Polymers and Neutron Scattering

, by Higgins, Julia S.; Benoît, Henry C.

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ISBN: 9780198500636 | 0198500637
Cover: Paperback
Copyright: 3/27/1997

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The application of neutron scattering to polymers provides information which cannot be obtained by other techniques and has allowed progress in our understanding of polymer conformation in the bulk. This book presents examples which are needed to understand how these techniques can be applied. It is specifically written to introduce the newcomer and non-expert in physics or chemistry to the experimental techniques and the basic theory necessary to understand the results.

GLOSSARY OF SYMBOLS

1 INTRODUCTION

(25)

1.1 Introduction

(1)

1.2 Possibilities--deuterium labelling

(3)

1.3 The components of a neutron experiment

(2)

1.4 Background--a little history

(1)

1.5 Scattering--the basics: introducing q

(4)

1.6 Relationship between scattering and structure

(3)

1.7 Elastic and inelastic scattering

(2)

1.8 Cross sections

(4)

1.9 Summary

(6)

1.9.1. Molecular shape

(2)

1.9.2. Sample morphology

(1)

1.9.3. Main chain segmental motion

(3)

2 NEUTRON PRODUCTION AND DETECTION: THE NUTS AND BOLTS

(25)

2.1 Introduction

(1)

2.2 Neutron sources

(7)

2.2.1 Reactor sources

(2)

2.2.2 Electron accelerator sources

(2)

2.2.3 Spallation neutron sources

(2)

2.2.4 Relative advantages of different sources

(1)

2.3 Moderators: wavelengths and pulse length

(2)

2.4 Neutron transportation

(4)

2.4.1 Guides

(2)

2.4.2 Collimation

(2)

2.5 Stopping neutrons--shielding, beamstops, diaphragms

(1)

2.6 Monochromation--choosing wavelengths

(3)

2.6.1 Crystal monochromators

(1)

2.6.2 Mechanical velocity selection

(2)

2.6.3 Measuring wavelengths

(1)

2.7 Detecting and monitoring neutron beams

(2)

2.8 Polarized neutron beams

(2)

2.9 Spectrometers

(3)

3 SPECTROMETERS AND WHAT THEY MEASURE

(30)

3.1 Introduction

(6)

3.1.1 Diffractometers for liquids and amorphous materials

(1)

3.1.2 Diffractometers for crystalline powders

(1)

3.1.3 Diffractometers with polarization analysis

(1)

3.1.4 Single crystal diffractometers

(1)

3.1.5 Spectrometers for measuring Sinc(q,w) (inelastic incoherent scattering) with modest resolution and high intensity

(2)

3.1.6 Spectrometers, for measuring Scoh (q,w) (inelastic coherent scattering)

(1)

3.1.7 High resolution quasielastic scattering spectrometers

(1)

3.2 Small angle scattering

(13)

3.2.1 Setting up the spectrometer

(2)

3.2.2 Samples and their environment

(2)

3.2.3 Ancilliary measurements--background, resolution, transmission

(3)

3.2.4 Data analysis and data reduction

(6)

3.3 Neutron reflection

(2)

3.4 Inelastic and quasielastic scattering

(9)

3.4.1 Vibrational spectroscopy

(1)

3.4.2 Rotational motion

(1)

3.4.3 The back-scattering spectrometer

(3)

3.4.4 Main chain motion

(1)

3.4.5 The spin-echo spectrometer

(4)

4 THEORETICAL BASIS OF SCATTERING

(35)

4.1 Introduction

(1)

4.2 Definition of the cross section

(1)

4.3 Elastic interaction between a neutron and a single nucleus

(3)

4.4 Coherent and incoherent elastic scattering

(2)

4.4.1 Mixtures of isotopes

(1)

4.4.2 The effect of nuclear spin

(1)

4.5 The static correlation function

(4)

4.6 Inelastic cross section

(4)

4.7 Coherent and incoherent inelastic scattering

(1)

4.8 Van Hove equation and correlation function

(6)

4.8.1 The correlation function

(2)

4.8.2 The autocorrelation function

(1)

4.8.3 The static approximation

(1)

4.8.4 Some remarks about notation

(3)

4.9 Some examples

102

(14)

4.9.1 Translational motion--free diffusion

103

(5)

4.9.2 Rotational motion

108

(3)

4.9.3 Vibrational motion--one-phonon inelastic scattering

111

(5)

5 LABELLING WITH DEUTERIUM--HOW, WHY, AND WHEN TO USE

116

(25)

5.1 Introduction

116

(3)

5.2 Scattering laws for incompressible systems

119

(5)

5.2.1 The basic scattering laws and the partial structure factors

119

(1)

5.2.2 Relationship between the partial structure factors

120

(1)

5.2.3 Generalization to a mixture of more than two species

121

(1)

5.2.4 Decomposition of S(q) into intra- and intermolecular interferences

122

(2)

