A new model to describe composition and temperature dependence of thermal conductivity for solution phases in binary alloys

doi:10.1016/j.jmst.2020.04.045

Abstract

Abstract:

Modelling temperature- and composition-dependent thermal conductivity in alloys is challengeable and is seldom studied systematically. In the present work, a new model is developed to describe the temperature and concentration dependence of thermal conductivity for binary alloys. In this new model, firstly thermal conductivity of pure metals was modelled as the function of temperature for each phase and each magnetic state by the corresponding physically sound model. Secondly, in order to describe the composition and temperature dependence of thermal conductivity for solid phases, the combination of the theories of Nordheim and Mott for electric conductivity of alloys with the Wiedemann-Franz law was performed. Thirdly, the reliability of the new model was verified by presently measured thermal conductivities for pure Co, Ni and Co-Ni alloys at 300, 600, 900 and 1100 K as well as for binary Al-Zn, Mg-Zn and U-Zr systems using the data taken from the literature. The calculated thermal conductivities can well reproduce the measured ones in one-phase regions of a series of Co-Ni alloys. The thermal conductivity in a two-phase region of the Co-Ni system is reasonably predicted as well. It is demonstrated that the new model can be utilized to evaluate the thermal conductivity over the whole investigated composition and temperature ranges for the first time and is expected to be extended to ternary and multicomponent systems by CALPHAD method, which contributes significantly to the development of computational design of materials.

Key words: Thermal conductivity, Co-Ni alloys, CALPHAD method, Measurement, Modelling

Shiyi Wen, Yuling Liu, George Kaptay, Yong Du. A new model to describe composition and temperature dependence of thermal conductivity for solution phases in binary alloys[J]. J. Mater. Sci. Technol., 2020, 59: 72-82.

Figures/Tables 15

Fig. 1. Comparison of calculated thermal conductivity of Al-Zn alloys at: (a) 298 K, (b) 398 K, (c) 448 K and (d) 498 K by Huang et al. [45] and that by the present model, along with the measured thermal conductivity.

Table 1 The equations for thermal conductivity of pure metals in different phases and in different magnetic states (T in K, k in W/mK).

Element	T-ranges, K	States	Equations	Average Deviation	Refs.
Ni	100 - 627	fcc- ferromagnetic	k_Ni,fcc-f⁰=67.91-0.0367·T + 9727·T^-1	2.18 %	The present work
Ni	627 - 1728	fcc- paramagnetic	k_Ni,fcc-p⁰ = 65.12 + 0.0122·T-9180·T^-1	7.34 %	The present work
Co	100 - 695	hcp- ferromagnetic	k_Co,hcp-f⁰ = 97.75-0.0836·T + 7284·T^-1	2.48 %	The present work
Co	695 - 1388	fcc- ferromagnetic	k_Co,fcc-f⁰ = 46.21-0.00918·T + 12,705·T^-1	7.39 %	The present work
Al	273- 933	fcc	k_Al,fcc⁰ = 325.52-0.110·T-19476.77·T^-1		[45]
Mg	273- 923	hcp	k_Mg,hcp⁰ = 179.67-0.0400·T-6062.38·T^-1		[45]
Zn	273- 693	hcp	k_Zn,hcp⁰ = 127.94-0.0400·T-99.61·T^-1		[45]
U	1049- 1408	bcc	k_U,bcc⁰ = 4.862 + 0.03284·T+5844·T^-1		[68]
Zr	1136- 2182	bcc	k_Zr,bcc⁰ = 0.5694 + 0.0150·T+5857·T^-1		[68]

Table 2 The primary experimental results obtained in this work.

No.	Compo-sition	Temperature (K)	Density* (g/cm³)	Heat⁺ capacity (J/gK)	Thermal diffusivity (mm²/s)	Thermal conductivity (W/mk)	State [63]
1	pure Co	300	8.576	0.454 (0.429)	26.0	101	hcp-ferro
		600	8.486	0.519 (0.499)	13.7	60.3	hcp-ferro
		900	8.375	0.675 (0.587)	9.92	56.1	fcc-ferro
		1100	8.308	0.680 (0.679)	7.85	44.3	fcc-ferro
2	Co-20 wt.%Ni	300	8.627	0.419 (0.430)	20.6	74.5	hcp-ferro
		600	8.526	0.527 (0.504)	11.3	50.8	**
		900	8.424	0.645 (0.607)	8.47	46.0	fcc-ferro
		1100	8.356	0.727 (0.725)	6.89	41.9	fcc-ferro
3	Co-40 wt.%Ni	300	8.716	0.429 (0.431)	19.7	73.7	fcc-ferro
		600	8.613	0.545 (0.510)	12.5	58.7	fcc-ferro
		900	8.509	0.682 (0.635)	8.43	48.9	fcc-ferro
		1100	8.441	0.761 (0.802)	6.49	41.7	fcc-ferro
4	Co-60 wt.%Ni	300	8.703	0.429 (0.432)	18.3	68.3	fcc-ferro
		600	8.599	0.541 (0.519)	11.6	54.0	fcc-ferro
		900	8.496	0.691 (0.688)	7.76	45.6	fcc-ferro
		1100	8.427	0.661 (0.638)	7.07	39.4	fcc-para
5	Co-80 wt.%Ni	300	8.710	0.439 (0.434)	16.9	64.6	fcc-ferro
		600	8.605	0.578 (0.540)	10.9	54.2	fcc-ferro
		900	8.501	0.682 (0.600)	8.03	46.6	fcc-para
		1100	8.431	0.653 (0.577)	8.93	49.2	fcc-para
6	pure Ni	300	8.762	0.450 (0.439)	21.8	86.0	fcc-ferro
		600	8.656	0.607 (0.618)	11.9	62.5	fcc-ferro
		900	8.551	0.631 (0.534)	12.8	69.1	fcc-para
		1100	8.480	0.579 (0.561)	12.9	63.3	fcc-para

