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Calculation of thermal expansion coefficients of pure elements and their alloys
Scripta Materialia 46 (2002) 557–562 www.actamat-journals.com
Calculation of thermal expansion coefficients of pure elements and their alloys Phillip Abel a, Guillermo Bozzolo a
a,b,*
NASA Glenn Research Center at Lewis Field, Mail Stop 23-2, Cleveland, OH 44135, USA b Ohio Aerospace Institute, 22800 Cedar Point Road, Cleveland, OH 44142, USA Received 6 August 2001; accepted 8 November 2001
Abstract A simple algorithm for computing the coefficient of thermal expansion of pure elements and their alloys, based on features of the binding energy curve, is introduced. The BFS method for alloys is used to determine the binding energy curves of intermetallic alloys and Ni-base superalloys. Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Thermal expansion; Alloys; Intermetallics; Quantum approximate methods; Nickel; Aluminium; Copper
1. Introduction Recent advances in computational materials science due to the introduction of powerful, computationally efficient quantum approximate methods have resulted in substantial growth in the field of finite temperature large-scale atomistic simulations. However, to properly describe finite temperature effects requires knowledge of the volume dependence on temperature of the computational cell, in order to achieve the same degree of efficiency and accuracy that already characterize existing methods for zero temperature calculations. We therefore introduce a simple and straightforward approach for extracting finite temperature
* Corresponding author. Tel.: +1-216-433-5824; fax: +1-440962-3057. E-mail address: [email protected] (G. Bozzolo).
lattice expansion out of readily available zero temperature theoretical input, for enhanced accuracy in the finite temperature simulations. Back when establishing the validity of the semiempirical universal binding energy relation (UBER) for metals, and using experimentally determined element properties for input, Guinea et al. [1] calculated thermal expansion coefficients at temperatures near the Debye temperature which were in reasonable agreement with experiment. In further developing a universal equation of state, Vinet et al. [2] calculated a number of parameters, including thermal expansion, as a function of temperature based on single temperature input parameters, again with good experimental agreement. Current quantum approximate methods can be based entirely on ab initio input, and Moruzzi et al. [3] did extensive calculation of finite temperature properties based on electronic structure calculations, achieving good agreement with experimentally determined parameter values for the
1359-6462/02/$ - see front matter Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 2 3 6 - 2
558
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
14 metals treated. In spite of the progress made in this area, however, the need for even more powerful algorithms remains, hence the computationally simple yet accurate method demonstrated here. The immediate goal is an efficient technique to determine the coefficient of thermal expansion of pure elements and their alloys, thus providing a simple method for dealing with vibrational degrees of freedom. It should be noted that this approach is limited to low temperatures, as it is not possible, for instance, to reduce the issue of thermal vacancies to such a simple framework as the one presented here. The approach is solely based on the readily available UBER [4], which provides a simple description of interatomic interactions as a function of volume. In this approach, thermal expansion coefficients from zero to well above room temperature are easily computed by relating changes in thermal energy to changes in lattice parameter via the asymmetry of the UBER. For an approximation of the amount that a crystal expands with temperature above absolute zero, we start with the notion of an atom oscillating in place with increasing amplitude as it gains energy. If that atom is in a potential well completely symmetric about the zero temperature equilibrium position, no matter how much energy it gains (until it escapes) the mean position around which that atom gyrates would not change. Following the standard argument, it is the asymmetry of the potential energy well, the difference between the repulsive and attractive forces between atoms, that causes a shift in equilibrium position with increasing thermal energies. By simply translating properly the thermal energy available to each atom onto the binding energy versus volume curve, the difference between potential distance moved into the repulsive regime versus the attractive region will scale the linear thermal expansion of the entire crystal. A simple difference is then computed between the ‘‘maximum excursion’’ allowed by the thermal energy available (attractive region of the energy well) and the corresponding distance into repulsion. That small difference, resulting from a simple subtraction, is the amount by which the equilibrium spacing between atoms increases for a given temperature. To lowest order, we can take ad-
vantage of the elegant simplicity of the universal potential implicit in the theory. The approach can be applied to alloys as well as to pure elements, provided that the corresponding zero temperature UBER is readily available. In the case of alloys, the Bozzolo–Ferrante–Smith (BFS) method [5] constitutes an ideal tool for a direct and accurate calculation of such curves. To illustrate graphically the simplicity of this approach, consider the binding energy versus separation curve in Fig. 1 for a single element or an alloy. With zero thermal energy, the probe atom can be thought of as ‘‘sitting at the bottom of the curve’’, i.e., the most probable separation between atoms corresponds to the distance at which the potential energy is most negative, denoted by ae in the figure. This is the (zero-temperature) equilibrium spacing between atoms. For any finite temperature then, we assign the appropriate average amount of energy per atom, denoted in the figure by the level Eth and note that the energy curve is intersected at two distances, one on either side of the equilibrium position. If we think of this region as the most probable set of distances at which the (thermally) excited probe atom is found, then the
Fig. 1. Binding energy curve for a pure element (or alloy). Ec denotes the cohesive energy and ae indicates the equilibrium lattice parameter. Eth stands for the vibrational (‘‘thermal’’) energy.
