*Johnson Matthey Technol. Rev.*, 2015, **59**, (3), 174

doi:10.1595/205651315x688091

## 《铂族金属的选定电阻率数值——第一部分：钯与铂》

### 获得钯和铂液相的改进值

该文评估了铂族金属钯和铂的固相和液相的电阻率数值，尤其是得到了这些金属的液相的改进值。以往针对电阻率，且对铂族金属进行评估的研究，包括米登(1)、巴斯(2)、萨维斯基等人(3)、 宾科勒与布鲁恩（4）的研究，以及其它独立研究。

## Selected Electrical Resistivity Values for the Platinum Group of Metals Part I: Palladium and Platinum

### Improved values obtained for liquid phases of palladium and platinum

- John W. Arblaster
- Wombourne, West Midlands, UK Email: jwarblaster@yahoo.co.uk

### Article Synopsis

Electrical resistivity values for both the solid and liquid phases of the platinum group metals (pgms) palladium and platinum are evaluated. In particular improved values are obtained for the liquid phases of these metals. Previous reviews on electrical resistivity which included evaluations for the pgms included those of Meaden (1), Bass (2), Savitskii *et al.* (3) and Binkele and Brunen (4) as well as individual reviews by Matula (5) on palladium and White (6) on platinum.

## 1. Introduction

Electrical resistivity (ρ) is defined in terms of the International System of Units (SI units) as:

ρ = *R A* / *l* (i)

where

*R* is the electrical resistance of a uniform specimen of material in ohms (Ω)

*A* is the cross-sectional area of the specimen in square metres (m^{2})

*l* is the length of the specimen in metres (m)

The units of ρ are therefore Ω m although practically the most useful units are μΩ cm.

The measured electrical resistivity (ρ) usually consists of a temperature dependent intrinsic resistivity, ρ_{i}, which is due to the pure metal and is caused by the scattering of the charge carriers (electrons or holes) by phonons (quantised vibrations of the lattice) and by their collisions with each other, and a residual resistivity (ρ_{0}) due to impurities which also scatter the carriers and increase the resistivity. The quantity ρ_{0} is considered to be a summation of the effects of different impurities and is also considered to be temperature independent. The two contributions to the total resistivity are combined according to Matthiessen's Rule: ρ = ρ_{0} + ρ_{i} and because ρ_{0} may vary from sample to sample then attempts are made to evaluate values of ρ_{i} which should be universal for a specific metal.

## 1.1 Correction for Thermal Expansion Effects

In order to obtain a reference value to which all other measurements are adjusted the electrical resistivity is evaluated at 273.15 K (0ºC).

In the low temperature region below about 30 K the resistivity can be represented by ρ = ρ_{0} + A T^{2} + B T^{5} where the temperature dependent terms represent the intrinsic resistivity, whilst up to room temperature the experimental values are generally given in such a form that interpolation can be achieved by using simple polynomials rather than using the complicated Bloch-Grüneisen formula (7–9). In the definition of resistivity as* *ρ = *R* *A* / *l* then *A *and *l *are usually measured at room temperature and therefore at different temperatures both *A* and *l* have to be corrected for thermal expansion effects. It is found below room temperature that for the level of accuracy given for ρ, thermal expansion corrections are generally negligible but at higher temperature the measurements have to be corrected, especially if they are based entirely on the room temperature values for *A* and *l* which are usually measured at 293.15 K, the accepted reference temperature for length change measurements:

ρ (corrected) = ρ (uncorrected) [(*A _{T}* /

*A*

_{293.15}) × (

*I*

_{293.15}/

*I*

_{T})] (ii)

= ρ (uncorrected) [1 + (*I*_{T} − *I*_{293.15}) / *I*_{293.15}] (iii)

where Equation (iii) can be considered to be a close approximation of Equation (ii). However since 273.15 K is the actual reference temperature then corrected values of ρ(T) should be further corrected for thermal expansion from 293.15 K to 273.15 K. Since this correction is usually negligible at the level of accuracy given then it is not applied.

