Co-Ni alloys are widely applied in cemented carbides as the binder phase and in the superalloys [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]]. Both the cemented carbides and superalloys mostly serve at the high temperatures, being a challenge to the performance and the service life of materials. For these applications, one of the most important thermophysical properties of these materials is thermal conductivity. It is a key input parameter to simulate the temperature field by finite element method [29], which is useful to understand the temperature distribution during service and finding the most vulnerable over-heated parts.

Moreover, thermal conductivity plays an important role in electronic equipment. Especially with the coming of 5 G era, the power consumption of 5 G mobile phones is more than doubled in comparison with that of 4 G mobile phones [30], which requires a higher thermal conductivity for the alloys utilized in mobile phones. It is well established that thermal conductivity is heavily dependent on temperature and composition [31]. Thus, modelling the temperature and composition dependence of thermal conductivity is necessary to computational design of materials with desired thermal conductivity.

Thermal conduction in metals and alloys is mainly due to the electrons [32]. Wiedemann-Franz [33] found that the product of the electric resistivity and thermal conductivity is the same constant value for each alloy at given temperature. Thus, modelling electric resistivity is one solution to model thermal conductivity. Nordheim [34] firstly developed the electric resistivity model, followed by Mott [35]. Several studies [[36], [37], [38], [39]] modelling thermal conductivity were based on the theories of Nordheim [34] and Wiedemann-Franz [33], which were actually not applicable to the transition metals [34]. Recently, Wei et al. [40] updated Mott’s two-band model [35] for electric/thermal conductivity which was applied to several binary alloys including transition metals. The above models can be applied to evaluating the composition-dependent electric resistivity and thus the thermal conductivity. As for the temperature dependence of thermal conductivity, Zhang and Andersson [41] combined the theories of Tsiovkin et al. [42] and Wiedemann-Franz [33] to evaluate thermal conductivity only for binary actinide alloys.

The CALPHAD (CALculation of PHAse Diagrams) method [43] can be utilized to evaluate thermophysical properties as a function of composition and temperature, relying on key experimental data and physically sound models. Recently, Zhang et al. [44] and Huang et al. [45] have developed a CALPHAD-type method for evaluating thermal conductivity, but this model does not have a strong physical basis and cannot be used to predict thermal conductivities over the whole investigated composition range, in particular in metastable composition range. To specify this deficiency, Fig. 1 is plotted to show the predicted thermal conductivity of Al-Zn system at 298, 398, 448 and 498 K along the whole composition range by Huang’s model [45]. It can be seen that there are unreasonable negative values at each temperature in the middle of the composition range. To the best of our knowledge, so far, no general physical model was developed to describe both the composition- and temperature-dependent thermal conductivity for binary and multi-component alloys. In the present work, a new physical model describing thermal conductivity dependent on temperature and composition is to be developed and incorporated into the CALPHAD framework.

Thermal conductivities of pure Co [[46], [47], [48], [49], [50], [51], [52], [53], [54]] and pure Ni [[54], [55], [56], [57], [58], [59]] have been investigated extensively. Although thermal conductivity of the Co-Ni alloys has also been studied previously [32,60,61], those measured values are controversial. Moreover, no model is available to describe the concentration and temperature dependence of thermal conductivity for the Co-Ni alloys with wide composition ranges. The major focus of the present work is to develop a model using a minimum number of model parameters, which can reproduce experimental values and predict thermal conductivity of the Co-Ni alloys over wide composition and temperature ranges. Another goal of the present work is to measure the composition- and temperature-dependence of thermal conductivity for Co-Ni alloys experimentally, which are used to test the reliability of the new model. To further verify the applicability of the present model, the available thermal conductivity data for the binary Al-Zn, Mg-Zn and U-Zr systems are compared with the presently model-predicted values.