Fig. 2. Comparison of temperature dependency of thermal conductivities of the same fcc-ferromagnetic phases for pure Ni (below 627 K) and for pure Co (above 695 K). The line is calculated by$k_{fcc-f}^{o}\cong 49.1-0.0116 \bullet T+\frac{12440}{T}$.

Fig. 3. The XRD results for: (a) Co with 20 w% Ni sample No 2 and (b) Co with 40 - 60 - 80 w% Ni samples Nos. 3-5. The phase diagram of Co-Ni binary system [63] is shown in (c) and the sample numbers are given in Table 2.

Fig. 4. Comparison of the measured data from the literature [[54], [55], [56], [57]], our measured data and the lines calculated by equations of Table 1 for the temperature dependence of thermal conductivity of pure Ni. Vertical dotted line shows the fcc-ferromagnetic - fcc-paramagnetic transition of Ni at 627 K.

Fig. 5. Comparison of the measured data from the literature [47,49,54], our measured data and the lines calculated by equations of Table 1 for temperature dependence of thermal conductivity of pure Co. Vertical dotted line shows the hcp-ferromagnetic - fcc-ferromagnetic phase transition of Co at 695 K.

Fig. 6. The temperature dependence of parameter rfcc-f calculated for the 11 measured points of fcc-ferromagnetic alloys using data of Table 2 and Eq. (13). The line is estimated by Eq. (14).

Fig. 7. The composition dependence of parameter rfcc-f calculated for the 11 measured points of fcc-ferromagnetic alloys using data of Table 2 and Eq. (13). The x-axes is selected in accordance with Eq. (12).

Table 3 Summary of parameters applied in Eq. (12) for modelling the 15 data points of Table 2 in 3 one-phase fields for the Co-Ni system.

State α	Data points	r_0,α, mK/W	r_1,α, mK/W
fcc-ferromagnetic	11	0.013	-0.0055
fcc-paramagnetic	3	0.038	---
hcp-ferromagnetic	1	(0.022)	---

Fig. 8. Comparison of measured (x-axis) and modelled (y-axis) thermal conductivity values for 11 fcc-ferromagnetic alloys and 3 fcc-paramagnetic alloys.

Fig. 9. The concentration dependence of thermal conductivity of the Co-Ni alloy at: (a) T =300 K, (b) T = 600 K, (c) T = 900 K and (d) T = 1100 K. Points: measured in this paper, lines: Eq. (12) with parameters of Table 3. Vertical broken lines show phase boundaries in agreement with the phase diagram [63] (see also Table 2).

Fig. 10. The three-dimensional planes of thermal conductivities of Co-Ni system over the whole investigated composition and temperature ranges along with the phase diagram.

Table 4 Model parameters of Al-Zn, hcp Mg-Zn and bcc U-Zr alloys based on Eq.(15).

Binary systems	a₀	b₀
Al-Zn	0.0544	-0.00007767
Mg-Zn	0.1493	-0.0001118
U-Zr	0.2249	-0.0002102

Fig. 11. Predicted three-dimensional planes of thermal conductivity for: (a) Mg-Zn and (b) U-Zr alloys, in comparison with the data from:(a) Huang et al. [45] and (b) Touloukian et al. [68].

