Calculation of Valence Electron Structures and Melting Points in Ti- Al Alloys With Interstitial Impurities
By Peng, J Z Yang, X Z; Zhou, F C; Gray, M F
According to the average lattice and atom model of the empirical electron theory (EET) of solids and molecules, effects of interstitial impurities on valence electron structures (VESs) and phase transformation of Ti-Al alloys are analysed. In particular, melting points or allotropic transformation in Ti-Al phase diagram affected by interstitial impurities are calculated using the bond energy formula of the EET. It is demonstrated that because of the effects of interstitial impurities, the atom states of VESs increase, and the chemical bonds are seriously anisotropic. Such changes in VESs and chemical bonds make the melting points decrease. The decrease in the melting points can be further estimated by the EET, which is basically consistent with the available experimental results. Keywords: Ti-Al alloys, Valence electron structure, Interstitial impurity, Melting point
The intermetallic compounds in Ti-Al alloys have received considerable attention as candidate materials for relatively high temperature uses such as turbine engine components. The major disadvantages of these materials are their low ductility and toughness at room temperatures. Considerable improvements on these properties have been achieved by alloying and thermal mechanical processing.1,2 However, the complete success on the elimination of the embrittlement has never been reached due to the lack of a thorough understanding on the phase transformation behaviour in this alloy system. It was indicated that besides the accuracy in experimental observations and the limitation in measuring techniques, the strong interactions between different interstitial impurities (Us) such as O, N, C, H and Ti, Al atoms affect dramatically the transformation behaviour in Ti-Al alloys.3-6 For example, a small amount of Us can result in the concurrence of unclear or uncertain phases, implying a shift in the position of the phase boundaries of an equilibrium diagram. This therefore makes it difficult to achieve a completely satisfactory phase diagram in this system. The main reason for this is that essentially, phase transformation behaviour should be attributed to chemical bond nature or electronic structures. For a fundamental understanding, it is necessary to obtain electronic structures; however, this has never been easy matter. In principle, electronic structures can be calculated using first principles (e.g. local density functional method).7-9 In practice, however, such calculations are not only physically difficult, but also computationally expensive, in particular, for studies of the effect of Hs in a complex alloy system.
The empirical electron theory (EET) of solids and molecules, which was first proposed and further developed by Yu,10 provides a concise and practical bond length difference (BLD) method to calculate electron structures and is especially suitable for the study of such a complex alloy system with various Us. The main body of EET was based on Pauling’s electron theory of metals and the quantum theory. The general conclusions drawn from the investigation were summarised in three hypotheses and a special computational method. For detailed information on the method and further procedures for the calculation of valence electron structures (VESs), refer to Refs. 11-14.
In the present study, according to the method and the average crystal lattice and atom model of the EET, the VESs of Ti-Al alloys with and without of Us were analysed. Based on the analysis of VES, the melting points or allotropie transformation in Ti-Al phase diagram due to Hs are further calculated using the bond energy formula of the EET.
Analysis of VESs and model for melting points
Valence electron structures of various phases in Ti-Al alloys without Us
In Ti-Al alloys, stable phases mainly includes alpha-Ti, beta- Ti, Ti^sub 3^Al (alpha^sub 2^), TiAl (gamma), TiAl^sub 3^ and pure Al. The VESs of these phases have been analysed in detail based on the BLD method of the EET. The calculated results are listed in Table I.15
Valence electron structures of various phases in Ti-Al alloys with Us
Valence electron structures of solid solution phases with O, N, C and H can be calculated using the same procedure based on the average crystal lattice model of the EET.10,16 However, it should be indicated in the present paper that because of the low solubility of Hs into Ti-Al alloys (also see below), the interstitial solid solution may be thought as an ideal mixture of crystal lattices with and without interstitial impurity atoms. Thus, the present study will mainly focus on the VESs of various crystal lattices with Us since the VESs without Ils have been well documented before.15 For the calculation of the VESs of various crystal lattices with Us, the following issues on the crystal lattice with Us should be addressed.
Positions of Us
O, N, C and H should locate in the largest interstitial positions of every crystal lattice. For alpha-Ti, its crystal structure and most interstitial positions are A3 and octahedron, respectively; for beta-Ti, they are A2 and tetrahedron respectively; for gamma, they are L^sub 10^ and approximate tetrahedron, respectively; for alpha^sub 2^, they are DO^sub 19^ and approximate octahedron respectively; and for TiAl^sub 3^, they are DO^sub 22^ and approximate tetrahedron respectively.
Solubility of Us
Ti-Al alloys are very susceptible to Hs, but it is known from Refs. 17-19 that the real solubility of Ti-Al alloys is not more than 2-0 mol.-%. Thus, each single crystal lattice can contain only one interstitial atom.
