Influence of composition and microstructure on the electrical resistivity of binary magnesium alloys

electrical resistivity of binary magnesium alloys

vorgelegt von M.-Ing.

Xiao Zhang

ORCID: 0000-0003-2719-3159

an der Fakultät III – Prozesswissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften - Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr.-Ing. Aleksander Gurlo Gutachter: Priv.-Doz. Dr.-Ing. Sören Müller Gutachter: Prof. Dr.-Ing. Norbert Hort

Tag der wissenschaftlichen Aussprache: 21. Juli 2021

Berlin 2021

I

Acknowledgement

First of all, I would like to acknowledge the financial support of the China Scholarship Council.

I sincerely thank Priv.-Doz. Dr.-Ing. Sören Müller for being my supervisor in Technische Universität Berlin. His valuable suggestions and advice help me to complete my work. I also gratefully thank Prof. Dr.-Ing. Norbert Hort as my supervisor in Helmholtz-Zentrum Hereon.

His support and guide me through the whole Ph.D duration. I especially thank Prof. Dr.

Aleksander Gurlo for being the chairman of my Ph.D defense committee.

I would like to express my gratitude to Dr. Yuanding Huang for his guidance and fruitful discussions on my work. Many thanks also go to Dr. Serge Gavras for operating the TEM and high resolution SEM. I also want to thank Dr. Veronika Kodetová for helping to measure resistivity. Ms. Yuhui Zhang is acknowledged for offering the thermal database for precipitation simulation.

Mr. Günter Meister helps prepare the materials; Mrs. Sabine Schubert and Mr. Daniel Strerath help analyze the chemical composition. Many thanks to them.

Thank Dr. Weimin Gan and Dr. Xiaohu Li for their help with lab facilities and experiments during my beam time in MLZ.

I would like to thank all the staffs of MBF department: Dr. Domonkos Tolnai, Dr. Hajo Dieringa, Dr. Yiyi Lu, Dr. Yuling Xu, Dr. Sihang You, Dr. Hong Yang and Dr. Yaping Zhang, etc. for their help. I would also like to thank all the colleagues and friends in MagIC for their help and assistance during my Ph.D study.

Finally, I would like to thank my parents for their love and support. I would like to thank my lovely wife Kerong Shi and daughter Sike Zhang for unreserved assistance. Only with the love and support from my family I can finish my Ph.D.

III

Abstract

Electrical resistivity is one characteristic and important physical property of a metal, and it is sensitive to the composition and microstructure. The relationship between resistivity and composition and microstructure makes resistivity a useful tool in materials research, such as non-destructive evaluation and monitoring precipitation kinetics. However, this needs a good understanding of how composition and microstructure influence resistivity, which is currently lack in Mg alloys. Therefore, a systematic investigation of the resistivity of Mg alloys is necessary.

Mg-Al, Mg-Gd, Mg-Sn and Mg-Zn series alloys with different solute content are prepared for the current investigation. The resistivity of these alloys in the as-cast, solution treated, and aged status are measured at different temperatures to study the influence of temperature, composition and microstructure on the resistivity. In situ measurements are also conducted to study the resistivity changes during isothermal ageing of Mg alloys.

The results show that Mg alloys have a positive temperature coefficient of resistivity (TCR).

The TCR varies from different solute content, which demonstrate the deviation from Matthiessen’s rule in Mg alloys. When the alloys are solution treated, the following equation can describe the relationship between resistivity and solute contents:

ρ(T)=ρMg(T)+δ(T)×c

ρ(T) is the resistivity of the alloy under a certain temperature, ρMg(T) is the resistivity of pure Mg, δ(T) is the coefficient, and c is the concentration of the solute. δ(T) depends on both the temperature and the type of solute. The reason for the increment is the lattice distortion caused by the solute elements. When the alloys are aged, a phenomenological formula can describe the relationship between the resistivity and the volume fraction of precipitates：

𝜌_𝑒𝑓𝑓 = 𝜌_𝛼1 +1 2 𝑉^𝛽 1 − 𝑉_𝛽

𝜌_𝑒𝑓𝑓 is the effective resistivity, 𝑉_𝛽 is the volume fraction of the precipitates, 𝜌_𝛼 is the resistivity of the 𝛼-Mg matrix. With the help of this formula, resistivity can be used to quantify the precipitation kinetics of binary magnesium alloys.

IV

Zusammenfassung

Der spezifische elektrische Widerstand ist eine charakteristische und wichtige physikalische Eigenschaft eines Metalls und er ist empfindlich gegenüber Zusammensetzung und Mikrostruktur. Die Beziehung zwischen spezifischem Widerstand, Zusammensetzung und Mikrostruktur macht den spezifischen Widerstand zu einem nützlichen Werkzeug in der Materialforschung, wie z. B. der zerstörungsfreien Bewertung und Überwachung der Ausscheidungskinetik. Dies erfordert jedoch ein gutes Verständnis dafür, wie Zusammensetzung und Mikrostruktur den spezifischen Widerstand beeinflussen. Dies fehlt zurzeit jedoch bei Mg-Legierungen. Daher ist eine systematische Untersuchung des spezifischen Widerstands von Mg-Legierungen erforderlich.

Für die aktuelle Untersuchung werden Legierungen der Mg-Al-, Mg-Gd-, Mg-Sn- und Mg-Zn- Reihe mit unterschiedlichen Gehalten an Legierungselementen verwendet. Der spezifische Widerstand dieser Legierungen im gegossenen, lösungsbehandelten und gealterten Zustand wird bei verschiedenen Temperaturen gemessen, um den Einfluß von Temperatur, Zusammensetzung und Mikrostruktur auf den spezifischen Widerstand zu untersuchen. In situ- Messungen werden auch durchgeführt, um die spezifischen Widerstandsänderungen während der isothermen Alterung von Mg-Legierungen zu bestimmen.

Die Ergebnisse zeigen, dass Mg-Legierungen einen positiven Temperaturkoeffizienten des spezifischen Widerstands (TCR) aufweisen. Der TCR variiert mit verschiedenen Gehalten an gelösten Legierungselement, was die Abweichung von der Matthiessen-Regel in Mg- Legierungen zeigt. Wenn die Legierungen lösungsbehandelt werden, kann die folgende Gleichung die Beziehung zwischen dem spezifischen Widerstand und dem Gehalt an gelösten Stoffen beschreiben:

ρ(T)=ρMg(T)+δ(T)×c

ρ(T) ist der spezifische Widerstand der Legierung bei einer bestimmten Temperatur, ρMg(T) ist der spezifische Widerstand von reinem Mg, δ(T) ist der Koeffizient und c ist die Konzentration des gelösten Stoffes. δ(T) hängt sowohl von der Temperatur als auch von der Art des gelösten Stoffes ab. Der Grund für die Zunahme ist die durch die gelösten Elemente verursachte Gitterverzerrung. Wenn die Legierungen gealtert werden, kann eine phänomenologische Formel die Beziehung zwischen dem spezifischen Widerstand und dem Volumenanteil der Ausscheidungen beschreiben:

V 𝜌_𝑒𝑓𝑓 = 𝜌_𝛼1 +1

2 𝑉^𝛽 1 − 𝑉_𝛽

𝜌_𝑒𝑓𝑓 ist der effektive spezifische Widerstand, 𝑉_𝛽 ist der Volumenanteil der Ausscheidungen, 𝜌_𝛼 ist der spezifische Widerstand der α-Mg-Matrix. Mit Hilfe dieser Formel kann der spezifische Widerstand verwendet werden, um die Ausscheidungskinetik binärer Magnesiumlegierungen zu quantifizieren.

