An analysis of the quasi-chemical model of a ternary solution:
On the counting of pairs
Analiza kvazikemičnega modela ternarne raztopine:
O štetju parov
Jiawen Chen1- *
University of Cambridge, Materials Science and Metallurgy, Cambridge CB2 3QZ,
U. K.
Corresponding author. E-mail: jiawen.chenn@gmail.com Received: September 5, 2011	Accepted: September 6, 2011
Abstract: The quasi-chemical model of a ternary solid solution was proposed in 1971 by Alex and McLellan. The model begins with the counting of different kinds of pairs between nearest atoms in a ternary crystal. The formulae of two pairs used in the model has been shown to be incorrect. A solution is suggested. A new quasi-chemical model of the solid solution developed from this work can be useful in physical metallurgy.
Izvleček: Kvazikemični model ternarne trdne raztopine sta predlagala Alex in McLellan leta 1971. Začne se s štetjem različnih vrst parov med najbližjimi atomi v ternarnem kristalu. Prikazana je nepravilna formulacija dveh parov, uporabljenih v modelu, in predlagana pravilna rešitev. Nov kvazikemični model za trdne raztopine, razvit v tem delu, je lahko uporaben v fizikalni metalurgiji.
Key words: theory; statistical mechanics; ferritic steels
Ključne besede: teorija, statistična mehanika, feritna jekla
Introduction
A statistical treatment of ternary metals, e.g. Fe-Mn(Cr,Al)-C(N,H) or Nb-
Ti(V,Mo,Zr,Ta)-O(C,N), can yield thermodynamic properties which could be diffcult to obtain experimentally. As an example, the carbon-carbon interac-
tion energy in the ferrite phase in steel can be calculated by fitting the experimentally determined Gibbs free energy to a function deduced from a simple quasi-chemical model.[1, 2] Many recent publications[3, 4] use data that are derived from the quasi-chemical model of a ternary system proposed by Alex & McLellan.[5] The current work demonstrates a fundamental problem in the model and suggests a solution, which will lead to a correct quasi-chemical model of the ternary system.
A solid phase of two or more kinds of atoms is a solid solution. The forces between the atoms in solid solution are short-ranged. Guggenheim showed that the second nearest neighbour interaction energy is negligible, if it varies as r_6.[6] Most solution models consider only the nearest neighbour interactions.
The energy of a solid solution is the sum of the interaction energy of all kinds of pairs. Let us consider a binary model containing A and B atoms, and we interchange an A atom and a B atom on any two sites. Whatever the initial arrangement around the two sites, if the number of A-A pairs increases or decreases by N, then the number of B-B pairs will change by the same amount N, and the number of A-B pairs will change by -2N. Therefore, the properties of the solid solution depends only on the change of combined energy
N(eAA + eBB - 2eAB)
For a non-ideal solution, i.e. eAA + eB
. *
if eAA + eB
< 2e o, the attractive
AB
force between unlike atoms is stronger, therefore there will be a tendency for each atom to be surrounded by unlike atoms. On the other hand, if + eDD
AA	BB
> 2eAB, the solution tends to segregate into A-rich and B-rich regions.
A quasi-chemical solution model accounts for the non-random distribution of atoms. It starts with the partition function , which equals the sum of all microstates the system can occupy.
Q
=Xexp{-
where k is the Boltzmann constant and UkT is the inverse temperature; E. is the energy level of microstate i. Each microstate has a unique arrangement
of atoms. Hence, E is the function of
5 i
numbers of different pairs and associated interaction energies. Thermodynamic quantities like the Gibbs free energy and the activity of a solute can be deduced from the partition function. An exact solution of the partition function in two dimensions was obtained by Onsager.[7] The partition function has not been evaluated exactly for a three dimensional lattice.
The system researched in the work by Alex and McLellan contains A, B and c atoms. A and B form a substitutional solution, and may interchange positions on the sites. Whereas c atom occupy
sites in the space between the A and B atoms. It is an interstitial solute of the solvent A and B. Figure 1 shows a body-centered cubic (BCC) lattice model.
any interstice are designated as Z1 and Z2 respectively. In the BCC structure, Z1 equals 4, and Z 2equals 2 (Figure 3).
The number of A, B and c atoms are designated as NA, NB and Nc respectively. The ratio of interstitial sites per lattice atom is p. Therefore the number of empty sites e equals (NA + NB) p -Nc. The pairs of nearest neighbouring atoms are separated by half the lattice parameter a/2 and can be divided into two groups (Figure 2).
The number of nearest interstices and the number of nearest lattice atoms to
The difficulty
Alex and McLellan proposed a set of formulae counting the numbers of the seven kinds of pairs for the construction of a quasi-chemical model.
For the first group of pairs, it starts with the number of pairs between atom c and site e, which is simply designated as Z]Xy Then the number of c-c pairs equals the total interstitial pairs con-
Figure 1. The ternary system consists of three types of atoms: type A, type B and type c. Atoms A and B form the BCC lattice, and atom c and empty sites e are the octahedral sites in BCC structure.