Co buttons (99.99 wt.%) and Ni buttons (99.99 wt.%) were utilized as raw materials. Pure Co, pure Ni and four Co-Ni alloys with 20, 40, 60 and 80 wt. % Ni were prepared in an arc melting furnace (WKDHL-1, Optoelectronics Co., Ltd., Beijing, China) under an Ar atmosphere using a non-reactive tungsten electrode. All the samples were melted four times in order to achieve good homogeneity. The weight loss of each alloy after melting was less than 0.4 wt.% and thus no composition analysis was performed. Subsequently, the samples were cut into cylinders of about 10 mm of diameter and 2∼3 mm of thickness (height) for the thermal conductivity measurements. The samples were then ground by sandpapers and sealed into quartz cubes under vacuum for solid-solution treatment at 1373 K for 168 h, followed by water quenching. After annealing, the samples were polished for the following measurements.

Phase identification of the four alloys was conducted by X-ray diffraction (XRD, Bruker D8, Advanced A25, Germany). The density of the alloys was measured at 300 K by dividing the measured mass with the measured volume of the sample. The density values for pure Co and pure Ni were re-calculated to other temperatures using known heat expansion data [62]. The density values of alloys were re-calculated to other temperature using average heat expansion data supposing zero excess molar volume of the Co-Ni alloy. Thermal diffusivity and heat capacity for all the samples were measured at 300, 600, 900 and 1100 K in an Ar atmosphere by laser flash method (Netzsh LFA 457 laser conductometer, ASTM E1461 Standard, Germany). The above mentioned Co-Ni alloys at the specified temperatures and compositions are in solid state, according to the phase diagram [63]. Before the measurement, the samples were painted by a carbon powder for efficiently absorbing the laser pulse. Thermal diffusivity a (mm²/s) was obtained as [64]:

where l (mm) is the thickness of the specimen, and t_1/2 (s) is the half-rise time, which is defined as the interval required for the rear surface temperature of the sample to reach 50 % of the maximum temperature increase. This latter parameter can be obtained according to the temperatures recorded every 3 min. Heat capacity of the sample can be obtained by the following equation [64]:

where C_p (J/gK) is the heat capacity of the sample with the mass of m (g) and measured temperature rise ΔT (K) while C_p,ref (J/gK) is the heat capacity of the reference sample with the mass of m_ref (g) and measured temperature rise ΔT_ref (K). Thermal conductivity (k, W/mK) can be obtained by substituting the density (ρ, g/cm³), thermal diffusivity (a, mm²/s) and heat capacity (C_p, J/gK) into the following equation [64]:

In order to ensure the accuracy, four different points are chosen along the surface of each sample for measuring the thermal diffusivity and the heat capacity, and the average values are used for the calculation of thermal conductivity.

The thermal conductivities of pure Co and Ni are described as a function of temperature using the following equation [45]:

where k_X,α⁰ (W/mK) is the thermal conductivity of the pure element X in the α state, T (K) is the temperature, and A, B and C are the parameters to be fitted. Using the measured literature data for pure Co [47,49,54] and Ni [[54], [55], [56], [57]], the parameters in Eq. (4) were obtained (see Table 1). As one can see, different equations are applied in this work not only to different phases (fcc and hcp), but also to different magnetic states (ferromagnetic and paramagnetic). Data are also added to Table 1 for other pure metals (Al, Mg, Zn, U, Zr) studied in this paper.

As electric resistivity of metals is much easier to measure compared to their thermal conductivity, the first model was developed for electric resistivity of metallic solid solutions. The first equation was due to Nordheim [34]:

where ρ_α (Ωm) is the electric resistivity of solid solution α, C_α (Ωm) is the coefficient (of positive sign) for the given alloy of state α in the A-B system, x_A (dimensionless) is the mole fraction of component A in alloy, x_B (dimensionless) is the same for component B. The Nordheim equation was quite successful in the “symmetrical” systems, composed of two pure components that are chemically similar and the concentration of free conducting electrons in the alloy is not dependent on alloy composition. Eq. (5) was later modified by Mott [35] taking into account two types of conducting electrons with their concentrations being functions of the alloy composition. The theory of Mott is quite complex, but it can be boiled down to the following simple equation:

where C_0,α and C_1,α (Ωm) are semi-empirical coefficients dependent on the given material, which are related to the electron transfer. Eqs. (5) and (6) can be unified as:

