Preview Carrahers Polymer Chemistry, 10th Edition by Charles E. Carraher Jr. (2018)

Carraher’s Polymer Chemistry
Tenth Edition



Carraher’s Polymer Chemistry
Tenth Edition

Charles E. Carraher, Jr.


CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2018 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-4987-3738-8 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the
validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the
copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to
publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let
us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted,
or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including ­photocopying, microfilming, and recording, or in any information storage or retrieval system, without written
­permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://

www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA
01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users.
For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been
arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only
for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Names: Carraher, Charles E., Jr., 1941Title: Carraher’s polymer chemistry.
Other titles: Polymer chemistry
Description: Tenth edition. | Boca Raton : CRC Press, 2017.
Identifiers: LCCN 2017017296 | ISBN 9781498737388 (hb)
Subjects: LCSH: Polymers. | Polymers--Textbooks. | Polymerization. |
Polymerization--Textbooks.
Classification: LCC QD381 .S483 2017 | DDC 547/.7--dc23
LC record available at />Visit the Taylor & Francis Web site at

and the CRC Press Web site at



Contents
Preface............................................................................................................................................xvii
Acknowledgments............................................................................................................................xix
Author..............................................................................................................................................xxi
Chapter 1 Introduction to Polymers...............................................................................................1
1.1 History of Polymers............................................................................................ 1
1.2 Why Polymers?................................................................................................. 11
1.3 Today’s Marketplace......................................................................................... 15
1.4 Environmental Assessment.............................................................................. 18
1.5Summary.......................................................................................................... 23

Glossary.......................................................................................................................24
Exercises......................................................................................................................24
Answers.......................................................................................................................25
Additional Reading.....................................................................................................25
General Encyclopedias and Dictionaries.....................................................................26
Chapter 2 Polymer Structure (Morphology)................................................................................ 27
2.1

Stereochemistry of Polymers............................................................................28
2.1.1 Chain Length...................................................................................... 29
2.1.2 Pendant Groups and Branching.......................................................... 31
2.1.3Tacticity.............................................................................................. 31
2.2 Molecular Interactions......................................................................................34
2.2.1 Glass Transition and Melt Transition................................................. 39
2.2.2 Secondary Structure........................................................................... 42
2.3 Polymer Crystals.............................................................................................. 42
2.4 Amorphous Bulk State.....................................................................................46
2.5 Polymer Structure–Property Relationships......................................................46
2.6Crosslinking..................................................................................................... 50
2.7 Crystalline and Amorphous Combinations...................................................... 51
2.8Summary.......................................................................................................... 53
Glossary....................................................................................................................... 54
Exercises...................................................................................................................... 55
Answers....................................................................................................................... 56
Additional Reading..................................................................................................... 58
Chapter 3 Molecular Weight of Polymers.................................................................................... 59
3.1Introduction...................................................................................................... 59
3.2Solubility.......................................................................................................... 61
3.2.1 Polymer Solubility: Actual................................................................. 65
3.3 Average Molecular Weight Values................................................................... 68

3.4 Fractionation of Polydisperse Systems............................................................. 71
3.5Chromatography............................................................................................... 71
3.6 Colligative Molecular Weights......................................................................... 75
3.6.1Osmometry......................................................................................... 75
v


vi

Contents

3.6.2 End-Group Analysis........................................................................... 77
3.6.3 Ebulliometry and Cryometry............................................................. 77
3.7 Light Scattering Photometry............................................................................ 77
3.8 Other Techniques.............................................................................................. 83
3.8.1Ultracentrifugation............................................................................. 83
3.8.2 Mass Spectrometry............................................................................. 85
3.9Viscometry....................................................................................................... 86
3.10Summary..........................................................................................................92
Glossary....................................................................................................................... 93
Exercises...................................................................................................................... 95
Answers.......................................................................................................................97
Additional Reading..................................................................................................... 98
Chapter 4 Polycondensation Polymers (Step-Reaction Polymerization)................................... 101
4.1 Comparison between Polymer Types and Kinetics of Polymerization.......... 101
4.2Introduction.................................................................................................... 106
4.3 Stepwise Kinetics........................................................................................... 106
4.4 Polycondensation Mechanisms....................................................................... 111
4.5Polyesters........................................................................................................ 113
4.6Polycarbonates................................................................................................ 122

