Metal Nanopowders -  - ebook

Metal Nanopowders ebook

0,0
599,99 zł

Opis

Written with both postgraduate students and researchers in academia and industry in mind, this reference covers the chemistry behind Metal Nanopowders, including production, characterization, oxidation and combustion. The contributions from renowned international scientists working in the field detail applications in technologies, scale-up processes and safety aspects surrounding their handling and storage.

Ebooka przeczytasz w aplikacjach Legimi na:

Androidzie
iOS
czytnikach certyfikowanych
przez Legimi
Windows
10
Windows
Phone

Liczba stron: 767




Table of Contents

Cover

Related Titles

Title Page

Copyright

Foreword

List of Contributors

Introduction

Chapter 1: Estimation of Thermodynamic Data of Metallic Nanoparticles Based on Bulk Values

1.1 Introduction

1.2 Thermodynamic Background

1.3 Size-Dependent Materials Data of Nanoparticles

1.4 Comparison of Experimental and Calculated Melting Temperatures

1.5 Comparison with Data for the Entropy of Melting

1.6 Discussion of the Results

1.7 Conclusions

1.A Appendix: Zeros and Extrema of the Free Enthalpy of Melting

Chapter 2: Numerical Simulation of Individual Metallic Nanoparticles

2.1 Introduction

2.2 Molecular Dynamics Simulation

2.3 Size-Dependent Properties

2.4 Sintering Study of Two Nanoparticles

2.5 Oxidation of Nanoparticles in the Presence of Oxygen

2.6 Heating and Cooling of a Core–Shell Structured Particle

2.7 Chapter Summary

Chapter 3: Electroexplosive Nanometals

3.1 Introduction

3.2 Electrical Explosion of Wires Technology for Nanometals Production

3.3 Conclusion

Acknowledgments

Chapter 4: Metal Nanopowders Production

4.1 Introduction

4.2 EEW Method of Nanopowder Production

4.3 Recondensation NP-Producing Methods: Plasma-Based Technology

4.4 Characteristics of Al Nanopowders

4.5 Nanopowder Chemical Passivation

4.6 Microencapsulation of Al Nanoparticles

4.7 The Process of Producing Nanopowders of Aluminum by Plasma-Based Technology

Chapter 5: Characterization of Metallic Nanoparticle Agglomerates

5.1 Introduction

5.2 Description of the Structure of Nanoparticle Agglomerates

5.3 Experimental Techniques to Characterize the Agglomerate Structure

5.4 Mechanical Stability

5.5 Thermal Stability

5.6 Rate-Limiting Steps: Gas Transport versus Reaction Velocity

5.7 Conclusions

Acknowledgments

Chapter 6: Passivation of Metal Nanopowders

6.1 Introduction

6.2 Theoretical and Experimental Background

6.3 Characteristics of Passivated Particles

6.4 Conclusion

Acknowledgment

Chapter 7: Safety Aspects of Metal Nanopowders

7.1 Introduction

7.2 Some Basic Phenomena of Oxidation of Nanometal Particles in Air

7.3 Determination of Fire Hazards of Nanopowders

7.4 Sensitivity against Electrostatic Discharge

7.5 Ranking of Nanopowders According to Hazard Classification

7.6 Demands for Packing

Chapter 8: Reaction of Aluminum Powders with Liquid Water and Steam

8.1 Introduction

8.2 Experimental Technique for Studying Reaction Al Powders with Liquid and Gaseous Water

8.3 Oxidation of Aluminum Powder in Water Vapor Flow

8.4 Nanopowders Passivated with Coatings on the Base of Aluminum Carbide

8.5 Study of Al Powder/H

2

O Slurry Samples Heated Linear in “Open System” by STA

8.6 Ultrasound (US) and Chemical Activation of Metal Aluminum Oxidation in Liquid Water

8.7 Conclusion

Acknowledgments

Chapter 9: Nanosized Cobalt Catalysts for Hydrogen Storage Systems Based on Ammonia Borane and Sodium Borohydride

9.1 Introduction

9.2 A Study of Nanosized Cobalt Borides by Physicochemical Methods

9.3 Conclusions

Acknowledgment

Chapter 10: Reactive and Metastable Nanomaterials Prepared by Mechanical Milling

10.1 Introduction

10.2 Mechanical Milling Equipment

10.3 Process Parameters

10.4 Material Characterization

10.5 Ignition and Combustion Experiments

10.6 Starting Materials

10.7 Mechanically Alloyed and Metal–Metal Composite Powders

10.8 Reactive Nanocomposite Powders

10.9 Conclusions

Chapter 11: Characterizing Metal Particle Combustion In Situ: Non-equilibrium Diagnostics

11.1 Introduction

11.2 Ignition and Combustion of Solid Materials

11.3 Aluminum Reaction Mechanisms

11.4 The Flame Tube

11.5 Flame Temperature

11.6 Conclusions

Acknowledgment

Chapter 12: Characterization and Combustion of Aluminum Nanopowders in Energetic Systems

