Materials Transactions, Vol. 45, No. 10 (004) pp. 987 to 993 #004 The Japan Institute of Metals Measurement of the Densities of Nickel-Based Ternary, Quaternary and Commercial Alloys Kusuhiro Mukai 1, Zushu Li and Liang Fang 1; * 1 Department of Materials Science and Engineering, Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan Department of Materials, Imperial College London, Exhibition Road, London SW7 AZ, UK The modified sessile drop method () and the modified pycnometric method () have been employed to measure precisely the densities of liquid nickel-based ternary (Ni-Co-Al) and quaternary (Ni-Co-Al-Cr, Ni-Co-Al-Mo, Ni-Cr-Al-Mo) alloys and four commercial superalloys in liquid and mushy states. There was a good agreement between the values measured by the two methods. The measured densities of liquid model alloys and commercial superalloys in liquid and mushy states decreased linearly with increasing temperature for the experimental temperature range. The temperature coefficient of the density of liquid ternary Ni-Co-Al alloys measured in this work can be represented as a quadratic function of the aluminium concentration in the alloys. The density of liquid Ni-Co-Al alloys can be expressed as a function of both temperature and aluminium concentration in the alloys. The recommended equations for the densities of liquid nickel-based ternary and quaternary alloys and four commercial superalloys in liquid and mushy states were obtained by analysing density values measured by both and. The calculated values from the recommended density equations show good agreement with those measured by both and. (Received June 17, 004; Accepted August 5, 004) Keywords: nickel-based alloy, density, measurement, modified sessile drop method, modified pycnometric method 1. Introduction The density () and its temperature dependency (d=dt) are important input data for simulation of the solidification of Ni-based superalloys, e.g. the prediction of defects such as gas porosity and microsegregation, as well as the evaluation of the convective contribution to heat transport. Accurate experimental values of the densities of nickel and Ni-based model and commercial alloys (including their dependencies on composition and temperature) are needed to develop models to predict the densities of Ni-based superalloys. In order to predict the densities of Ni-based commercial alloys, it is necessary to have an understanding of the effects of the various alloy elements (such as Cr, Co, Ta, W, Al, Mo, Re, Hf, Nb, Ti, etc.) on the densities of liquid nickel alloys. This can be obtained from accurate measurements of the densities of Ni-based binary alloys in relation to the concentration and temperature. Comparisons of the densities of liquid Ni-based ternary alloys with those for the corresponding binary alloys provide useful information on the effects of the second alloy element on the properties of liquid alloys. For some alloy elements, such as Re, their concentrations in the Ni-based superalloys are relatively low but they are very reactive, which may make the accurate density measurement of their Ni-based binary alloys difficult. The effect of these elements on the densities of Ni-based superalloys can be obtained by comparing the densities of liquid Ni-based commercial alloys with those of liquid Nibased multi-component model alloys with similar chemical compositions to those of the Ni-based commercial alloys. Furthermore, accurate density values of both Ni-based model and commercial alloys can be used to verify the validity of the models developed to predict the densities of liquid Nibased superalloys. *Present address: Department of Applied Physics, Chongqing University, Chongqing 400044, P.R. China. However, there is a large scatter in the reported density values for liquid nickel and its temperature dependence because of the experimental uncertainties associated with the different techniques. Furthermore, there are very few reports on the measurement of the density of Ni-based alloys in relation to the effects of composition and temperature. 