4.7 Synthetic Polyamides..................................................................................... 126
4.8Polyimides...................................................................................................... 135
4.9Polyamide-Imide............................................................................................ 137
4.10 Polybenzimidazoles and Related Polymers.................................................... 138
4.11 Polyurethanes and Polyureas.......................................................................... 141
4.12Polysulfides..................................................................................................... 145
4.13 Polyethers and Epoxys.................................................................................... 145
4.14Polysulfones.................................................................................................... 150
4.15 Poly(Ether Ether Ketone) and Polyketones.................................................... 151
4.16 Phenolic and Amino Plastics.......................................................................... 152
4.17 Furan Resins................................................................................................... 155
4.18 Synthetic Routes............................................................................................. 156
4.19 Liquid Crystals............................................................................................... 157
4.20Microfibers..................................................................................................... 160
4.21Summary........................................................................................................ 162
Glossary..................................................................................................................... 163
Exercises.................................................................................................................... 164
Answers..................................................................................................................... 166
Additional Reading................................................................................................... 167
Chapter 5 Ionic Chain-Reaction and Complex Coordination Polymerization (Addition
Polymerization)......................................................................................................... 169
5.1 Chain Growth Polymerization: General......................................................... 170
5.2 Cationic Polymerization................................................................................. 171
5.3 Anionic Polymerization.................................................................................. 177
5.4Stereoregularity.............................................................................................. 182
5.5 Polymerization with Complex Coordination Catalysts.................................. 183
5.6 Soluble Stereoregulating Catalysis................................................................. 184
5.7Polyethylenes.................................................................................................. 188

Contents

vii

5.8Polypropylene................................................................................................. 196
5.9 Polymers from 1,4-Dienes..............................................................................202
5.10Polyisobutylene...............................................................................................205
5.11 Metathesis Reactions......................................................................................206
5.12 Zwitterionic Polymerization...........................................................................207
5.13 Isomerization Polymerization.........................................................................208
5.14 Precipitation Polymerization..........................................................................208
5.15Summary........................................................................................................209
Glossary..................................................................................................................... 210
Exercises.................................................................................................................... 210
Answers..................................................................................................................... 213
Additional Reading................................................................................................... 215
Chapter 6 Free Chain Polymerization (Addition Polymerization)............................................. 217
6.1 Initiators for Free Radical Chain Polymerization.......................................... 217
6.2 Mechanism for Free Radical Chain Polymerization...................................... 221
6.3 Chain Transfer................................................................................................ 228
6.4 Polymerization Techniques............................................................................ 230
6.5 Fluorine-Containing Polymers....................................................................... 234
6.6Polystyrene..................................................................................................... 239
6.7 Poly(Vinyl Chloride)...................................................................................... 241
6.8 Poly(Methyl Methacrylate)............................................................................. 245
6.9 Poly(Vinyl Alcohol) and Poly(Vinyl Acetals)................................................ 250
6.10Polyacrylonitrile............................................................................................. 251
6.11 Solid-State Irradiation Polymerization........................................................... 252
6.12 Plasma Polymerizations.................................................................................. 253
6.13 Controlled Radical Polymerizations............................................................... 253