where C_j,α (Ωm) are semi-empirical coefficients dependent on the given material. It should be noted, however, that all Eqs. (5), (6), (7) do not obey the obvious boundary condition, as they lead to zero electric resistivity for pure metals. This is probably the case because those values are close to negligible compared to the values for alloys. In order to describe the electric resistivities of binary alloy in the whole composition range from pure A to pure B, the present authors suggest that the electric resistivities of the two pure metals should be added to Eq. (7) as:

where ρ_A,α^o (Ωm) and ρ_B,α^o (Ωm) are electric resistivities of pure A and B components in state α, respectively, at the same temperature. Eq. (8) is a reasonable equation for the electric resistivity of solid and liquid binary metallic solid solutions. The last term of Eq. (8) resembles the Redlich-Kister polynomial [65], frequently used to describe the concentration dependences of the excess Gibbs energy of solutions in CALPHAD modelling.

Combining Eq. (8) with the Wiedemann-Franz law [33], which claims that the product of the electric resistivity and thermal conductivity is the same constant value for each alloy at given temperature, the following equation is obtained for thermal conductivity of binary solid or liquid metallic solutions:

where r_j,A-B,α (mK/W) is the jth order heat resistivity interaction parameter for phase α between components A-B and it can have only positive values. Eq. (9) is a new equation for thermal conductivity of binary solid and liquid solutions (but see also [36] for a similar equation). Eq. (9) leads to reasonable boundary conditions for pure components. This equation can be extended to ternary solutions in a usual CALPHAD way as:

where r_j,B-C,α, r_j,A-C,α and r_j,A-B-C,α (mK/W) are the heat resistivity interaction parameters in B-C, A-C and A-B-C solutions, respectively.

As follows from Eq. (9), the thermal conductivity of any solid solution in the Co-Ni binary alloy can be modelled using the following equation:

where k_α (W/mK) is the thermal conductivity of the solid solution α in the given magnetic state, x_Co (dimensionless) is the mole fraction of Co in the same solid solution, x_Ni (dimensionless) is the same for Ni, k_Co,α^o (W/mK) is the thermal conductivity of pure Co in the same state and same temperature, k_Ni,α^o (W/mK) is the thermal conductivity of pure Ni in the same state and same temperature, r_j,α (mK/W) is the jth order heat resistivity interaction parameter in the Co-Ni alloy in the same state α. The practical application of Eq. (11) requires the values for the thermal conductivities of both pure components in the same state in which the given solid solution is modelled. However, as follows from Table 2, thermal conductivities of pure Ni in hcp-ferromagnetic state and that of pure Co in fcc-paramagnetic state are missing. The only state for which there are thermal conductivity values for both pure components is the fcc-ferromagnetic state, but even for this state there is no overlapping temperature range for the two pure metals, as follows from Table 2. For this state, the temperature dependence of thermal conductivities of the two pure metals are shown in a joint Fig. 2. The major message of Fig. 2 is that the thermal conductivity values of pure Co and pure Ni in the same state can be described by a single equation. We presume that it is due to the fact that Co and Ni are neighbors in the Periodic Table of the elements and many of their properties are very similar. Therefore, the following simplification will be used in this paper:$k_{Co,\alpha }^{o}\cong k_{Ni,\alpha }^{o}\equiv k_{\alpha }^{o}$. Therefore, Eq. (11) is simplified as follows:

Eq. (12) will be used for each 3 states in this paper, using the measured k_α^o for Co or for Ni upon convenience at the same temperature and in the same state (see Table 2). The task in the next section is to assess the values of r_j,α for the 3 different states as the function of temperature and composition. For this purpose, parameters r_α (which is defined as the summation term of the last term of Eq. (12)) are first calculated for the alloys by the following equation, expressed from Eq. (12) as:

Next, the concentration and temperature dependence of r_α is described. The concentration dependence is already involved in Eq. (12). The temperature dependence can be estimated for each parameter r_j,α as:

where r_j,α,0 (mK/W) is the value of r_j,α extrapolated to T = 0 K, τ_j,α (K) is a temperature-parameter expected to be within 1000 … 5000 K (average of 3000 K). The form of Eq. (14) ensures that the interaction parameters of heat resistivity weaken with temperature. This is expected as molar volumes of all solid solutions increase with temperature and so the atomic distances in all solid solutions increase with temperature. As a result, any interaction between the components weaken with the increase of temperature (see also [66]). In more simple cases the following linear equation can be also used:

where a_j,α,0 (mK/W) is the value of r_j,α extrapolated to T = 0 K and b_j,α,0 (m/W) is the slope of r_j,α (mK/W) when temperature increases.