1 7) There are no reports on the measurement of the densities of liquid Ni-based multi-component model alloys such as ternary and quaternary alloys. Therefore, in order to develop a model to predict the densities of commercial Ni-based superalloys, we have used two accurate methods, namely, the modified sessile drop method () and the modified pycnometric method () to measure the densities of liquid nickel and Ni-based model alloys and commercial alloys in the liquid state and in the mushy (ie liquid-solid coexistence ) state for some alloys. The principles, advantages and experimental procedures of the two methods have been described previously. 5 7) The densities of liquid nickel and liquid Ni-Cr, Ni-Co, Ni-Ta, Ni-W and Ni-Al alloys have been measured precisely with both methods, and density values for Ni-Cr and Ni-Al alloys in mushy state have also been determined. 5 7) In this paper, both and were employed to measure the densities of liquid ternary model alloys (Ni-Co- Al, molar fraction x Ni :x Co 86:14, Al mass% = 011), liquid quaternary model alloys (Ni-Co-Al-Cr, Ni-Co-Al-Mo and Ni-Cr-Al-Mo), and four commercial Ni-based superalloys in liquid and mushy states. The chemical compositions of these model alloys were similar to those of commercial Nibased superalloys. The recommended density values for these alloys have been obtained by combining the measured values determined with the two methods.. Experimental Details of the experimental methods of and can be found in the previous papers. 5 7)
988 K. Mukai, Z. Li and L. Fang The samples of Ni-based ternary and quaternary model alloys were composed of high purity Ni-X (0 mass%, X ¼ Co, Al, Cr, Mo) master alloy discs, and the further nickel was added from high purity nickel disc. The samples of commercial alloys were cut from the ingots of the commercial Ni-based superalloys. The chemical compositions of the alloy elements in the alloys were analysed by radio-frequency, inductively-coupled plasma (I.C.P.) emission spectrometry. Oxygen concentration in the alloys was determined using infrared adsorption and thermal conductimetric methods after fusion in a He gas flow. Sulphur concentration in the alloys was measured by the infrared adsorption method after combustion in an induction furnace. The chemical compositions of the post-measurement samples for ternary alloys are shown in Table 1, for quaternary alloys in Table and for commercial alloys in Table 3. The molar ratio of nickel to cobalt (x Ni :x Co ) in the ternary Ni-Co-Al alloys was fixed as 86:14 which is close to the average value for commercial superalloys investigated in this work. 3. Density of Liquid Ni-Co-Al Alloy The densities of liquid Ni-Co-Al alloys for Al concentrations below 11 mass% are given in Fig. 1 and Table 4. The density of liquid Ni-Co-Al alloy (x Ni :x Co = 86:14 and Table 1 Chemical compositions of Ni-Co-Al alloy sample after the experiments (molar ratio: x Ni :x Co 86:14) (mass%). 