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
asymmetry of the curve suggests the average location of the atom should shift ‘‘outward’’ from the equilibrium position ae . To lowest order then, we can take for the change in equilibrium position, Da, the displacement from ae ‘‘outward’’ (dao ) allowed by the thermal energy, minus the magnitude of the displacement from ae ‘‘inward’’ (dai ), as noted in Fig. 1. The equilibrium spacing between atoms as a function of temperature then simply becomes ae þ Da, with the temperature dependence of the second term expressing the physical characteristics of each atomic type, as carried in the BFS parameterization of the binding energy curves in the case of alloys. Assuming equipartition of energy among the phonon modes available, and given the degrees of freedom in a solid, within the standard Debye model, the vibrational energy for an N-atom homogenous solid is given by [6] Eth ¼ 9NkT
T HD
3 Z
HD =T 0
ex
x3 dx; 1
ð1Þ
where HD is the Debye temperature, T the temperature and k is Boltzmann’s constant. The value of Da is extracted by finding the two roots of the equation EC 1 ð1 þ a Þea ¼ Eth ;
ð2Þ
where EC is the cohesive energy per atom and a is the BFS scaled lattice parameter [5]. Taking the positive difference Da between the two roots, the temperature-dependent lattice spacing becomes aðT Þ ¼ ae þ Da. The use of a zero temperature UBER to describe finite temperature interactions is not, in spite of its appearance, an incorrect formulation. The underlying concept in the argument above is that of an instantaneous ‘snapshot’ of the lattice in its thermally excited state. In this framework, by describing the interactions in terms of static configurations (i.e., by instantaneously freezing the vibrational degrees of freedom when determining dao and dai ) it becomes necessary to describe the interactions with zero temperature potentials. To check whether this simplistic approximation is adequate, comparison with experimentally determined thermal expansion coef-
559
ficients for pure elements and alloys should provide the acceptance test.
2. Results and discussion In this section we apply the streamlined calculation of thermal expansion coefficients to a number of single elements and alloys, and compare with experiment [7–9]. The results for single elements shown in Fig. 2(a)–(c) clearly establish the validity of the proposed approach for low temperatures (i.e., up to and above room temperature) but also indicate that at higher temperatures the inclusion of defects, e.g. thermal vacancies, is necessary in order to provide a consistent description of the thermal behavior below the melting point (T ¼ TM ). The limiting temperature of this approach can then be related to features in the UBER since our treatment of vibrational effects should be valid roughly up to the inflection point in the attractive (tensile) region of the UBER. As shown by Cotterill [10], deviation beyond the inflection point leads to formation and growth of defects, considered to lead ultimately to melting. For the material examples used here, Table 1 compares the values of Eth that are obtained from Eq. (1) for the melting temperature (T ¼ TM ) with the value of the energy (Einfl: ) that corresponds to the inflection point in the zero temperature UBER of the material (cf. Guinea et al. [1]). The usefulness of this approach for alloys is governed by the availability of the corresponding parameters needed to construct the corresponding UBER’s. In this sense, the BFS method for alloys provides an efficient basis for such tasks as it readily allows for the calculation of zero temperature UBER’s for arbitrary multicomponent systems. The BFS method for alloys [5] has been proven to be highly effective for the study of multicomponent systems. With the proper parameterization, it allows for an extremely economical, computationally simple, and physically sound description of large collections of atoms. The BFS method is a quantum approximate technique based on the assumption that the heat of formation, DH , of a given collection of atoms is the sum of the contributions of each atom in the
560
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
Fig. 2. Coefficient of thermal expansion for (a) Ni, (b) Al, (c) Cu, (d) NiAl, and the (e) Astelloy alloy S and (f) Astroloy Ni-base superalloys. In (a)–(c), BFS predictions are based on different choices for the first-principles input parameters: linear muffin tin orbital method (LMTO) [11] (solid curve) and linearized augmented plane wave method (LAPW) [12] (dashed curve). Experimental results are indicated with circles [7]. For (d), the predictions are based on LMTO Ni-bcc and Al-bcc parameters. The experimental results [8] are summarized with the fit indicated by the dashed curve. In (e) and (f), the predictions correspond to LAPW parameters for the fcc phases of the constituents. The triangles indicate the ramped results (average between room temperature values and a given value of T) obtained from the solid curve, and the circles denote the experimental values [9].