6.14Summary........................................................................................................ 254
Glossary..................................................................................................................... 254
Exercises.................................................................................................................... 255
Answers..................................................................................................................... 257
References................................................................................................................. 258
Chapter 7 Copolymerization...................................................................................................... 261
7.1 Kinetics of Copolymerization........................................................................ 262
7.2The Q–e Scheme............................................................................................266
7.3 Commercial Copolymers................................................................................ 268
7.4 Block Copolymers.......................................................................................... 268
7.5 Graft Copolymers........................................................................................... 270
7.6Elastomers...................................................................................................... 271
7.7 Thermoplastic Elastomers.............................................................................. 273
7.8Blends............................................................................................................. 274
7.8.1 Immiscible Blends............................................................................ 274
7.8.2 Miscible Blends................................................................................ 276
7.9Fluoroelastomers............................................................................................ 278
7.10 Nitrile Rubber................................................................................................. 278
7.11 Acrylonitrile–Butadiene–Styrene Terpolymers............................................. 279
7.12 EPDM Rubber................................................................................................280
7.13 Networks: General..........................................................................................280


viii

Contents

7.14 Polymer Mixtures........................................................................................... 282
7.15Dendrites........................................................................................................ 282
7.16Ionomers.........................................................................................................284

7.17 Laboratory Gloves.......................................................................................... 286
7.18Summary........................................................................................................ 286
Glossary..................................................................................................................... 288
Exercises.................................................................................................................... 289
Answers..................................................................................................................... 290
Additional Reading................................................................................................... 291
Chapter 8 Composites and Fillers.............................................................................................. 293
8.1Fillers.............................................................................................................. 293
8.2 Types of Composites....................................................................................... 294
8.3 Long Fiber Composites: Theory..................................................................... 296
8.4 Fibers and Resins............................................................................................ 298
8.5 Long Fiber Composites: Applications............................................................300
8.6Nanocomposites.............................................................................................302
8.7Fabrication......................................................................................................306
8.7.1 Processing of Fiber-Reinforced Composites....................................306
8.7.2 Structural Composites......................................................................306
8.7.3Laminating.......................................................................................307
8.7.4Particulate.........................................................................................307
8.8 Summary........................................................................................................308
Glossary.....................................................................................................................308
Exercises....................................................................................................................309
Answers.....................................................................................................................309
Additional Reading................................................................................................... 310
Chapter 9 Naturally Occurring Polymers—Plants.................................................................... 313
9.1Polysaccharides.............................................................................................. 314
9.2Cellulose......................................................................................................... 316
9.3Paper............................................................................................................... 319
9.4 Cellulose-Regenerating Processes.................................................................. 323
9.4.1Rayon................................................................................................ 324
9.4.2Cellophane........................................................................................ 326

9.5 Esters and Ethers of Cellulose........................................................................ 326
9.5.1 Wrinkle-Free Fabric......................................................................... 327
9.5.2 Inorganic Esters................................................................................ 327
9.5.3 Organic Esters.................................................................................. 328
9.5.4 Organic Ethers.................................................................................. 330
9.6Starch.............................................................................................................. 332
9.7Homopolysaccharides..................................................................................... 335
9.7.1Fructans............................................................................................ 337
9.7.2 Chitin and Chitosan.......................................................................... 338
9.7.3Others............................................................................................... 339
9.8Heteropolysaccharides.................................................................................... 339
9.9 Synthetic Rubbers........................................................................................... 343
9.10 Naturally Occurring Polyisoprenes................................................................ 349
9.11Resins............................................................................................................. 354


Contents

ix

9.12Balloons.......................................................................................................... 355
9.13Lignin............................................................................................................. 356
9.14Melanins......................................................................................................... 358
9.15Asphalt............................................................................................................ 359
9.16 Castor Oil.......................................................................................................360
9.17Summary........................................................................................................ 361
Glossary..................................................................................................................... 362
Exercises.................................................................................................................... 363
Answers.....................................................................................................................364
Additional Reading................................................................................................... 365