In the case of rapid pulse heating to high temperatures, because of inertia *l* generally is unaltered and it is *A *that changes. If *D *is the diameter of the wire then:

ρ (T) = ρ (measured) (*D*_{T}^{2} / *D*_{293.15}^{2}) = ρ (measured) (*V*_{T} / *V*_{293.15}) (iv)

where *V*_{T} is the volume of the sample at temperature T and *V*_{293.15} is the volume at 293.15 K. These are essentially *D*_{T}^{2} and *D*_{293.15}^{2} respectively since *l* is assumed to be unaltered.

## 2. Palladium

Palladium has a face-centred cubic structure and the melting point is a secondary fixed point on the International Temperature Scale of 1990 (ITS-90) at 1828.0 ± 0.1 K (10).

## 2.1 Solid

Electrical resistivity values for solid palladium at 273.15 K are given in **Table I**. The selected value is an average of the last three determinations. The ρ_{0} correction to the measurement of Laubitz and Matsumura (14) was suggested by Matula (5) who also appears to have selected this value as the reference value.

##### Table I

Authors | Ref. | ρ_{i}, μΩ cm | Temperature of data |
---|---|---|---|

Powell et al. | 11 | 9.79 | At 273.15 K. Corrected for ρ_{0} 0.144 μΩ m |

Powell et al. | 12 | 9.75 | Interpolated 200 – 400 K. Corrected for ρ_{0} 0.143 μΩ m |

White and Woods | 13 | 9.70 | At 273.15 K. Average of three samples |

Laubitz and Matsumura | 14 | 9.760 | Interpolated 250–300 K. Corrected for ρ_{0} 0.020 μΩ m |

Williams and Weaver | 15 | 9.751 | At 273.15 K. Corrected for ρ_{0} 0.007 μΩ m |

Khellar and Vuillemin | 16 | 9.765 | Calculated. Fit 17–300 K |

Selected | 9.76 ± 0.01 | At 273.15 K |

From 71 data sets for solid palladium Matula (5) selected only the measurements of Schriempf (17) (1.6 K–10.6 K), White and Woods (13) (10 K–295 K) and Laubitz and Matsumura (14) (90 K–1300 K). However it is considered that the values of White and Woods have been superseded by the later high precision measurements of Williams and Weaver (15) (0 K–300 K) and Khellar and Vuillemin (16) (17 K–300 K), with the latter given only in the form of an equation which was evaluated at 17 K and then at 10 K intervals from 20 K to 270 K. The measurements of Williams and Weaver were interpolated above 100 K so as to also obtain a full evaluation at 10 K intervals from 20 K to 270 K. The measurements of Schriempf and of Williams and Weaver agree satisfactorily and were averaged to 10 K with the measurements of Williams and Weaver being extended to 16 K. The measurements of the latter and of Khellar and Vuillemin do not agree below 35 K. However the equation of Khellar and Vuillemin showed peculiar behaviour below this temperature with derived values being 6% higher than those of Williams and Weaver at 17 K but 31% lower at 20 K. Therefore the latter measurements were given preference up to 35 K. At this temperature and above values from the two sets of measurements were averaged. Overall agreement is to within 0.5% between 60 K and 180 K and to within 0.1% above 180 K. The selected values of Matula below 273.15 K are based on a combination of the measurements of White and Woods and of Laubitz and Matsumura and on average the intrinsic values show a bias of 0.02 μΩ cm above the more recently selected values. Other measurements in the low temperature region were discussed by Matula.

In the high temperature region Matula (5) selected only the measurements of Laubitz and Matsumura (14) (90 K–1300 K). After correction for ρ_{0} = 0.020 μΩ cm the values were calculated at 50 K intervals from 350 to 1300 K. In the present evaluation these measurements were combined with the more recent measurements of Khellaf *et al.* (18) (295 K–1700 K) which were given in the form of an equation which was also evaluated at 50 K intervals but over the range 350 K to 1750 K. After correction of both sets of measurements for thermal expansion using the values selected by the present author (19) they were fitted to Equation (v) which has an overall accuracy as a standard deviation of ± 0.13 μΩ cm. The two sets of measurements show a maximum disagreement of 1.0% at 1300 K. The equation was extrapolated to the melting point and selected values are given in **Table II**.