P sample, DH ¼ ei . Each contribution ei consists of two terms: a strain energy (eSi ) which accounts for the change in geometry with respect to a single monatomic crystal of the reference atom, and a chemical energy (eCi ), linked by a coupling function (gi ) so that ei ¼ eSi þ gi eCi . Three parameters for each of the constituent atoms are needed (equilibrium lattice parameter, cohesive energy and bulk modulus) in order to calculate these terms. It should be noted that the parameters used for the
pure elements were obtained from first-principles calculations [5,11,12], thus ensuring that the parameters used to describe the zero temperature UBER of every element were obtained from zero temperature calculations. The chemical energy accounts for the corresponding change in composition, considered as a defect in an otherwise pure crystal. The chemical ‘defect’ deals with pure and mixed bonds; therefore, two additional parameters are needed to describe the perturbation on the
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562 Table 1 Debye temperature (HD ), melting temperature TM (in K), thermal energy (see Eq. (1)) corresponding to the melting temperature, Eth ðT ¼ TM Þ (in eV) and value of the UBER corresponding to the inflection point, Einfl: (in eV) Element
HD
TM
Eth ðT ¼ TM Þ
Einfl:
Au Ag Pt Pd Cu Al Ni
165 225 233 275 343 420 465
1337 1235 2045 1825 1356 933 1726
0.33 0.298 0.506 0.446 0.318 0.203 0.403
0.332 0.261 0.515 0.347 0.308 0.294 0.391
electron density in the vicinity of one atom due to the presence of an atom of a different type. All the alloy parameters used are also determined via firstprinciples methods. We refer the reader to Ref. [5] for a detailed description of the BFS method and the determination of the parameters used in this work. Once the zero temperature UBER for the alloy is computed, it is a straightforward process (as discussed above) to determine the temperature dependence of the lattice parameter. Fig. 2 shows three alloy examples. First, we consider the ordered intermetallic NiAl (Fig. 2(d)). The predicted aðT Þ is compared with an empirical expression valid for the range 300 < T < 1300 K [8] a bit above, unfortunately, the temperature range for which the proposed approximation is valid. In this case, the results are compared to experimentally accepted mean coefficients of thermal expansion for the range 300 K to a given temperature T. aNiAl ðK1 Þ ¼ 1:16026 105 þ 4:08531 109 T 1:58368 1012 T 2 þ 4:18374 1016 T 3 :
561
tures. Still, in spite of its approximate nature, the methodology provides enough accuracy and sufficient simplicity to make it a good candidate for its ultimate application, which is to provide the necessary input for large-scale temperature-dependent atomistic simulations.
3. Conclusions We introduce a simple algorithm for the determination of the thermal expansion coefficients of pure elements. For alloys, the BFS method [5] is used for the calculation of zero temperature UBER’s for multicomponent systems, and the corresponding estimates of the coefficient of thermal expansion. Finite temperature information extracted from zero temperature properties, coupled with the availability of the BFS method for the calculation of zero temperature properties for arbitrary alloys, allows for the determination of finite temperature properties in simulations of complex systems. As a proof of concept, we illustrated the approach with examples for single elements, an intermetallic alloy and Ni-base superalloys.
