If Bt

where X is the neck diameter, D is the particle diameter, t is the isothermal sintering time, and B is a term that collects material and geometric constants that are specified in Table 3. The values of n, m, and B change with time and degree of sintering, but for many analyses, they have been assumed constant with the mechanism of mass transport. Many experiments have been analyzed using the exponent n to identify the sintering mechanism. However, in cases where the grain boundary energy is relatively high (dihedral angle is small), neck growth will be paced by grain growth (X = G sin

: J/2) and will not follow Eq 11. In these cases, there will be a groove at the neck.

Table 3 Initial stage sintering equation for spheres

Table 3 Initial stage sintering equation for spheres

Mechanism n m B

Viscous flow 2 1 3 - /(2¡)

Plastic flow 2 1 9 TfO'b DJ{kT)

Evaporation-condensation 3 2 (3_p " /i'-12) (~~/2)'/2 (M/(kT)3/2

Volume diffUsion 5 3 80Dv" f "'./(kT)

Grain boundary diffusion 6 4 20$ Db70/(kT)

Surface diffusion 7 4 56Ds" H4/3(kT)

Symbol Definition

b Burgers vector

Db Grain boundary diffusivity

Ds Surface diffusivity

Dv Volume diffusivity

k Boltzmann's constant

M Molecular weight

P Vapor pressure

T Absolute temperature

- Surface energy

.■■> Grain boundary width

fj Viscosity

i> Theoretical density

( ) Atomic volume

There are many different solutions to the initial stage sintering problem, so the forms given in Table 3 are only representative of the efforts. For example, depending on the assumptions, various models for surface diffusion controlled initial stage sintering give n values of 3, 5, 6, 7, and 7.5. The simplified model represented by Eq 11 gives a rough approximation up to a neck size ratio of 0.3. The diffusion coefficient is embedded in the parameter B and follows an Arrhenius temperature dependence. The required frequency factor and activation energy for several common materials are collected in Table 4 for surface, volume (lattice), and grain boundary diffusion. Equations such as those introduced here for initial stage solid-state sintering are typically in error by 10 to 20% over actual numerical solutions. Hence, little precision can be assigned to calculations based on these integral models.

Table 4 Physical properties and diffusion factors for various materials

Material

Density,

Crystal

Atomic

Melting

Boiling

Heat

Heat

Heat of

Solid-

VD

VD

GBD

GBD

SD

SD

g/cm3

type(a)

diameter,

point,

point,

capacity,

of

vaporization,

vapor

frequency®,

activation

frequency(c),

activation

frequency(d),

activation

nm

°C

°C

J/(kg ■

fusion,

kJ/mol

surface

m2/s

energy ,

m3/s

energy ,

m2/s

energy ,

°C)

kJ/mol

energy,

kJ/mol

kJ/mol

kJ/mol

J/m2

Alumina

3.98

H

2054

2980

775

1.12

0.19

636

3 x 10-3

477

3 x 10-3

536

(MO3)

Aluminum

2.70

fcc

0.286

660

2467

917

10.7

294

1.14

2 x 10-4

142

3 x 10-14

60

5 x 10-2

142

Antimony

6.69

R

0.364

630

1635

209

20.9

68

0.45

1 x 10-4

165

93

Beryllia (BeO)

2.86

H

2570

3787

1046

1.42

530

Beryllium

1.85

HCP

0.228

1289

2970

1880

9.8

308

1

6 x 10-5

161

Brass (Cu-

8.50

fcc

965

377

1.6 x 10-6

126

30Zn)

Bronze (Cu-

8.88

fcc

957

377

1.7

1.6 x 10-10

207

6 x 10-10

105

2.5

205

10Sn)

Chromia

5.21

H

2440

3000

670

16 x 10-3

423

1 x 10-15

240

(Cr2O3)

Chromium

7.23

bcc

0.250

1875

2665

461

15.0

342

2.2

2 x 10-5

308

1 x 10-13

198

83

215

Cobalt

8.90

HCP

0.250

1495

2870

423

15.0

382

2.1

8 x 10-5

292

2 x 10-14

117

4 x 10-5

130

Cobalt oxide

6.46

NaCl

0.426

1935

703

5 x 10-3

398

1 x 10-13

200

(CoO)

