Abstract
Liquid NiTi alloys were processed in a containerless way using the technique of electromagnetic levitation in order to determine their densities. An improved optical method was utilized where, in addition to recording shadowgraph images from the side, a second camera recorded images of the sample from the top. A correction factor for the density was calculated from the topview images. This method yields measurements insensitive to droplet rotation and static deformation which removes the need to assume axial symmetry. The measured densities are discussed in terms of the molar volume. A negative molar excess volume was obtained, indicating that NiTi is a highly nonideal system. These measurements were then used to test a recently proposed relationship between the molar excess volume, the excess free energy, and the isothermal compressibility. For the first time, the excess volume of a binary alloy, i.e., NiTi, is adequately predicted by a thermodynamic model.
Similar content being viewed by others
Abbreviations
 ρ :

Mass density (g cm^{−3})
 Δρ :

Uncertainty of the density (g cm^{−3})
 ρ _{Ni} :

Mass density of liquid Ni (g cm^{−3})
 ρ _{Ti} :

Mass density of liquid Ti (g cm^{−3})
 ρ _{L} :

Mass density at liquidus (g cm^{−3})
 ρ _{T} :

Temperature coefficient of mass density (g cm^{−3}K^{−1})
 ρ _{T,Ni} :

Temperature coefficient of the mass density of liquid Ni (g cm^{−3}K^{−1})
 ρ _{T,Ti} :

Temperature coefficient of the mass density of liquid Ti (g cm^{−3}K^{−1})
 T :

Temperature [K (°C)]
 T _{L} :

Liquidus temperature [K (°C)]
 T _{P} :

Pyrometer signal [K(°C)]
 T _{L,P} :

Pyrometer signal at liquidus temperature [K(°C)]
 ^{E} G :

Excess free energy (kJmol^{−1})
 P :

Pressure (Pa)
 κ _{T} :

Isothermal compressibility coefficient (Pa^{−1})
 κ _{e} :

Effective isothermal compressibility coefficient (Pa^{−1})
 κ _{T,Ni} :

Isothermal compressibility coefficient of liquid Ni (Pa^{−1})
 κ _{T,Ti} :

Isothermal compressibility coefficient of liquid Ti (Pa^{−1})
 κ _{e,0} :

1st linear coefficient for the dependence of κ_{e} on ^{E}G (Pa^{−1})
 κ _{e,1} :

2nd linear coefficient for the dependence of κ_{e} on ^{E}G (Pa^{−1})
 R _{m} :

Molar gas constant (8.314 kJmol^{−1})
 u _{S} :

Ultrasonic sound velocity (m s^{−1})
 c _{P} :

Isobaricspecific heat (Jg^{−1}K^{−1})
 R :

Radius of an edge point in the sideview image represented in polar coordinates (pixel)
 R ^{top} :

Radius of an edge point in the topview image represented in polar coordinates (pixel)
 φ :

Azimuthal angle of an edge point in the sideview image represented in polar coordinates
 ϕ :

Polar angle of an edge point in the topview image represented in polar coordinates
 X ^{top} :

Cartesian edgepoint “x”component in the topview image (pixel)
 Y ^{top} :

Cartesian edgepoint “y”component in the topview image (pixel)
 \( X_{\hbox{max} }^{\text{top}} \) :

Maximum value of X^{top} (pixel)
 \( Y_{\hbox{max} }^{\text{top}} \) :

Maximum value of Y^{top} (pixel)
 a _{i,X} :

ith expansion coefficient of X^{top} (pixel)
 a _{i,Y} :

ith expansion coefficient of Y^{top} (pixel)
 b _{i,X} :

ith expansion coefficient of X^{top} (pixel)
 b _{i,Y} :

ith expansion coefficient of Y^{top} (pixel)
 Π _{i} :

Legendre polynomial of the order i
 a _{i} :

Coefficient associated with Π_{i}
 V _{P,Circle} :

Volume of a sample with an assumed circular cross section (pixel^{3})
 V _{P,real} :

Real volume of a sample (pixel^{3})
 ΔV _{P,real} :

Uncertainty of the real volume (pixel^{3})
 ^{S} V :

Calibrated volume of a sample (cm^{3})
 ^{S} M :

Mass of a sample (g)
 M :

Molar mass (gmol^{−1})
 M _{Ni} :

Molar mass of Ni (gmol^{−1})
 M _{Ti} :

Molar mass of Ti (gmol^{−1})
 Q _{Circle} :

Area of the circular cross section (pixel^{2})
 h :

Position (height) on the vertical axis of the droplet (pixel)
 Q _{real} :