Figure 2. The left shows the pairs between interstices, i.e. c-e, c-c and e-e, and the right shows the pairs between interstice and a main lattice atom, i.e. c-A, c-B, e-A and e-B.
Figure 3. Zt is the number of nearest interstices to any interstice. Z2 is the number of main lattice atoms to any interstice.
Table 1. Formulae of the number of pairs used in a ternary quasi-chemical model[5]
Type of pair	Count
c-e and e-c	Vi
c-c	(z^-z^yi
e-e	[Z, (Na + Nb)p - (Z,Nc ~ Z&) ~ 2 Z^J/2
c-A	ZA
c-B	Z2N-Z2X2
e-A	ZiK-ZA
e-B	Z2Nb-(Z2N-Z2X2)
necting to c, ZxNc, minus those connecting c to e, and then divided by 2 as the same pair is double counted by either c of the pair. The same logic applies to the e-e pairs.
For the second group of pairs, first the number of pairs between c and A is designated as Z2^2. Then the number of c-B pairs equals all the pairs connected to c minus the c-A pairs. The formulae of the numbers of e-A and e-B pairs are problematic, which will be discussed below. The list of the formulae of the counting of the pairs used by Alex & McLellan is shown on Table 1.
Let us take a closer look at the number of e-A pair and e-B pair. A simple sum of the formulae of e-A and e-B pairs in Table 1 yields the equation for the total number of pairs connecting e to A or B atoms.
(ZNc - Wl
z2(Na+Np)P-Z2N
(2)
- Z^J + ZNB = Z2 Na + Nb) - ZNC
(1)
However, clearly the two equations are not equal, as equals 3 for a BCC lattice.
The solution
To solve the problem, a new variable, Z3 is needed. It is the number of nearest interstices to an A atom or a B atom (Figure 4). For both body-centered cubic metal, e.g. ferrite in steel and face-centered cubic metal, e.g. austenite in steel, Z3 equals 6.
Equation 1, in principle, should equal the number of all pairs connecting to both c and e, Z2 (NA + NB)fi, minus c-A Figure 4. Z3 is the number of nearest in-and c-B pairs	terstices to a lattice atom.
Table 2. The comparison of the formulae of e-A and e-B pairs.
Type of pair	Old count	New count
e-A	ZJV, - z±	ZJV, - z±
		
e-B	Z3Vb-(Z2Vc-ZA)	z2(vA+vB)ye-z3vA-(z2vc-ZA)
The number of e-A pairs equals the total number of pairs connecting A to interstices, Z3Na, minus c-A pairs
Z3NA - ZÂ
And the number of e-B pairs equals all the pairs connecting e to lattice atoms, Z2[(Na + NB)fi - Nc], minus e-A pairs
WA +	- Nc] - (Z3NA - Z2^2)
or
Z2N + NbW - Z3NA - ZN - ZA)
The new and old formulae are compared in Table 2
Conclusion
A careful investigation of the quasi-chemical model of a ternary solid solution proposed by Alex & McLellan[5] has revealed a mistake in two formulae counting the number of pairs. The solution to the flaw proposed in this work will give a new quasi-chemical model of the solid solution, which will produce different functions of thermodynamic properties. It would be interesting to investigate the compatibility of the new quasi-chemical model of the ternary system with the model of a binary system[2] at the limits where NB ^ 0, or the interaction energy eBX ^ eAX, where X represents the interstitial atom in the system.
References
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Thermodynamics and Solubility of Carbon in Ferrite and Ferri-tic Fe-Mo Alloys. Metallurgical Transactions A; 7A:1347.
[2]	R. B. McLellan & W. W. Dunn
(1969): A Quasi-Chemical Treatment of Interstitial Solid Solutions: Its Application to Carbon Austenite. J. Phys. Chem. Solids; 30:2631-2637.
[3]	L. M. Yu, F. X. Yin & D. H. Ping
(2007): Natural Mechanism of the Broadened Snoek Relaxation Profile in Ternary Body-cen-tered-cubic Alloys. Phys. Rev. B; 75(17):174105.
[4]	M. Grujicic & X. W. Zhou (1993):
Monte-carlo Analysis of Short-range Order in Nitrogen-strengthened Fe-Ni-Cr-N Austen-itic Alloys.Materials Science and Engineering A, 169:103-110.
[5]	K. Alex & R. B. McLellan (1971):
A Quasi-Chemical Approach to the Thermodynamics of Ternary Solid Solutions Containing Both Substitutional and Interstitial Solute Atoms. J. Phys. Chem. Solids; 32:449-457.
[6]	E. A. Guggenheim (1952): Mixtures.
Oxford University Press, Oxford.
[7]	L. Onsager (1944): Crystal Statis-
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