Copper

8.96

fcc

0.256

1083

2578

386

13.0

307

1.75

6 x 10-5

213

2 x 10-14

107

2.6

205

Erbium

9.07

HCP

0.356

1529

2863

168

17.2

293

4.5 x 10-4

302

Europium

5.24

bcc

0.398

822

1597

182

10.5

176

1 x 10-4

144

Gadolinium

7.90

HCP

0.358

1313

3266

300

15.5

312

1 x 10-6

138

Germanium

5.32

Diamond

0.244

937

2830

310

34.7

224

0.7

7.8 x 10-4

287

1 x 10-1'

172

Glass-

2.40

1140

335

1 x 10-7

84

borosilicate

(SiO2-13B2O3)

Gold

19.30

fcc

0.288

1063

2807

343

13.0

343

1.37

6 x 10-6

172

3 x 10-16

110

1 x 106

234

Hafnium

13.31

HCP

0.334

2227

5197

147

25.5

661

1.21 x 10-7

163

Hematite

5.24

H

1570

628

2 x 10-4

326

4 x 10-13

210

(Fe2O3)

Ice

0.92

H

0

100

4217

0.1

9 x 10-4

59

8.3 x 10-13

38

3.6 x 102

59

Indium

7.31

fct

0.332

156

2080

243

3.3

226

0.6

3 x 10-4

79

Invar (Fe-36Ni)

8.05

fcc

1425

515

2.5 x 10-3

310

Iron

7.87

bcc

0.248

1536

2750

456

15.0

340

1.95

2 x 10-4

251

1 x 10-17

128

1.1 x 101

239

Iron oxide

5.70

NaCl

1370

3414

803

1 x 10-2

326

1 x 10-13

195

(FeO)

Kovar (Fe-

8.36

1450

460

1.5 x 10-3

315

29Ni-17Co)

Lanthanum

6.15

HCP

0.374

921

3457

200

10.0

400

0.7

1 x 10-6

103

Lead

11.10

fcc

0.350

328

1740

130

5.0

178

0.61

1 x 10-4

109

2 x 10-13

68

2.6 x 10-4

109

Lithium

0.53

fcc

0.304

180

1347

3517

4.6

135

0.4

2.3 x 10"5

55

Lithium

2.64

NaCl

846

1717

1631

7.4 x 10"3

214

fluoride (LiF)

Magnesia

3.58

NaCl

2825

3260

920

0.5

1.4 x 10-6

460

1.4 x 10-15

230

1.5 x 103

460

(MgO)

Magnesium

1.74

HCP

0.320

649

1090

1038

9.0

129

0.57

1 x 10-4

135

5 x 10-12

92

Molybdenum

10.22

bcc

0.272

2610

4612

251

28.0

590

2.2

5 x 10-5

418

6 x 10-14

263

10

241

Mullite

3.17

O

1828

765

0.66

3.6 x 106

810

0.0036

434

1 x 107

405

(3Al2O3-2SiO2)

Nichrome (Ni-

8.50

fcc

1400

430

1.6 x 10-4

285

2.8 x 10-15

115

20Cr)

Nickel

8.90

fcc

0.250

1453

2732

452

18.0

375

1.86

1.4 x 10-4

276

4 x 10-16

108

2 x 10-2

164

Nickel

5.86

Ordered

1647

843

4.8 x 10-5

335

1.6 x 10-15

290

2.2

325

aluminide

(NiAl)

Nickel

7.25

Ordered

1380

123

2

4 x 10"4

306

2 x 10-13

152

1 x 104

306

trialuminide

(NiaAl)

Niobium

8.57

bcc

0.284

2468

4927

268

27.0

697

2.3

1 x 10-4

402

5 x 10-14

263

1

235

Palladium

12.02

fcc

0.274

1552

3140

247

17.2

393

2.1

2 x 10"5

274

Platinum

21.45

fcc

0.278

1769

3827

134

19.7

511

1.3

3 x 10"5

294

4 x 10-7

108

Potassium

0.86

bcc

0.462

63

774

754

2.4

78

0.12

3.2 x 10-5

41

Rhenium

21.02

HCP

0.274

3180

5625

138

33.1

707

9 x 10-5

221

Rhodium

12.41

fcc

0.268

1966

3727

243

21.6

495

4 x 10-6

173

Silicon

2.33

Diamond

0.234

1410

2355

729

39.6

383

0.9

0.9

496

1 x 10-15

300

Silicon carbide

3.15

H

2700

2972

628

3

0.05

696

2.2 x 10-11

557

7 x 104

696

(SiC)