1 3 4 5 6 Ni 83.99 84.19 80.00 81.08 79 76.6 Co 13.73 13.76 1.60 11.76 11.66 13.38 Al.8.05 0 7.16 10.45 10.00 Method Table Chemical compositions of Ni-based quaternary alloy samples after the experiments (mass%). Alloy Method Ni Co Al Cr Mo Ni-Co-Al-Cr Ni-Co-Al-Mo Ni-Cr-Al-Mo 75.36 11.78 6 5.81 - - 75.55 11.76 6.81 5.88 - - 75.73 1.0 7.11 - - 5.14 75.58 1.10 6 - - 5.5 75.47 - - 6.9 13.18 5.06 75.69 - - 6.18 13.1 5.0 Table 3 Chemical compositions of commercial alloy samples after the experiments (mass%). Alloy Ni Cr Mo Al Co W Ta Ti INCO713C 74.6 1.50 4.0 6.10 - - - - - - 0.80 TMS75 60.00 3.00.00 6.00 1.00 6.00 6.00 - - CM47LC 61.77 8.10 0.50 5.60 9.0 9.50 3.0 0.70 CMSX-4 60.65 6.40 0.61 5.67 9.60 6.40 6.60 1.04 (continued) Alloy Re Hf Nb Fe B Zr C INCO713C - - 1.40.00 - - 0.01 0.01 0.1 TMS75 5.00 0.1 - - - - - - - - - - CM47LC - - 1.40 - - - - 0.015 0.015 - - CMSX-4.90 0.10 - - 0.03 - - - - - - : Ni85.95-Co14.05 Mukai et al. 7) Ni81.89-Co13.8-Al.8 Ni80.00-Co1.60-Al0 Ni79-Co11.66-Al10.45 Ni85.95-Co14.05 Mukai et al. 7) Ni81.43-Co13.8-Al.05 Ni81.08-Co11.76-Al7.16 Ni76.6-Co13.38-Al10.00 1700 1750 1800 1850 1900 Fig. 1 The densities of liquid Ni-Co-Al alloys measured by and as a function of temperature. mass%al ¼ 0) as a function of temperature was obtained from the results for liquid Ni-Co alloys determined previously. 7) The only reported liquidus temperatures of Ni-Co-Al alloys are those due to Kudoh 8) for Ni-x mass% Co-7 mass% Al (x ¼ 5, 10, 15) alloys using differential thermal analysis. The measured liquidus temperatures are 1697 K, 1693 K and 1703 K for these alloys. In this work, the liquidus temperatures of Ni-Co-Al alloys were estimated as follows: (i) The liquid Ni-Co-Al alloy was considered as a (Ni- Co)-Al pseudo-binary alloy since the Ni-Co alloy is close to ideal. (ii) The effect of Co content on the liquidus temperatures of Ni-Co alloys was estimated from the Ni-Co binary phase diagram 9) by taking out Al from Ni-Co-Al alloy. The effect is expressed as T Co ; (iii) The effect of Al content on the liquidus temperatures of (Ni-Co)-Al pseudo-binary alloys was estimated from the Ni-Al binary phase diagram. 10) The effect is expressed as T Al. (iv) Therefore, the liquidus temperatures of Ni-Co-Al alloys can be estimated as ðt Ni þ T Co ÞþT Al, where T Ni is the melting point of nickel, 178 K. Using this estimation method, the liquid temperatures for Nix mass% Co-7 mass% Al (x ¼ 5, 10, 15) alloys were estimated to be 1700 K, 170 K and 1704 K. These are in good agreement with those determined by Kudoh. 8) The liquidus temperatures calculated for Ni-Co-Al alloys (x Ni :x Co ¼ 86:14, Al mass% = 0,.,, 10.0) were found to be 1733 K, 179 K, 1704 K and 1695 K (shown in Table 4). The densities of liquid Ni-Co-Al alloys were found to decrease linearly with increasing temperature for the experimental temperature range. The volume thermal expansion coefficient,, is in the range of 1:06 10 4 1:74 10 4 K 1 for various Al concentrations. The temperature coefficients for the densities of liquid Ni-Co-Al alloys, k ¼ d=dt, were found to change with aluminium concen-
Measurement of the Densities of Nickel-Based Ternary, Quaternary and Commercial Alloys 989 Table 4 Densities of liquid Ni-Co-Al alloys as a function of temperature at different Al concentrations. Method T/K C Al (mass%) =Mgm 3 /10 4 K 1 1733 1833 0 ¼ 7:80 1:33 10 3 ðt 1733Þ 1.71 1733 1834.8 ¼ 7:5 1:03 10 3 ðt 179Þ 1.37 1713 1834 0 ¼ 7:07 7:90 10 4 ðt 1704Þ 1.1 1704 1833 10.45 ¼ 6:81 7:8 10 4 ðt 1695Þ 1.15 1733 1833 0 ¼ 7:80 1:36 10 3 ðt 1733Þ 1.74 1733 1893.05 ¼ 7:45 9:6 10 4 ðt 179Þ 1.4 1713 1893 7.16 ¼ 7:0 7:40 10 4 ðt 1704Þ 1.05 1703 1893 10.00 ¼ 6:77 7:5 10 4 ðt 1695Þ 1.07 C Al : Al concentration in the alloy, mass% Temperature Coefficient, k, 10 4 / Mg m -3 K -1-4 -6-8 -10-1 -14 0 4 6 8 10 Aluminum Concentration, C Al (mass%) : 1733K 1833K 1733K 1833K 0 4 6 8 10 Al concentration, C Al (mass%) Fig. The temperature coefficients for the densities of liquid Ni-Co-Al alloy as a function of Al concentration. Fig. 3 