Thermal Spreading and Contact Resistance

Yuri S. Muzychka is a Professor of Mechanical Engineering at Memorial University of Newfoundland, Canada. He is a Fellow of ASME, CSME, and the Engineering Institute of Canada (EIC) and has published over 250 journal and conference proceedings papers, in addition to three handbook chapters. M. Michael Yovanovich is a Distinguished Professor Emeritus at the University of Waterloo, Canada. He is a fellow of ASME, CSME, AIAA, AAAS, and RSC. He has published seven handbook chapters and over 350 journal and conference proceedings papers, and has given over 150 keynote lectures.

About the Authors xv

Preface xvi

Acknowledgments xix

Nomenclature xx

1 Fundamental Principles of Thermal Spreading Resistance 1

1.1 Applications 2

1.2 Semi-Infinite Regions, Flux Tubes, Flux Channels, and Finite Spreaders 4

1.3 Governing Equations and Boundary Conditions 6

1.3.1 Source Plane Conditions 6

1.3.2 Sink Plane Conditions 7

1.3.3 Interface Conditions 8

1.4 Thermal Spreading Resistance 8

1.4.1 Half-Space Regions 8

1.4.2 Semi-Infinite Flux Tubes and Channels 10

1.4.3 Finite Disks and Channels 11

1.5 Solution Methods 11

1.6 Summary 12

References 12

2 Thermal Spreading in Isotropic Half-Space Regions 15

2.1 CircularAreaonaHalf-Space 15

2.1.1 Isothermal Circular Source 16

2.1.2 Isoflux Circular Source 17

2.1.3 Parabolic Flux Circular Source 19

2.1.4 Summary of Circular Source Thermal Spreading Resistance 20

2.2 Elliptical Area on a Half-Space 20

2.2.1 Isothermal Elliptical Source 20

2.2.2 Isoflux Elliptical Source 22

2.2.3 Parabolic Flux Elliptical Source 23

2.3 Method of Superposition of Point Sources 25

2.3.1 Application to a Circular Source 26

2.3.2 Application to Triangular Source Areas 28

2.4 Rectangular Area on a Half-Space 29

2.4.1 Isothermal Rectangular Area 29

2.4.2 Isoflux Rectangular Source 30

2.5 Spreading Resistance of Symmetric Singly Connected Areas: The Hyperellipse 33

2.6 Regular Polygonal Isoflux Sources 34

2.7 Additional Results for Other Source Shapes 36

2.7.1 Triangular Source 36

2.7.2 Rhombic Source 36

2.7.3 Rectangular Source with Rounded Ends 37

2.7.4 Rectangular Source with Semicircular Ends 37

2.8 Model for an Arbitrary Singly Connected Heat Source on a Half-Space 38

2.9 Circular Annular Area on a Half-Space 40

2.9.1 Isothermal Circular Annular Ring Source 40

2.9.2 Isoflux Circular Annular Ring Source 40

2.10 Other Doubly Connected Areas on a Half-Space 41

2.11 Problems with Source Plane Conductance 42

2.11.1 Isoflux Heat Source on a Convectively Cooled Half-Space 42

2.11.2 Effect of Source Contact Conductance on Spreading Resistance 44

2.12 Circular Area on Single Layer (Coating) on Half-Space 45

2.12.1 Equivalent Isothermal Circular Contact 45

2.12.2 Isoflux Circular Contact 47

2.12.3 Isoflux, Equivalent Isothermal, and Isothermal Solutions 47

2.12.3.1 Isoflux Contact Area 47

2.12.3.2 Equivalent Isothermal Contact Area 48

2.12.3.3 Isothermal Contact Area 48

2.13 Thermal Spreading Resistance Zone: Elliptical Heat Source 48

2.14 Temperature Rise of Multiple Isoflux Sources 52

2.14.1 Two Coplanar Isoflux Circular Sources 52

2.15 Temperature Rise in an Arbitrary Area 56

2.15.1 Temperature Rise at Arbitrary Point 56

2.15.2 Average Temperature Rise 57

2.16 Superposition of Isoflux Circular Heat Sources 58

2.16.1 Nine Coplanar Circles on Square Cluster 61

2.16.2 Five Coplanar Circles on Square Cluster 62

2.16.3 Four Coplanar Circles on Triangular Cluster 63

2.17 Superposition of Micro- and Macro-Spreading Resistances 64

References 68

3 Circular Flux Tubes and Disks 71

3.1 Semi-Infinite Flux Tube 71