References 37

[1]	E. Fletcher, Belgium: Centre D’Information Du Cobalt, 1960, pp. 345-361.
[2]	L.E. Toth, Transition Metal Carbides and Nitrides, Academic Press, New York, 1971.
[3]	W.S. Williams, Mater. Sci. Eng.A 105-106 (1988) 1-10.
[4]	X. Zhang, G. Liu, J. Tao, Y. Guo, J. Wang, G. Qiao, J. Mater. Sci. Technol. 34 (2018) 1180-1188. DOI URL
[5]	G.S. Upadhyaya, Mater. Des. 22 (2001) 483-489. DOI URL
[6]	S. Nogren, J. Garcia, A. Blomqvist, L. Yin, Int. J. Refract. Met. Hard. Mater. 48 (2015) 31-45. DOI URL
[7]	G. Zhang, Y.i. Liu, Y. Wang, F. Guo, X. Liu, Y. Wang, J. Mater. Sci. Technol. 33 (2017) 1346-1352.
[8]	M.A. Lagos, I. Agota, T. Schubert, T. Weissgaerben, J.M. Gallardo, J.M. Montes, L. Prakash, C. Andreouli, V. Oikonomou, D. Lopez, J.A. Calero, Int. J. Refract. Met. Hard. Mater. 66 (2017) 88-94.
[9]	H. Wang, M. Gee, Q. Qiu, H. Zhang, X. Liu, H. Nie, X. Song, Z. Nie, J. Mater. Sci. Technol. 35 (2019) 2435-3446. DOI URL
[10]	M. He, J. Wang, R. He, H. Yang, J. Ruan, J. Alloys Compd. 766 (2018) 556-563. DOI URL
[11]	J. Tong, J. Zhang, Y. Wang, F. Min, X. Wang, H. Zhang, J. Ma, Adv. Powder Technol. 10 (2019) 2311-2319.
[12]	B. Guimares, D. Fiugeriedo, C.M. Fernandes, F.S. Silva, G. Miranda, O. Carvalho, Int. J. Refract. Met. Hard. Mater. 81 (2019) 316-324. DOI URL
[13]	J. Li, S. Xiang, H. Luan, A. Amar, X. Liu, S. Lu, Y. Zeng, G. Le, X. Wang, F. Qu, C. Jiang, G. Yang, J. Mater. Sci. Technol. 35 (2019) 2430-2434. DOI URL
[14]	V.B. Voitovich, V.V. Sverdel, R.F. Voitovich, E.I. Golovko, Int. J. Refract. Met. Hard. Mater. 14 (1996) 289-295. DOI URL
[15]	M. Tokita, Mater. Sci. Forum. 492-493 (2005) 711-718. DOI URL
[16]	I.B. Panteleev, T.V. Lukashova, S.S. Ordan’yan, Powder Metall. Metal. Ceram. 45 (2006) 342-345. DOI URL
[17]	A. Tahir, G. Li, M. Liu, G. Yang, C. Li, Y. Wang, C. Li, J. Mater. Sci. Technol. 37 (2020) 1-8. DOI URL
[18]	M. Aristizabal, L.C. Ardila, F. Veiga, M. Arizmendi, J. Fernandez, J.M. Sanchez, Wear 280-281 (2012) 15-21. DOI URL
[19]	L. Emanueli, M. Pellizzaru, A. Molinari, F. Castellani, E. Zinutti, Int. J. Refract. Met. Hard. Mater. 60 (2016) 118-124. DOI URL
[20]	M. Erfanmanesh, H. Abdollah-Pour, H. Mohammadian-Senmani, R. Shoja-Razavi, Ceram. Int. 44 (2018) 12805-12814. DOI URL
[21]	B.L. Ezquerra, L. Lozada, H. van den Berg, M. Wolf, J.M. Sanchez, Int. J. Refract. Met. Hard. Mater. 72 (2018) 89-96. DOI URL
[22]	C. Zhang, H. Yin, J. Lv, Y. Du, Z. Tan, Y. Liu, Calphad 67 (2019), 101664. DOI URL
[23]	X. Zhang, J. Zhou, K. Liu, W. Shen, Z. Lin, Z. Li, Y. He, N. Lin, Int. J. Refract. Met. Hard. Mater. 80 (2019) 123-129. DOI URL
[24]	C.W. Liu, M.D. Jean, Q.T. Wang, B.S. Chen, Strength Mater. 51 (2019) 95-101. DOI URL
[25]	R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge University Press, Cambridge, UK, 2006.
[26]	T.M. Pollock, S. Tin, J. Propul. Power 22 (2006) 361. DOI URL
[27]	R. Arroyave, D.L. McDowell, Ann.Rev. Mater. Res. 49 (2019) 108-126.
[28]	W. Xia, X. Zhao, L. Yue, Z. Zhang, J. Mater. Sci. Technol. (2020), in press. DOI URL PMID
[29]	J.I. Kapusta, C. Gale, J. Phys. G - Nucl. Part. Phys. (2008). URL PMID
[30]	Y. Kabalci, in: E. Kabalci, Y. Kabalci (Eds.), Smart Grids and Their Communication Systems. Energy Systems in Electrical Engineering, Springer, Singapore, 2019.
[31]	Y. Du, B. Sundman, J. Phase Equilib. Diffus. 38 (2017) 601-602. DOI URL
[32]	X. Zheng, D.G. Cahill, P. Krasnochtchekov, R.S. Averback, J.-C. Zhao, Acta Mater. 55 (2007) 5177-5185. DOI URL
[33]	D.R. Poirier, G.H. Geiger, Transport Phenomena in Materials Processing, TMS, Wearrendale, 1994.
[34]	L. Nordheim, I. Ann. Physik 401 (1931) 607-640.
[35]	N.F. Mott, Proc. R. Soc. A 153 (1936) 699-717.
[36]	Y. Terada, K. Ohkubo, T. Mohri, T. Suzuki, J. Appl. Phys. 81 (1997) 2263-2268. DOI URL
[37]	A. Rudajevova, F. von Buch, B.L. Mordike, J. Alloys. Compd. 292 (1999) 27-30. DOI URL