5.3 Applications of the general formulae to deuteration

124

(6)

5.3.1 Two identical polymers, one deuterated the other not, in a melt or a glass

124

(2)

5.3.2 The Babinet principle and some geometric justifications

126

(1)

5.3.3 The case of two different polymers

127

(1)

5.3.4 Deuterated-hydrogenous mixture in a solvent

128

(1)

5.3.5 Mixture of deuterated and hydrogenous polymer in any system

129

(1)

5.4 The symmetries of Q(q)

130

(1)

5.4.1 Linear polymers

130

(1)

5.4.2 The case of a ring molecule

131

(1)

5.5 Approximate methods for the evaluation of the scattered intensity

131

(6)

5.5.1 A mixture of two polymers of different molecular dimensions, one deuterated the other not

131

(2)

5.5.2 The case of a symmetrical block copolymer in the bulk

133

(3)

5.5.3 Copolymers of any composition and structure in bulk

136

(1)

5.6 Deuteration effects on thermodynamics

137

(4)

5.6.1 Observation and explanation

137

(2)

5.6.2 Consequences of the isotope effect--when is deuteration safe?

139

(2)

6 FORM FACTORS

141

(51)

6.1 Introduction

141

(1)

6.2 The behaviour at small q values

142

(9)

6.2.1 Series expansion of P(q)

142

(2)

6.2.2 The radius of gyration

144

(1)

6.2.3 The radius of gyration for various geometrical shapes

145

(1)

6.2.4 The radius of gyration for a Gaussian chain

145

(3)

6.2.5 The ring Polymer

148

(1)

6.2.6 The case of copolymers

149

(2)

6.3 The complete form factor

151

(14)

6.3.1 The mathematical methods

151

(1)

6.3.2 The sphere

152

(1)

6.3.3 Other objects with spherical symmetry (for example, shells)

153

(3)

6.3.4 Other simple shapes-discs and rods

156

(2)

6.3.5 Gaussian chains

158

(3)

6.3.6 Chains of different architecture

161

(4)

6.4 The intermediate and high q range

165

(9)

6.4.1 Qualitative interpretation of the different q domains

165

(3)

6.4.2 The use of scaling arguments for the determination of exponents at high q values

168

(2)

6.4.3 The case of objects with rough surfaces

170

(2)

6.4.4 Scattering by fractals

172

(1)

6.4.5 Surface fractals

173

(1)

6.5 Practical methods for characterization of the scattering curves

174

(10)

6.5.1 The plots at low angles

174

(1)

6.5.2 The plots at large angles

175

(1)

6.5.3 The Gaussian chain

176

(1)

6.5.4 Star molecules

177

(1)

6.5.5. Ring polymers

177

(2)

6.5.6 Determination of the persistence length

179

(3)

6.5.7 Rod-like particles

182

(1)

6.5.8 The case of two-dimensional objects

183

(1)

6.5.9 The case of three-dimensional objects

184

(1)

6.6 The effect of polydispersity

184

(8)

6.6.1 Some definitions for polydisperse macromolecular systems

184

(1)

6.6.2 An example of the effect of polydispersity

185

(2)

6.6.3 The radius of gyration in polydisperse systems

187

(1)

6.6.4 The effect of polydispersity at high q values

188

(3)

6.6.5 Polydisperse spheres

191

(1)

7 INTERACTING SYSTEMS

192

(53)

Part 1 Zero angle scattering

192

(17)

7.1 Density and concentration fluctuations--one or two components

192

(6)

7.1.1 Introduction

192

(1)

7.1.2 A one-component system

193

(2)

7.1.3 Incompressible solutions

195

(2)

7.1.4 The example of a gas or a dilute solution

197

(1)

7.2 Fluctuations in multicomponent systems

198

(5)

7.2.1 Introduction

198

(1)

7.2.2 Evaluation of

199

(2)

7.2.3 Application to a one-component system

201

(1)

7.2.4 Application to a two-component system

201

(2)

7.3 A general theory for zero angle scattering

203

(6)

7.3.1 The exchange chemical potential

203

(2)

7.3.2 The scattering equation

205

(2)

7.3.3 Application to polymer solutions

207

(1)

7.3.4 Divergence of the intensity at zero angle

208

(1)

Part 2 Finite angle scattering

209

(22)

7.4 Dilute solution scattering

209

(5)

7.4.1 The radial distribution function

209

(2)

7.4.2 Properties of the function g(r)

211

(2)

7.4.3 Relationship with the second virial coefficient

213

(1)

7.4.4 Geometrical interpretation of the excluded volume parameter

214

(1)

7.5 The Zimm formula (single contact approximation)

214

(2)

7.6 The Ornstein-Zernike formula

216

(6)

7.6.1 The case of small molecules

216

(3)

7.6.2 Application of the O-Z method to macromolecules

219

(2)

