Polymers and Neutron Scattering

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

• ISBN13: 9780198500636
• ISBN10: 0198500637
• Edition: Reprint
• Format: Paperback
• Copyright: 1997-03-27
• Publisher: Clarendon Press

Summary

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.

Table of Contents

GLOSSARY OF SYMBOLS	xv
1 INTRODUCTION	1	(25)
1.1 Introduction	1	(1)
1.2 Possibilities--deuterium labelling	2	(3)
1.3 The components of a neutron experiment	5	(2)
1.4 Background--a little history	7	(1)
1.5 Scattering--the basics: introducing q	7	(4)
1.6 Relationship between scattering and structure	11	(3)
1.7 Elastic and inelastic scattering	14	(2)
1.8 Cross sections	16	(4)
1.9 Summary	20	(6)
1.9.1. Molecular shape	20	(2)
1.9.2. Sample morphology	22	(1)
1.9.3. Main chain segmental motion	23	(3)
2 NEUTRON PRODUCTION AND DETECTION: THE NUTS AND BOLTS	26	(25)
2.1 Introduction	26	(1)
2.2 Neutron sources	27	(7)
2.2.1 Reactor sources	27	(2)
2.2.2 Electron accelerator sources	29	(2)
2.2.3 Spallation neutron sources	31	(2)
2.2.4 Relative advantages of different sources	33	(1)
2.3 Moderators: wavelengths and pulse length	34	(2)
2.4 Neutron transportation	36	(4)
2.4.1 Guides	36	(2)
2.4.2 Collimation	38	(2)
2.5 Stopping neutrons--shielding, beamstops, diaphragms	40	(1)
2.6 Monochromation--choosing wavelengths	41	(3)
2.6.1 Crystal monochromators	41	(1)
2.6.2 Mechanical velocity selection	42	(2)
2.6.3 Measuring wavelengths	44	(1)
2.7 Detecting and monitoring neutron beams	44	(2)
2.8 Polarized neutron beams	46	(2)
2.9 Spectrometers	48	(3)
3 SPECTROMETERS AND WHAT THEY MEASURE	51	(30)
3.1 Introduction	51	(6)
3.1.1 Diffractometers for liquids and amorphous materials	52	(1)
3.1.2 Diffractometers for crystalline powders	53	(1)
3.1.3 Diffractometers with polarization analysis	54	(1)
3.1.4 Single crystal diffractometers	54	(1)
3.1.5 Spectrometers for measuring Sinc(q,w) (inelastic incoherent scattering) with modest resolution and high intensity	54	(2)
3.1.6 Spectrometers, for measuring Scoh (q,w) (inelastic coherent scattering)	56	(1)
3.1.7 High resolution quasielastic scattering spectrometers	56	(1)
3.2 Small angle scattering	57	(13)
3.2.1 Setting up the spectrometer	57	(2)
3.2.2 Samples and their environment	59	(2)
3.2.3 Ancilliary measurements--background, resolution, transmission	61	(3)
3.2.4 Data analysis and data reduction	64	(6)
3.3 Neutron reflection	70	(2)
3.4 Inelastic and quasielastic scattering	72	(9)
3.4.1 Vibrational spectroscopy	72	(1)
3.4.2 Rotational motion	73	(1)
3.4.3 The back-scattering spectrometer	74	(3)
3.4.4 Main chain motion	77	(1)
3.4.5 The spin-echo spectrometer	77	(4)
4 THEORETICAL BASIS OF SCATTERING	81	(35)
4.1 Introduction	81	(1)
4.2 Definition of the cross section	82	(1)
4.3 Elastic interaction between a neutron and a single nucleus	83	(3)
4.4 Coherent and incoherent elastic scattering	86	(2)
4.4.1 Mixtures of isotopes	86	(1)
4.4.2 The effect of nuclear spin	87	(1)
4.5 The static correlation function	88	(4)
4.6 Inelastic cross section	92	(4)
4.7 Coherent and incoherent inelastic scattering	96	(1)
4.8 Van Hove equation and correlation function	96	(6)
4.8.1 The correlation function	96	(2)
4.8.2 The autocorrelation function	98	(1)
4.8.3 The static approximation	99	(1)
4.8.4 Some remarks about notation	99	(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