Silicon nitride

3.18

Cubic

2564

400

0.5

0.05

636

2 x 10-11

509

7.6 x 10-2

636

(S13N4)

Silver

10.50

fcc

0.288

961

2212

234

11.3

255

1.14

4 x 10"5

185

6 x 10-15

90

5 x 103

266

Sodium

0.92

HCP

0.380

98

883

1240

2.6

89

0.19

2.4 x 10"5

44

Sodium

2.16

NaCl

801

1465

870

14.0

323

0.28

0.025

217

6.2 x 10-10

155

5.1 x 104

217

chloride (NaCl)

Spinel

3.58

Cubic

2135

812

8.9 x 10"5

439

1 x 10-14

264

(MgAl2O4)

Stainless steel,

8.00

fcc

1400

500

2.2

4 x 10"5

280

2 x 10-13

167

0.5

220

304L

Stainless steel,

8.05

fcc

1375

500

2

4 x 10"5

280

2 x 10-13

167

0.4

250

316L

Steel 1018

7.86

bcc

0.252

1525

464

15.0

340

2.1

2 x 10"5

270

8 x 10-14

159

300

220

Superalloy (Ni-

8.52

Mixed

1400

413

1.7

2 x 10-4

285

2 x 10-6

115

4.8

285

base)

Superconductor

1200

8 x 10"4

278

5 x 10-18

139

(YBa2Cu3O7)

Tantalum

16.65

bcc

0.228

2996

5427

142

31.0

758

2.8

1.2 x 10-4

425

5.5 x 10-12

300

1.1 x 10-3

326

Thoria (ThO2)

10.00

CaF2

3250

4400

230

3.5 x 10"5

625

1 x 10-14

375

Thorium

11.72

fcc

0.360

1750

3850

100

19.2

544

10-5

349

Tin

7.30

Tetra.

0.316

232

2270

226

7.2

290

0.68

9 x 10"4

105

3 x 10-15

40

0.07

64

Titania (TiO2)

4.17

H

1830

799

4.6 x 10"6

240

Titanium (hcp)

4.50

HCP

0.294

1668

3286

528

20.9

429

1.7

6 x 10-12

123

4 x 10-16

97

4 x 10-3

150

Titanium (bcc)

4.00

bcc

0.294

1668

3286

418

20.9

429

1.7

1 x 10-4

251

5 x 10-17

153

6 x 10-3

153

Titanium

3.83

Ordered

1457

650

1.76

8 x 10"7

200

8 x 10-13

180

2.4

150

aluminide

(TiAl)

Titanium

4.93

NaCl

0.300

3065

564

83.6

501

1.19

4.4

737

10-13

543

2.2 x 10-4

288

carbide (TiC)

Ti-6Al-4V

4.46

1650

544

1.3 x 10"6

180

Tool steel (Fe-

8.00

1410

450

2.2

4 x 10-5

280

2 x 10-13

167

0.5

220

4Cr-6W-5Mo-

2V-0.9C)

Tungsten

19.30

bcc

0.274

3410

5657

138

35.0

824

2.8

8 x 10"6

520

3 x 10-13

385

10-3

293

Urania (UO2)

10.96

CaF2

2827

243

0.74

10-5

452

2 x 10-15

293

0.1

452

Uranium

19.10

O

0.276

1132

3745

117

15.5

423

2 x 10-7

123

3 x 10-11

184

Vanadium

6.11

bcc

0.264

1887

3377

498

18.0

460

2.5

4 x 10-5

317

5 x 10-14

209

Zinc

7.13

HCP

0.266

420

906

394

6.7

115

0.8

2 x 10-5

94

10-14

60

9 x 10-6

27

Zirconia-yttria

6.08

Mono.

