Density of Liquid Niobium and Tungsten and the Estimation of Critical Point Data

The surface radiance of the sample is monitored throughout the experiment by using a pyrometer with a central wavelength of \(\lambda = {650} \,{\hbox {nm}}\) and a full-width-at-half-maximum of 27 nm. Data points are collected every 100 ns. Neutral density filters were employed to break down the pyrometer signal by a constant fraction and thus enlarge the measurable temperature region. After concluding the experiment, the known radiance temperature at melting is assigned to the visible inflection in the thermogram, i.e., the melting plateau, to derive a radiance temperature \(T_{\mathrm {r}}(t)\) as a function of time t. The radiance temperature at melting is for this purpose calculated using the literature value for the true melting temperature \(T_{\mathrm {m}}\) and the normal spectral emissivity at melting \(\varepsilon _{\mathrm {m}}\). Together with the temperature-dependent normal spectral emissivity \(\varepsilon (\lambda ,T_{\mathrm {r}})\) of the metal, the pyrometer signal is then converted into a true temperature T(t) following Eq. [1]:where \(c_2\) is the second radiation constant. Table I sums up the utilized values and parameters to derive the true temperature T(t) for tungsten and niobium. For a more detailed description of the temperature deduction as well as an exemplary thermogram, please refer to a previous publication.[6]

Thermal radial expansion is investigated by means of fast shadow-imaging. During the experiment, a high-power photoflash (Multiblitz X10, 1000 Ws) provides intense background illumination to produce shadow images of the expanding wire at specific instants in time. These instants are time-synchronized with the pulse-heating experiment. By this means, a temperature can be assigned to each shadow image taken. The CCD-camera system (Co. PCO imaging with controller unit by Co. Theta System and Graz Univ. of Technol.) is capable of acquiring one image every 2.5 μs, compare e.g., References 6 and 7. After the experiment, summing over the pixel lines of each image gives a cup-shaped intensity profile. The full-width-at-half-maximum of these intensity profiles obtained by this means corresponds to the diameter d of the wire at a specific time and thus temperature T. In relating this temperature-dependent diameter d(T) to the diameter at room temperature \(d_0\), density can be derived by using the literature value for the room-temperature density \(\rho _0\). Note that longitudinal expansion of the wire is inhibited thanks to the short timescales of the experiment. Thus, the measured relative radial expansion squared \((d(T)/d_0)^2\) equals the relative volume expansion \(V(T)/V_0\) of the sample. The density as a function of temperature \(\rho (T)\) can thus be calculated viaFor niobium, we used a room-temperature density of \(8.57 \times 10^{3} \,\hbox {kg} \,\hbox {m}^{-3}\) as given in the CRC Handbook of Chemistry and Physics.[8] For tungsten, a room-temperature density \(\rho _0\) of \(19 256 \,\hbox {kg}\,\hbox {m}^{-3}\) was adopted from Ming and Manghnani,[9] found in the NIST Alloy data web application.[10] Note that this density value is only 0.2 pct lower than that given in the CRC Handbook of Chemistry and Physics.[8]

The measured liquid-phase density as a function of temperature was taken to estimate the critical temperature \(T_{\mathrm {c}}\), critical density \(\rho _{\mathrm {c}}\) , as well as the phase diagram in the (\(\rho \),T)-plane. The estimation is done by an extrapolation algorithm following the method in the publication of Schröer and Pottlacher.[4] In this approach, the measured liquid-phase density is extrapolated according to simplified Ising- and mean-field behavior to estimate critical temperature and critical density.

A simplified phase diagram, given by the Eqs. [3] and [4], is then also constructed from the measured liquid-phase density \(\rho _{+,{\mathrm {meas}}}(T)\) by extrapolating up to the critical temperature according to Eq. [3],In this equation, the subscript ‘+’ indicates the saturated liquid line and ‘−’ indicates the saturated vapor line of the phase diagram—the equation describes how the density changes as a function of temperature up to the critical point. \(\rho _{\mathrm {diam}}\) is the so-called phase-diagram diameter, i.e., the mean value between the saturated liquid density and the saturated vapor density at a given isotherm, \((\rho _++\rho _-)/2\). It is extrapolated up to the critical temperature \(T_{\mathrm {c}}\) according to Eq. [4], where \(\rho _{\mathrm {diam}}\) itself can be calculated from the measured liquid-phase density \(\rho _{+,{\mathrm {meas}}}(T)\) that is located in the low-temperature branch of the binodal, i.e., \(\rho _{\mathrm {diam}} \approx \rho _{+,{\mathrm {meas}}}/2,\)The extrapolations described above yield the fitting coefficients b, \(b_2\), a and c. For a more detailed description of the formalism, please refer to the original publication[4] or to a previous publication.[6]

Note, that this algorithm delivered remarkably good results when compared to experimentally obtained critical point data of the alkalis.[4] In addition, this approach was tested on the transition metal tantalum, which also delivered good concordance compared to the literature, see Reference 6.

The measured liquid density of niobium and tungsten are plotted and compared to experimental data given in the literature. The liquid-phase density regressions are tabulated in Table II. The experimentally obtained data points are listed in Tables III (niobium) and IV (tungsten).

The critical point of niobium and tungsten was estimated according to the publication of Schröer and Pottlacher.[4] Starting from the critical temperature \(T_{\mathrm {c}}\) and critical density \(\rho _{\mathrm {c}}\), the phase diagram was estimated as described in Section II−C. Table V gives the parameters needed to plot these phase diagrams in the (\(\rho \),T)-plane. On the way to estimate the phase diagram, a simplified Ising and mean-field approach is used, compare Reference 4. The resulting phase diagrams are also plotted in Figures 3 and 4. In addition, (\(T_{\mathrm {c}},\rho _{\mathrm {c}}\))-pairs were plotted when found in the literature, see Minakov et al.[24] as well as Hess and Schneidenbach.[25]