4. Acknowledgements Fruitful discussions with N. Bozzolo are gratefully acknowledged. We wish to thank J. Garces and H. Mosca for contributions in preparing this manuscript. This work was supported by the HOTPC project at NASA Glenn Research Center, Cleveland, Ohio.
ð3Þ
The lack of limitations in BFS regarding the number of elements in the alloy results in the possibility of estimating the coefficient of thermal expansion for Ni-base superalloys (Fig. 2(e)–(f)), containing a large number of elements. In all cases, whether it is a single element, a binary ordered alloy or a multicomponent Ni-base superalloy, the agreement is good for low temperatures only, just providing a rough estimate for higher tempera-
References [1] Guinea F, Rose JH, Smith JR, Ferrante J. Appl Phys Lett 1984;44:53. [2] Vinet P, Smith JR, Ferrante J, Rose JH. Phys Rev B 1987;35:1945. [3] Moruzzi VL, Janak JF, Schwarz K. Phys Rev B 1988;37:790. [4] Rose JH, Smith JR, Ferrante J. Phys Rev B 1983;50:1835. [5] Bozzolo G, Noebe RD, Ferrante J, Amador C. J Comput Aided Mater Design 1999;6:1, and references therein.
562
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
[6] Burns G. Solid state physics. New York: Academic Press Inc.; 1985. [7] Gel’man EB. In: Grigoriev IS, Meilikhov EZ, editors. Handbook of physical quantities. Boca Raton: CRC Press; 1997. [8] Clark RW, Whittenberger JD. In: Hahn TA, editor. Thermal Expansion, vol 8. New York: Plenum Press; 1984. p. 189.
[9] Properties of metals and alloys. third ed. New York: The International Nickel Company Inc.; 1968. [10] Cotterill RMJ. Physica Scripta 1978;18:37. [11] Andersen OK. Phys Rev B 1975;12:3060. [12] Blaha P, Schwartz K, Luitz J. WIEN97. Vienna University of Technology. Updated Unix version of the copyrighted WIEN code Blaha P, Schwartz P, Sorantin P, Trickey SB. Comput Phys Commun 1990;59:399.
Calculation of thermal expansion coefficients of pure elements and their alloys Phillip Abel a, Guillermo Bozzolo a
a,b,*
NASA Glenn Research Center at Lewis Field, Mail Stop 23-2, Cleveland, OH 44135, USA b Ohio Aerospace Institute, 22800 Cedar Point Road, Cleveland, OH 44142, USA Received 6 August 2001; accepted 8 November 2001
Abstract A simple algorithm for computing the coefficient of thermal expansion of pure elements and their alloys, based on features of the binding energy curve, is introduced. The BFS method for alloys is used to determine the binding energy curves of intermetallic alloys and Ni-base superalloys. Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Thermal expansion; Alloys; Intermetallics; Quantum approximate methods; Nickel; Aluminium; Copper
1. Introduction Recent advances in computational materials science due to the introduction of powerful, computationally efficient quantum approximate methods have resulted in substantial growth in the field of finite temperature large-scale atomistic simulations. However, to properly describe finite temperature effects requires knowledge of the volume dependence on temperature of the computational cell, in order to achieve the same degree of efficiency and accuracy that already characterize existing methods for zero temperature calculations. We therefore introduce a simple and straightforward approach for extracting finite temperature
* Corresponding author. Tel.: +1-216-433-5824; fax: +1-440962-3057. E-mail address: [email protected] (G. Bozzolo).