VI

Table of contents

Acknowledgement ... I Abstract ... III Zusammenfassung... IV Table of contents ... VI List of figures ... X List of tables ... XII List of abbreviations ... XIII List of symbols ... XIV

1 Introduction ... 1

2 Literature review... 3

2.1 Magnesium and its alloys ... 3

2.1.1 Common alloying elements in magnesium alloys ... 4

2.1.1.1 Aluminium ... 4

2.1.1.2 Zinc ... 4

2.1.1.3 Manganese ... 4

2.1.1.4 Silver ... 5

2.1.1.5 Zirconium ... 5

2.1.1.6 Rare Earth Metals ... 5

2.1.1.7 Iron, Nickel and Copper ... 5

2.1.2 Precipitation in magnesium alloys ... 6

2.1.2.1 Mg-Al based alloys ... 7

2.1.2.2 Mg-Ca based alloys ... 8

2.1.2.3 Mg-Sn based alloys ... 8

2.1.2.4 Mg-Zn based alloys ... 9

2.1.2.5 Mg-RE based alloys... 10

2.1.2.5.1 Mg-Gd based alloys ... 11

2.1.2.5.2 Mg-Nd based alloys ... 12

2.2 Electrical resistivity of metals and its application ... 13

2.2.1 Drude model... 13

2.2.2 Matthiessen’s rule ... 17

2.2.2.1 Matthiessen’s rule ... 17

2.2.2.2 Deviations from Matthiessen’s Rule ... 19

2.2.3 Influencing Factors of electrical resistivity... 19

VII

2.2.3.1 Effect of lattice imperfection ... 20

2.2.3.1.1 Vacancy ... 20

2.2.3.1.2 Dislocations and grain boundaries ... 20

2.2.3.2 Influence of solution element ... 20

2.2.3.3 Effect of temperature ... 22

2.2.4 Application of electrical resistivity ... 22

2.2.4.1 Non-destructive testing ... 23

2.2.4.2 Phase transformation monitor ... 23

2.2.4.3 Purity evaluation of metals ... 24

3 Motivation ... 25

4 Materials and Experimental Details ... 26

4.1 Materials ... 26

4.1.1 Casting procedure ... 26

4.1.2 Solution treatment ... 27

4.1.3 Ageing treatment ... 28

4.2 Experimental Details ... 28

4.2.1 Microstructure characterizations ... 28

4.2.1.1 Optical microscopy (OM) ... 28

4.2.1.2 Scanning electron microscopy (SEM) ... 29

4.2.1.3 X-ray diffraction analysis ... 29

4.2.1.4 Synchrotron radiation diffraction analysis ... 30

4.2.2 Hardness test ... 30

4.2.3 Electrical resistivity measurements... 30

4.2.3.1 Resistivity measurement at low and room temperatures ... 30

4.2.3.2 Resistivity measurement at high temperature and in situ measurements ... 31

5 Results ... 32

5.1 Microstructure characterization ... 32

5.1.1 As-cast alloys ... 32

5.1.1.1 Mg-Al alloys ... 32

5.1.1.2 Mg-Gd alloys ... 35

5.1.1.3 Mg-Sn alloys... 37

5.1.1.4 Mg-Zn alloys ... 39

5.1.2 As-extruded alloy ... 41

5.1.3 Solution treated alloys... 42

5.1.3.1 Cast alloys... 42

5.1.3.2 Extruded alloy... 45

5.1.4 Aged alloys ... 46

VIII

5.1.4.1 Mg-Al alloys ... 47

5.1.4.2 Mg-Gd alloys ... 49

5.1.4.3 Mg-Sn alloys... 51

5.1.4.4 Mg-Zn alloys ... 53

5.2 Age hardening behaviour ... 55

5.2.1 Mg-Al alloys ... 55

5.2.2 Mg-Gd alloys ... 56

5.2.3 Mg-Sn alloys ... 58

5.2.4 Mg-Zn alloys ... 59

5.3 Electrical resistivity ... 61

5.3.1 The resistivity of the as-cast alloys ... 61

5.3.1.1 Low and room temperatures ... 61

5.3.1.2 Moderate temperatures ... 61

5.3.2 The resistivity of the solution treated alloys ... 62

5.3.2.1 Cast alloys... 62

5.3.2.2 Extruded alloys ... 63

5.3.3 The resistivity of aged alloys ... 64

5.3.4 In situ resistivity measurements during isothermal ageing ... 65

6 Discussion ... 67

6.1 Microstructure ... 67

6.1.1 As-cast alloys ... 67

6.1.1.1 Intermetallic phases ... 67

6.1.1.2 Grain size ... 69

6.1.2 Solution treated alloys... 71

6.1.2.1 Cast alloys... 71

6.1.2.2 Extruded alloy... 73

6.1.3 Aged alloys ... 73

6.2 Age hardening mechanism ... 74

6.2.1 Initial state ... 74

6.2.2 Peak-aged condition ... 76

6.2.3 Over-aged condition... 77

6.3 Influencing factor of resistivity in binary magnesium alloys ... 78

6.3.1 Grain size ... 78

6.3.2 Temperature ... 79

6.3.3 Alloying elements and their contents ... 81

6.3.4 Heat treatment ... 83

IX

6.3.4.1 Solution treatment... 83

6.3.4.2 Ageing treatment ... 85

6.4 Precipitation kinetics quantified by resistivity ... 87

7 Conclusion ... 90

Reference ... 91

X

List of figures

Fig. 2-1 Unit cell and slip planes of magnesium [29]. ... 3

Fig. 2-2 Schematic diagrams of the Drude Model. ... 14

Fig. 2-3 Electrical resistivity of annealed and cold-worked (deformed) copper alloys. ... 18

Fig. 2-4 Deviation from Matthiessen’s rule in SrRuO3 and CaRuO3 alloys. ... 19

Fig. 2-5 Resistivity of Cu-Au alloy in different states. ... 21

Fig. 2-6 Relationship between the resistivity and yield stress. ... 24

Fig. 4-1 Casting system. (a) furnace, (b) direct chill casting system. ... 27

Fig. 4-2 Resistivity sample and room temperature measurements apparatus. ... 31

Fig. 4-3 High-temperature resistivity measurement apparatus and schematic of the delta method measurements. ... 31

Fig. 5-1 OM and SEM (BSE) micrographs of as-cast Mg-Al alloys. ... 33

Fig. 5-2 X-ray diffraction patterns of as-cast Mg-Al alloys. ... 34

Fig. 5-3 OM and SEM (BSE) micrographs of as-cast Mg-Gd alloys. ... 35

Fig. 5-4 Enlarged BSE micrograph of as-cast Mg-2.5Gd alloy. ... 36

Fig. 5-5 X-ray diffraction patterns of as-cast Mg-Gd alloys. ... 37

Fig. 5-6 OM and SEM (BSE) micrographs of as-cast Mg-Sn alloys... 38

Fig. 5-7 BSE micrograph of as-cast Mg-2.5Sn alloy. ... 38

Fig. 5-8 X-ray diffraction patterns of as-cast Mg-Sn alloys. ... 39

Fig. 5-9 OM and SEM (BSE) micrographs of as-cast Mg-Zn alloys. ... 40

Fig. 5-10 X-ray diffraction patterns of as-cast Mg-Zn alloys. ... 41

Fig. 5-11 OM and SEM (BSE) micrographs of as-extruded Mg-0.8Gd alloy. ... 42

Fig. 5-12 OM and SEM (BSE) micrographs of solution treated alloys. ... 43

Fig. 5-13 X-ray diffraction of solution treated alloys. ... 44

Fig. 5-14 Synchrotron diffraction pattern of the solution treated Mg-Gd alloy. ... 45

Fig. 5-15 Composition of the intermetallic phase in as-cast and solution treated Mg-Zn alloy. ... 45

Fig. 5-16 OM of extruded Mg-0.8Gd alloy with different solution times. ... 46

Fig. 5-17 Microstructures of Mg-8Al alloy aged at 175 °C. ... 47

Fig. 5-18 Microstructures of Mg-8Al alloy aged at 200 °C. ... 48

Fig. 5-19 Microstructures of Mg-8Al alloy aged at 225 °C. ... 48

Fig. 5-20 X-Ray diffraction of Mg-8Al alloys aged at different conditions. ... 49

