p,

(28)

containing T m(1)and T m(2) as melting temperatures of the

1

st

or doping element and of the 2

nd

or host element,

re-spectively, which compose the binary alloy

Therefore, the melting temperature of binary alloys has

been obtained from our calculated ratio of RMSF in atomic

positions on the equilibrium lattice positions and nearest

neighbour distance Eq (19)

The eutectic point is calculated using the condition for the

minimum of the melting curve, i.e.,

dT m

Focusing on a thermodynamic model for a binary alloy,

we will show further that the Gibbs energy of the

eutec-tic binary alloy system described by the present theory is

always at minimum, satisfying the condition of

thermody-namic equilibrium

The total Gibbs energy G of a system can be written

for-mally as

G= X

i

n i g i+ ∆mix, (30)

where n i is the number of moles, g iis the Gibbs energy

per mole of phase i, and ∆Gmixthe change in Gibbs energy

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 = nAliqgAliq+ nAsolgAsol+ nBliqgBliq+ nBsolgBsol+ ∆Gmix.

(31)

According to the definition in thermodynamics, the Gibbs

energy G of a binary alloy has the following form

G = U + P V − T S, (32)

where U , P , V , T , S are internal energy, pressure, volume,

temperature, and entropy of the system, respectively

Tak-ing the differentiation of the Gibbs energy Eq (32) we

obtain

dG = dU + P dV + V dP − T dS − SdT (33)

Since dU = T dS − P dV the differential of Eq (33) is

changed into

In our approach, the time-dependence of pressure is not

taken into account so that dP = 0 The direct relation

between the DWF and the mean module of atomic

vibra-tion described by Eq (3) is time-independent, so that the

time-differential of our derived DWF is equal to zero

dW

dt = 1

2 X

q

d

dt |K ¯u q |2

Alternatively, the time-differential of DWF of Eq (13) can

be written as

dW

dt = C

dT

where C is a symbol denoting all the time constants in

Eq (13)

Hence, comparing Eq (36) to Eq (35) we obtain

dT

which shows that in our model if the system is

ener-getically isolated, the temperature does not change with

time This is consistent with the meaning of temperature

being that the mean atomic vibration amplitude is

time-independent

Therefore, in Eq (34) we have dP = 0 and dT = 0, so

that dG = 0 This means that the Gibbs energy in our

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

al-loys 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 T E.

Figure 1. Possible typical phase diagrams of a binary alloy formed

by components A and B.

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

1

8atom on the vertex and

1

2 atom

on the surface of the fcc are localized in the elementary

cell Hence, the total number of atoms in a fcc elementary

cell is p(fcc)=4 Similarly,

1

8 atom on the vertex and one atom in the center of the bcc are localized in the

elemen-tary cell Therefore, the total number of atoms in a bcc

elementary cell is p(bcc)=2

According to the phenomenological theory for the phase

diagrams of the binary eutectic systems [8] Figure1shows

qualitative schemes of typical possible phase diagrams of

a binary alloy formed by the components A and B, i.e.,

the dependence of the temperature T on the proportion

x of element B doped in the host element A. Below the

isotropic liquid mixture L, the liquidus or melting curve

beginning from the melting temperature T Aof the host

el-ement passes through a temperature minimum T E known

as the eutectic point E and ends at the melting

tempera-ture T B of the doping element. The phase diagrams

con-tain two solid crystalline phases α and β The eutectic

isotherm T = T E passes through the eutectic point. The

eutectic temperature T E can have a value either lower T A

and T B(Figure1a) or equaling T A(Figure1b) or T B

(Fig-ure 1c) The mass proportion x appears to characterise

the proportion of the doping element mixed in the host

element to form the binary alloy

Our numerical calculations using the derived theory are

focused mainly on the melting curves giving the

Linde-mann’s melting temperatures with respect to any

pro-portion of the constituent elements, and eutectic points

of binary alloys composed by fcc or bcc elements The

eutectic isotherm is apparently T = T E. All input

data have been taken from Ref [6] The calculated

melting temperatures T E and their respective proportions

x E of doping elements for eutectic points of binary al-loys Cu1−xAgx, Cu1−xAlx, Cu1−xNix (fcc) and Cr1−xRbx,

Cs1−xRbx, Cr1−xMox (bcc) are presented in Table1, com-pared to experiment [7,16,17,24]

Alloys Cu1−xAgx Cu1−xAlx Cu1−xNix Cr1−xRbx Cs1−xRbx Cr1−xMox

x E, Present 0.7107 0.7089 0.0 1.0 0.3212 0.1977

x E, Expt 0.719 [ 24 ] 0.672 [ 16 ] 0.0 [ 7 ] 0.357 [ 7 ] 0.17 [ 7 ]

T E, Present 1170.0 887.0 1358.0 312.6 288.0 2127.0

TE, Expt 1123.5 [ 17 ] 870.0 [ 7 ] 1356.0 [ 7 ] 285.8 [ 7 ] 2127.0 [ 7 ]

Table 1. Calculated eutectic melting temperatures T E (K ) and their respective proportions x Eof doping elements for binary alloys Cu1−xAgx,

Cu1−xAlx, Cu1−xNix, (fcc) and Cr1−xRbx, Cs1−xRbx, Cr1−xMox(bcc) compared to experiment [ 7 , 16 , 17 , 24 ].