The densities of liquid Ni-Co-Al alloys as a function of Al concentration. tration as shown in Fig.. The relationship between temperature coefficient and aluminium concentration in the alloys was obtained using a least-squares method as follows: : Table 5 Densities of liquid Ni-Co-Al alloys as a function of Al concentration at different temperatures. Method T/K /Mgm 3 1733 ¼ 7:79 1:18 10 1 C Al þ :11 10 3 CAl kð10 3 Mgm 3 K 1 Þ¼ 1:3 þ 0:13C Al 7:80 10 3 C Al ð1þ 1753 ¼ 7:76 1:16 10 1 C Al þ 1:96 10 3 C Al 1773 ¼ 7:74 1:13 10 1 C Al þ 1:80 10 3 C Al 1793 ¼ 7:71 1:10 10 1 C Al þ 1:65 10 3 C Al kð10 3 Mgm 3 K 1 Þ¼ 1:31 þ 0:17C Al 1:09 10 CAl ðþ The recommended relationship between temperature coefficient and Al concentration (eq. (3)) was obtained by analysing all the obtained temperature coefficients using the least-squares method. : kð10 3 Mgm 3 K 1 Þ¼ 1:31 þ 0:15C Al 9:7 10 3 CAl ð3þ The relationship between the density of liquid Ni-Co-Al alloy and aluminium concentration at different temperatures obtained in this work is shown in Fig. 3 and Table 5. The relationship between the densities of liquid Ni-Co-Al alloys with both temperature and aluminum concentration is expressed by eq. (4) for results and eq. (5) for determinations. : T 1833 K 1813 ¼ 7:68 1:08 10 1 C Al þ 1:49 10 3 CAl 1833 ¼ 7:66 1:05 10 1 C Al þ 1:33 10 3 CAl 1733 ¼ 7:77 1:51 10 1 C Al þ 5:00 10 3 CAl 1753 ¼ 7:75 1:45 10 1 C Al þ 4:5 10 3 CAl 1773 1793 ¼ 7:7 1:39 10 1 C Al þ 4:04 10 3 CAl ¼ 7:70 1:34 10 1 C Al þ 3:56 10 3 CAl 1813 ¼ 7:67 1:8 10 1 C Al þ 3:08 10 3 CAl 1833 ¼ 7:64 1: 10 1 C Al þ :60 10 3 CAl ðmgm 3 Þ¼ð7:79 1:15 10 1 C Al þ :06 10 3 CAl Þ ð1:3 0:13C Al þ 7:80 10 3 CAl Þ 10 3 ðt Þ ð4þ T 1893 K ðmgm 3 Þ¼ð7:77 1:41 10 1 C Al þ 4:31 10 3 CAl Þ ð1:31 0:17C Al þ 1:09 10 CAl Þ 10 3 ðt Þ ð5þ
990 K. Mukai, Z. Li and L. Fang where is the liquidus temperature of the Ni-Co-Al alloy. A least-squares analysis of the density values obtained by the two methods as a function of both temperature and aluminum concentration are carried out to minimize the effects of systematic errors specific to the two methods. Equation (6) is recommended for the densities of liquid Ni- Co-Al alloys (x Ni :x Co 86:14). : T 1833 K ðmgm 3 Þ¼ð7:78 1:9 10 1 C Al þ 3:30 10 3 CAl Þ ð1:31 0:15C Al þ 9:7 10 3 CAl Þ 10 3 ðt Þ ð6þ The difference between the values calculated from eq. (6) and the measured values by either or was found to be less 0:68%. 4. Density of Liquid Quaternary Alloy The densities of liquid Ni-Co-Al-Cr, Ni-Co-Al-Mo and Ni- Cr-Al-Mo alloys were measured by both and. The chemical compositions of these quaternary alloys are close to those of commercial alloys measured in this work. The experimental results are given in Table 6, and are also shown in Fig. 4 (Ni-Co-Al-Cr alloy), Fig. 5 (Ni-Co-Al-Mo alloy) and Fig. 6 (Ni-Cr-Al-Mo alloy). The densities of liquid quaternary alloys investigated in this work were found to decrease with increasing temperature. The density data for these systems obtained by the two methods in this work were found to be in good agreement. Table 6 Densities of liquid Ni-based quaternary alloys measured by and. Alloy T/K Method /Mgm 3 Ni-Co-Al-Cr Ni-Co-Al-Mo Ni-Cr-Al-Mo 1693 1833 ¼ 8:3 7:54 10 4 T 1693 1873 ¼ 8:39 8:11 10 4 T 1693 1833 ¼ 8:67 8:36 10 4 T 1693 1873 ¼ 8:68 8:68 10 4 T 1693 1833 ¼ 8:58 8:78 10 4 T 1693 1873 ¼ 8:49 8:57 10 4 T : Ni75.36-Co11.78-Al6-Cr5.81 Ni75.55-Co11.76-Al6.81-Cr5.88 1650 1700 1750 1800 1850 1900 Fig. 4 The densities of liquid Ni-Co-Al-Cr alloys measured by and. : 1650 1700 1750 1800 1850 1900 Least-squares analysis of the data obtained by the two methods yielded the recommended equations for liquid Ni- Co-Al-Cr (eq. (7)), Ni-Co-Al-Mo (eq. (8)) and Ni-Cr-Al-Mo (eq. (9)) alloys. Ni-Co-Al-Cr, 1693 K T 1833 K: ðmgm 3 Þ¼8:35 7:8 10 4 T ð7þ ðk 1 Þ¼9:34 10 5 Ni-Co-Al-Mo, 1693 K T 1833 K: ðmgm 3 Þ¼8:70 8:65 10 4 T ðk 1 Þ¼9:89 10 5 Ni-Cr-Al-Mo, 1693 K T 1833 K: ðmgm 3 Þ¼8:5 8:58 10 4 T ðk 1 Þ¼1:01 10 4 5. Density of Commercial Alloy Ni75.73-Co1.0-Al7.11-Mo5.14 Ni75.58-Co1.10-Al6-Mo5.5 Fig. 5 The densities of liquid Ni-Co-Al-Mo alloys measured by and. : Ni75.47-Cr13.18-Al6.9-Mo5.06 Ni75.69-Cr13.1-Al6.18-Mo5.0 1650 1700 1750 1800 1850 1900 Fig. 6 The densities of liquid Ni-Cr-Al-Mo alloys measured by and. The densities of commercial alloys INCO713C, CMSX-4, TMS75 and CM47LC were measured by both and. The experimental results are shown in Table 7, and ð8þ ð9þ
Measurement of the Densities of Nickel-Based Ternary, Quaternary and Commercial Alloys 991 Table 7 Densities of commercial superalloys measured by and. Alloy States T/K Methods =Mgm 3 /10 4 K 1 1611 1833 ¼ 7:15 8:30 10 4 ðt 1610Þ 1.16 Liquid 1609 1873 ¼ 7:09 9:77 10 4 ðt 1610Þ 1.38 INCO713C 1571 1611 ¼ 7:33 4:64 10 3 ðt 1571Þ 6.33 Mushy 1571 1609 ¼ 7:4 3:6 10 3 ðt 1571Þ 5.00 CMSX-4 Liquid/ 1513 1833 ¼ 11:41 : 10 3 T Mushy 1603 1873 ¼ 11:30 :0 10 3 T TMS75 CM47LC Liquid Mushy Liquid Mushy 169 1836 ¼ 7:75 4:3 10 4 ðt 1696Þ 0.56 1696 1873 ¼ 7:70 4: 10 4 ðt 1696Þ 0.55 1643 169 ¼ 7:78 3:01 10 4 ðt 1643Þ 0.39 1643 1696 ¼ 7:77 1:1 10 3 ðt 1643Þ 1.44 1645 183 ¼ 7:51 1:70 10 3 ðt 1647Þ.6 1648 1873 ¼ 7:44 1:50 10 3 ðt 1647Þ.0 161 1645 ¼ 7:51 9:81 10 4 ðt 161Þ 1.31 161 1648 ¼ 7:47 1:90 10 3 ðt 161Þ.54 7. 6.8 6.6 1600 1700 1800 1900 Liquid, Liquid, Liquid, Mushy, Mushy, Mushy, Fig. 7 The densities of INCO713C alloys measured by and as a function of temperature. also in Fig. 7 (INCO713C), Fig. 8 (CMSX-4), Fig. 9 (TMS75) and Fig. 10 (CM47LC). The liquidus and solidus temperatures of commercial alloys used below are determined by the present work combining with the values by Kudoh. 8) The density of INCO713C alloy was found to decrease with increasing temperature. The temperature coefficient was found to change at 16091611 K, consequently, this point can be considered to be the liquidus temperature for INCO713C superalloy. There was a good agreement between the density values obtained by the two methods. A leastsquare analysis of the density values obtained with the two methods yielded the recommended equation for the density of liquid INCO713C alloy as eq. (10). Liquid, 16101833 K ðmgm 3 Þ¼7:1 9:04 10 4 ðt 1610Þ ð10þ ðk 1 Þ¼1:7 10 4 Since the measurement accuracy in the mushy state of the alloy is higher for than that for, 6,7) the influence of the systematic error of the two methods on the 8. 7. 1500 1600 1700 1800 1900 Fig. 8 The densities of CMSX-4 alloys measured by and as a function of temperature. measured density values was minimized in the following manner. The recommended density of liquid INCO713C alloys at the liquidus temperature was calculated from the recommended equation (eq. (10)) and then combined with the temperature coefficient of INCO713C alloys obtained by for the mushy state to give the following equation (eq. (11)). The same treatments were also carried out for the densities of TMS75 and CM47LC alloys in mushy states as shown below. Mushy, 15711610 K ðmgm 3 Þ¼7:6 3:6 10 3 ðt 1571Þ ð11þ ðk 1 Þ¼4:97 10 4 The density of CMSX-4 alloy was also found to decrease with increasing temperature in the temperature from 1513 to 1873 K. However, no transition point was detected for the temperature coefficient. There was a good agreement between the results by the two methods. A least-square analysis of the density data by the two methods yielded the recommended equation for the density of liquid CMSX-4 alloy (eq. (1)).