XI

Fig. 5-21 Microstructures of Mg-2.5Gd alloys aged at 200 °C and 225 °C. ... 49

Fig. 5-22 Microstructures of Mg-2.5Gd alloys aged at 250 °C. ... 50

Fig. 5-23 Synchrotron diffraction of Mg-2.5Gd alloys aged at different conditions. ... 51

Fig. 5-24 Microstructures of Mg-2.5Sn alloy aged at 200 °C. ... 51

Fig. 5-25 Microstructures of Mg-2.5Sn alloy aged at 225 °C. ... 52

Fig. 5-26 Microstructures of Mg-2.5Sn alloy aged at 250 °C. ... 52

Fig. 5-27 X-Ray diffraction of Mg-2.5Sn alloys aged at different conditions. ... 53

Fig. 5-28 Microstructures of Mg-2.5Zn alloy aged at 175 °C. ... 53

Fig. 5-29 Microstructures of Mg-2.5Zn alloy aged at 200 °C. ... 54

Fig. 5-30 Microstructures of Mg-2.5Zn alloy aged at 225 °C. ... 54

Fig. 5-31 X-Ray diffraction of Mg-2.5Zn alloys aged at different conditions ... 55

Fig. 5-32 Age hardening curves of Mg-Al alloys. ... 56

Fig. 5-33 Age hardening curves of Mg-Gd alloys. ... 57

Fig. 5-34 Age hardening curves of Mg-Sn alloys. ... 59

Fig. 5-35 Age hardening curves of Mg-Zn alloys. (a) 175 °C; (b) 200 °C; (c) 225 °C. ... 60

Fig. 5-36 The resistivity of as-cast alloys. ... 61

Fig. 5-37 The resistivity of as-cast alloys. ... 62

Fig. 5-38 The resistivity of solution treated alloys. ... 62

Fig. 5-39 The resistivity of solution treated alloys. ... 63

Fig. 5-40 Resistivity of Mg-0.8Gd alloy after solution treatment. ... 64

Fig. 5-41 The resistivity of alloys aged at 225 °C. ... 65

Fig. 5-42 Resistivity changes during isothermal ageing. ... 66

Fig. 6-1 Phase diagrams of binary Mg alloys in the Mg-rich corner. ... 68

Fig. 6-2 BSE micrographs of as-cast alloys. ... 69

Fig. 6-3 Microstructure of solution treated Mg-1.5Zn alloy. ... 72

Fig. 6-4 Hardness of alloys in the as-solution treated states. ... 74

Fig. 6-5 Calculated diffusion rate of alloying elements in Mg matrix at 225 °C. ... 77

Fig. 6-6 Resistivity of solution treated alloys. ... 81

Fig. 6-7 Resistivity before and after solution treatment. ... 83

Fig. 6-8 Resistivity of alloys aged at 225 °C. ... 86

Fig. 6-9 Volume fraction of the precipitates during isothermal aged at 225 °C. ... 88

XII

List of tables

Table 2-1 The solubility data calculated by Pandat software (wt. %). ... 6

Table 2-2 The conduction electron density, relaxation time and resistivity of selected elements at 273 K [152]. ... 16

Table 4-1 Nominal chemical compositions of the alloys (at. %). ... 26

Table 4-2 Solution treatment parameters of the as-cast alloys. ... 27

Table 4-3 Ageing parameters of different alloys. ... 28

Table 5-1 Chemical compositions of the alloys (at. %). ... 32

Table 5-2 Grain sizes of as-cast Mg-Al alloys. ... 33

Table 5-3 Amount of the Mg17Al12 in as-cast Mg-Al alloys and corresponding GOF. ... 34

Table 5-4 Grain sizes of as-cast Mg-Gd alloys. ... 36

Table 5-5 Amount of the Mg5Gd in as-cast Mg-Gd alloys and corresponding GOF. ... 36

Table 5-6 Grain sizes of as-cast Mg-Sn alloys. ... 37

Table 5-7 Amount of the Mg2Sn in as-cast Mg-Sn alloys and corresponding GOF. ... 39

Table 5-8 Grain sizes of as-cast Mg-Zn alloys. ... 39

Table 5-9 Amount of the Mg7Zn3 in as-cast Mg-Zn alloys and corresponding GOF. ... 41

Table 5-10 Grain sizes of the as-extruded alloy in different directions. ... 42

Table 5-11 Grain sizes of the cast alloys after solution treated. ... 43

Table 5-12 Grain sizes of the extruded Mg-0.8Gd alloy after solution treatment. ... 46

Table 5-13 Age hardening data of Mg-8Al alloy. ... 56

Table 5-14 Age hardening data of Mg-Gd alloys. ... 58

Table 5-15 Age hardening data of Mg-Sn alloys. ... 58

Table 5-16 Age hardening data of Mg-Zn alloys. ... 60

Table 5-17 Resistivity of different alloys aged at 225 °C. ... 64

Table 6-1 Amount of alloying elements and intermetallic phases. ... 68

Table 6-2 Parameters for calculating GRFs of different alloying elements ... 70

Table 6-3 Local strains of the nearest neighbouring Mg atoms from alloying elements [210]. ... 75

Table 6-4 Alloys aged at 225 °C. ... 76

Table 6-5 TCR of different alloys. ... 80

Table 6-6 Specific resistivity increase and some physical properties of alloying elements. ... 82

Table 6-7 RRR values of the alloys. ... 84

XIII

List of abbreviations

AJ Mg-Al-Sr

AM Mg-Al-Mn

AS Mg-Al-Si

AZ Mg-Al-Zn

BCC Body centred cubic BSE Backscattered electron

BF Bright field

DESY Deutsches Elektronen-Synchrotron FCC Face centred cubic

Gd Gadolinium

G.P. Guinier–Preston

GRF Growth restriction factor HAADF High-angle annular dark-field HCP Hexagonal close packed

HV Vickers hardness

OM Optical microscopy

OPS Oxide polishing suspensions

RE Rare Earth metals

RRR Residual resistivity ratio SEM Scanning electron microscopy T4 Solution treatment

T6 Solution treatment and then artificially aged TEM Transmission electron microscopy

TCR Temperature coefficient of resistivity UTS Ultimate tensile strength

XRD X-Ray diffraction

XIV

List of symbols

R Resistance, see Eq.(2-1).

I Electrical current, see Eq.(2-1).

V Voltage, see Eq.(2-1),

Also, the unit of voltage: Volt.

E Electric field, see Eq.(2-2).

 Electrical resistivity, see Eq.(2-2).

J Current density, see Eq.(2-2).

 Unit of Resistance: Ohm.

A Unit of Electrical current: Ampere.

τ Relaxation time.

e Proton charge.

𝑚_𝑒 Mass of the electron, see Eq.(2-3).

𝒗 Average electronic velocity, see Eq.(2-3).

𝒗_𝐸 Average electronic velocity in an external electric field, see Eq.(2-4).

𝑁_𝐴 Avogadro’s number, see Eq.(2-8).

𝜌_𝑚 Mass density, see Eq.(2-8).

M Atomic mass of the clement, see Eq.(2-8).

𝜌_𝑎 Electrical resistivity of an alloy, see Eq.(2-9).

𝜌_𝑝 Electrical resistivity of pure metal, see Eq.(2-9).

𝜌₀ Residual resistivity, see Eq.(2-10).

𝜌_𝑇 Resistivity caused by thermal vibrations.

𝜌_𝐼 Resistivity caused by impurity solute.

𝜌_𝑐𝑤 Resistivity caused by cold-word.

α0 Temperature coefficient of resistivity, see Eq.(2-13).

1

1 Introduction

Magnesium is a promising structural material due to its low density and high specific strength [1-4]. In order to expand its application, current researches on magnesium are mainly focused on improving the strength [5, 6], ductility [7, 8], creep resistance [9-11] and corrosion resistance [12-14]. There are also investigations on the biomedical use of magnesium alloys [15, 16]. However, studies lying on the physical properties of magnesium alloys are much less compared to those on mechanical properties.

One of the most characteristic and important physical properties of a metal is the ability to conduct electricity. Metals are good conductors because the valence electrons can move freely through the whole metal and act as the charge carriers to conduct electricity. The free movement of these electrons is affected by scatterers such as lattice imperfection, solution elements and thermal vibration, so the conductivity is also affected by these factors [17-19].