2600

4548

456

0.3

2.7 x 10-5

423

2 x 10-12

309

0.04

430

(ZrO2-3Y2O3)

Zirconium

6.50

bcc

0.316

1852

4377

264

23.0

567

1.9

1.3 x 10-4

272

6 x 10-16

184

0.06

180

Zirconium

6.73

NaCl

3532

5100

368

0.1

720

5 x 10-12

468

carbide (ZrC)

Crystal types: bcc, body-centered cubic; H, hexagonal; fcc, face-centered cubic; R, rhombohedral; O, orthorhombic; fct, face-centered tetragonal, Tetra., tetragonal; Mono., monoclinic.

VD, volume diffusion.

GBD, grain boundary diffusion.

SD, surface diffusion.

Source: Ref 15

Although not very exact, Eq 11 does illustrate some key processing factors in sintering. A high sensitivity to the inverse particle size means smaller particles give more rapid sintering. Surface diffusion and grain boundary diffusion have the highest sensitivity to particle size changes; thus they are enhanced relative to the other processes by a smaller particle size. In all cases, temperature appears in an exponential term, which means small temperature changes can have a large effect. Finally, time has a relatively small effect in comparison to temperature and particle size.

Bulk transport processes decrease the interparticle spacing as neck growth processes, which results in compact shrinkage and the formation of additional, new necks. It is easier to measure the compact dimensional change instead of the neck size. Shrinkage is approximately related to the neck size by a simple model as follows:

where the shrinkage AL/L0 is the compact length change divided by the initial length. Shrinkage is actually a negative value, but the sign is usually ignored. Actual measurements of both shrinkage and neck size reveal that the shrinkage is slightly larger than predicted by Eq 12. On this basis, shrinkage during initial stage sintering follows a kinetic law similar to Eq 11:

The parameter B is exponentially dependent on temperature:

where k is Boltzmann's constant, T is the absolute temperature, and B0 consists of material parameters (such as surface energy, atomic size, atomic vibration frequency, and system geometry as defined in Table 3).

One of the benefits of such computer simulations is recognition that strength and other mechanical, thermal, or physical properties are dependent on the neck size. For example, strength data are directly related to the neck size ratio squared. A major complication arises due to multiple mechanisms contributing simultaneously to particle bonding. Accurate calculation of the sintering rate is necessary to build a basis for sintering predictions. Unfortunately, most of the models are only approximations to an underlying complexity that cannot be treated by simple geometric applications. Consequently, computer simulation is enlisted to address the sintering problem. This is an important direction to provide realistic modeling capabilities. Much of the early work focused on accurate predictions of the sintering geometry and mass flow kinetics by a single transport mechanism, for example grain boundary diffusion.

Only recently, have computer simulations embraced realistic conditions. A typical shortfall of the early models was a failure to incorporate a grain boundary dihedral angle in the sinter neck. Other problems are associated with the assumption of monosized spheres in ideal packing geometries with instantaneous heating. A serious deficiency in early simulations was a loss of mass or volume. For example, simulating the coalescence of spheres by surface diffusion resulted in nearly a 5% gain in volume. These problems limit the usefulness of the early simulations. In contrast, actual sintering experiments involve a wide range of particle sizes, packing inhomogeneities, and multiple sintering mechanisms during slow heating to the final temperature. A real difficulty is in using too few initial particles in the simulation. It is not uncommon in practice to see a final grain size that is ten times larger than the initial particle size. Thus, the initial simulation must start with over 1000 particles for every final grain in the sintered structure. This is a major computational burden. Furthermore, there are many situations where 109 particles are consumed to form one sintered grain, which is far beyond current simulation capabilities.

The variety of computer simulations for sintering processes is large. Approaches include numerical solution to the different equations, finite element analysis, finite difference solutions, and Monte Carlo simulations. A review of the field is provided in the Selected References . Although there are over 200 sintering simulations in circulation, many focus on single mechanisms; even so they have provided important checks on the models and a basis for multiple mechanism simulations.

Multiple mechanism assessments of sintering are useful in determining dominant processes. For example, volume diffusion versus surface diffusion controlled sintering can be assessed by taking the ratio of neck growth rates. These assessments show that early sintering favors surface diffusion and grain boundary diffusion, while the latter stages tend to shift to volume diffusion control.

Most of the modern simulations recognize that sintering involves the delivery of mass to the interparticle bond by several simultaneous transport mechanisms. A typical assumption is that the instantaneous sintering mass flux is simply the sum of the contributions from each mechanism. At any instant, the rate of neck growth is the sum of the individual fluxes:

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