12.1 Fuels in Energetic Systems: Introduction and Literature Survey

12.2 Thermochemical Performance of Energetic Additives

12.4 Mechanical and Rheological Behavior with Nanopowders

Index

End User License Agreement

Pages

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

58

57

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

187

188

186

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

359

358

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

Guide

Table of Contents

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 1.6

Figure 1.7

Figure 1.8

Figure 1.9

Figure 1.10

Figure 1.11

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Figure 2.17

Figure 2.18

Figure 2.19

Figure 2.20

Figure 2.21

Figure 2.22

Figure 2.23

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

Figure 8.12

Figure 8.13

Figure 8.14

Figure 8.15

Figure 8.16

Figure 8.17

Figure 8.18

Figure 8.19

Figure 8.20

Figure 8.21

Figure 8.22

Figure 8.23

Figure 8.24

Figure 8.25

Figure 8.26

Figure 8.27

Figure 8.28

Figure 8.29

Figure 8.30

Figure 8.31

Figure 9.1

Figure 9.2

Figure 9.3

Figure 9.4

Figure 9.5

Figure 9.6

Figure 9.7

Figure 9.8

Figure 9.9

Figure 9.10

Figure 9.11

Figure 9.12

Figure 9.13

Figure 9.14

Figure 9.15

Figure 9.16

Figure 9.17

Figure 9.18

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 10.6

Figure 10.7

Figure 10.8

Figure 10.9

Figure 10.10

Figure 10.11

Figure 10.12

Figure 10.13

Figure 10.14

Figure 10.15

Figure 10.16

Figure 10.17

Figure 10.18

Figure 10.19

Figure 10.20

Figure 10.21

Figure 10.22

Figure 10.23

Figure 10.24

Figure 10.25

Figure 10.26

Figure 10.27

Figure 10.28

Figure 10.29

Figure 10.30

Figure 10.31

Figure 10.32

Figure 10.33

Figure 10.34

Figure 10.35

Figure 10.36

Figure 10.37

Figure 10.38

Figure 10.39

Figure 10.40

Figure 10.41

Figure 10.42

Figure 10.43

Figure 10.44

Figure 10.45

Figure 10.46

Figure 10.47

Figure 11.1

Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5

Figure 11.6

Figure 11.7

Figure 12.1

Figure 12.2

Figure 12.3

Figure 12.4

Figure 12.5

Figure 12.10

Figure 12.11

Figure 12.12

Figure 12.13

Figure 12.14

Figure 12.15

Figure 12.16

Figure 12.17

Figure 12.18

Figure 12.19

Figure 12.20

Figure 12.21

Figure 12.22

Figure 12.23

Figure 12.24

Figure 12.25

Figure 12.26

Figure 12.27

Figure 12.28

Figure 12.29

Figure 12.30

Figure 12.31

Figure 12.32

Figure 12.33

Figure 12.34

Figure 12.35

List of Tables

Table 1.1

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 3.1

Table 4.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 6.6

Table 6.7

Table 6.8

Table 6.9

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Table 7.5

Table 8.1

Table 8.2

Table 9.1

Table 9.2

Table 9.3

Table 9.4

Table 9.5

Table 10.1

Table 11.1

Table 12.1

Table 12.2

Table 12.3

Table 12.4

Table 12.5

Table 12.6

Table 12.7

Table 12.8

Table 12.10

Table 12.12

Table 12.13

Table 12.14

Table 12.15

Table 12.16

Table 12.17

Table 12.18

Table 12.19

Related Titles

Koch, E.

Metal-Fluorocarbon Based Energetic Materials

2012

ISBN: 978-3-527-32920-5 (Also available in digital formats)

Agrawal, J.P.

High Energy Materials

Propellants, Explosives and Pyrotechnics

2010

ISBN: 978-3-527-32610-5 (Also available in digital formats)

Tjong, S.C.

Polymer Composites with Carbonaceous Nanofillers

Properties and Applications

2012

ISBN: 978-3-527-41080-4 (Also available in digital formats)

Teipel, U. (ed.)

Energetic Materials

Particle Processing and Characterization

2005

Print ISBN: 978-3-527-30240-6 (Also available in digital formats)

Metal NanopowdersProduction, Characterization, and Energetic Applications

Edited by

Alexander Gromov and Ulrich Teipel

 

 

 

 

 

The Editors

Prof. Dr. Alexander Gromov

Tomsk Polytechnic University

30 Lenin Prospekt

634050 Tomsk

Russia

and

Technical University Nürnberg

Georg-Simon-Ohm

Process Engineering Department

Wassertorstr. 10

90489 Nürnberg

Germany

Prof. Dr. Ulrich Teipel

Technical University Nürnberg

Georg-Simon-Ohm

Process Engineering Department

Wassertorstr. 10

90489 Nürnberg

Germany

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are \hbox{available} on the Internet at <http://dnb.d-nb.de>.

© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Print ISBN: 978-3-527-33361-5

ePDF ISBN: 978-3-527-68072-6

ePub ISBN: 978-3-527-68071-9

mobi ISBN: 978-3-527-68070-2

oBook ISBN: 978-3-527-68069-6

Foreword

Interest in studying the combustion of metal powders dramatically raised since Russian scientists Kondratyuk and Tsander suggested the use of metals as energetic additives to rocket fuels at the beginning of the twentieth century. Since that time, it is obvious that an increase in the dispersion of flammable substances participating in heterogeneous combustion processes leads to an increase in rate and heat of combustion. The major energy contribution belongs to the process of oxidation, which is also bound up with powder dispersion and purity. Burning of metal nanopowders is accompanied by new physical and chemical laws (such as high reactivity under heating, threshold phenomena, formation of nitrides in air), which allow to fully appreciate the advantages and disadvantages of nanopowders when used in fuel systems.

Widespread use of metal nanopowders is currently hampered by the lack of enough advanced technology for their preparation, certification, and standardization procedures, instability during storage, and subjective factors: the possible toxicity of nanopowders, investment risks, cost of nanotechnologies, and so on. Therefore, the main objective for the authors is to inform a wide readership of fundamental and applied studies on the processes of oxidation and combustion of metal nanopowders.

Prof. Dr.-Ing. George Manelis, Prof. Dr.-Ing. Hiltmar Schubert,Institute of Problems of Chemical Physics

,

Fraunhofer Institute of

Russian Academy of Science, Chernogolovka, Chemical Technology,

RussiaPfinztal, Germany

List of Contributors

Giovanni Colombo

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Luigi T. De Luca

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Stefano Dossi

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Edward L. Dreizin

University Heights

Otto H. York Department of Chemical, Biological, and Pharmaceutical Engineering

New Jersey Institute of Technology

Warren St

Newark

NJ 07102-1982

USA

N. Eisenreich

Institute of Problems of Chemical and Energetic Technologies

Russian Academy of Science

Socialisticheskaya str., 1

Byisk

Russia

Cory Farley

Texas Tech University

Mechanical Engineering Department

Corner of 7th and Boston Ave.