7.6.3 Relationship with thermodynamics

221

(1)

7.7 Mixture of two polymers

222

(6)

7.7.1 In the presence of solvent

222

(2)

7.7.2 The case of a mixture without solvent

224

(2)

7.7.3 The case of copolymers

226

(1)

7.7.4 Generalization of the Zimm equation

226

(2)

7.8 General equation

228

(2)

7.9 The case of polydisperse systems

230

(1)

Part 3 Systems existing in more than one phase

231

(14)

7.10 Introduction

231

(1)

7.11 Critical opalescence

231

(5)

7.11.1 The Ornstein-Zernike approach

231

(2)

7.11.2 Geometrical representation

233

(2)

7.11.3 Screening length in polymer solutions

235

(1)

7.12 The two-density model

236

(3)

7.12.1 The Porod law

236

(1)

7.12.2 The Porod invariant

236

(3)

7.13 The Debye-Bueche equation

239

(1)

7.14 Scattering by spherical particles

240

(2)

7.15 The correlation hole

242

(3)

8 EXPERIMENTAL EXAMPLES OF STRUCTURAL STUDIES

245

(52)

8.1 Introduction

245

(1)

8.2 Single chain conformation

245

(24)

8.2.1 H-D mixtures of identical polymers in melts or glasses

245

(9)

8.2.2 H-D mixtures in the presence of a third component--solutions and blends

254

(6)

8.2.3 Strong interactions--polyelectrolytes and block copolymer solutions

260

(3)

8.2.4 Systems with mesomorphic phases

263

(6)

8.3 Thermodynamic parameters

269

(10)

8.3.1 Solutions

269

(2)

8.3.2 Blends

271

(3)

8.3.3 Copolymers

274

(1)

8.3.4 Phase separation in polymer blends

275

(2)

8.3.5 Transesterification

277

(2)

8.4 Structure and morphology in multiphase systems

279

(10)

8.4.1 Form factors of particles of known shape

279

(3)

8.4.2 Structural arrangements of particles

282

(3)

8.4.3 Complex structures

285

(1)

8.4.4 Fractal systems

286

(3)

8.5 Anisotropic structures

289

(8)

8.5.1 Effect on the scattering formulae of orienting the samples

289

(2)

8.5.2 Aligned or stretched single chains

291

(2)

8.5.3 Anisotropic structures

293

(4)

9 DYNAMICS

297

(45)

9.1 Introduction

297

(5)

9.1.1 Types of motion

297

(3)

9.1.2 Practicalities of separating translation, rotation, and vibration

300

(2)

9.2 Vibrations

302

(12)

9.2.1 The scattering laws for vibrational motion

302

(2)

9.2.2 Vibrational motion in polymeric samples

304

(1)

9.2.3 Inelastic scattering from torsional vibrations

305

(4)

9.2.4 Acoustic phonons in crystalline samples

309

(5)

9.3 Rotational motion of side groups

314

(8)

9.3.1 The scattering laws

314

(2)

9.3.2 Neutron quasielastic scattering from rotating side groups

316

(4)

9.3.3 Window scans and quasielastic scattering

320

(1)

9.3.4 Effects of multiple scattering

321

(1)

9.4 Main chain motion

322

(20)

9.4.1 The correlation time and its q dependence

322

(5)

9.4.2 The scattering laws

327

(3)

9.4.3 Motion in polymer solutions

330

(5)

9.4.4 Contrast matching in dynamic studies of polymer solutions

335

(3)

9.4.5 Dynamics in the melt

338

(2)

9.4.6 Incoherent scattering from polymer melts

340

(2)

10 NEUTRON REFLECTION FOR STUDYING SURFACES AND INTERFACES

342

(25)

10.1 Introduction

342

(3)

10.2 Neutron specular reflection

345

(2)

10.3 Reflectivity profiles

347

(9)

10.3.1 General

347

(3)

10.3.2 Approximate scattering functions

350

(5)

10.3.3 Multilayer optical method for calculating reflectivity

355

(1)

10.4 A Model reflection experiment

356

(1)

10.5 Examples of reflection data from polymeric samples

357

(10)

10.5.1 Polymers adsorbed at a liquid interface

357

(4)

10.5.2 Polymer--polymer interdiffusion

361

(3)

10.5.3 Block copolymer organization

364

(2)

10.5.4 Closing remarks

366

(1)

APPENDIX 1 The Fourier transform and some of its applications

367

(15)

APPENDIX 2 Thermodynamics

382

(16)

APPENDIX 3 Some remarks about the use of theoretical formulae for the interpretation of experimental data; comparison with light and X-ray scattering

398

(20)

APPENDIX 4 Chemical formulae of some of the polymers used in this book

418

(3)

REFERENCES

421

(8)

NAME INDEX

429

(4)

SUBJECT INDEX

433

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Polymers and Neutron Scattering

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