##### Table II

Measurements of Milošević and Babić (20) (250 K–1800 K) were independently corrected for thermal expansion. Their equation differs from the selected equation sinusoidally by trending from initially 0.3% high to 1.7% high at 400 K to 0.9% low at 1400 K to 0.4% high at 1800 K. **Figure 1** shows the deviations of the selected values of Matula (which are considered as incorporating the measurements of Laubitz and Matsumura) and the experimental values of Khellaf *et al.* and Milošević and Babić from the fitted curve. Measurements of Binkele and Brunen (4) (273–1423 K) which were also independently corrected for thermal expansion, showed systematic biases of 1.3% high for runs 1 and 2 and 1.7% high for run 3.

### Fig. 1.

Also in the high temperature region there are a number of other measurements which were published after the review of Matula. After correction for thermal expansion (19) the electrical resistivity measurements of Miiller and Cezairliyan (21) (1400 K–1800 K) trend from 4.0% to 6.9% high whilst the measurement of Pottlacher (22) at the melting point is 5.9% high. Resistivity ratio measurements of García and Löffler (23) (295 K–1100 K) were corrected from R_{T}/R_{295} to R_{T}/R_{273.15} and were also corrected for thermal expansion. On this basis the differences reached a maximum of 4.1% high at 450 K but then showed some scatter varying between 1.0% low at 800 K and 1.6% high at 1100 K. **Figure 2** shows the deviations of these three sets of measurements from the fitted curve where the resistivity ratios of García and Löffler were converted to electrical resistivity values for comparison purposes.

### Fig. 2.

## 2.2 Liquid

Electrical resistivity values for palladium at the melting point are given in **Table III**. In the liquid state* *neither Dupree *et al.* (24) (1832 K–1924 K) nor Güntherodt *et al.* (25) (1864 K–2019 K) obtained evidence for any variation of resistivity with temperature. Although Seydel and Fischer (26) (1825 K–3000 K) did obtain evidence of such a variation, the values of Pottlacher (22) (1828 K–2900 K) were selected and fitted to Equation (vi) with selected values for the electrical resistivity of the liquid and are also given in **Table II**.

##### Table III

Authors | Reference | ρ_{S}, μΩ cm | ρ_{L}, μΩ cm | ρ_{L} /ρ_{S} | Notes |
---|---|---|---|---|---|

Dupree et al. | 24 | (48.8) | 83.0 | 1.700 | (a) |

Güntherodt et al. | 25 | 47.3 | 78.8 | 1.666 | |

Seydel and Fischer | 26 | 50.2 | 79.1 | 1.576 | |

Khellaf et al. | 18 | (45.2) | 77.3 | 1.710 | (b) |

Pottlacher | 22 | 49.6 | 81.4 | 1.641 | |

Present assessment | – | 46.84 | 81.4 | 1.738 |

Notes to Table III

(a) Solid value based on (ρL – ρS)/ ρS = 0.70 ± 0.05

(b) Solid value based on ρL /ρS = 1.71

## 3. Platinum

Platinum has a face-centred cubic structure and the melting point is a secondary fixed point on ITS-90 at 2041.3 ± 0.4 K (10).

## 3.1 Solid

The resistance ratio of platinum, W_{T} = R_{T}/R_{273.15}, forms the basis of the International Temperature Scale which White (6) extended to 1300 K and calculated values of intrinsic resistivity using the fixed reference value of 9.82 ± 0.01 μΩ cm at 273.15 K. Above 1300 K White combined the selected values to this temperature with the electrical resistivity measurements of Righini and Rosso (27) (1000 K–2000 K), Laubitz and van der Meer (28) (300 K–1500 K), and Flynn and O’Hagan (29) (273 K–1373 K) and the resistance ratios of Roeser (30) (73 K–1773 K) and Kraftmakher (31) (1000 K–2000 K) together with resistivity measurements given by Martin *et al.* (32) (300 K–1200 K). White fitted all selected values from 100 K to 2000 K to Equation (vii) which was extrapolated to the melting point. Differences between values derived from this equation and the tabulated values of White as given in **Table IV** do not exceed 0.01 μΩ cm. An abridged version of the values for the solid phase as given in **Table IV** was originally given in *Platinum Metals Review* by Corti (33).