Chapter 10 Naturally Occurring Polymers—Animals................................................................ 367
10.1Proteins........................................................................................................... 368
10.2 Levels of Protein Structure............................................................................. 371
10.2.1 Primary Structure............................................................................. 372
10.2.2 Secondary Structure......................................................................... 372
10.2.2.1Keratins.............................................................................. 373
10.2.2.2 Silk and Spider Webs......................................................... 375
10.2.2.3Wool................................................................................... 378
10.2.2.4Collagen............................................................................. 380
10.2.2.5Elastin................................................................................ 381
10.2.3 Tertiary Structure............................................................................. 381
10.2.3.1 Globular Proteins............................................................... 382
10.2.3.2 Fibrous Proteins................................................................. 383
10.2.3.3 Membrane Proteins............................................................ 383
10.2.4 Quaternary Structure........................................................................ 383
10.3 Nucleic Acids.................................................................................................. 386
10.4 Flow of Biological Information...................................................................... 391
10.5 RNA Interference........................................................................................... 393
10.6 Polymer Structure........................................................................................... 394
10.7 Protein Folding............................................................................................... 399
10.8 Genetic Engineering....................................................................................... 401
10.9 DNA Profiling................................................................................................403
10.10 The Human Genome—General.....................................................................405
10.11Chromosomes................................................................................................. 410
10.11.1 The Next Steps.................................................................................. 422
10.12Spliceosomes.................................................................................................. 423
10.13Proteomics...................................................................................................... 423
10.14 Protein Site Activity Identification................................................................. 425
10.15Summary........................................................................................................ 425
Glossary..................................................................................................................... 426

Exercises.................................................................................................................... 429
Answers..................................................................................................................... 430
Additional Reading................................................................................................... 431
Chapter 11 Organometallic and Inorganic–Organic Polymers................................................... 433
11.1Introduction.................................................................................................... 433
11.2 Inorganic Reaction Mechanisms.................................................................... 433


x

Contents

11.3 Condensation Organometallic Polymers........................................................ 437
11.3.1Polysiloxanes.................................................................................... 438
11.3.2 Organotin and Related Condensation Polymers............................... 441
11.4 Coordination Polymers...................................................................................444
11.4.1 Platinum-Containing Polymers........................................................446
11.5 Addition Polymers.......................................................................................... 447
11.5.1 Ferrocene-Containing and Related Polymers..................................449
11.5.2 Polyphosphazenes and Related Polymers......................................... 451
11.5.3 Boron-Containing Polymers............................................................. 452
11.6 Ion-Exchange Resins...................................................................................... 453
11.7Summary........................................................................................................ 455
Glossary..................................................................................................................... 455
Exercises.................................................................................................................... 455
Answers..................................................................................................................... 456
Additional Reading................................................................................................... 457
Chapter 12 Inorganic Polymers................................................................................................... 459
12.1Introduction.................................................................................................... 459
12.2 Portland Cement............................................................................................. 459

12.3 Other Cements................................................................................................ 462
12.4Silicates........................................................................................................... 463
12.4.1Network............................................................................................465
12.4.2Layer.................................................................................................466
12.4.3Chain................................................................................................ 467
12.5 Silicon Dioxide (Amorphous)......................................................................... 467
12.6 Kinds of Amorphous Glass............................................................................ 470
12.7 Safety Glass.................................................................................................... 473
12.7.1 New Applications of Glass............................................................... 475
12.8Lenses............................................................................................................. 475
12.9Sol–Gel........................................................................................................... 478
12.10Aerogels.......................................................................................................... 479
12.11 Silicon Dioxide (Crystalline Forms): Quartz Forms...................................... 481
12.12 Silicon Dioxide in Electronic Chips............................................................... 482
12.13 Silicon Dioxide in Optical Fibers................................................................... 483
12.14Asbestos..........................................................................................................484
12.15 Fly Ash and Aluminosilicates........................................................................ 485
12.16 Polymeric Carbon: Diamond.......................................................................... 487
12.17 Polymeric Carbon: Graphite........................................................................... 487
12.18Internal Cyclization: Carbon Fibers and Related Materials........................... 489
12.19 Carbon Nanotubes.......................................................................................... 490
12.19.1Structures.......................................................................................... 491
12.20Bitumens......................................................................................................... 496
12.21 Carbon Black.................................................................................................. 497
12.22Polysulfur....................................................................................................... 499
12.23Ceramics......................................................................................................... 499
12.24 High-Temperature Superconductors............................................................... 501
12.24.1 Discovery of the 123-Compound..................................................... 501
12.24.2 Structure of the 123-Compound....................................................... 501
12.25Zeolites........................................................................................................... 502