Lattice constant of crystal lattice with IIs
Table 1 Valence electron structures of various phases in Ti-AI system without IIs*
Here a few remarkable points in the above analysis should be pointed out:
(i) it is assumed that Hs exist in the form of solid solution but not in compound because the content of IIs in real alloys is very low
(ii) IIs have no or negligible effects on liquid phase because the reaction between atoms in liquid phase is weaker than that in solid phases, and the variation of bond structures of solid phases affected by IIs is essentially different from that of liquid phases
(iii) entropy of crystal increases very largely if a small amount of Hs is dissolved in a crystal.2 However, whether melting or crystallising is related to liquids phase equilibrium, Hs can weaken the main bonds of solid phases and make a ‘nucleating’ mechanism operate. This leads to that the effect of Us on the enthalpy of solid phases is similar to that of a ‘stress concentration’, and the average or statistical effect of Us on entropy is smaller than that of Us on enthalpy. Thus, it can be believed that Us only affect enthalpy and have no effect on the entropy of solid phases
(iv) ‘linear revision’ of ‘average IIs’ and real solubility is used for simplifying calculations. This approximation may excessively simplify the real situation, but as a quantitative estimation it is rational; this has been supported by the following calculated results (see below).
Table 2 Valence electron structures of various phases TI-Al system with average IIs
Calculations of melting points
Results and discussion
The calculated VESs of various phases in Ti-Al alloys without and with IIs are listed in Tables 1 and 2. Comparing Table 2 with Table 1, one can see that IIs enhance the hybridisation states of Ti and Al atoms, which leads to changes in VES of crystal lattice with IIs. Generally, the strongest bond of crystal lattice with IIs is formed between IIs and metal atoms where the number of covalent electron is much more than that without IIs, while A’, B’ and C’ bonds formed between Ti and Al atoms are weakened by IIs, and their numbers of covalent electrons are decreased. At the same time, the number of lattice electrons of a crystal lattice with IIs is also decreased very much. From the above results, it can be concluded that IIs make the bonds stronger in some directions and weaker in others, implying a quiet anisotropic bonding structure. Because of the difference in crystalline and bond structure of the different phases, the difference in the effects of Us is very large. Generally, original A and B bonds of alpha-Ti and beta-Ti are weakened most dramatically, followed by those of Ti^sub 3^Al and TiAl, whereas those of TiAl^sub 3^ are affected most slightly. Furthermore, if comparing the VESs of solidified alpha-Ti with beta-Ti as well as ordered Ti^sub 3^Al with TiAl, there is a little difference in bond structures. However, it is the difference, especially with cooperation with the effects of compositions and temperature, which results in very seriously anisotropic bond structures in these phases, which may further affect the corresponding phase transformations. This is the essential reason why IIs affect phase transformation in Ti-Al alloys, in particular, the melting points (see below).
Table 3 Experimental and calculated phase transformation temperatures
The calculated melting point and allotropie transformation temperatures are listed in Table 3. As shown, all the calculated transformation temperatures of alloy phases without IIs are higher than the experimental ones, but the calculated values with IIs are much lower than those without IIs. This result is consistent with the experimental investigation on liqudus curves at high temperatures and has been theoretically explained well in term of the effect of IIs.15 Furthermore, it is seen in Table 3 that the descendant degree of the melting point is much larger than that of the allotropic transformation temperature because of the effect of IIs. This can also be understood based on the above analysis of effect of IIs on VES. As seen in Tables 1 and 2, IIs can weaken the main bonds of all solid phases, including alpha, alpha and gamma. Since different phases are involved in different phase transformations, changes in main bonds are different. For the calculation of the melting points, only one solid phase is needed to be taken into account. However, for the calculation of the allotropie transformation, two phases are needed, which leads to a counteraction in the effect of IIs since the main bonds of both solid phases are weakened simultaneously. As a result, seemingly, the calculated allotropic transformation temperature is not decreased very largely. Conclusion
Effects of IIs on VESs and melting points in Ti-Al alloys have been investigated theoretically based on the EET. It is demonstrated that IIs make atom states increased and VESs considerably anisotropic. The calculated melting points are basically consistent with experimental results. Based on the analysis of VES with and without Us, the discrepancy of melting points between theoretical and experimental results can be further interpreted.
Project supported by Hunan Provincial Natural Science Foundation of China under grant no. 05JJ30078.
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J. Z. Peng*1[dagger], X. Z. Yang2, F. C. Zhou2 and M. F. Gray3
1 College of Physics Science and Information Engineering, JiShou University, JiShou 416000, China
2 Department of Mechamisal Engineering, Changchun University, Changchun, Jilin, China 130022
3 Materials Science and Engineering Department, The Ohio State University, 650 Ackerman Road, Suite 255, Columbus, OH, 43202, USA
* Corresponding author, email firstname.lastname@example.org
[dagger] see Acknowledgement
Copyright Institute of Materials May 2008
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