The conductivity of a metal can also be characterized by its inverse, electrical resistivity.

Electrical resistivity, which is independent of the geometry of the sample, has shown sensitivity to the microstructure changes. Therefore, it offers a possibility to study the microstructure change of an alloy by monitoring the electrical resistivity changes.

Some investigations had been performed on electrical resistivity in magnesium alloys.

Salkovitz et al. [20, 21] investigated the resistivity of some dilute magnesium alloys. The results showed that the resistivity increased linearly with alloying content. Pan et al. [22]

studied the electrical resistivity of some binary magnesium alloys and found that increment of resistivity due to the solute element was in the sequence Zn<Al<Ca<Sn<Mn<Zr. Ying et al.

[23] discussed the influence of temperature on the resistivity of magnesium alloys range from 2 to 300 K. Their results indicated that the electrical resistivity of magnesium alloys was not temperature-dependent in the temperature range 2 - 40 K. In the temperature range 40 - 300 K, the electrical resistivity increased sharply due to the enhanced thermal vibration of the lattice.

Nevertheless, the materials used in these investigations are either in as-cast or T4 states; the influence of microstructure change due to the heat treatment on the electrical resistivity has not been well investigated.

Therefore, in order to have a full understanding of resistivity in magnesium, it is necessary to investigate the influence of microstructure on resistivity. In this work Mg-Al, Mg-Gd, Mg-Sn and Mg-Zn alloys had been chosen to study the influence of both the alloying elements and the

2

microstructure on the resistivity. Resistivity in as-cast and T4 states was measured to study the influence of alloying elements and their concentrations on resistivity. In situ measurements of the resistivity during ageing were performed to investigate the influence of microstructure on resistivity.

The findings in the current work are expected to clarify the relationship between electrical resistivity and the microstructure.

3

2 Literature review

2.1 Magnesium and its alloys

Over the years, with the increasing demand for economical use of energy resources and ever- stricter control over emissions to lower environmental impact, industries are constantly searching for new, advanced materials as alternatives to “conventional” materials. Magnesium is such a promising lightweight metal due to its low density and high specific strength [1-3].

Magnesium has a hexagonal close packed (hcp) crystallographic structure, and the lattice parameters are a=0.31954 nm and c=0.51872 nm [24]. As shown in Fig. 2-1, there are four slip planes in magnesium. At room temperature, slip mainly occurred in the basal plane (0001) along the most occupied direction <11-20>. At high temperatures, non-basal prismatic and pyramidal slip planes can be activated [25-28].

Fig. 2-1 Unit cell and slip planes of magnesium [29].

As a metal, magnesium is formed through the metallic bond. Each magnesium atom contributes its valence electrons to form the electron cloud, known as the metallic bond. The metallic bond makes magnesium a good conductor of electricity. The resistivity of pure magnesium is about 43 nΩ∙m at 295 K [30].

a

c

a

a

Basal {0001}

Pyramidal I {01-11}

Prismatic {01-10}

Pyramidal II {11-22}

Unit cell

4

2.1.1 Common alloying elements in magnesium alloys

Pure magnesium is rarely used for engineering applications due to its low mechanical properties. Therefore, it is necessary to improve the properties of magnesium before its usage in engineering applications. The strengthening mechanisms of magnesium include grain boundary strengthening, solid solution strengthening and precipitation strengthening [31, 32].

The addition of alloying elements is a normal and effective way to improve the properties of magnesium since the alloying elements can provide the solution strengthening and have the potential to offer precipitation and grain boundary strengthening. Aluminium, zinc, manganese, silver, zirconium and RE (Rear Earth) elements are commonly used in commercial Mg alloys.

2.1.1.1 Aluminium

Al is the most commonly used alloying element in magnesium. It is the major element in the AZ, AM, AS and AE series alloys. Among them, AZ91 is the most widely used die casting magnesium alloy [33].

When the content of aluminium is low, the alloy is strengthened by solid solution strengthening.

At a higher concentration of aluminium, the alloy can be strengthened by the precipitation of the Mg17Al12 phase.

2.1.1.2 Zinc

Zn is another commonly used alloying element in commercial magnesium alloys. The ZK, ZE and ZC series alloys are designed with the primary alloying element of Zn. It is the secondary alloying element in the AZ series alloys.

Zn can effectively refine the grain size and provide the grain boundary strengthening [34];

Additionally, Zn contained alloys, such as ZK60 [35], ZE41 [36] and ZC63 [37], can always be heat treated to obtain precipitation strengthening.

2.1.1.3 Manganese

M1A and M1C are alloys that contain Mn as the primary alloying element [38]. Instead of a primary alloying element, Mn is more likely to be added as a subordinate alloying element in Mg alloys. Such as in the Al contained AZ, AM and AS series alloys, and the Zn contained ZC series alloys [38]. Mn can reduce iron content by formatting the Fe-Mn compound hence improve the corrosion resistance of the alloys.

5 2.1.1.4 Silver

Ag is the primary alloying element in the QE series alloys. The QE22 alloy has been employed for several aerospace applications, including landing wheels, gearbox housings and rotor heads for helicopters [39]. The QE22 alloy is heat treatable when it is in the peak-aged condition; it has superior creep resistance over many other magnesium alloys [5].

2.1.1.5 Zirconium

Zr is the most effective grain refiner in magnesium alloys [46], a small amount of Zr can significantly reduce the grain sizes of the cast alloys.

In many Mg alloys, such as the ZK, ZE, WE and QE series alloys, Zr is added [38]. However, it is incompatible with Al or Mn containing alloys because these alloying elements will interact with Zr and form stable compounds, which will eliminate the effect of grain refinement [47].

2.1.1.6 Rare Earth Metals

RE metals are a set of seventeen chemical elements grouped in the periodic table. The most successful commercial Mg-RE alloys are the WE54 and WE43 alloys, which contain yttrium, neodymium and heavy RE elements consist of Yb, Er, Dy and Gd [40]. RE elements are also added as subordinate alloying elements in ZE, QE and AE series alloys [38].

Despite the commercial Mg alloys, high-performance Mg-RE alloys are also developed [41, 42]. One example is the Mg-RE-Zn alloys, with different compositions and appropriate heat treatment; these alloys can have a high strength [43] or high ductility [44].

2.1.1.7 Iron, Nickel and Copper

Except for the above elements that can improve the performance of Mg alloys, there are also impurities, mainly Fe, Ni and Cu, which are detrimental to the corrosion performance that should be noted in Mg alloys [45, 46]. The corrosion rate of Mg alloys is usually insignificant when the concentrations of impurities are under the tolerance limits, but it will substantially increase when the impurity concentrations exceed the tolerance limits. Normally, the tolerance limits of Fe, Ni and Cu are 170 weight ppm, 5 weight ppm and 1000 weight ppm [45].

The corrosion mechanism of these impurities is galvanic corrosion since Mg has the lowest standard corrosion potential of all the engineering metals [47, 48]. Fe, Ni and Cu with a higher standard corrosion potential and combined with low hydrogen overvoltage can constitute efficient cathodes for magnesium and cause severe galvanic corrosion [47].

6

2.1.2 Precipitation in magnesium alloys

Precipitation strengthening is an effective method to enhance the strength of Mg alloys and it largely depends on the amount, morphology and distribution of the precipitates. Precipitation strengthening of an alloy is normally achieved through three steps:

1) The alloy is heated up to a high temperature to get a supersaturated 𝛼-Mg phase;

2) Quench the alloy to room temperature to maintain the supersaturated 𝛼-Mg phase;

3) Subsequent ageing the alloy at a relatively low temperature to achieve a controlled decomposition of the supersaturated solid solution into a fine distribution of precipitates in the magnesium matrix [5, 49].

The supersaturated 𝛼-Mg single-phase is thermodynamically unstable; it tends to decompose and reduces the internal energy of the system. Energy reduction is the driving force for precipitation. In theory, the more exceeding solute, the higher the driving force and a larger amount of the precipitates. Table 2-1 shows the maximum solubility and solubility at 200 °C (common ageing temperature) of some alloying elements calculated by Pandat software.

