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Density of Liquid Ni-Ti and a New Optical Method for its Determination

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Abstract

Liquid Ni-Ti 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 top-view 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 Ni-Ti is a highly non-ideal 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., Ni-Ti, is adequately predicted by a thermodynamic model.

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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−3K−1)

ρ T,Ni :

Temperature coefficient of the mass density of liquid Ni (g cm−3K−1)

ρ T,Ti :

Temperature coefficient of the mass density of liquid Ti (g cm−3K−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 κe on EG (Pa−1)

κ e,1 :

2nd linear coefficient for the dependence of κe on EG (Pa−1)

R m :

Molar gas constant (8.314 kJmol−1)

u S :

Ultrasonic sound velocity (m s−1)

c P :

Isobaric-specific heat (Jg−1K−1)

R :

Radius of an edge point in the side-view image represented in polar coordinates (pixel)

R top :

Radius of an edge point in the top-view image represented in polar coordinates (pixel)

φ :

Azimuthal angle of an edge point in the side-view image represented in polar coordinates

ϕ :

Polar angle of an edge point in the top-view image represented in polar coordinates

X top :

Cartesian edge-point “x”-component in the top-view image (pixel)

Y top :

Cartesian edge-point “y”-component in the top-view image (pixel)

\( X_{\hbox{max} }^{\text{top}} \) :

Maximum value of Xtop (pixel)

\( Y_{\hbox{max} }^{\text{top}} \) :

Maximum value of Ytop (pixel)

a i,X :

ith expansion coefficient of Xtop (pixel)

a i,Y :

ith expansion coefficient of Ytop (pixel)

b i,X :

ith expansion coefficient of Xtop (pixel)

b i,Y :

ith expansion coefficient of Ytop (pixel)

Π i :

Legendre polynomial of the order i

a i :

Coefficient associated with Πi

V P,Circle :

Volume of a sample with an assumed circular cross section (pixel3)

V P,real :

Real volume of a sample (pixel3)

ΔV P,real :

Uncertainty of the real volume (pixel3)

S V :

Calibrated volume of a sample (cm3)

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 (pixel2)

h :

Position (height) on the vertical axis of the droplet (pixel)

Q real :

Area of the real cross section (pixel2)

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 (cm3 pixel−3)

V :

Molar volume (cm3 mol−1)

E V :

Excess molar volume (cm3 mol−1)

id V :

Molar volume of an ideal solution (cm3 mol−1)

V Ti :

Molar volume of Ti (cm3 mol−1)

V Ni :

Molar volume of Ni (cm3 mol−1)

0 V :

Volume interaction constant (cm3 mol−1)

0 A :

Coefficient for the temperature dependence of 0V (cm3 mol−1)

0 B :

Coefficient (slope) for the temperature dependence of 0V (cm3 mol−1K−1)

x Ti :

Mole fraction of Ti (at. pct)

x Ni :

Mole fraction of Ni (at. pct)

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Acknowledgment

Many thanks to Dr. F. Yang, Dr. A Rawson, and Priv.-Doz. Dr. D. Holland-Moritz for their critical reviews of the manuscript.

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Correspondence to J. Brillo.

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 = idV + EV, that Eq. [17] also holds for the excess volume EV:

$$ {}^{\text{E}}V \cdot \kappa_{\text{e}} (^{\text{E}} G) = \frac{{\partial {}^{\text{E}}G}}{\partial P} \cdot \kappa_{\text{e}} (^{\text{E}} G) = \frac{{\partial {}^{\text{E}}V}}{\partial P} . $$
(A1)

In Eq. [A1], it is shown that the excess volume can be obtained by differentiation of the excess free energy EG. Moreover, κe is assumed to be a function of EG. To the first degree of approximation, this function would be linear with κe,0 and κe,1 being the corresponding coefficients:

$$ \kappa_{\text{e}} (^{\text{E}} G) \approx \kappa_{\text{e,0}} + \kappa_{\text{e,1}} {}^{\text{E}}G . $$
(A2)

Combining Eqs. [A1] and [A2] yields:

$$ \kappa_{\text{e,0}} \frac{{\partial {}^{\text{E}}G}}{\partial P} + \kappa_{\text{e,1}} {}^{\text{E}}G\frac{{\partial {}^{\text{E}}G}}{\partial P} \approx \frac{{\partial {}^{\text{E}}V}}{\partial P}. $$
(A3)

The pressure is eliminated by integration of Eq. [A3]:

$$ \left[ {\kappa_{\text{e,0}} {}^{\text{E}}G + \frac{1}{2}\kappa_{\text{e,1}} {}^{\text{E}}G^{2} } \right] \approx {}^{\text{E}}V. $$
(A4)

This expression can be transformed further by expanding the linear approximation into an exponential and letting the ratio of κe,0 to κe,1 be written as λ/RmT:

$$ {}^{\text{E}}V \approx 2\kappa_{\text{e,0}} {}^{\text{E}}G\exp (\lambda \frac{{{}^{\text{E}}G}}{{R}_m{T}}). $$
(A5)

In reality, this relation is not necessarily true generally as there are also systems where the signs of EV and EG are different.[18,21] However, Eq. [A5] may be true for a large number of systems.

Mathematically, Eq. [A5] exhibits a minimum at EG = EGmin, where EGmin is obtained as

$$ {}^{\text{E}}G_{\hbox{min} } = - \frac{{R}_m{T}}{\lambda }. $$
(A6)

However, the internal energy U = 3RmT is a lower limit for EG 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

$$ \lambda = \frac{1}{3}. $$
(A7)

Combining Eqs. [A5] and [A7] yields Eq. [18].

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Brillo, J., Schumacher, T. & Kajikawa, K. Density of Liquid Ni-Ti and a New Optical Method for its Determination. Metall Mater Trans A 50, 924–935 (2019). https://doi.org/10.1007/s11661-018-5047-8

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