Lubbock

TX 79409-1021

USA

Marco Fassina

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Franz Dieter Fischer

Montanuniversität Leoben

Institute of Mechanics

Franz-Josef-Straße 18

A-8700 Leoben

Austria

Luciano Galfetti

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Alexander Gromov

Tomsk Polytechnic University

Lenin prospekt, 30

Tomsk

Russia

and

George Simon Ohm University of Applied Sciences

Processing Department

Wassertorstr 10

Nürnberg

Germany

Sh. Guseinov

State Research institute for Chemistry and Technology of Organoelement Compounds (GNIIChTEOS)

Shosse Entuziastov str., 38

Moscow

Russia

Alexander Il'in

Tomsk Polytechnic University

Lenin prospekt, 30

Tomsk

Russia

Keerti Kappagantula

Texas Tech University

Mechanical Engineering Department

Corner of 7th and Boston Ave.

Lubbock

TX 79409-1021

USA

Oksana V. Komova

Boreskov Institute of Catalysis SB RAS

Pr Akademika Lavrentieva 5

Novosibirsk

Russia

Larichev Mikhail Nikolaevich

V.L Talrose Institute for Energy Problems for Chemical Physics Russian Academy of Science

Leninsky prospect

bl 38/2:, 119334 Moscow

Russia

M. Lerner

Institute of Strength Physics and Material Science

Russian Academy of Science

8/2 Academicheskiy St.

Tomsk

Russia

Filippo Maggi

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Olga Nazarenko

Tomsk Polytechnic University

Lenin prospekt, 30

Tomsk

Russia

Olga V. Netskina

Boreskov Institute of Catalysis SB RAS

Pr Akademika Lavrentieva 5

Novosibirsk

Russia

Michelle Pantoya

Texas Tech University

Mechanical Engineering Department

Corner of 7th and Boston Ave.

Lubbock

TX 79409-1021

USA

Christian Paravan

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Julia Pautova

Tomsk Polytechnic University

Lenin prospekt, 30

Tomsk

Russia

Alice Reina

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

Mirko Schoenitz

University Heights

Otto H. York Department of Chemical, Biological, and Pharmaceutical Engineering

New Jersey Institute of Technology

Warren St

Newark

NJ, 07102-1982

USA

Valentina I. Simagina

Boreskov Institute of Catalysis SB RAS

Pr Akademika Lavrentieva 5

Novosibirsk

Russia

P.X. Song

Institute of Particle Science and Engineering

University of Leeds

Leeds, LS2 9JU UK National Institute of Clean-and-Low-Carbon Energy

Future Science & Technology Park

Changping District

Beijing 102209

China

Andrea Sossi

Dipartimento di Scienze e Tecnologie Aerospaziali, SPLab

Politecnico di Milano Campus Bovisa Sud

Via La Masa

I-20156 Milan

Italy

P. Storozhenko

State Research institute for Chemistry and Technology of Organoelement Compounds (GNIIChTEOS)

Shosse Entuziastov str., 38

Moscow

Russia

Ulrich Teipel

George Simon Ohm University of Applied Sciences

Processing Department

Wassertorstr 10

Nürnberg

Germany

Dmitry Tikhonov

Tomsk Polytechnic University

Lenin prospekt, 30

Tomsk

Russia

Dieter Vollath

Nano Consulting

Primelweg 3

D-76297 Stutensee

Germany

A. Vorozhtsov

Tomsk State University

Lenin str., 36

Tomsk

Russia

and

Institute of Problems of Chemical and Energetic Technologies

Russian Academy of Science

Socialisticheskaya str., 1

Byisk

Russia

Alfred P. Weber

Technical University of Clausthal

Institute of Particle Technology

Leibnizstrasse 19

D-38678 Clausthal-Zellerfeld

Germany

D.S. Wen

Institute of Particle Science and Engineering

University of Leeds

Leeds, LS2 9JU UK National Institute of Clean-and-Low-Carbon Energy

Future Science & Technology Park

Changping District

Beijing 102209

China

Introduction

Stabilization of low-dimension structures, especially nanosized ones, and their use in the heterogeneous chemical reactions as nanopowders allow considering high specific surface as an independent thermodynamic parameter along with the temperature, pressure, concentration of reactants, and so on. New characteristics of 2D nanomaterials are well known – the thermal conductivity of graphene (5000 W (m K)−1) with 1000 m2 g−1 specific surface exceeds those for metals in a factor 10 [1]. The use of the advantages of high specific surface of 3D nanostructures – nanopowders in catalysis, oxidation, and combustion results in high rates of heterogeneous reactions and reduction in activation energies of ignition due to the small size of solid reactants. The laws of classical chemistry and physics are little applicable to the analysis of processes with metal nanopowders. An example of such a system is the burning of the composition nanoAl/nanoMoO3 at the rate of about 1 km s−1 [2].

In USSR, metal ultrafine (in fact, nano-) powders with reproducible properties were first obtained during World War II. In the 1960s and 1970s, numerous works were carried out on metal nanopowder production by electrical explosion of wires [3], evaporation-and-condensation method [4], and the technologies of metal nanopowder application for nuclear synthesis in the USSR and the US. In 1977, the result of these works was published in Morokhov's book [5], where the methods for metal nanopowder production by thermal decomposition of salts were viewed. In Western Europe and the US, the term nanocrystalline material appeared and spinned off after the Gleiter's publication in 1980 [6].