Table 2-1 The solubility data calculated by Pandat software (wt. %).

Element Maximum solubility Solubility at 200 °C Solubility changes

Ag 13.78 0.01 13.77

Al 12.71 2.80 9.91

Ca 1.34 0.03 0.67

Ce 0.80 ~0 0.80

Dy 30.37 2.96 27.41

Gd 23.67 1.64 22.03

Li 5.37 5.72 -0.35

Mn 2.15 0.01 2.14

Nd 3.68 0.01 3.67

Sn 13.41 0.38 13.03

Y 13.91 1.76 12.15

Zn 5.94 2.65 3.29

Zr 2.13 0.01 2.12

7 Despite the solubility changes, the precipitation process is another crucial factor that affects the precipitation strengthening of magnesium alloys. Precipitation processes vary a lot with different alloying elements. In some alloy systems, the only precipitates are the equilibrium phase, some systems will precipitate the metastable phase during ageing, and in some systems, there exist the G.P. zones [5]. In addition, precipitate shape and orientation also matter on the precipitation strengthening in Mg alloys [50, 51]. Therefore, the investigations of the precipitation process in Mg alloys have attracted researchers’ attention for a long time.

Alloying elements that can offer strong precipitation strengthening in magnesium alloys such as Al [52, 53], Ca [54, 55], Sn [56, 57], Zn [58, 59] and RE [60-62] have been studied.

2.1.2.1 Mg-Al based alloys

Aluminium is the most widely used alloying element in magnesium alloys [63]. The commercial AZ, AM and AJ series magnesium alloys are developed based on the binary Mg- Al alloy with the addition of Zn and Manganese [64]. The maximum solubility of Al decreases from 12.71 wt. % at eutectic temperature (437 °C) to about 2.8 wt. % at 200 °C.

The Mg17Al12 phase is believed to be the only precipitates formed during the ageing of Mg-Al alloy [65]. It has a Burgers orientation relationship with the matrix, the growth habit plane is (0001)Mg || (110)β with a coincident direction [1-210]Mg || [1-11]β [66]. The precipitation of the Mg17Al12 phase during isothermal ageing is either continuous or discontinuous. Normally, continuous and discontinuous precipitation can occur simultaneously. However, only discontinuous or continuous precipitation can be observed at the end of the ageing under a certain condition. Duly et al. [67]proposed a “Precipitation morphology map” to predict the occurrence of discontinuous and continuous precipitation at different Al contents and ageing temperatures. According to their results, continuous precipitation dominates at both high and low temperatures. At intermediate temperatures, only discontinuous precipitation occurs during isothermal ageing. However, the “Precipitation morphology map” is not consistent with the results obtained by other experiments. A research of AZ91 alloy conducted by Malik [68]

reported that discontinuous precipitation is favoured at low temperatures and continuous precipitation is favoured at high temperatures; at intermediate temperatures, discontinuous and continuous precipitation can occur simultaneously. Robson [69] proposed a classical kinetic theory based model to predict the continuous and discontinuous precipitation process in Mg- Al alloy. The model took the competition between discontinuous and continuous precipitation into account to predict the final microstructure. Their results demonstrated that continuous

8

precipitation could effectively suppress the discontinuous precipitation by reducing the supersaturation of Al in the matrix. In contrast, discontinuous precipitation had a weak influence in suppressing continuous precipitation.

2.1.2.2 Mg-Ca based alloys

The maximum solubility of Ca in magnesium is about 1.34 wt. % and it is almost 0 wt. % at 200 °C. The binary Mg-Ca alloy shows moderate precipitation strengthening during isothermal ageing at 200 °C [54]. The equilibrium precipitates in the Mg-Ca alloy is the Mg2Ca phase; it has the same P63/mmc space group of magnesium matrix, this similarity may result in a higher nucleation rate, therefore, a higher number density of precipitates [5]. However, due to the sparse distribution and coarse morphology of the Mg2Ca phase, the precipitation strengthening is quite weak in binary Mg-Ca alloy [70]. Therefore, efforts have been made in the past years to enhance the precipitation strengthening of Mg-Ca alloy.

Mendis et al. reported the enhancement of age hardening response in Mg-Ca alloy due to the microalloying with Al, In and Zn elements [71]. The addition of Al and Zn microalloying elements could cause the formation of the uniformly dispersed metastable plate-like internally ordered G.P. zones in the peak-aged condition. The addition of In could alter the habit plane of plate-like G.P. zones from the basal plane to the prismatic plane, which was expected to be more effective in hindering dislocation movement and hence enhanced the age response. Oh-ishi et al. [70] confirmed the formation of the ordered G.P. zones in Mg-0.3Ca-xZn alloys and considered the G.P. zones had excellent thermal stability since they remained even after overaged conditions. Oh-ishi also concluded that the optimum amount of the Zn addition was 0.6 at. %. Excess addition of Zn would change the precipitation process and form a Ca2Mg6Zn3

ternary phase. The Ca2Mg6Zn3 phase would suppress the formation of the ordered G.P. zones and reduce the peak hardness. Jayaraj et al. [72] verified that the peak-aged of Mg-0.5Ca-0.3Al alloy was attributed to the ordered G.P. zones and the subsequent formation of the Al2Ca phase caused the over-aged of Mg-0.5Ca-0.3Al.

2.1.2.3 Mg-Sn based alloys

The maximum solubility of Sn in Magnesium is 13.41 wt. % at eutectic temperature and 0.38 wt. % at 200 °C. The solubility varies over 13 wt. %, which makes the Mg-Sn alloy appropriate to precipitation strengthening. The only precipitate in Mg-Sn alloy is the equilibrium FCC Mg2Sn phase [5]. The characteristic property of the Mg2Sn phase is its high melting

9 temperature of 770 °C. This makes the Mg-Sn alloy has the potential as a creep resistant alloy [73, 74]. Nevertheless, the precipitation strengthening of binary Mg-Sn alloy is limited due to the direct precipitation of Mg2Sn without any metastable precipitate phases during the isothermal ageing [75]. Therefore, methods have been proposed to enhance its precipitation strengthening. Like the Mg-Ca alloy, microalloying with other elements effectively enhances the precipitation strengthening [76-79].

Microalloying with Zn had been proved to enhance the precipitation strengthening in Mg-Sn alloy by making the Mg2Sn phase finer and dispersed more uniformly [80]. In addition to that, the number density of the precipitates also increased compared to the non-Zn containing Mg- Sn alloy [80]. The influence of the Ca addition on Mg-3Sn and Mg-5Sn alloys has also been studied [81, 82]. The Ca addition would cause the formation of a high thermal stability phase, the CaMgSn phase, which resulted in improved hardness, strength and creep resistance.

Schmid-Fetzer et al. investigated phase formation in Mg-Sn-Ca alloys by combining the Calphad method with experimental investigations [83, 84]. They found that except for the CaMgSn phase, the precipitation of the Mg2Ca or Mg2Sn phase would happen. The type of the phase depends on the Sn/Ca weight ratio in the alloy.

For a long time, it is believed that the precipitation process of Mg-Sn alloy involves only the precipitation of the FCC Mg2Sn phase. However, Fu et al. [85] introduced a high-pressure ageing method and a novel hexagonal type Mg2Sn phase. This novel hexagonal Mg2Sn phase had an average grain size of 25 nm; the uniformly distributed hexagonal Mg2Sn particles significantly improved the strength and the ductility of Mg-Sn alloy. In addition, Kim et al.

[86] and Liu et al. [87, 88] found the metastable phase formation during the isothermal ageing at low temperatures. Wang et al. [89] used first principle to study the precipitation process in Mg-Sn alloy and proposed a new precipitation sequence: supersaturated solid solution (SSSS)

→ G.P. zones → HCP Mg3Sn → FCC Mg3Sn → 𝛽 Mg2Sn. However, the precipitation of G.P.

zones and metastable phase need to be confirmed by more experiments before they are formally accepted in the precipitation sequence of Mg-Sn alloys [89].