99 K. Mukai, Z. Li and L. Fang 8.4 8. 7. 1600 1700 1800 1900 Liquid, Liquid, Liquid, Mushy, Mushy, Mushy, Fig. 9 The densities of TMS75 alloys measured by and as a function of temperature. The densities of CM47LC alloy in liquid and mushy states were measured by both and in the temperature range from 161 to 1873 K. In the temperature from 161 to 1873 K, the density decreased with increasing temperature. The temperature coefficient of the density was found to change at a temperature between 1645 and 1648 K. The transition point of 1645-1648 K can be considered as the liquidus temperature. A least-square analysis gave the recommended equation for the density of CM47LC alloy. Liquid, 1647183 K ðmgm 3 Þ¼7:47 1:60 10 3 ðt 1647Þ ðk 1 Þ¼:14 10 4 Mushy, 1611647 K ðmgm 3 Þ¼7:54 1:90 10 3 ðt 161Þ ðk 1 Þ¼:54 10 4 ð15þ ð16þ 8. 7. 1600 1700 1800 1900 Liquid, Liquid, Liquid, Mushy, Mushy, Mushy, Fig. 10 The densities of CM47LC alloys measured by and as a function of temperature. ðmgm 3 Þ¼11:36 :1 10 3 T 16031833 K ð1þ The densities of TMS75 alloy in liquid and mushy states were measured by both and for the temperature range of 165 to 1873 K. There was a good agreement between the density values obtained with the two methods. The density was found to decrease with increasing temperature, and the temperature coefficient changed slope around 1691696 K, which can be considered as the liquidus temperature. A least-square analysis of the data obtained by the two methods gave the recommended equations of the densities of TMS75 alloy in liquid and mushy states. Liquid: 16961836 K ðmgm 3 Þ¼7:73 4:7 10 4 ðt 1696Þ ð13þ ðk 1 Þ¼5:5 10 5 Mushy: 16431696 K ðmgm 3 Þ¼7:79 1:1 10 3 ðt 1643Þ ðk 1 Þ¼1:44 10 4 ð14þ 6. Conclusions The modified sessile drop method () and the modified pycnometric method () were employed for precise measurement of the densities of liquid Ni-based ternary (Ni-Co-Al) and quaternary (Ni-Co-Al-Cr, Ni-Co-Al- Mo, Ni-Cr-Al-Mo) alloys and four commercial superalloys in liquid and mushy states. The following results were obtained. (1) There was a good agreement between the values measured by the two methods for the liquid ternary and quaternary alloys and commercial superalloys in liquid and mushy states investigated in this work. () The measured densities of liquid model alloys and commercial alloys in liquid and mushy states were found to decrease linearly with increasing temperature for the experimental temperature range. (3) The temperature coefficient of the density of liquid ternary Ni-Co-Al alloys measured in this work can be expressed as a quadratic function of the aluminum concentration in the alloys. The densities of liquid Ni- Co-Al alloys have been expressed as a function of both temperature and aluminum concentration. (4) The recommended equations for the densities of liquid nickel-based ternary and quaternary alloys and commercial superalloys in liquid and mushy states were obtained by analysing the density values measured with both and. The calculated values obtained from the recommended density equations show good agreement with those measured by the two methods. Acknowledgements This work was performed as a part of Research and Development of Innovative Casting Simulation supported by New Energy and Industrial Technology Development Organization, Japan. The authors would like to sincerely thank Prof. Kenneth C Mills of Imperial College London for his discussion of the manuscript.
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