Area of the real cross section (pixel^{2})
 a _{asy} :

Asymmetry coefficient
 a :

Half axis of an elliptic sample cross section (pixel)
 b :

Other half axis of an elliptic sample cross section (pixel)
 q :

Scaling factor for calibration (cm^{3} pixel^{−3})
 V :

Molar volume (cm^{3} mol^{−1})
 ^{E} V :

Excess molar volume (cm^{3} mol^{−1})
 ^{id} V :

Molar volume of an ideal solution (cm^{3} mol^{−1})
 V _{Ti} :

Molar volume of Ti (cm^{3} mol^{−1})
 V _{Ni} :

Molar volume of Ni (cm^{3} mol^{−1})
 ^{0} V :

Volume interaction constant (cm^{3} mol^{−1})
 ^{0} A :

Coefficient for the temperature dependence of ^{0}V (cm^{3} mol^{−1})
 ^{0} B :

Coefficient (slope) for the temperature dependence of ^{0}V (cm^{3} mol^{−1}K^{−1})
 x _{Ti} :

Mole fraction of Ti (at. pct)
 x _{Ni} :

Mole fraction of Ni (at. pct)
References
E. Akca, A. Gursel, Periodicals of Eng, and Nat. Sci. 2015, vol. 3, pp. 1527
M.T. Jovanovic, B. Lukic, Z. Miskovic, I. Bobic, I. Cvijovic, B. Dimcic, Metalurgija – Journal of Metallurgy 2007, vol. 13, pp. 91106
E.I. GalindoNava, W.M. Rainforth, P.E.J. RiveraDíazdelCastillo, Acta Mater. 2016, vol. 117, pp. 270285.
G.S. Firstov, J.V. Humbeeck, Y.N. Koval, Mater. Sci. Eng. 2004, vol. A 378, pp. 2–10.
K. Otsuka, X. Ren, Prog. Mater. Sci. 2005, vol. 50, pp. 511678.
X. Yi, K. Sun, W. Gao, X. Meng, W. Cai, L. Zhao, Journal of Alloys and Compounds 2018, vol. 735, pp. 12191226.
J. Brillo, G. Lohöfer, F. SchmidtHohagen, S. Schneider, I. Egry, J. of Mat. Prod. Tech. 2006, vol. 16, pp. 247273
R.A. Harding, R.F. Brooks, G. Pottlacher, J. Brillo: Thermophysical Properties of a Ti44Al8Nb1B alloy in the solid and molten conditions, Gamma Titanium Aluminides 2003, TMS (The Minerals, Metals & Materials Society), 75, 2003.
J. Brillo, S. Schneider, I. Egry, and R. Harding: Density, Thermal Expansion and Surface Tension of Liquid Titanium Alloys Measured by Noncontact Techniques, Proceedings of the 10th Titanium World Conference, Hamburg 2003, H. Lütjering, Hrsg., WileyVCh, 411, 2003
S. Amore, S. Delsante, H. Kobatake, J. Brillo, J. Chem. Phys. 2013, vol. 139, pp. 0645041.
S. Amore, J. Brillo, I. Egry, R. Novakovic, App. Surf. Sci. 2011, vol. 257, pp. 77397745
J.J. Wessing, and J. Brillo: Metall. Mater. Trans., 2017, vol. A 48, pp. 868–82
I. Egry, R. Brooks, D. HollandMoritz, R. Novakovic, T. Matsushita, E. Ricci, S. Seetharaman, R. Wunderlich, D. Jarvis, Int. J. Thermophys. 2007, vol. 28, pp. 10261036
S. Ozawa, K. Morohoshi, T. Hibiya, ISIJ International 2014, vol. 54, pp. 20972103
P.F. Paradis, T. Ishikawa, G.W. Lee, D. HollandMoritz, J. Brillo, W.K. Rhim, J. Okada, Materials Science and Engineering R: Reports 2014, vol. R76, pp. 153.
G. Pottlacher: High Temperature Thermophysical Properties of 22 Pure Metals, edition keiper, Graz, Austria, 2010.
K.C. Mills: Recommended Values of Thermophysical Properties for Selected Commercial Alloys, Woodhead Publishing Ltd., Cambridge, UK, 2002.
J. Brillo: Thermophysical properties of multicomponent liquid alloys, de Gruyter, Berlin, Germany, 2016.
P.F. Zhou, H.P. Wang, S.J. Wang, L. Hu, B. Wei, Metallurgical and Materials Transactions 2018, vol. 49A, pp. 54885496
H.L. Peng, T. Voigtmann, G. Kolland, H. Kobatake, J. Brillo: Phys. Rev. 2015, vol. B 97, pp. 1842011–18420113
S. Amore, J. Horbach, I. Egry, J. Chem. Phys. 2011, vol. 134, pp. 0445151  0445159
Y. Plevachuk, J. Brillo, and A. Yakymovych: Metall. Mater. Trans. 2018, vol. A, in press