2.1.2.4 Mg-Zn based alloys

Zn is another commonly used alloying element in commercial magnesium alloys. The ZK and ZC series alloys are developed based on the binary Mg-Zn alloy [38]. The maximum solid solubility of Zn in magnesium is 5.94 wt. % at eutectic temperature and decreases to 2.65 wt. %

10

at 200 °C. Compared to the Mg-Al alloy, the precipitation process of Mg-Zn alloy is quite complex. The commonly accepted precipitation sequence is: SSSS → G.P. zones → 𝛽₁^′ Mg4Zn7 → 𝛽₂^′ MgZn2 → 𝛽 MgZn.

Murakami et al. [90] first found the formation in Mg-Zn alloy; however, it is then realized that the formation of the G.P zones is restricted to the ageing temperature under 110 °C [91-93].

The precipitation of 𝛽₁^′ phase is also controversial. At the early stage, due to the limit of the equipment, the 𝛽₁^′ phase is determined as a hexagonal structure and the composition is the same as the 𝛽₂^′ phase MgZn2 [94, 95]. Later, a study conducted by Gao and Nie [96] claimed that the 𝛽₁^′ phase has a base-centred monoclinic structure and the composition is Mg4Zn7. Most researchers now accept their results. Nevertheless, researchers in Japan [97-100] showed different results to Gao and Nie, but coincident with the earlier study. They claimed the difference was due to different ageing temperatures. They believed a higher ageing temperature is favourable to the formation of the Mg4Zn7 type 𝛽₁^′ phase while a lower ageing temperature results in the formation of the MgZn2 type 𝛽₁^′ phase.

In contrast to the argument of 𝛽₁^′ phase, 𝛽₂^′ phase is widely accepted to be the hexagonal MgZn2

phase. Most 𝛽₂^′ phase has a basal plane plate morphology and provides much less of an obstruction to the movement of dislocations [50, 101]. Therefore the over-aged of Mg-Zn alloy is due to the formation of the 𝛽₂^′ phase. The formation of equilibrium 𝛽 phase needs longer period of time [92].

2.1.2.5 Mg-RE based alloys

The RE elements can significantly improve the mechanical properties of magnesium alloys.

An Mg-RE based ultra-high alloy has been reported with the properties of 610 MPa in tensile yield stress and 5 % in elongation [102]. Since the Mg-RE based alloys show the greater potential for developing ultra-high strength magnesium alloys via precipitation strengthening, many works have been conducted to study the Mg-RE based alloys [103-111].

The RE elements can be divided into two subgroups according to their atomic number, those from La to Sm (lower atomic numbers and masses) being referred to as the light RE elements and those from Gd to Lu (higher atomic numbers and masses) being referred to as the heavy RE elements [112]. According to Rokhlin [113] and Hadorn et al. [114], the intermediate phase formation amongst the Mg-RE alloy in the same sub-group shows great similarity.

11 2.1.2.5.1 Mg-Gd based alloys

Gadolinium belongs to the heavy RE elements; its maximum solubility in Mg is 23.67 wt. % and decreases to 1.64 wt. % at 200 °C. Many works have been devoted to investigating the precipitation process of binary Mg-Gd alloy [115-118]. The precipitation sequence of binary Mg-Gd alloy is considered as SSSS → 𝛽^′′ Mg3Gd → 𝛽^′ Mg7Gd → 𝛽₁ Mg3Gd → 𝛽 Mg5Gd [5].

Although both the 𝛽^′′ and the 𝛽₁ precipitates have the same Mg3Gd composition. Their lattice structure is different, the 𝛽^′′ phase has an HCP structure while the 𝛽₁ phase has an FCC structure. The 𝛽^′′ phase precipitates at the early stage of ageing, according to Gao et al.[116], it coexists in the matrix with the 𝛽^′ phase after ageing at 250 °C for 0.5 hours. When the ageing time is extended to 2 hours the only existing precipitates is the 𝛽^′ phase. The 𝛽^′ phase has a base-centred orthorhombic structure and it is the key strengthening precipitate phase [119-121].

Recent studies revealed that the 𝛽^′ phase includes two types of precipitates, the 𝛽_𝑆^′ and 𝛽_𝐿^′ phase [122-124]. They have the same base-centered orthorhombic structure but different in the lattice parameters, the lattice parameters of 𝛽_𝐿^′ phase are: a = 0.64 nm, b = 2.22 nm, c = 0.52 nm while a = 0.64 nm, b = 1.11 nm, c = 0.52 nm for 𝛽_𝑆^′ phase [123]. The precipitation of 𝛽₁ phase is somehow under debate, Nie et al. [125] and Gao et al. [116] believed that the 𝛽₁ phase nucleates at the necks of the decomposed 𝛽^′ precipitates and grows at the expense of 𝛽^′, it is supported by the fact that the 𝛽₁ phase is always attached to two 𝛽^′ particles; this is also consistent to other Mg-RE alloys [126]. However, Apps et al. [127] disagreed with that; they assumed that both 𝛽₁ and 𝛽^′ all nucleate on the 𝛽^′′ phase and further ageing caused the two 𝛽^′ particles attached to the 𝛽₁ phase. Additionally, Meng et al. [128] even concluded that the precipitation of 𝛽₁ phase is impossible in binary Mg-Gd alloy due to its high formation energy and low vibrational entropy according to their first-principles calculation. The formation of the equilibrium 𝛽 phase caused the over-aged of Mg-Gd alloy. It is believed to be transformed in situ from the 𝛽₁ phase and the orientation relationship between the 𝛽 phase and matrix is the same as that the orientation relationship between 𝛽₁ phase and the matrix.

Recently, a study on the precipitation of binary Mg-Gd alloy associated with HAADF-STEM (High-angle annular dark-field scanning transmission electron microscopy) was conducted by Zhang et al. [129]. They proposed a very different precipitation sequence as follows: SSSS → ordered solute clusters → G.P. zones → 𝛽^′ → 𝛽_𝑆^′ + tail-like hybrid structures → 𝛽₁ → 𝛽.

12

Additionally, an unusual FCC-structured Gd platelets was found to precipitate when Mg-Gd alloy was rapidly heated to 250 °C and held for 2 hours [130].

2.1.2.5.2 Mg-Nd based alloys

Nd is one of the light RE elements; its maximum solubility in Mg is 3.68 wt. % and decreases to 0.01 wt. % at 200 °C. The precipitation sequence in the Mg‒Nd alloy is suggested to be:

SSSS → G.P. zones → 𝛽^′′ Mg3Nd → 𝛽^′ Mg7Nd → 𝛽₁ Mg3Nd → 𝛽 Mg12Nd → 𝛽_𝑒 Mg41Nd5

[5]. Compared to the precipitation in binary Mg-Gd, the difference in binary Mg-Nd is the formation of G.P. zones.

The information on the precipitation of G.P. zones is limited due to the small size of these features and instrumentation restrictions [131], Saito et al. [132] and Lefebvre et al. [133]

found that the G.P. zones were needle-shaped with long axes parallel to the [0001]Mg, but the driving forces for G.P. zones remained a mystery. The 𝛽^′′ phase was determined by Lefebvre et al. [133], it had an FCC structure and the composition of Mg3Nd. It formed on the prismatic planes and was fully coherent with the matrix [134]. Ma et al. [134] thought the formation of the 𝛽^′′ phase was mainly responsible for the precipitation strengthening in Mg-Nd alloy.