Since the discovery by Yu. Kondratyuk and F. Tsander in 1910 [7], the possibilities of powdery metal being used as an additive in energetic materials and as the reagents for self-propagating high-temperature synthesis [8] were intensively studied. Several books (e.g., the work of Pokhil et al. [9] and Sammerfield [10]) were published, where the laws of combustion of micron-sized metal powders (5÷500 μm) in high-temperature oxidizing environments were discussed. The study of the laws of combustion of powdered metals was done mainly for Al, Be, Mg, Ti, Zr, and B. The lack of micron-sized metal powders were detected during the first test of metallized fuels in the 1940s: an agglomeration of particles (especially for aluminum and magnesium) in the heating zone of energetic material, a low degree of metal reaction in the vapor phase (incomplete combustion), significant biphasic loss of a specific impulse (15% or more for the compositions containing 20–25 wt% Al) [9].

In the 1970s, Zeldovich and Leipunsky et al. [11] showed one of the approaches to reduce this lack by using low-sized metallic particles for fuels and combustion catalysts, in particular, metal nanopowders. This book summarizes the efforts of several teams over the world to realize those ideas.

The revitalization of the use of metal nanopowders in materials science and engineering became further possible in the 1990s, when the technologies for the large-scale production of those materials became available. Nowadays, tons of rather inexpensive metal nanopowders are produced in several countries for different technological applications, while the problems of their standardization, storage, handling, toxicity, correct application, and so on, are still unsolved.

The idea of this book is also to show the true picture of the properties of metal nanopowders and, correspondingly, their application avenues. The “romantic atmosphere” around nanomaterials and metal nanopowders accordingly should be left in the twentieth century forever. Nanoparticles and, especially, metal nanoparticles are very “capricious” technological raw materials with metastable physical and chemical properties in many cases, because nanometals (in addition to small particle size) show very high reducing properties: nanoCu react similarly to bulk Zn – release the hydrogen from acids, nanoAl show the properties of bulky alkali metals – react with water under room temperature, and so on.

Special scientific and engineering interests represent the new fundamental laws of combustion for the metal nanopowders, analysis of the combustion regimes, and intermediate and final burning products reported in this book. Excited by the experimental works of Ivanov and Tepper [12], scientists worked in the direction of nanometals application in energetic materials intensively during the past decade and the most valuable results are presented in this book.

In conclusion, we want to underline that the study of industrially available metal nanopowders allowed opening previously unknown laws and they will open the significant application prospects in science and technology of the twenty-first century.

Alexander GromovUlrich Teipel

1. Seol, J.H., Jo, I., Moore, A.L., Lindsay, L., Aitken, Z.H., Pettes, M.T., Li, X., Yao, Z., Huang, R., Broido, D., Mingo, N., Ruoff, R.S., and Shi, L. (2010) Two-dimensional phonon transport in supported graphene.

Science

,

328

(5975), 213–216.

2. Bockmon, B.S., Pantoya, M.L., Son, S.F., and Asay, B.W. (2003) Burn rate measurements in nanocomposite thermites. Proceedings of the American Institute of Aeronautics and Astronautics Aerospace Sciences Meeting, Paper No. AIAA-2003-0241.

3. Chase, W.G. and Moore, H.K. (eds) (1962)

Exploding Wires

, Plenum Press, New York.

4. Gen, M.Ya. and Miller, A. (1981) A method of metal aerosols production. USSR Patent 814432. No. 11. p. 25.

5. Morokhov, I.D., Trusov, L.I., and Chizhik, S.P. (1977)

Ultradispersed Metal Medium

, Atomizdat, Moscow.

6. Gleiter, H. (1989) Nanocrystalline materials.

Prog. Mater. Sci.

,

33

(4), 223–315.

7. Gilzin, K.A. (1950) Rocket Engines, Moscow.

8. Merzhanov, A.G., Yukhvid, V.I., and Borovinskaya, I.P. (1980) Self-propagating high-temperature synthesis of cast refractory inorganic compounds.

Dokl. Akad. Nauk USSR

,

255

, 120.

9. Pokhil, P.F., Belyaev, A.F., Frolov, Y.V., Logachev, V.S., and Korotkov, A.I. (1972)

Combustion of Powdered Metals in Active Media

, Nauka, Moscow.

10. Sammerfield, M. (ed.) (1960)

Solid Propellant Rocket Research

, Academic Press, New York.

11. Zeldovich, Y.B., Leipunsky, O.I., and Librovich, V.B. (1975)

Theory of Non-Stationary Combustion of Powders

, Nauka, Moscow.

12. Ivanov, G.V. and Tepper, F. (1997) “Activated” aluminum as a stored energy source for propellants.

Int. J. Energetic Mater. Chem. Propul.

,

4

(1–6), 636–645.

1

Estimation of Thermodynamic Data of Metallic Nanoparticles Based on Bulk Values

Dieter Vollath and Franz Dieter Fischer

1.1 Introduction

It is a well-accepted fact: the temperature of phase transformation is particle-size dependent. In general, this dependency is described as

1.1

In Equation 1.1, the quantities and stand for the transformation temperature of nanoparticles and the bulk material, respectively, d for the particle diameter, and α is a constant value depending on the entropy of transformation and the difference of the surface energy in both phases [1]. The same description, as proved for phase transformations, was found to be valid for the enthalpy of phase transformations. As typical examples, experimental results obtained for aluminum particles were given by Eckert [2], for tin particles by Lai et al. [3], or by Suresh and Mayo [4, 5] on yttrium-doped zirconia particles.

The range of particle sizes where Equation 1.1 is valid is limited. In the case of larger particles, Coombes [6] has shown that these have a surface layer of about 3 nm, where melting starts. As long as this surface layer dominates the behavior of the particles, Equation 1.1 cannot be applied. The existence of such a surface layer was also shown by Chang and Johnson [7] by theoretical considerations, concluding that this surface layer is less ordered than the center of the particles. As it was shown by Kaptay [8], the thickness of this premelting layer can be estimated by the rules of classical thermodynamics. Therefore, the assumption of a surface layer where melting starts is well justified. Now, one may ask if there is also a lower limit of particle sizes, below which Equation 1.1 is not applicable. Experimental results suggest this. Figure 1.1 displays the melting temperature of gold nanoparticles according to Castro et al. [9]. In this graph, the melting temperature is plotted versus the inverse particle size. According to Equation 1.1, one has to expect a linear relation.