M. Watanabe, M. Adachi, H. Fukuyama, J. Mater. Sci. 2016, vol. 51, pp. 33033310
M. Adachi, T. Aoyagi, A. Mizuno, M. Watanabe, H. Kobatake, H. Fukuyama, Int. J. Thermophys. 2008, vol. 29, pp. 20062014
J. Brillo, I. Egry, I. Ho, Int. J. Thermophys. 2006, vol. 27, pp. 494506
S. Krishnan, G.P. Hansen, R.H. Hauge, J.L. Margrave, High Temp. Sci. 1990, vol. 29, pp. 1752
T.B. Massalski: Binary Alloy Phase Diagram, American Society for Metals, Ohio, USA, 1986.
J. Brillo, I. Egry, Int. J. Thermophys. 2003, vol. 24, pp. 11551170
F. Kargl, C. Yuan, and G.N. Greaves, Int. J. Microgravity Sci. Appl. 2015, vol. 32, pp. 3202121  3202125
H. Kobatake, J. Brillo, J. Mater. Sci. 2013, vol. 48, pp. 49344941
J. Brillo, I. Egry, Z. Metallkd. 2004, vol. 95, pp. 691697
J. Brillo, I. Egry, J. Westphal, Int. J. Mat. Res. 2008, vol. 99, pp. 162167
J. Brillo, I. Egry, T. Matsushita, Int. J. Mat. Res. 2006, vol. 97, pp. 15261532
Y. Plevachuk, I. Egry, J. Brillo, D. HollandMoritz, I. Kaban, Int. J. Mat. Res. 2007, vol. 98, pp. 107111
W. Tang, B. Sundman, R. Sandstroem, C. Qiu, Acta Mater 1999, vol. 47, pp. 34573468
Y. Kawai, Y. Shiraishi: Handbook of Physicochemical Properties at High Temperatures, The Iron and Steel Institute of Japan ISIJ, Osaka, Japan, 1988.
J.B. Walter, K.L. Telschow, and R.E. Haun: https://www.osti.gov/servlets/purl/9233, 1999, Accessed 27 March 2018
Acknowledgment
Many thanks to Dr. F. Yang, Dr. A Rawson, and Priv.Doz. Dr. D. HollandMoritz for their critical reviews of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Manuscript submitted May 2, 2018.
Appendix
Appendix
A detailed discussion of the model, Eq. [18], is given in Reference 22. In the following, it is briefly outlined how Eq. [18] is obtained in Reference 22 from Eq. [17].
Formally, Eq. [17] is the same as the definition of the isothermal compressibility, except that the meaning of the pressure is different and that the minus sign is omitted. It is easily seen by multiplying both sides by V = ^{id}V +^{ E}V, that Eq. [17] also holds for the excess volume ^{E}V:
In Eq. [A1], it is shown that the excess volume can be obtained by differentiation of the excess free energy ^{E}G. Moreover, κ_{e} is assumed to be a function of ^{E}G. To the first degree of approximation, this function would be linear with κ_{e,0} and κ_{e,1} being the corresponding coefficients:
Combining Eqs. [A1] and [A2] yields:
The pressure is eliminated by integration of Eq. [A3]:
This expression can be transformed further by expanding the linear approximation into an exponential and letting the ratio of κ_{e,0} to κ_{e,1} be written as λ/R_{m}T:
In reality, this relation is not necessarily true generally as there are also systems where the signs of ^{E}V and ^{E}G are different.[18,21] However, Eq. [A5] may be true for a large number of systems.
Mathematically, Eq. [A5] exhibits a minimum at ^{E}G = ^{E}G_{min}, where ^{E}G_{min} is obtained as
However, the internal energy U = 3R_{m}T is a lower limit for ^{E}G if it is assumed that in addition to translations also a rotational degree of freedom exists. This holds at least as an approximation, as the excess entropy was neglected. Hence, it follows from Eq. [A6] that
Rights and permissions
About this article
Cite this article
Brillo, J., Schumacher, T. & Kajikawa, K. Density of Liquid NiTi and a New Optical Method for its Determination. Metall Mater Trans A 50, 924–935 (2019). https://doi.org/10.1007/s1166101850478
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s1166101850478