3.1.1 Isothermal Source on a Flux Tube 76

3.2 Finite Disk with Sink Plane Conductance 77

3.2.1 Distributed Heat Flux over Source Area 81

3.3 Compound Disk 82

3.3.1 Special Limits in the Compound Disk Solution 85

3.4 Multilayered Disks 85

3.5 Flux Tube with Circular Annular Heat Source 88

3.6 Flux Tubes and Disks with Edge Conductance 90

3.7 Spreading Resistance for an Eccentric Source on a Flux Tube 93

3.8 Thermal Spreading with Variable Conductivity Near the Contact Surface 94

3.9 Effect of Surface Curvature on Thermal Spreading Resistance in a Flux Tube 97

References 99

4 Rectangular Flux Channels 103

4.1 Two-Dimensional Semi-Infinite Flux Channel 104

4.1.1 Variable Heat Flux Distributions 106

4.2 Three-Dimensional Semi-Infinite Flux Channel 108

4.2.1 Correlation Equations for Various Combinations of Source Areas and Boundary Conditions 110

4.3 Finite Two- and Three-Dimensional Flux Channels 111

4.4 Compound Two- and Three-Dimensional Flux Channels 115

4.4.1 Special Limiting Cases for Rectangular Flux Channels 118

4.5 Finite Two- and Three-Dimensional Flux Channels with Eccentric Heat Sources 120

4.6 Rectangular Flux Channels with Edge Conductance 124

4.7 Multilayered Rectangular Flux Channels 126

4.8 Rectangular Flux Channel with an Elliptic Heat Source 128

4.9 Spreading in a Curved Flux Channel (Annular Sector) 130

4.10 Effect of Surface Curvature on Thermal Spreading Resistance in a Two-Dimensional Flux Channel 134

References 135

5 Orthotropic Media 137

5.1 Heat Conduction in Orthotropic Media 137

5.2 Circular Source on a Half-Space 141

5.3 Single-Layer Flux Tubes 143

5.3.1 Circular Flux Tubes with Edge Cooling 144

5.4 Single-Layer Rectangular Flux Channel 144

5.4.1 Rectangular Flux Channels with Edge Cooling 146

5.5 Multilayered Orthotropic Spreaders 147

5.5.1 Circular Flux Tubes 148

5.5.2 Multilayered Orthotropic Flux Channels 151

5.5.3 Multilayered Orthotropic Flux Channels with an Eccentric Source 153

5.6 General Multilayered Rectangular Orthotropic Spreaders 153

5.6.1 Coordinate Transformations for Fully Orthotropic Media 155

5.6.2 General Solution for K X ≠ K Y ≠ K Z 156

5.6.3 Total Thermal Resistance 159

5.7 Measurement of Orthotropic Thermal Conductivity 160

References 163

6 Multisource Analysis for Microelectronic Devices 167

6.1 Multiple Heat Sources on Finite Isotropic Spreaders 168

6.1.1 Single Source Surface Temperature Distribution 169

6.1.2 Multisource Surface Temperature Distribution 170

6.2 Influence Coefficient Method 172

6.2.1 Thermal Resistance 174

6.2.2 Source Plane Convection 174

6.3 Extension to Compound, Orthotropic, and Multilayer Spreaders 175

6.3.1 Compound Media 175

6.3.2 Orthotropic Spreaders 177

6.3.3 Multilayer Isotropic/Orthotropic Spreaders 178

6.4 Non-Fourier Conduction Effects in Microscale Devices 181

6.5 Application to Irregular-Shaped Heat Sources 185

References 187

7 Transient Thermal Spreading Resistance 189

7.1 Transient Spreading Resistance of an Isoflux Source on an Isotropic Half-Space 189

7.1.1 Transient Spreading Resistance of an Isoflux Circular Area 190

7.1.2 Transient Spreading Resistance of an Isoflux Strip on a Half-Space 193

7.1.3 Transient Spreading Resistance of an Isoflux Hyperellipse 194

7.1.4 Transient Spreading Resistance of Isoflux Regular Polygons 194

7.1.5 Universal Time Function 195

7.2 Transient Spreading Resistance of an Isothermal Source on a Half-Space 195

7.3 Models for Transient Thermal Spreading in a Half-Space 199

7.4 Transient Spreading Resistance Between Two Half-Spaces in Contact Through a Circular Area 201

7.5 Transient Spreading in a Two-Dimensional Flux Channel 202