Figure 1.1Experimental data for the melting temperature of gold nanoparticles, according to Castro et al. [9], together with linear fits plotted versus the inverse particle size. This graph shows clearly two separated ranges of the melting temperature: at larger particles, a range following Equation 1.1 (Range I) and a second range with particle-size-independent temperature (Range II).

The experimental data of Castro et al. may be separated into two ranges: Range I, which follows Equation 1.1 and a separated Range II, which is far off from the expected value. A linear fit of the experimental data in both ranges delivers an intersection at an inverse particle size of 0.62 nm−1, which is equivalent to a particle size of 1.6 nm. Obviously, for particle sizes below this intersection, Equation 1.1 is no longer valid. Such a phenomenon or similar ones are quite often described; as, for example, in the case of sodium particles [10]. Well in line with the above-described phenomenon, found for gold and sodium particles, are experimental results of Shvartsburg and Jarrold [11] on small tin particles consisting of 19–31 atoms, exhibiting melting points significantly higher than those of the bulk material. Besides a reduction of the melting temperature, close to melting or crystallization, additional phenomena are observed. Oshima and Takayanagi [12] found in 6 nm tin particles crystalline embryos with sizes around 1.5 nm. It is remarkable that this size is in the range of the limitation of Equation 1.1, as was found in the case of the melting of gold particles.

1.2 Thermodynamic Background

A general analysis of these phenomena needs detailed quantum mechanical studies. However, in most cases, one is interested in just a first approach using thermodynamic data of metallic nanoparticles. It is the aim of this chapter to show a simplified approach in this direction. The most important tool for any analysis of this kind is classical thermodynamics. Certainly, as this tool describes continuous systems, such an approximation cannot deliver phenomena depending on the quantum nature of the cohesion energy of small particles, or, in other words, magic particle sizes, superatoms, or jellium shell concepts cannot be expected as the result. These phenomena are excluded.

To analyze phase transformations, a detailed knowledge of the thermodynamic data of the materials in question is necessary. In addition, in the case of nanoparticles, knowledge of the surface energy in both phases is of great importance. As typical and well-studied examples for a phase transformation, melting, and crystallization were selected. In the following considerations, for reasons of simplicity, the minor changes of geometry and density are neglected. Generally, in the proximity of the melting point, the difference of the free enthalpy

1.2a

at the temperature T are

1.2b

The quantity is the enthalpy and the entropy of melting, both with the additional subscript “nano” or “bulk.” The term stands for the difference in the surface energy in the liquid and solid states. The quantity represents the surface area per mol of nanoparticles. It is important to note that the quantities in Equations 1.2 are the differences of the thermodynamic quantities observed during the melting process.

1.3

In the case of bulk materials, the surface energy term of Equation 1.2b is generally neglected but it is of relevance in the case of nanoparticles.

For lack of better data, in most cases, the material data of the bulk material and are used for nanoparticles, too, yielding

1.4

Setting leads to the well-known reduction in the melting point of nanoparticles in comparison with the one of the bulk material, , as was described for the first time more than a hundred years ago by Pawlow [13] using , , and neglecting for the bulk material, and more recently in [14–16] as

1.5

In Equation 1.5, M stands for the molecular weight and ρ for the density of the particles. To visualize the general trend in the reduction of the melting temperature with decreasing particle size, the use of the absolute value of the fraction is the only correct way in the case of melting and crystallization.

As already mentioned, the derivations leading to Equation 1.5 assume that it is allowed to use bulk data for nanomaterials: however, this is problematic in the case of nanoparticles. The following gives a series of indications.

In a review on the melting of solids, Mei and Lu [17] devote a whole chapter to abnormal size effects on melting. Most interesting in this context are experimental findings of Shvartsburg and Jarrold [11] reporting that tin clusters consisting of 19–31 atoms exhibit melting points significantly above that of the bulk material. Molecular dynamic (MD) simulations for clusters of Cn, Sin, Gen, and Snn clusters for by Lu et al. [18] also revealed melting points significantly above that of the bulk materials.

The material data of nanoparticles differ from those in bulk materials. For example, Vollath

et al

. [19] found drastic changes in the thermodynamic data of phase transformations in nanoparticulate zirconia, leading in the case of small particles to a change in the phase sequence with respect to temperature being reversed. An analogous phenomenon was found by Ushakov

et al

. [20] for pure and La-doped zirconia and hafnia nanoparticles with a diameter of 5–6 nm, where, at room temperature, the amorphous phase was more stable than the tetragonal one, in contrast to the general opinion that amorphous particles are the least stable ones.

Even more dramatic is the influence on surface energy. There are experimental indications for a six times larger surface energy for nanoparticles of gold [21] and silver [22] compared to the bulk values. However, the model leading to such an evaluation of these experiments is seriously questioned [23, 24]. An increase in the surface energy by a factor of roughly two for aluminum nanoparticles was predicted in a theoretical study by Medasani and Vasiliev [25]. Both results contradict theoretical estimates that find a reduction in the surface energy with decreasing particle size [23, 24].

To estimate the thermal behavior of nanoparticles, knowledge of thermodynamic quantities and surface energy is essential. Therefore, it is the goal of this contribution to present proper and reliable approaches to estimate the thermodynamics of nanoparticles based on bulk data.