lattice expansion out of readily available zero temperature theoretical input, for enhanced accuracy in the finite temperature simulations. Back when establishing the validity of the semiempirical universal binding energy relation (UBER) for metals, and using experimentally determined element properties for input, Guinea et al. [1] calculated thermal expansion coefficients at temperatures near the Debye temperature which were in reasonable agreement with experiment. In further developing a universal equation of state, Vinet et al. [2] calculated a number of parameters, including thermal expansion, as a function of temperature based on single temperature input parameters, again with good experimental agreement. Current quantum approximate methods can be based entirely on ab initio input, and Moruzzi et al. [3] did extensive calculation of finite temperature properties based on electronic structure calculations, achieving good agreement with experimentally determined parameter values for the
1359-6462/02/$ - see front matter Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 2 3 6 - 2
558
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
14 metals treated. In spite of the progress made in this area, however, the need for even more powerful algorithms remains, hence the computationally simple yet accurate method demonstrated here. The immediate goal is an efficient technique to determine the coefficient of thermal expansion of pure elements and their alloys, thus providing a simple method for dealing with vibrational degrees of freedom. It should be noted that this approach is limited to low temperatures, as it is not possible, for instance, to reduce the issue of thermal vacancies to such a simple framework as the one presented here. The approach is solely based on the readily available UBER [4], which provides a simple description of interatomic interactions as a function of volume. In this approach, thermal expansion coefficients from zero to well above room temperature are easily computed by relating changes in thermal energy to changes in lattice parameter via the asymmetry of the UBER. For an approximation of the amount that a crystal expands with temperature above absolute zero, we start with the notion of an atom oscillating in place with increasing amplitude as it gains energy. If that atom is in a potential well completely symmetric about the zero temperature equilibrium position, no matter how much energy it gains (until it escapes) the mean position around which that atom gyrates would not change. Following the standard argument, it is the asymmetry of the potential energy well, the difference between the repulsive and attractive forces between atoms, that causes a shift in equilibrium position with increasing thermal energies. By simply translating properly the thermal energy available to each atom onto the binding energy versus volume curve, the difference between potential distance moved into the repulsive regime versus the attractive region will scale the linear thermal expansion of the entire crystal. A simple difference is then computed between the ‘‘maximum excursion’’ allowed by the thermal energy available (attractive region of the energy well) and the corresponding distance into repulsion. That small difference, resulting from a simple subtraction, is the amount by which the equilibrium spacing between atoms increases for a given temperature. To lowest order, we can take ad-
vantage of the elegant simplicity of the universal potential implicit in the theory. The approach can be applied to alloys as well as to pure elements, provided that the corresponding zero temperature UBER is readily available. In the case of alloys, the Bozzolo–Ferrante–Smith (BFS) method [5] constitutes an ideal tool for a direct and accurate calculation of such curves. To illustrate graphically the simplicity of this approach, consider the binding energy versus separation curve in Fig. 1 for a single element or an alloy. With zero thermal energy, the probe atom can be thought of as ‘‘sitting at the bottom of the curve’’, i.e., the most probable separation between atoms corresponds to the distance at which the potential energy is most negative, denoted by ae in the figure. This is the (zero-temperature) equilibrium spacing between atoms. For any finite temperature then, we assign the appropriate average amount of energy per atom, denoted in the figure by the level Eth and note that the energy curve is intersected at two distances, one on either side of the equilibrium position. If we think of this region as the most probable set of distances at which the (thermally) excited probe atom is found, then the
Fig. 1. Binding energy curve for a pure element (or alloy). Ec denotes the cohesive energy and ae indicates the equilibrium lattice parameter. Eth stands for the vibrational (‘‘thermal’’) energy.
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
asymmetry of the curve suggests the average location of the atom should shift ‘‘outward’’ from the equilibrium position ae . To lowest order then, we can take for the change in equilibrium position, Da, the displacement from ae ‘‘outward’’ (dao ) allowed by the thermal energy, minus the magnitude of the displacement from ae ‘‘inward’’ (dai ), as noted in Fig. 1. The equilibrium spacing between atoms as a function of temperature then simply becomes ae þ Da, with the temperature dependence of the second term expressing the physical characteristics of each atomic type, as carried in the BFS parameterization of the binding energy curves in the case of alloys. Assuming equipartition of energy among the phonon modes available, and given the degrees of freedom in a solid, within the standard Debye model, the vibrational energy for an N-atom homogenous solid is given by [6] Eth ¼ 9NkT
T HD
3 Z
HD =T 0
ex
x3 dx; 1
ð1Þ
where HD is the Debye temperature, T the temperature and k is Boltzmann’s constant. The value of Da is extracted by finding the two roots of the equation EC 1 ð1 þ a Þea ¼ Eth ;
ð2Þ
where EC is the cohesive energy per atom and a is the BFS scaled lattice parameter [5]. Taking the positive difference Da between the two roots, the temperature-dependent lattice spacing becomes aðT Þ ¼ ae þ Da. The use of a zero temperature UBER to describe finite temperature interactions is not, in spite of its appearance, an incorrect formulation. The underlying concept in the argument above is that of an instantaneous ‘snapshot’ of the lattice in its thermally excited state. In this framework, by describing the interactions in terms of static configurations (i.e., by instantaneously freezing the vibrational degrees of freedom when determining dao and dai ) it becomes necessary to describe the interactions with zero temperature potentials. To check whether this simplistic approximation is adequate, comparison with experimentally determined thermal expansion coef-
559
ficients for pure elements and alloys should provide the acceptance test.