Nevertheless, Satio et al. [132] disagreed with that. They reported that 𝛽^′′ phase was not formed in Mg-Nd alloy was when ageing at temperatures ranging from 170 °C to 250 °C, the peak-aged was due to the coexistence of G.P. zones and the 𝛽^′ phase. They also concluded that when the Mg-Nd alloy was over-aged, both the G.P. zones and the 𝛽^′ phase disappeared and coarse stable 𝛽₁ phase was precipitated. Therefore, they assumed that the 𝛽₁ phase was harmful to the precipitation strengthening in Mg-Nd alloy. However, a study by Zhu et al. [135]

concluded that the 𝛽₁ phase was the key strengthening phase in Mg-Nd alloy. They believed that the 𝛽₁ phase had six variants and formed on the {01-10} planes. They also found an unreported phase designated as 𝛽₂, the 𝛽₂ phase always formed in connection points of two 𝛽₁ particles of the same variant or different variants but having opposite shears directions. The 𝛽 phase coexisted with the 𝛽₁ phase and was considered to be the equilibrium phase by Zaden et al. [131]. However, it is confirmed that the equilibrium phase is in fact the 𝛽_𝑒 phase, but it is formed only at high heat treatment temperatures and sufficiently long durations [136].

The precipitation process in magnesium alloys has been extensively studied with the help of TEM [95, 137], HAADF-STEM [138, 139], DSC (Differential scanning calorimetry) [140], synchrotron radiation [141] and dilatometry [142]. Despite the traditional methods, electrical

13 resistivity has been successfully introduced in investigating the precipitation process in steel [143] and Al alloys [144, 145]. However, limited work has been performed to study electrical resistivity in Mg alloys. Therefore, the current study targets understanding the electrical resistivity changes in Mg alloys and exploring the possible use of the electrical resistivity in Mg alloys.

2.2 Electrical resistivity of metals and its application

In 1827, Ohm published his work on resistance, the Ohm’s law:

𝐼 =𝑉

𝑅 (2-1)

Where I is the electrical current through the conductor, V is the voltage measured across the conductor and R is the resistance of the conductor. The term of Ohm’s law could be changed to [30, 146, 147]:

𝑬 = 𝜌𝐽 (2-2)

Where E is the electric field, J is current density, ρ is electrical resistivity. The electrical resistivity is defined as the electrical resistance per unit length and unit of cross-sectional area.

It is an intrinsic property of metals independent of the shape of the sample and the applied electric field. The resistivity is believed to be brought by the incoherent scattering of conduction electrons [148-151]. Therefore, anything that increases the incoherent scattering, such as impurity atoms, lattice defects and temperature, will raise the resistivity. Considerable efforts have been devoted to understanding the electrical resistivity in metals since the early nineteenth century.

2.2.1 Drude model

In 1897, Thomson discovered the electron and it was quickly realized that electrons contribute to the electric currents. Three years later, Drude proposed a model based on the classic kinetic theory of gases to calculate resistivity. In his model, he treated the free electrons as a classical ideal gas; although the Drude model has many shortcomings, it is still used today as a quick practical way to form simple pictures and rough estimates of the electrical properties of metals.

He made four basic assumptions in the model [152, 153]:

1) Free electron approximation. Drude assumed that when atoms of a metallic element are brought together to form a metal, the valence electrons become detached and wander

14

freely through the metal. They are the so-called free electrons and they are the charge carriers. The positively charged ion cores remain intact and are immobile. The free electrons can collide with the ion cores while they are moving. These collisions instantaneously change their velocity. Besides the collisions, the electrons do not interact with the ion cores.

2) Independent electron approximation. The free electrons do not interact with each other at all: There is no coulomb interaction, and as opposed to a classical gas model, they do not collide with each other either. Thus in the absence of externally electromagnetic fields, each electron moves in a straight line until it collides with the ion cores. When an external field is applied, each electron is taken to move as determined by Newton’s laws of motion. Fig. 2-2 illustrates the free electron approximation and independent electron approximation of the Drude Model.

Fig. 2-2 Schematic diagrams of the Drude Model.

a) without external electric field, b) with external electric field E.

3) Relaxation time and mean free path. Drude assumed that the probability that an electron experiences a collision with the ion cores per unit time is 1/τ. It means that an electron picked at random will, on average, travel for a time τ since its last collision. The time τ is variously known as the relaxation time. It is assumed independent of the electron position and is independent of time. In between collisions, the electrons move freely.

The mean length of this free movement is called the mean free path.

4) Electrons will achieve thermal equilibrium with their surroundings through collisions with the ion cores. Drude presumed the electrons maintain local thermodynamic equilibrium in a particularly simple way: immediately after each collision, an electron obtains a velocity uncorrelated to its velocity before the collision, this velocity is

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

a) b)

+

Free electrons

𝒗_𝐸 = − 𝑬 𝑚_𝑒

v=0 E

15 randomly directed and its speed is related to the temperature where the collision occurred.

With the basic assumptions of the Drude Model, we can now explain the resistivity of metals.

Consider the movement of an electron when an electric field E is applied. The equation of motion, according to Newton’s law, is [152-154]:

𝑚_𝑒𝑑𝒗

𝑑𝑡 = − 𝑬 (2-3)

𝑚_𝑒 is the mass of the electron, 𝒗 is the average electronic velocity and is the proton charge.

According to the fourth assumption of Drude Model, the electrons are at thermal equilibrium with their surroundings before the electric field is applied. Since there is no transfer of the electrical current in the absence of externally applied electromagnetic fields, the average electronic velocity 𝒗 equals zero. When an electric field E is applied, the electrons will achieve a new thermal equilibrium with their surroundings in time τ. Therefore the average electronic velocity in an external electric field E is:

𝒗_𝐸 =− 𝑬

𝑚_𝑒 (2-4)

With the average electronic velocity, we can calculate the current density J. Consider an area A perpendicular to the electric field. The amount of charge passing through the area per unit time is:

− 𝑛𝒗_𝐸𝐴

n is the conduction electron density, that is, the number of conduction electrons per unit volume.

Therefore, the current density J is:

𝐽 = − 𝑛𝒗_𝐸 (2-5)

Now with the Eq.(2-4) we can get:

𝐽 =𝑛 ²

𝑚_𝑒 𝑬 (2-6)

Consider the Eq.(2-2) we know that:

𝜌 = 𝑚_𝑒

𝑛 ² (2-7)

16

The Drude model thus explains the electrical resistivity quantitatively. In Eq.(2-7), 𝑚_𝑒 and are physical constants, so the electrical resistivity is determined by the conduction electron density n and relaxation time . n is calculated by assuming that every atom contributes Z conduction electrons (the valence electrons) [153] when atoms are brought together to form metal. Then n is calculated:

𝑛 = 𝑁_𝐴𝑍𝜌_𝑚

𝑀 (2-8)

𝑁_𝐴 is the Avogadro’s number, 𝜌_𝑚 is the density of the solid in kg/m³, M is the atomic mass in kilograms per atom. The relaxation time is calculated by applying the experimental data into the Eq.(2-7). Table 2-2 gives the conduction electron density n, relaxation time and resistivity 𝝆 of selected elements. The Drude model does not seem to make any real predictions, because it determines resistivity only at the expense of introducing another unknown parameter, the relaxation time . It does, however, frame electrical resistivity in the terms that will be used later for more detailed calculation, as a balance between the force -eE causing electrons to accelerate, with the scattering events encoded in that causes them to decelerate [155].

Table 2-2 The conduction electron density, relaxation time and resistivity of selected elements at 273 K [152].

Element n (10²²/cm³) (10^-14 s) 𝜌 (mΩ∙cm)

Li 4.70 0.88 8.55

Na 2.65 3.20 4.20

K 1.40 4.10 6.10

Rb 1.15 2.80 11.00

Cs 0.91 2.10 18.8

Cu 8.47 2.70 1.56

Ag 5.86 4.00 1.51

Au 5.90 3.00 2.04

Mg 8.61 1.10 3.90

Fe 17.00 0.24 8.90

Zn 13.20 0.49 5.50

Al 18.10 0.80 2.45

17 The Drude Model was a great success at that time; it could explain Ohm’s law and the Hall effect. Drude also “successfully” explained the Wiedemann–Franz law quantitatively using his model. Despite its great success, it fails to explain the disparity between the expected heat capacities of metals compared to insulators. In addition, the Drude model fails to explain the existence of apparently positive charge carriers, as demonstrated by the positive Hall effect of certain materials [153]. Sommerfeld put forth the free electron model by combining the Drude model and quantum mechanical Fermi–Dirac statistics. It resolves the major problems of the classical Drude model [156, 157]. Nevertheless, the successes of the Drude model were considerable and it is still used today as a quick practical way to form simple pictures and rough estimates of properties.