7.6 Transient Spreading in a Circular Flux Tube from an Isoflux Source 203

7.7 Transient Spreading in a Circular Flux Tube from an Isothermal Source 205

7.8 Models for Transient Thermal Spreading in Circular Flux Tubes 207

References 211

8 Applications with Nonuniform Conductance in the Sink Plane 213

8.1 Applications with Nonuniform Conductance 213

8.1.1 Distributed Heat Transfer Coefficient Models 214

8.1.2 Mixed-Boundary Conditions in the Source Plane 216

8.1.3 Least Squares Approximation 217

8.2 Finite Flux Channels with Variable Conductance 218

8.2.1 Two-Dimensional Flux Channel 218

8.2.2 Three-Dimensional Flux Channel 221

8.3 Finite Flux Tube with Variable Conductance 225

References 228

9 Further Applications of Spreading Resistance 231

9.1 Moving Heat Sources 231

9.1.1 Governing Equations 232

9.1.2 Asymptotic Limits 233

9.1.3 Stationary and Moving Heat Source Limits 234

9.1.3.1 Stationary Heat Sources (Pe → 0) 234

9.1.3.2 Moving Heat Sources (Pe → ∞) 236

9.1.4 Analysis of Real Contacts 238

9.1.4.1 Effect of Contact Shape 238

9.1.4.2 Models for All Peclet Numbers 240

9.1.5 Prediction of Flash Temperature 241

9.2 Problems Involving Mass Diffusion 243

9.2.1 Mass Transport from a Circular Source on a Half-Space 244

9.2.2 Diffusion from Other Source Shapes 245

9.2.2.1 Elliptic Source 246

9.2.2.2 Rectangular Source 246

9.3 Mass Diffusion with Chemical Reaction 246

9.3.1 Diffusion from a 2D Strip Source with Chemical Reaction 247

9.3.2 Circular Source on a Disk with Chemical Reaction 249

9.3.3 Diffusion from a Rectangular Source with Chemical Reaction 251

9.4 Diffusion Limited Slip Behavior: Super-Hydrophobic Surfaces 254

9.4.1 Circular and Square Pillars 256

9.4.1.1 Circular/Square 256

9.4.1.2 Ridges 257

9.4.2 Rectangular and Elliptical Pillars for φ s → 0 258

9.4.3 Effect of Meniscus Curvature 261

9.5 Problems with Phase Change in the Source Region (Solidification) 261

9.6 Thermal Spreading with Temperature-Dependent Thermal Conductivity 263

9.6.1 Kirchoff Transform 263

9.6.2 Thermal Conductivity Models 265

9.6.3 Application for Thermal Spreading Resistance in a Rectangular Flux Channel 266

9.7 Thermal Spreading in Spherical Domains 268

9.7.1 Thermal Spreading in Hollow Spherical Shells 268

9.7.2 Thermal Spreading in a Hollow Hemispherical Shell with Convection on the Interior Boundary 271

References 272

10 Introduction to Thermal Contact Resistance 275

10.1 Thermal Contact Resistance 275

10.2 Types of Joints or Interfaces 278

10.2.1 Conforming Rough Solids 279

10.2.2 Nonconforming Smooth Solids 281

10.2.3 Nonconforming Rough Solids 281

10.2.4 Single Layer Between Two Conforming Rough Solids 281

10.3 Parameters Influencing Contact Resistance or Conductance 282

10.4 Assumptions for Resistance and Conductance Model Development 283

10.5 Measurement of Joint Conductance and Thermal Interface Material Resistance 283

References 285

11 Conforming Rough Surface Models 287

11.1 Conforming Rough Surface Models 288

11.2 Plastic Contact Model for Asperities 290

11.2.1 Vickers Micro-hardness Correlation Coefficients 293

11.2.2 Dimensionless Contact Conductance: Plastic Deformation 293

11.3 Elastic Contact Model for Asperities 294

11.3.1 Dimensionless Contact Conductance: Elastic Deformation 295

11.4 Conforming Rough Surface Model: Elastic–Plastic Asperity Deformation 296

11.4.1 Correlation Equations for Dimensionless Contact Conductance: Elastic–Plastic Model 297

11.5 Radiation Resistance and Conductance for Conforming Rough Surfaces 300

11.6 Gap Conductance for Large Parallel Isothermal Plates 302