1.3 Size-Dependent Materials Data of Nanoparticles

For quite some time, there have been approaches to estimate the thermodynamic data of nanoparticles as a function of their size. Tolman [26, 27] presented such a relation for the surface energy as

1.6

The quantity is the so-called Tolman length. Das and Binder [28] generalized the Tolman relation to a wider applicable equation of type

1.7

In Equation 1.7, the quantity is again a characteristic length. However, Equation 1.7 was developed not for a free surface but an interface between two coexisting phases. A further relation, appraised as a “universal” relation, was reported by Guisbiers [29] in the form of

1.8

Guisbiers argues that this relation is valid for the material property , which may be the melting temperature, Debye temperature, superconducting temperature, Curie temperature, cohesive energy, activation energy of diffusion, or vacancy formation energy. The quantity is a material constant with the dimension of a length, and s is a positive number depending on the material property. For and Equation 1.6 and Equation 1.8 are practically equivalent. Furthermore, Equation 1.5 for the melting temperature obeys relation 1.8 with .

A more sophisticated relation for compared to Equation 1.8, based on the cohesive energy of nanocrystals, was reported by Lu and Jiang [23, 24] and Ouyang et al. [30], as

1.9a

The quantity is the “smallest size” of d if this equation is valid. For , Equation 1.9a can be rewritten as

1.9b

and as a further approximation as

1.9c

Equation 1.9c agrees again with Equation 1.8 for .

A different physical approach was reported by Li [31], using a layer-by-layer structure of the reference crystal from which the nanoparticle is cut out. This concept leads to an extremely complicated relation, which yields for the same approximation as Equation 1.9c.

With respect to the melting point of nanomaterials , the thermodynamic approach of Letellier et al. [15, 16] should be noted. These authors have concluded the same tendency as shown in this chapter (Equation 1.5)

1.9d

However, they introduce also as a conceptual extension a positive exponent s to the quantity as with being a reference quantity and c, a positive constant factor. They report a value of for lead nanoparticles and 1.20 for tin nanoparticles.

Safaei and Attarian Shandiz [32] published a model for the melting entropy of metallic nanoparticles, based on the methods of statistical physics. It is important to emphasize that their final formulae for the entropy of melting and the melting temperature confirm the earlier work by Jiang and Shi [33, 34], who developed, on the basis of Mott's equation for the melting entropy, a model for size-dependent melting temperature and entropy. Using an earlier approach to calculate the melting temperature of nanoparticles [35], Attarian Shandiz and Safaei [36] derived the same relations as Jiang and Shi [33, 34] for the melting temperature of metallic nanoparticles. The special feature of these derivations may be found in the fact that the final formulae depend on the thermodynamics and the crystal structure of the bulk material only. Furthermore, this approach inherently incorporates the influence of the surface energy. Therefore, this term no longer appears explicitly in the further equations.

Neglecting the electronic contribution, according to Attarian Shandiz and Safaei [32, 36], the ratio of melting temperatures is given by

1.10

In Equation 1.10, stands for a critical particle size, where the particle consists of surface atoms only. It is important to point out that, from its derivation, Equation 1.10 already contains the influence of the difference of the surface energy. From its definition, depends on the crystal structure of the particle. Attarian Shandiz and Safaei [36] give a table of this quantity for different lattices, for example, for the fcc structure , where δ stands for the diameter of one atom in the metallic environment. It is obvious that a table as in [36] for d0 is applicable only in exceptional cases where the structure of the smallest particle is identical to the bulk structure. This assumption is not necessarily correct; for example, for gold, see Tian et al. [37]. Therefore, in many cases, it may be necessary to fit the parameter d0 with experimental data. The quantity is the ratio of the coordination numbers at the surface, , and in the volume, , of the bulk material. For bulk materials and larger particles, is valid. Comparing calculated values with different results from the literature led Attarian Shandiz and Safaei [35, 36] to the conclusion that in the case of very small particles, a value is more appropriate. This finding is well in line with a study on coordination numbers as a function of particle size and structure by Montejano et al. [38]. Therefore, a fit function was developed, which gives for the bulk material a value 0.5 and which decreases to 0.25 for the particle size d0. Hence, it may be appropriate to select an expression such as

1.11

to fit experimental data of melting temperatures of nanoparticles. Equation 1.10 and Equation 1.11 contain the same and only one fit parameter d0. Obviously, d0 is the lower limit for d when this approximation may be applied. Certainly, this fitting process does not fulfill the claim of the original intention with respect to Equation 1.10, namely, to describe the melting properties of nanoparticles free of fitting parameters, depending on well-known bulk properties only. However, it will be shown later on in this contribution that the parameter d0 can be expressed by bulk values, making fitting processes unnecessary.

For the entropy and enthalpy of melting, Attarian Shandiz et al. [32, 35, 36] and Jiang et al. [33, 34] derived the following equation:

1.12a

Equation 1.12a does not take into account the electronic contribution to the entropy of melting, for which Safaei and Attarian Shandiz [32] proposed the term

1.12b

including the electronic contribution to the enthalpy of melting, described by the second term using the parameter . However, Safaei and Attarian Shandiz [32] pointed out that this term is zero in cases where the numbers of free electrons per ion in the solid and liquid phase are equal and not size dependent. According to Safaei and Attarian Shandiz [32], this may be correct in the case of metals, but certainly not for semiconductors.

The enthalpy of melting as a function of the particle size is given by

1.13

Equation 1.13 stems from the relation , valid both for the bulk and nanoparticles (see the additional subscripts “bulk” and “nano”); recall that is incorporated in the current version of .

From Equation 1.12 and Equation 1.13, the free enthalpy of melting is calculated as

1.14

Lastly, Equation 1.2b, containing the surface energy explicitly, and Equation 1.10, incorporating the surface energy, must lead to identical results. Therefore, these equations are well suited to calculate as a function of the particle diameter, yielding

1.15a

and, consequently

1.15b

This leads to as a function of d, using Equation 1.11, as

1.15c

Certainly, one may criticize that this approach does not take care of the influence of the possible faceting of the surface, especially for the smallest particles. However, this may be justified in a first approximation and by the fact that, experimentally, faceting is rarely observed in the case of very small metallic particles.

