Temperature and Orientation Dependence of Austenite/B1 Type Compound Interfacial Energy
Zhi-Gang Yang* and Masato Enomoto
Dept. of Materials Science, Ibaraki University, Hitachi, Ibaraki 316-8511, Japan
*
On leave from Dept. of Materials Science, Tsinghua University, Beijing 100084, China
Keywords: interface, interfacial energy, discrete lattice plane, nearest neighbor broken bond model, austenite, inclusion, steel
Abstract
A discrete lattice plane, nearest neighbor broken bond model is applied to calculate the interfacial energy of a coherent austenite(g )/B1 compound interphase boundary. The composition profiles, temperature and orientation dependence of interfacial energy are investigated. The composition profiles are found to be fairly sharp and the variation of interfacial energy with temperature does not appear to be large below 1500K. In contrast to coherent f.c.c./f.c.c. interfaces, a (111) interface is likely to have the largest chemical interfacial energy.
Introduction
The discrete lattice plane (DLP), nearest neighbor broken bond (NNBB) model has been utilized to investigate the basic characteristics of the surface energy [1,2] and the interfacial energy of coherent f.c.c./f.c.c.[3] and b.c.c./b.c.c.[4] interphase boundaries. This method was applied to an f.c.c./h.c.p. heterophase interface in binary system[5]. It was also extended to a ternary substitutional system [6] and to a ternary substitutional-interstitial system[7].
Some carbides and nitrides in steel such as TiC, TiN, VC and VN have a B1 type structure, which is isomorphous to austenite (g ). The interfacial energy of these compounds with g may play a significant role in controlling microstructure of steels.
In this report, the interfacial energy of B1 compound with g was calculated by means of DLP/NNBB method using the grand potential, which is convenient for analyzing the free energy of an open system. This treatment is in principle the same as a previous method[3,7], and the relevant equations are derived straight forwardly.
Calculation method
The system is regarded as an f.c.c. solid solution of Fe, and Ti or V (denoted as M) as a substitutional solute, and carbon or nitrogen as an interstitial solute(denoted as I). The two phases, g and the compound (x ), are assumed to have a coherent interface with cube-cube orientation relationship [8]. The thermodynamic potential W (grand potential) for the present Fe-M-I substitutional-interstitial system is defined as,
where E is the internal energy, T is the temperature, S is the entropy, m q and Nq (q =1, 2 and 3 for Fe, M and I) are the chemical potential and the number of q atoms in the system, respectively.
The internal energy, E, is composed of the standard energy and the formation energy of the phases. The formation energy is calculated as the sum of the energy of interaction of the atoms in a given plane (Ei,i) and the energy of interaction of the atoms in the i’th plane with nearest neighbor atoms in other planes (Ei,i+j). They are written as,
and
where 
and 
are the numbers of substitutional and interstitial sites per unit area of the interface, respectively, and 
is the energy of an Fe-M atom pair etc. (I-I interaction is assumed to be negligible). 
and 
are the numbers of Fe (or M) and I nearest neighbors to an Fe (or M) atom, respectively, in the j-th layer from the i’th plane, and 
and 
are the nearest neighbor numbers within the i’th plane. 
and 
are the atomic fractions of M and I atoms in the i’th plane in substitutional and interstitial sublattices, respectively. 
’s are summed up to the maximum j-th layer in which nearest neighbor atoms are located.
The entropy S is calculated from the equation,
where 
is Boltzmann’s constant.
The equilibrium concentrations of M and I atoms in the i’th plane of the interface region are determined from the condition that W is stationary with respect to the compositions as,
(5)
which become (noting that = 
),
where 
and 
, 
and 
are the concentration of M atoms and I atoms in homogeneous g
(while the i’th plane belongs to g
) or the compound (while the i’th plane belongs to the compound), Z and Z Eare the coordination numbers of Fe (or M) and I nearest neighbors to an Fe (or M) atom in the bulk, respectively.
From the equilibrium concentration determined from eqs.(6), the interfacial free energy is calculated as,
where superscript h stands for the homogenous phase prior to the formation of interface. From eqs. (2) ~ (4), eq. (7) can be rewritten as,
When T=0K, the entropy terms vanish and the solute profiles change abruptly across the interface. Then, eq. (8) is simplified to,
Practically, 
and 
are ~1, 
and 
are ~ 0, accordingly, eq. (9) becomes,
The interfacial coordination numbers 
and 
vary with the interface orientation, and are determined by a vector method [3,7]. 
and 
are computed from thermodynamic data. For the g
/TiN interface, they are -0.042ev and 0.483ev, respectively [7, 9].
The chemical potentials for each component atom are expressed by,
and
The phase boundary compositions are calculated prior to the interfacial energy. The results at 1500K are shown in Figs. 1 (a) and (b), which are in fair agreement with the reported solubility product of TiN in g [10]. Practically, the mole fractions of Ti and N in g are of the order of 10-3 and 10-4, respectively. It is seen from Fig.1(a) that TiN at equilibrium with g of such a composition is almost stoichiometric as long as the concentration of Ti is larger than that of N in the bulk. In the calculation, a tie line which passes through the bulk composition of 0.01 for both Ti and N was chosen (though the N content is somewhat too large from practical interest).
(a) (b)
Fig. 1 (a) Calculated equilibrium tie-lines in g /TiN system and (b) the logarithmic plot of the phase boundary compositions compared with the data from solubility product at 1500K.
Results and Discussion
The calculated composition profiles in (100) g
/TiN interface region are shown in Figs. 2(a) and (b). It is seen that the composition varies over one atom layer at 1500K; even at 3000K it does so within three layers. This is resulted from the relatively large value of , i.e. the strong affinity of Ti-N compared with Fe-N. Thus, Fe atoms tend to segregate in g
, whereas Ti and N atoms tend to segregate in the compound. As a result, the temperature dependence of g
/TiN interfacial energy is small, i.e. only a slight (~ 5%) decrease from 0K to 1500K for (100), (110) and (111) interfaces, as shown in Fig.3.
As mentioned above, the interfacial energy is not sensitive to temperature in the range of practical interest (below 1500K). Hence, eq. (10) can be used conveniently to calculate the orientation dependence of g /B1 compound interfacial energy. The contour plot of the chemical interfacial energy is shown in Fig. 4 for g /TiN interphase boundary. It is seen that a (100) interface has the smallest energy among all interfaces, while a (111) interface has the largest energy, in contrast to the ordinary f.c.c./f.c.c. interfaces where a (111) interface has the smallest energy. This is due to the strong influence of interstitial-substitutional interaction. It is noted that similar temperature and orientation dependence of interfacial energy are obtained for other B1 type compounds, such as TiC, VN.
(a) (b)
Fig. 2 (a) Titanium and (b) nitrogen concentration profiles at (100) g /TiN interface.
Conclusions
The coherent g /B1 compound interfacial energy is calculated using DLP/NNBB model. The results show that because of the strong interstitial-substitutional interaction in this system, very steep composition profiles are predicated at g /B1 interfaces at temperatures of practical interest. As a result, the temperature dependence of interfacial energy is insignificant at temperatures below 1500K. The anisotropy of interfacial energy is very different from that of ordinary f.c.c./f.c.c. interfaces. A (111) interface is predicted to have the largest chemical interfacial energy, whereas a (100) interface has the smallest energy.
References