Here the calculated results for Cu1−xNix agree well with

the measured values [7], and those for theother binary

alloys are found to be in a reasonable agreement with

experiment [7,16,17,24]

Figure2illustrates the calculated melting curves

provid-ing information on the Lindemann’s meltprovid-ing temperatures

and eutectic points of binary alloys Cu1−xAgx (fcc) and

Cs1−xRbx (bcc) compared to experiment [7]. They

corre-spond to their experimental phase diagrams [7] and

be-long to the types presented in Figure1a, i.e., their

eutec-tic temperatures are lower than the melting temperature

of the host element Cu and Cs and those of the doping

elements Ag and Rb, respectively Figure 3 illustrates

the calculated melting curves of Cu1−xNix and Cr1−xRbx.

The results for Cu1−xNix agree well with experiment [7]

and belong to the types presented in Figure 1b, where

its eutectic temperature is equal to the melting

temper-ature of the host element Cu The results for Cr1−xRbx

belong to those of Figure1c, where its eutectic point is

equal to the melting temperature of the doping element

Rb The calculated melting curves represented in Figures

2and3correspond to their experimental phase diagrams

[7], showing that the Lindemann’s melting temperatures of

the considered binary alloys vary, with respect to

increas-ing proportions x of the dopincreas-ing elements Ag, Rb, Ni and

Rb, between the melting temperatures of the pure host

elements, when the whole elementary cell is occupied by

the host atoms, and the pure dopant elements, where the

whole elementary cell is occupied by the dopant atoms

Figure2also 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

Figure 2. Calculated melting curves providing information on

Linde-mann’s melting temperature and eutectic points of binary alloys Cu1−xAgx(fcc) and Cs1−xRbx (bcc), where the re-sults for Cs1−xRbxare compared to experiment [ 7

melting temperature decreases up to the eutectic point,

and more tightly bonded after the eutectic point because

the melting temperature increases Figure3for Cu1−xNix

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−xRbxit shows

the rate that the atoms become more weakly bonded after

the host element Cr was doped by Rb, because its melting

temperature decreases

Table 2 shows the good agreement of the Lindemann’s

melting temperatures of Cu1−xNix(fcc) and Cs1−xRbx(bcc),

taken from the calculated melting curves of these binary

alloys, with experiment [7] for different proportions x (it is

also for any proportion x ) of Ni and Rb doped in Cu and

Cs, respectively, to form these binary alloys The present

calculation has been carried out for the binary alloys

com-posed of the constituent elements with the same fcc or bcc

Figure 3. Calculated melting curves of binary alloys Cu1−xNix(fcc)

and Cr1−xRbx (bcc), where the results for Cu1−xNixare compared to experiment [ 7

x 0.10 0.30 0.50 0.70 0.90

Cu1−xNix, Present 1396 1468 1538 1611 1687

Cu1−xNix, Expt [ 7 ] 1388 1461 1531 1605 1684

Cs1−xRbx, Present 292.6 287.5 290.0 295.0 305.0

Cs1−xRbx, Expt [ 7 ] 291.4 286.0 287.4 293.5 304.0

Table 2. Calculated Lindemann’s melting temperatures T m (K ) of

Cu1−xNix(fcc) and Cs1−xRbx(bcc) with respect to different

proportions x of Ni and Rb doped in Cu and Cs, respectively,

compared to experiment [ 7

structure, but it also can be generalized to alternate

struc-tures for those composed of constituent elements with the

same other structures by calculation of the atomic number

in their elementary cells

4 Conclusions

In this work a thermodynamic lattice theory on the

melt-ing curves, eutectic points and eutectic isotherms of

bi-nary alloys composed by the constituent elements with

the same structure has been derived Ourdevelopment

is derivation of the analytical expressions for the ratio of

RMSF in atomic positions on the equilibrium lattice

po-sitions and the nearest neighbor distance, which lead to

expressions for the melting curve providing information on

the Lindemann’s melting temperatures of the binary alloys

composed of any proportions of constituent elements, and

on their eutectic points

Focusing ona thermodynamicmodel for a binary alloy

system, it has resulted that the Gibbs energy of the

eu-tectic binary alloy system described by the present theory

is always at minimum, so that the system is in a state of

thermodynamic equilibrium

The theory has been derived based on averaging several

physical quantities with the hope of getting an estimative

mean melting curve, yet provides quantitative results for

the melting of binary alloys The calculated melting curves

of binary alloys correspondto their experimental phase

diagrams, where the results for Cu1−xNix 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

the-ory may prove to be a simple and effective method for

pre-diction 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

Acknowledgments

The authors thankJ.J Rehr and P Fornasini for useful

comments One of the authors (N.V H.) appreciates the

DFG-projects KI 768/5-1, KI 768/5-2 under SPP 1191 and

the Wilhelm-Ostwald-Institute for Physical and

Theoret-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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Thermal expansion, Metallic, Elements and Alloys

Thermophys properties of matter, Vol 12 (IFI/Plenum,

New York-Washington, 1975)

alloys It is apparent that

1

8atom on the vertex and

1

2 atom...

8 atom on the vertex and one atom in the center of the bcc are localized in the

elemen-tary cell Therefore, the total number of atoms in a bcc

elementary cell is p (bcc) =2... 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