11.7 Gap Conductance for Joint Between Conforming Rough Surfaces 303

11.8 Joint Conductance for Conforming Rough Surfaces 306

11.9 Joint Conductance for Conforming Rough Surfaces: Scale Analysis Approach 310

11.10 Joint Conductance Enhancement Methods 317

11.10.1 Metallic Coatings and Foils 317

11.10.2 Ranking Metallic Coating Performance 325

11.10.3 Elastomeric Inserts 326

11.10.4 Thermal Greases and Pastes 328

11.10.5 Phase Change Materials (PCM) 332

11.11 Thermal Resistance at Bolted Joints 332

References 332

12 Contact of Nonconforming Smooth Solids 337

12.1 Joint Resistances of Nonconforming Smooth Solids 338

12.2 Point Contact Model 338

12.3 Local Gap Thickness 341

12.4 Contact Resistance of Isothermal Elliptical Contact Area 341

12.5 Elastogap Resistance Model 342

12.6 Joint Radiative Resistance 344

12.7 Joint Resistance of Sphere-Flat Contact 345

12.7.1 Joint Resistance for Sphere-Flat in Vacuum 345

12.7.2 Effect of Gas Pressure on Joint Resistance of a Sphere-Flat Contact 346

12.8 Joint Resistance for Contact of a Sphere and Layered Substrate 349

12.9 Joint Resistance for Elastic–Plastic Contact of Hemisphere and Flat in Vacuum 352

12.9.1 Alternative Constriction Parameter for Hemisphere 353

12.10 Ball Bearing Resistance 356

12.11 Line Contact Models 356

12.11.1 Contact Strip and Local Gap Thickness 356

12.11.2 Contact Resistance at Line Contact 357

12.11.3 Gap Resistance at Line Contact 358

12.11.4 Joint Resistance at Line Contact 358

12.12 Joint Resistance of Nonconforming Rough Surfaces 359

12.13 System for Nonconforming Rough Surface Contact 360

12.13.1 Vickers Micro-hardness Model 360

12.13.2 Scale Analysis Results 361

12.13.3 Contact of Smooth Hemisphere and Rough Flat 363

12.13.4 General Micro–Macro Spreading Resistance Model 364

12.13.5 Comparisons of Nonconforming Rough Surface Model with Vacuum Data 365

12.13.6 General Model Obtained from Scaling Analysis and Data 366

12.14 Joint Resistance of Nonconforming Rough Surface and Smooth Flat Contact 370

12.14.1 Micro-gap Thermal Resistance 371

12.14.2 Macro-gap Thermal Resistance 372

References 374

Appendix A Special Functions 379

A. 1 Gamma and Beta Function 379

A.. 1 Gamma Function 379

A.1. 2 Beta Function 382

A. 2 Error Function 382

A. 3 Bessel Functions 384

A.3. 1 Bessel Functions of the First and Second Kind 385

A.3. 2 Zeroes of the Bessel Functions 387

A.. 3 Modified Bessel Functions of the First and Second Kind 387

A. 4 Elliptic Integrals 389

A. 5 Legendre Functions 391

A. 6 Hypergeometric Function 392

A.6. 1 Relationship to Other Functions 393

References 393

Appendix B Hardness 395

B. 1 Micro- and Macro-hardness Indenters 395

B.. 1 Brinell and Meyer Macrohardness 395

B.1. 2 Rockwell Macro-hardness 397

B.1. 3 Knoop Micro-hardness Indenter and Test 398

B.1. 4 Vickers Micro-hardness Indenter and Test 399

B.1. 5 Berkovich Micro and Nano Hardness Indenter and Nano Hardness Tests 399

B. 2 Micro- and Macro-hardness Tests and Correlations 400

B.2. 1 Direct Approximate Method 402

B.. 2 Vickers Micro-hardness Correlation Equations 403

B. 3 Correlation Equations for Vickers Coefficients 406

B. 4 Temperature Effects on Vickers and Brinell Hardness 407

B.4. 1 Temperature Effects on Yield Strength and Vickers Micro Hardness of SS 304L 407

B.4. 2 Temperature Effect on Brinell Hardness 407

B.4. 3 Temperature Effect on Vickers Micro-hardness and Correlation Coefficients 409

B. 5 Nanoindentation Tests 411

References 416

Appendix C Thermal Properties 419

C.1 Thermal Properties of Solids 420

C.2 Thermal Conductivity of Gases 420

C.3 Resistance of Thermal Interface Materials (TIMs) 423

References 423

Index 425