##### Table IV

For comparison between these measurements and the selected values as given in **Figure 3**, the resistivity ratios of Roeser (30) and Kraftmakher (31) were converted to electrical resistivity values and all measurements except those of Flynn and O’Hagan (29) were corrected for thermal expansion using values selected by the present author (34). In addition the measurements of Martin *et al. *(32) were corrected to correspond to the selected electrical resistivity value at 273.15 K. Because of their larger deviations values of Righini and Rosso (27) are compared with the selected values in **Figure 4**.

### Fig. 3.

### Fig. 4.

In the case of additional electrical resistivity measurements of Birkele and Brunen (4) (273–1497 K), combined runs 1 and 5 trend from initially 0.8% high to 0.1% high at 1200 K to 0.4% high at 1373 K whilst combined runs 2, 3 and 4 trend to an average of 0.5% low above 1000 K. These trends are also shown in **Figure 3**.

Electrical resistivity measurements of Pottlacher (22) (473 K–1573 K and 1740 K–2042 K in the solid range) are initially 1% higher then trend to an average of 3% higher between 900 and 1573 K before trending to 1.2% higher and then to 0.5% higher between 1740 K and the melting point. These differences are also shown in **Figure 5**.

### Fig. 5.

## 3.2 Liquid

Electrical resistivity values of platinum at the melting point are given in **Table V**. In the liquid state electrical resistivity measurements of Pottlacher (22) (2042 K–2900 K) were selected as Equation (viii) since in the overlap region they are closely confirmed by measurements of Gathers *et al.* (36) (2100 K–7300 K) obtained at a pressure of 0.3 GPa which trend from 0.5% low at 2100 K to 1.0% high at 2900 K. Measurements of Hixson and Winkler (37) (2042 K–5100 K) are initially 7% low at the melting point and trend 1% low to 1% high between 2100 K and 2900 K but above 3000 K, in direct comparison with the measurements of Gathers *et al.*, the trend is to an average of 2% low. Selected values for the electrical resistivity of liquid platinum from the melting point to 2900 K are also given in **Table IV**.

##### Table V

Authors | Reference | ρ_{S}, μΩ cm | ρ_{L}, μΩ cm | ρ_{L} /ρ_{S} |
---|---|---|---|---|

Martynyuk and Tsapkov | 35 | 62.1 | 92.6 | 1.491 |

Pottlacher | 22 | 64.2 | 102.8 | 1.601 |

Present assessment | – | 63.87 | 102.8 | 1.610 |

#### High Temperature Intrinsic Resistivity of Solid Palladium (273.15 to 1828 K)

ρ_{i} (μΩ cm) = 4.58639 × 10^{–2} T – 1.39098 × 10^{–5} T ^{2} + 1.84118 × 10^{–9} T ^{3} – 1.76742 (v)

#### Intrinsic Resistivity of Liquid Palladium (1828 to 2900 K)

ρ_{i} (μΩ cm) = 2.058 × 10^{–3} T + 77.7 (vi)

#### Intrinsic Resistivity of Solid Platinum (100 to 2041.3 K)

ρ_{i} (μΩ cm) = 4.681197 × 10^{–2} T – 3.258075 × 10^{–5} T ^{2} + 8.554023 × 10^{–8} T ^{3} – 1.594242 × 10^{–10} T ^{4} + 1.837342 × 10^{–13} T ^{5} - 1.316886 × 10^{–16} T ^{6} + 5.678222 × 10^{–20} T ^{7} – 1.340980 × 10^{–23} T ^{8} + 1.329896 × 10^{–27} T ^{9} – 1.621733 (vii)

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## The Author

John W. Arblaster is interested in the history of science and the evaluation of the thermodynamic and crystallographic properties of the elements. Now retired, he previously worked as a metallurgical chemist in a number of commercial laboratories and was involved in the analysis of a wide range of ferrous and non-ferrous alloys.