The phase identifications by XRD of all the alloys prepared in the present work are shown in Fig. 3. As indicated in the diagrams, the samples with 40-80 wt.% of Ni contain only the fcc_A1 phase, while the sample with 20 w% Ni shows only the hcp_A3 phase. This is in accordance with the binary Co-Ni phase diagram [63] shown in Fig. 3(c). It should be mentioned that the heat capacities can be calculated by using our established thermodynamic database [67], which are listed in Table 2 and compared with the experimental ones. Good agreements between calculation and experiment are obtained. The crystalline and magnetic states of all the 6 samples at the 4 temperatures are given in the last column of Table 2, in accordance with Ref. [63].

Table 2 shows the experimental results. Altogether 24 measured thermal conductivity values are given in Table 2 for the 2 pure metals and for the 4 alloys at 4 different temperatures.

4.2.1. Assessment of parameters for pure Co and Ni

In Fig. 4, Fig. 5 the experimental points from the literature, our equations from Table 1 and our measured values from Table 2 are shown together for pure Ni (Fig. 4) and for pure Co (Fig. 5). As follows from Fig. 4, Fig. 5, all the presently measured data points are close enough to measured literature data and also to our equations of Table 1. For pure Ni the deviations of our 4 measured data points from the equations of Table 1 are (in order of increasing temperature): -3.7 % / +0.6 % / +4.9 % / -9.8 %. For pure Co the deviations of our 4 measured data points from the equations of Table 1 are (in order of increasing temperature): + 4.0 % / +0.9 % / +7.2 % / -7.6 %. Thus, the largest deviation in thermal conductivity values for pure metals is found below ±10 % between our measured values and the average of the reliable literature values. Thus, we presume that the uncertainty of measured values for Co-Ni alloys is also within ±10 %.

4.2.2. Modelling strategy for Co-Ni solid solutions

There are 8 measured data points for the two pure metals in Table 2, but to reproduce them we need 12 parameters within equations of Table 1. Although those 12 parameters reproduce much more experimental points (see Fig. 4, Fig. 5), but from the point of view of this paper it does not look good using 12 model parameters to reproduce 8 measured values. Therefore, in this paper the equations of Table 1 will not be used during the assessment of the concentration-dependent parameters. Instead, during the assessment of the parameters the 8 measured values of Table 2 will be used for the two pure metals and during modelling we will only deal with the concentration and temperature dependence of thermal conductivity of the 16 measured alloys. However, when the assessment of the parameters is done, then in further practical calculations the parameters of Table 1 can be used together with the newly assessed parameters.

During the assessment our task is to describe their thermal conductivity with a minimum possible number of parameters with as little uncertainty, as possible. Above we have seen that the uncertainty of experiments is less than ± 10 %. On the other hand, the uncertainty of simple models is usually also around ± 10 %. Combining these two independent uncertainties we can conclude that for at least about 75 % of data points (i.e. for at least 12 samples out of 16) the model is expected to reproduce the measured values within ± 10 %, while for maximum about 25 % of data points (i.e. for maximum of 4 samples out of the 16 samples) the model is expected to reproduce the data points within ± (10 … 20) %.

Any model for a solid solution works best, if the two pure elements and the solid solution itself are all in the same state (i.e. in the same phase and in the same magnetic state). From the last column of Table 2 the following conclusions can be drawn:

(i) alloy (at 300 K and at 20 wt.% Ni) is in the hcp-ferromagnetic state,

(ii) alloy (at 600 K and at 20 wt.% of Ni) is in a two-phase region = mixture of hcp-ferromagnetic and fcc-ferromagnetic states,

(iii) alloys are in fcc-ferromagnetic state,

(iv) alloys (at high temperatures and high Ni-content) are in fcc-paramagnetic state.