2. Results and discussion In this section we apply the streamlined calculation of thermal expansion coefficients to a number of single elements and alloys, and compare with experiment [7–9]. The results for single elements shown in Fig. 2(a)–(c) clearly establish the validity of the proposed approach for low temperatures (i.e., up to and above room temperature) but also indicate that at higher temperatures the inclusion of defects, e.g. thermal vacancies, is necessary in order to provide a consistent description of the thermal behavior below the melting point (T ¼ TM ). The limiting temperature of this approach can then be related to features in the UBER since our treatment of vibrational effects should be valid roughly up to the inflection point in the attractive (tensile) region of the UBER. As shown by Cotterill [10], deviation beyond the inflection point leads to formation and growth of defects, considered to lead ultimately to melting. For the material examples used here, Table 1 compares the values of Eth that are obtained from Eq. (1) for the melting temperature (T ¼ TM ) with the value of the energy (Einfl: ) that corresponds to the inflection point in the zero temperature UBER of the material (cf. Guinea et al. [1]). The usefulness of this approach for alloys is governed by the availability of the corresponding parameters needed to construct the corresponding UBER’s. In this sense, the BFS method for alloys provides an efficient basis for such tasks as it readily allows for the calculation of zero temperature UBER’s for arbitrary multicomponent systems. The BFS method for alloys [5] has been proven to be highly effective for the study of multicomponent systems. With the proper parameterization, it allows for an extremely economical, computationally simple, and physically sound description of large collections of atoms. The BFS method is a quantum approximate technique based on the assumption that the heat of formation, DH , of a given collection of atoms is the sum of the contributions of each atom in the
560
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
Fig. 2. Coefficient of thermal expansion for (a) Ni, (b) Al, (c) Cu, (d) NiAl, and the (e) Astelloy alloy S and (f) Astroloy Ni-base superalloys. In (a)–(c), BFS predictions are based on different choices for the first-principles input parameters: linear muffin tin orbital method (LMTO) [11] (solid curve) and linearized augmented plane wave method (LAPW) [12] (dashed curve). Experimental results are indicated with circles [7]. For (d), the predictions are based on LMTO Ni-bcc and Al-bcc parameters. The experimental results [8] are summarized with the fit indicated by the dashed curve. In (e) and (f), the predictions correspond to LAPW parameters for the fcc phases of the constituents. The triangles indicate the ramped results (average between room temperature values and a given value of T) obtained from the solid curve, and the circles denote the experimental values [9].