Next, one may ask about the existence and size of stable nuclei for crystallization within the melted particles. To calculate the critical diameter of the stable nuclei, , one needs to determine the extremum of the free enthalpy of melting yielding

1.16

with as solution. (The lengthy derivation with its details is given in Appendix 1.A.)

Equation 1.16 leads to a transcendent expression; the roots can be found numerically only. Nuclei or particles with diameter d below are fluctuating; they develop and disappear. Nuclei or particles with larger diameter are stable; they do not fluctuate, and, if possible, have a tendency to grow (see also [39, 40]). Looking at the critical diameter as a function of temperature, one may expect the following cases:

: In this stable case, one may expect a crystallization process of the particles starting from the interior. This process was already assumed to describe experimental results at the melting point of nanosized lead particles [6]. Furthermore, in this case, one expects the formation and decay of nuclei, which has been proved experimentally [12].

: This is the unstable, fluctuating case. As the formation of nuclei is spontaneous, instantaneous melting or crystallization of the whole particle may be expected. Therefore, one has to expect perpetual fluctuations between the solid and the liquid state.

1.4 Comparison of Experimental and Calculated Melting Temperatures

It is quite difficult to find reliable experimental or calculated data for the melting process of nanoparticles. In the first example, melting data of silver calculated by MD methods [41] were selected. Figure 1.2 shows the calculated and the fitted data by application of Equation 1.10 in connection with Equation 1.11. The fitting procedure led to a value of

Figure 1.2Melting temperature of silver nanoparticles. The MD data were taken from Luo et al. [41]. Fitting was performed using Equation 1.10 and Equation 1.11.

In the second example, experimental data for melting of tin nanoparticles obtained by Lai et al. [3] were fitted. As shown in Figure 1.3, in this case also, the fit is nearly perfect, resulting in the fitting parameter . In this figure, experimental data of Oshima and Takayanagi [12] were plotted as well, but not used in the fitting procedure. These data are of special interest because they split into two different paths at the particle size of about 5 nm. This phenomenon is better visible in Figure 1.4, where the data shown in Figure 1.3 are plotted versus the inverse particle size.

Figure 1.3 Melting temperature of tin nanoparticles. The experimental data used for fitting were taken from Lai et al. [3]. In addition, experimental data of Oshima and Takayanagi [12] are plotted. The inset shows, enlarged, the particle size–temperature range, where the experimental data are found. Fitting was performed using Equation 1.10 and Equation 1.11.

Figure 1.4Melting temperature of tin nanoparticles plotted versus the inverse particle size. In this graph, the same data as in Figure 1.2 were used [3, 12].

In Figure 1.4, in the field limited by the two different paths in the experimental data of Oshima and Takayanagi [12], liquid particles with crystallized inclusions can be observed in the electron micrographs. As the inclusions are unstable, this field was denominated as “pseudocrystalline.”

As a last example, the melting of gold nanoparticles is discussed. For gold, the experimental results of Castro et al. [9], which were already applied for demonstration in Section 1.1, are used. Figure 1.5 displays the melting data for gold nanoparticles, as discussed. For reasons of clarity, similar to Figure 1.4, the melting temperature of gold nanoparticles is plotted versus the inverse particle size.

Figure 1.5 Experimental and fitted data for the melting temperature of gold nanoparticles versus the inverse particle size. In this graph, experimental results of Castro et al. [9] and Dick et al. [42] are plotted. The results of these authors are different because Castro et al. used free and Dick et al. particles enclosed in silica.

Fitting of the experimental data for gold, denoted as Castro et al. I [9] in Figure 1.5, resulted in the fit parameter . In addition, the experimental results of Dick et al. [42] are plotted in this graph. Unfortunately, the experimental conditions of these two groups were so different that a common fitting is impossible. This is, because in the experiments of Dick et al. [42], the gold particles were mechanically confined in a silica matrix. Therefore, the experimental results of Dick et al. lead, in extrapolation, to a significantly higher melting temperature for the bulk material, . Interestingly, the value is practically identical in both experiments. Similar to the experimental results of Oshima and Takayanagi [12] for tin, the results obtained by Castro et al. [9] show, in the case of small particles (marked as Castro et al. II in Figure 1.5), a remarkable deviation from the fitted results.

Besides the data used earlier, there are more data obtained by MD simulations published in the literature. However, these data were not used because they predict melting points for the bulk material that are far off from the well-known ones.

Calculating the free enthalpy of melting as a function of temperature and particle size by application of Equation 1.14, with respect to Equation 1.10, Equation 1.11, and Equations 1.12 and , sheds more light on the melting behavior of metallic nanoparticles. On the basis of data calculated from the experimental results shown in Figure 1.1, Figure 1.2, Figure 1.3, Figure 1.4, and Figure 1.5, the diagrams in Figure 1.6a–c demonstrates the free enthalpy of melting for the examples explained. In the case of gold, the data obtained from the experiments by Castro et al. [9] were applied.

Figure 1.6 Free enthalpy of melting for (a) silver, (b) tin, and (c) gold versus particle size and temperature as parameter.

The graphs in Figure 1.6a–c have a few features in common: each of the curves shows two “zero-crossings” (roots with respect to Equation 1.14 using Equation 1.10 and Equation 1.11, one at a particle size showing a temperature-dependent root and the second a temperature-independent root.

The temperature-independent root is possible only if the condition

1.17

is fulfilled; for mathematical details, see Appendix 1.A. By combining Equation 1.10, Equation 1.11, and Equation 1.12, the condition in Equation 1.17 leads to the equation

1.18a

with the abbreviation . Equation 1.18a, a quadratic one, has two roots; however, only the positive root is physically reasonable, which is denominated as the “characteristic particle size” and Equation (1.10) with Equation (1.11) as

1.18b

This characteristic particle size is related to a characteristic temperature by combining Equation 1.12a with Equation 1.17 as

1.19

Table 1.1 displays the characteristic parameters evaluated for the three examples shown in Figure 1.6a–c.