As follows, instead of a single model as planned, we should develop 4 models for the 4 different regions / states. As there are 3 types of one-phase alloys in Table 2, a minimum of 3 model parameters are needed (at least one model parameter per alloy type). Our goal is not to increase the actual number of model parameters too much above 3.

4.2.3. Assessment of a parameter for the hcp-ferromagnetic state in the Co-Ni system

As shown in Table 2, only one alloy at T =300 K and 20 wt.% Ni is in the hcp-ferromagnetic state, so “assessment” here is a very simple “point-fitting”: r_0,hcp-f≅ 0.022 mK/W is found by using Eq. (13). As there is only a single data point in this state, there is no chance to study the concentration or temperature dependence of this parameter.

4.2.4. Assessment of a parameter for the fcc-paramagnetic state in the Co-Ni system

As follows from Table 2, there are 3 alloy samples at two different temperatures (900 and 1100 K) and at two different Ni-contents (60 and 80 wt. %) in this state, so modeling is somewhat more complex than that in the case above. Nevertheless, only a single parameter is found sufficient for this case (r_0,fcc-p≅ 0.037 mK/W), ensuring less than 10 % deviation for the 3 measured points. These 3 measured points do not allow a more detailed analysis on the concentration and temperature dependence of parameter r_fcc-p.

4.2.5. Assessment of parameters for the fcc-ferromagnetic state in the Co-Ni system

As follows from Table 2, there are 11 alloy samples distributed within all four studied temperatures and all four studied compositions in this state, so more complex modeling is possible in this case. The temperature dependence of the calculated r_fcc-f parameters is shown in Fig. 6. The line calculated by Eq. (14) is not convincing (it has very small R²), although the deduced parameter of τ_0,α =3125 K has a reasonable value (see after Eq. (14)). The average value from the 11 points is r_0,fcc-f≅ 0.013. Using this only parameter, all the 11 measured thermal conductivity data are reproduced within ±10 rel %, except for two points, which differ by +12.9 % and -10.9 % from their measured values. This result can be considered satisfactory compared with the requirements of section 4.2.2 above.

Once we concluded from Fig. 6 that the parameters for the fcc-ferromagnetic state of Co-Ni alloys are not really T-dependent, let us show their dependence on concentration (neglecting their T-dependence) in Fig. 7 as function of $\left( 1-2 \bullet {{x}_{Ni}} \right) $, as dictated by Eq. (12). The parameters that follow from Fig. 7 are: r_0,fcc-f≅0.013 mK/W and r_1,fcc-f≅-0.0055 mK/W. Substituting these parameters into Eq. (12) together with the values from Table 2, ten out of the 11 measured values for the 11 alloys are reproduced within ±10 rel %, and only one alloy is left with 12.2 % of deviation. This is an improvement compared to the above case with a single parameter.

4.2.6. Summary of the results for one-phase fields in the Co-Ni system

The model parameters used to describe the 15 experimental points in 3 different states for one-phase alloys are given in Table 3. One can see that 4 parameters are used to describe the concentration dependence of thermal conductivity in these 3 different states (which requires the minimum of 3 parameters). These 4 parameters combined with Eq. (12) and the equations of Table 1 are believed to describe reasonably well the thermal conductivity of any solid Co-Ni alloys at any temperature in the three 1-phase solid fields below the solidus line.

To demonstrate how Eq. (12) and parameters of Table 3 work, in Fig. 8 the experimental vs calculated thermal conductivity values are compared. In Fig. 9 the concentration dependence of thermal conductivity for the Co-Ni alloys is shown at the four measured temperatures. As one can see the calculated lines reproduce the measured values satisfactorily.