2.2.2 Matthiessen’s rule 2.2.2.1 Matthiessen’s rule

Before Drude proposed his model, studies on electrical resistivity were focused on the influence of impurity concentration and temperature. Matthiessen and Vogt measured the resistivity of a series of two-component mixtures of metals between 0 °C and 100 °C [158].

Based on the results, they concluded that the temperature derivative of the resistivity ρp(T) of an ideally pure metal could be closely approximated using the relation:

𝑑𝜌_𝑎(𝑐, 𝑇)

𝑑𝑇 =𝑑𝜌_𝑝(𝑇)

𝑑𝑇 (2-9)

where 𝜌_𝑎(𝑐, 𝑇) is the resistivity of a dilute alloy containing a concentration c of impurity.

Matthiessen and Vogt suggested that Eq.(2-9) can be applied at any temperature, then they integrated it, starting at 0 K, to obtain:

𝜌_𝑎(𝑐, 𝑇) = 𝜌_𝑝(𝑇) + 𝜌₀(𝑐) (2-10) where 𝜌₀(𝑐) is the impurity produced resistivity at 0 K. Eq.(2-10) is the so-called

“Matthiessen’s Rule”. According to Matthiessen’s Rule, the total resistivity of a dilute alloy is divided into two parts. The first part, ρp(T) called the ideal resistivity, is temperature-dependent and is independent of the impurity concentration. It is arising from the scattering of electrons by lattice waves or phonons. The second part, 𝜌₀(𝑐) called the residual resistivity, is a constant that relies on the impurity concentration. It is caused by the scattering of electrons by impurity atoms. Another way to express Matthiessen’s Rule is that the total resistivity is produced by

18

the two scatters of electrons, lattice vibrations and impurities, independent and additive [151].

In fact, Matthiessen’s Rule is never exactly valid. There are always deviations from Matthiessen’s Rule. However, in many cases, the deviations are small compared to either 𝜌_𝑝(𝑇) or 𝜌₀(𝑐), so Matthiessen’s Rule represents quite a good approximation to the experimental data [159, 160]. Fig. 2-3 is the resistivity of annealed and cold-worked (deformed) copper-containing various amounts of Ni in atomic percentage. According to Matthiessen’s rule, 𝜌_𝐼 is caused by the concentration of Ni in the Cu matrix and is temperature-independent, so it simply shifts up the 𝜌 versus T curve of pure Cu by an amount proportional to the Ni content, 𝜌_𝐼∝ NNi, where NNi is the Ni impurity concentration. 𝜌_𝑇 is resistivity caused by thermal vibrations and therefore is temperature-dependent, as shown in Fig. 2-3, from 80 K to 300 K, the resistivity increases with the temperature.

0 50 100 150 200 250 300

0 10 20 30 40 50 60

10-9 Ohmzm

_CW

_

_

pure Cu (annealed) pure Cu (deformed) 1.12 at.% Ni (annealed) 1.12 at.% Ni (deformed) 2.16 at.% Ni (annealed) 3.32 at.% Ni (annealed)

T / K

Fig. 2-3 Electrical resistivity of annealed and cold-worked (deformed) copper alloys.

𝜌_𝑇 is resistivity caused by thermal vibrations, 𝜌_𝐼 caused by Ni solute, 𝜌_𝑐𝑤 caused by cold- work [161, 162].

It should be noted that in Fig. 2-3, the deformed alloy, including pure Cu and Cu-1.12 % Ni, has a higher resistivity compared to the annealed alloy with the same composition. This is due to the higher concentration of dislocations in the deformed alloy. During the deformation, large numbers of dislocations are introduced into the alloy and dislocations can also scatter the conduction electrons and increase the resistivity, 𝜌_𝑐𝑤. Therefore, compared to the well- annealed alloy, the deformed alloy has a higher resistivity. If we treat the dislocations as a type of the “impurity” then the total residual resistivity, 𝜌₀(𝑐), is the sum of 𝜌_𝐼 and 𝜌_𝑐𝑤 and the Matthiessen’s rule is still correct in this term.

19 2.2.2.2 Deviations from Matthiessen’s Rule

Despite the good approximation of Matthiessen’s rule to many experimental data, deviations from Matthiessen’s Rule also exist in many alloys [163-167]. Fig. 2-4 shows the deviation of Matthiessen’s Rule in SrRuO3 and CaRuO3 alloys [168]. ∆𝜌_𝑖𝑟𝑟 is the resistivity caused by electron irradiation introduced point defects. According to the Matthiessen’s Rule, the point defects could be treated as some kind of “impurities” and the resistivity caused by them should be temperature-independent. However, Fig. 2-4 shows a negative deviation from Matthiessen’s rule in these two alloys. According to the author [168], this deviation from Matthiessen’s Rule was due to the marked anisotropic scattering of the electrons together with their relatively short mean free path. Deviation from Matthiessen’s Rule due to the interacting dislocations [169]

also has been observed. Generally, Matthiessen’s Rule is correct when assuming the electron- impurity scattering is temperature-independent. However, this could be wrong for some reasons [170]:

1) The addition of the impurities changed the phonon spectrum of the matrix;

2) The added impurities perturbed the phonon distribution;

3) The anisotropy of the relaxation times for phonon and impurities scattering.

Fig. 2-4 Deviation from Matthiessen’s rule in SrRuO3 and CaRuO3 alloys.

Reprinted from reference [168] with permission from EDP Science.

2.2.3 Influencing Factors of electrical resistivity

In Eq.(2-7), the parameters 𝑚_𝑒, and n are constants for a certain metal. Therefore, the resistivity is mainly determined by the relaxation time. The relaxation time, which means the average time between two collisions of an electron, is influenced by the lattice defects, impurity atoms and temperature. Consequently, resistivity is also affected by these factors.

20

2.2.3.1 Effect of lattice imperfection

Lattice imperfections are classified into point defects, linear defects and planar defects according to the geometry or dimensionality of the defect [171]. In general, any kind of lattice imperfection will destroy the periodicity of an ideal crystal and increase the scattering of the electrons. Therefore, the resistivity is increased by any kind of lattice imperfection. Among all the lattice imperfections, vacancy, dislocation and grain boundaries are the most common ones.

2.2.3.1.1 Vacancy

A vacancy is expected to cause a lattice relaxation in its immediate vicinity, which gives rise to a change of the crystal lattice parameter [172]. This relaxation leads to the detriment of the periodicity of the lattice and hence increases the scattering of the conduction electrons. The resistivity is then increased due to the increased scattering. The influence of vacancies on the resistivity is considered temperature-independent; however, the vacancy itself is temperature- dependent, the number density of vacancies increases with temperature. Therefore, the total contribution of vacancies to resistivity is temperature-dependent. The vacancy contribution to the resistivity is negligibly small at room and cryogenic temperatures and becomes significant at temperatures approaching the melting point [173].

2.2.3.1.2 Dislocations and grain boundaries

In contrast to vacancies, dislocations and grain boundaries are thought to be temperature- independent factors. Dislocations, both the edge and screw dislocations, will introduce a strain field due to the stretching or compressing of bonds [161]. This strain field destroys the periodicity of the crystal and therefore increase the resistivity. Grain boundaries can be treated as an aggregation of broken bonds, voids, vacancies, strained bonds and interstitial-type atoms [161]. Therefore, the arrangement of atoms in the grain boundary region is disordered and the disordered arrangement will increase the scattering probability of the electrons. Hence, the resistivity is increased.

2.2.3.2 Influence of solution element

The effects of alloying elements on resistivity vary with the contents. In a dilute alloy, where the content of the alloying element is restricted to less than 1 or 2 at. %, the increment of resistivity is almost proportional to the atomic percentage. According to Linder’s rule, ∆ρ is proportional to the square of the excess charge on the impurity,