Table 1.1Characteristic parameters calculated from the experimental data.

Metal and literature data used for fitting

d

0

(nm)

Equation 1.10

and

Equation 1.11

d

char

(nm)

Equation 1.18b

T

char

(K)

Equation 1.19

Ag [41]

1.76

607

Sn [4]

1.71

164

Au [9]

1.73

626

It is not too astonishing that the d0 values for gold and silver are nearly identical. More striking is the fact that dchar is nearly identical for the three metals in discussion.

Table 1.1 teaches that the characteristic particle size dchar is larger than d0, which is the smallest one, where these considerations are valid. In the interval of particle sizes between and , one may expect special phenomena, as the free enthalpy of melting (Figure 1.6a–c) changes sign at .

The graphs in Figure 1.6a–c exhibits the remarkable fact that, because of the change in the sign of the free enthalpy of melting, particles with sizes below dchar are liquid if their temperature is lower than Tchar and crystallized at higher temperatures.

Obviously, this is a case of reverse behavior of the phases, a rather strange phenomenon, which is quite often described. Very early, this phenomenon was predicted by Tammann [43]. Experimentally, the existence of this strange behavior was proved by findings of pressure-induced amorphization at constant temperature by Mishima et al. [44] for ice and by Hemley et al. [45] for silica. Crystallization at constant pressure with increasing temperature was found by Rastogi et al. [46, 47] for organic compounds. Recently, a theoretical basis for Tammann's generalized phase diagram was given by Stillinger and Debenetti [48] and Schupper and Shnerb [49].

With reference to the examples selected for this chapter, the phenomenon of reverse melting is better visible in a graph, where the free enthalpy of melting is plotted versus the temperature. Using silver as an example, this is shown for four different particle sizes in Figure 1.7, where the phenomenon of reverse melting and crystallization is perfectly visible. Particles with sizes of 1.5 nm are liquid at lower temperatures and solid at higher ones. In contrast, particles with 2 and 3 nm diameters behave in the usual way. In addition, the zero-valued, temperature-independent free enthalpy of melting for the characteristic particle size, 1.76 nm, is also plotted.

Figure 1.7 Free enthalpy of melting for silver particles versus temperature and particle size as parameter. Particles are solid in the case of positive values of the free enthalpy of melting and liquid in the case of negative values. Comparing the courses for the particle sizes, one realizes the reverse behavior of melting and crystallization for the particle size 1.5 nm.

From Figure 1.6a–c, one learns that at each temperature level, there are two particle sizes, where the solid and the liquid state are in equilibrium, . These equilibrium sizes, calculated by , according to Equation 1.14, are plotted in Figure 1.8a–c (See also Appendix 1.A.) for the three examples – silver, tin, and gold. In these graphs, the sizes of the nuclei, fulfilling Equation 1.16, are plotted, too. The graphs in Figure 1.8 show a few remarkable features. There is, as it is also visible in the earlier figures, one temperature-independent equilibrium particle size (Equilibrium 1), the “characteristic particle size,” dchar. The plot of the second temperature-dependent equilibrium particle size (Equilibrium 2) intersects the temperature independent equilibrium size at the “characteristic temperature.” The plot of the size of the stable nuclei , according to Equation 1.16, intersects at the same point; see the proof in Appendix 1.A. Obviously, dchar and are identical. The existence of the intersection of these three sizes at one point is a strong indication that this particle size characterizes the material.

Figure 1.8Equilibrium particle sizes (two roots with respect to Equation 1.14 using Equation 1.10) versus temperature for (a) silver, (b) tin, and (c) gold. In addition, the sizes of the stable nuclei (nucleus size) for crystallization according to Equation 1.16 are plotted. Note that these three curves have one point in common.

The consequences of the relative position of the sizes of the stable nuclei and the equilibrium particle sizes are obvious: At a particle size larger than the critical one, dchar, the size of the particles is larger than the size of the critical nuclei and the temperature-independent equilibrium particle size, which is smaller than the size of the critical nuclei . Therefore, at any temperature, these small particles repeatedly crystallize and melt. This is a continuing process of fluctuation, especially as these particles are smaller than the stable nuclei. Possibly, these nuclei are responsible for the crystallization process in melted metals. For a long time, these nuclei were postulated (see, e.g., [39, 40, 50]) and finally experimentally verified [51]. The situation is different for particle sizes smaller than . In this case, the size of the nuclei is larger than the one of the temperature-dependent equilibrium particles. Therefore, a priori, one cannot expect that these particles are stable, whereas particles with the characteristic particle size may be considered, with respect to their phase, as stable ones.

To analyze the stability of the particles against fluctuation, Figure 1.9a,b displays the free enthalpy of melting per particle as a function of the temperature for different particle sizes. In addition, the range where fluctuation is possible, limited by the lines (see, e.g., [52, 53]), is marked as shaded areas in Figure 1.9a,b.

Figure 1.9Free enthalpy of melting per particle for (a) silver and (b) gold versus particle size as parameter. Within the shaded area, limited by the thermal energy , fluctuation is possible.

Particles with size and temperature in this range may always alternate between the solid and the liquid state. Most interesting are the particles with the size dchar because for these particles the free enthalpy of melting is zero, independent of the temperature. Therefore, these particles are never stable: they always fluctuate between the two possible phases. Furthermore, looking at Figure 1.5, one realizes the strange phenomenon that melting of particles with sizes below 1 nm occurs at a constant temperature around 600 K. Analyzing Figure 1.6c and Figure 1.9b carefully makes it clear that for crystalline particles in this range of size and temperature inverse melting may be expected.

Equation 1.15c gives the difference in the surface energy between the solid and the liquid state for the bulk material and yields immediately for

1.20