4.2.7. Estimation of the thermal conductivity for the Co-Ni alloy in the two-phase region

As follows from Table 2, at 600 K and at 20 wt. % Ni content the Co-Ni alloy is in a 2-phase field, being a mixture of hcp-ferromagnetic and fcc-ferromagnetic phases. Calculation by Eq. (12) and parameters of Table 1, Table 3 provides thermal conductivity values for these two phases extrapolated to the given composition as: k_hcp-f≅ 49.7 W/mK and k_fcc-f≅ 57.0 W/mK. As follows from the phase diagram [63] shown in Fig. 3(c), this alloy consists of approximately 50 % of hcp and 50 % of fcc phases. Thus, as the first approximation the average of the above two thermal conductivity values can be taken as: k_hcp+fcc≅ 53.4 W/mK. Compared to the experimental value of 50.8 W/mK (see Table 2), the model over-estimates the measured value only by 5.1 %. This small discrepancy is due to an effect of the interface on the thermal conductivity. It is generally believed that the interface decreases the thermal conductivity. So, using our models for the 1-phase fields, even the thermal conductivity of the 2-phase field can be reasonably obtained without any additional parameters if the effect of interface is not significant. Let us note, however, that modelling 2-phase fields is not always that simple (see [45]).

In order to further study the composition- and temperature-dependent thermal conductivity of Co-Ni alloys, three-dimensional planes were obtained by Eq. (12) and were plotted in Fig. 10. The corresponding phases were also shown in the x-y plane. One can clearly see the variation tendency of thermal conductivity along the temperature or composition.

4.2.8. Application of the new model to Al-Zn, Mg-Zn and U-Zr alloys

To verify the applicability of the present model to other systems, two systems of light alloys (Al-Zn and Mg-Zn) and one of actinide alloys (U-Zr) were selected in the present work. The thermal conductivities of pure fcc-Al, hcp Mg and hcp Zn were directly taken from Ref. [45] while the ones of pure bcc U and bcc Zr were assessed by using the experimental data from Ref [68] (see Table 1). The evaluation process was similar as that of Co-Ni system.

Thermal conductivities of Al-Zn alloys at 298, 398, 448 and 498 K calculated by the present model were also plotted in Fig. 1. It can be concluded that the present model is reasonable when extrapolated to the whole investigated composition and temperature ranges, avoiding the drawback of the previous model [45]. The model parameters are shown in Table 4 and the results of calculated thermal conductivities for U-Zr and hcp Mg-Zn are presented in Fig. 11 along with the experimental data from Huang et al. [45] and Touloukian et al. [68]. As already shown in Fig. 1, the presently developed model can describe the thermal conductivity over the whole composition range in the Al-Zn system satisfactorily. This is not the case for the previous model [45], which leads to negative thermal conductivities in the composition range of 31-81 at. % Zn.

As a summary, the new model was successfully utilized to describe the thermal conductivities over the wide composition and temperature ranges in the Co-Ni, Al-Zn, Mg-Zn and U-Zr systems. The present model is equally valid for describing the thermal conductivity of other solution phases, and it can be applied to ternary and even multicomponent systems in the form of Eq. (10).

(1) A new model was developed in the present work to describe the concentration and temperature dependence of thermal conductivity and was successfully applied to several binary solid solutions, including Co-Ni, Al-Zn, Mg-Zn and U-Zr.

(2) For binary Co-Ni solid solutions, the model reproduced the presently measured 15 thermal conductivity values in 3 one-phase fields for the Co-Ni alloys with relative uncertainty below 10 % except one case, where the disagreement is 12.2 %. Even the measured data point for the two-phase field was estimated within 5 % of error without applying further model parameters.

(3) Three-dimensional planes for thermal conductivity of Co-Ni alloys were obtained according to the presently developed model. It can be concluded that the present model can be utilized to evaluate the thermal conductivity over the whole investigated composition and temperature ranges for the first time.

(4) The reliability of the present model was further verified by applying it to the measured thermal conductivities of fcc Al-Zn, hcp Mg-Zn and bcc U-Zr alloys in literature. It is highly expected that this model can be extended to ternary and even multicomponent systems coupled with CALPHAD method.

The financial support from National Natural Science Foundation of China (Grant No. 51671219) is greatly acknowledged. Prof. George Kaptay thanks the distinguished guest Professor Program from July 14 to August 14, 2019 released by Central South University of China. His work was also supported by nano-Ginop ProjectGINOP-2.3.2-15-2016-00027 in the framework of the Szechenyi 2020 program, supported by the European Union.