P sample, DH ¼ ei . Each contribution ei consists of two terms: a strain energy (eSi ) which accounts for the change in geometry with respect to a single monatomic crystal of the reference atom, and a chemical energy (eCi ), linked by a coupling function (gi ) so that ei ¼ eSi þ gi eCi . Three parameters for each of the constituent atoms are needed (equilibrium lattice parameter, cohesive energy and bulk modulus) in order to calculate these terms. It should be noted that the parameters used for the
pure elements were obtained from first-principles calculations [5,11,12], thus ensuring that the parameters used to describe the zero temperature UBER of every element were obtained from zero temperature calculations. The chemical energy accounts for the corresponding change in composition, considered as a defect in an otherwise pure crystal. The chemical ‘defect’ deals with pure and mixed bonds; therefore, two additional parameters are needed to describe the perturbation on the
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562 Table 1 Debye temperature (HD ), melting temperature TM (in K), thermal energy (see Eq. (1)) corresponding to the melting temperature, Eth ðT ¼ TM Þ (in eV) and value of the UBER corresponding to the inflection point, Einfl: (in eV) Element
HD
TM
Eth ðT ¼ TM Þ
Einfl:
Au Ag Pt Pd Cu Al Ni
165 225 233 275 343 420 465
1337 1235 2045 1825 1356 933 1726
0.33 0.298 0.506 0.446 0.318 0.203 0.403
0.332 0.261 0.515 0.347 0.308 0.294 0.391
electron density in the vicinity of one atom due to the presence of an atom of a different type. All the alloy parameters used are also determined via firstprinciples methods. We refer the reader to Ref. [5] for a detailed description of the BFS method and the determination of the parameters used in this work. Once the zero temperature UBER for the alloy is computed, it is a straightforward process (as discussed above) to determine the temperature dependence of the lattice parameter. Fig. 2 shows three alloy examples. First, we consider the ordered intermetallic NiAl (Fig. 2(d)). The predicted aðT Þ is compared with an empirical expression valid for the range 300 < T < 1300 K [8] a bit above, unfortunately, the temperature range for which the proposed approximation is valid. In this case, the results are compared to experimentally accepted mean coefficients of thermal expansion for the range 300 K to a given temperature T. aNiAl ðK1 Þ ¼ 1:16026 105 þ 4:08531 109 T 1:58368 1012 T 2 þ 4:18374 1016 T 3 :
561
tures. Still, in spite of its approximate nature, the methodology provides enough accuracy and sufficient simplicity to make it a good candidate for its ultimate application, which is to provide the necessary input for large-scale temperature-dependent atomistic simulations.
3. Conclusions We introduce a simple algorithm for the determination of the thermal expansion coefficients of pure elements. For alloys, the BFS method [5] is used for the calculation of zero temperature UBER’s for multicomponent systems, and the corresponding estimates of the coefficient of thermal expansion. Finite temperature information extracted from zero temperature properties, coupled with the availability of the BFS method for the calculation of zero temperature properties for arbitrary alloys, allows for the determination of finite temperature properties in simulations of complex systems. As a proof of concept, we illustrated the approach with examples for single elements, an intermetallic alloy and Ni-base superalloys.
4. Acknowledgements Fruitful discussions with N. Bozzolo are gratefully acknowledged. We wish to thank J. Garces and H. Mosca for contributions in preparing this manuscript. This work was supported by the HOTPC project at NASA Glenn Research Center, Cleveland, Ohio.
ð3Þ
The lack of limitations in BFS regarding the number of elements in the alloy results in the possibility of estimating the coefficient of thermal expansion for Ni-base superalloys (Fig. 2(e)–(f)), containing a large number of elements. In all cases, whether it is a single element, a binary ordered alloy or a multicomponent Ni-base superalloy, the agreement is good for low temperatures only, just providing a rough estimate for higher tempera-
References [1] Guinea F, Rose JH, Smith JR, Ferrante J. Appl Phys Lett 1984;44:53. [2] Vinet P, Smith JR, Ferrante J, Rose JH. Phys Rev B 1987;35:1945. [3] Moruzzi VL, Janak JF, Schwarz K. Phys Rev B 1988;37:790. [4] Rose JH, Smith JR, Ferrante J. Phys Rev B 1983;50:1835. [5] Bozzolo G, Noebe RD, Ferrante J, Amador C. J Comput Aided Mater Design 1999;6:1, and references therein.
562
P. Abel, G. Bozzolo / Scripta Materialia 46 (2002) 557–562
[6] Burns G. Solid state physics. New York: Academic Press Inc.; 1985. [7] Gel’man EB. In: Grigoriev IS, Meilikhov EZ, editors. Handbook of physical quantities. Boca Raton: CRC Press; 1997. [8] Clark RW, Whittenberger JD. In: Hahn TA, editor. Thermal Expansion, vol 8. New York: Plenum Press; 1984. p. 189.
[9] Properties of metals and alloys. third ed. New York: The International Nickel Company Inc.; 1968. [10] Cotterill RMJ. Physica Scripta 1978;18:37. [11] Andersen OK. Phys Rev B 1975;12:3060. [12] Blaha P, Schwartz K, Luitz J. WIEN97. Vienna University of Technology. Updated Unix version of the copyrighted WIEN code Blaha P, Schwartz P, Sorantin P, Trickey SB. Comput Phys Commun 1990;59:399.