12.26Summary........................................................................................................ 503


Contents

xi

Glossary.....................................................................................................................504
Exercises.................................................................................................................... 505
Answers.....................................................................................................................506
Further Reading......................................................................................................... 508
Chapter 13 Testing and Spectrometric Characterization of Polymers.........................................509
13.1Introduction....................................................................................................509
13.2 Spectronic Characterization of Polymers....................................................... 510
13.2.1 Infrared Spectroscopy...................................................................... 510
13.2.2 Raman Spectroscopy........................................................................ 511
13.2.3 Nuclear Magnetic Resonance Spectroscopy.................................... 512
13.2.4 NMR Applications........................................................................... 513
13.2.5 Electron Paramagnetic Resonance Spectroscopy............................ 514
13.2.6 X-Ray Spectroscopy......................................................................... 514
13.3 Surface Characterization................................................................................ 514
13.3.1 Auger Electron Spectroscopy and X-Ray Photoelectron
Spectroscopy..................................................................................... 515
13.3.2 Near-Field Scanning Optical Microscopy........................................ 516
13.3.3 Electron Microscopy........................................................................ 516
13.3.4 Scanning Probe Microscopy............................................................ 517
13.3.5 Superresolution Fluorescence Microscopy....................................... 519
13.3.6 Secondary Ion Mass Spectroscopy................................................... 520
13.4 Amorphous Region Determinations............................................................... 520
13.5 Mass Spectrometry......................................................................................... 521

13.6 Thermal Analysis........................................................................................... 521
13.7 Thermal Property Tests.................................................................................. 523
13.7.1 Softening Range............................................................................... 523
13.7.2 Heat Deflection Temperature............................................................ 524
13.7.3 Glass Transition Temperatures......................................................... 524
13.7.4 Thermal Conductivity...................................................................... 525
13.7.5 Thermal Expansion.......................................................................... 526
13.8Flammability.................................................................................................. 526
13.9 Electrical Properties: Theory......................................................................... 527
13.10 Electric Measurements................................................................................... 529
13.10.1 Dielectric Constant........................................................................... 530
13.10.2 Electrical Resistance........................................................................ 530
13.10.3 Dissipation Factor and Power Loss.................................................. 530
13.10.4 Electrical Conductivity and Dielectric Strength.............................. 531
13.11 Optical Properties Tests.................................................................................. 532
13.11.1 Index of Refraction........................................................................... 532
13.11.2 Optical Clarity.................................................................................. 533
13.11.3 Absorption and Reflectance............................................................. 533
13.12Weatherability................................................................................................ 534
13.13 Chemical Resistance...................................................................................... 534
13.14 Measurement of Particle Size......................................................................... 534
13.15 Measurement of Adhesion.............................................................................. 536
13.16 Permeability and Diffusion............................................................................ 537
13.17Summary........................................................................................................ 539
Glossary..................................................................................................................... 539
Exercises.................................................................................................................... 541


xii

Contents

Answers..................................................................................................................... 541
Additional Reading................................................................................................... 542
Chapter 14 Rheology and Physical Tests..................................................................................... 545
14.1Rheology........................................................................................................ 545
14.1.1 Rheology and Physical Tests............................................................ 548
14.1.2 Response Time................................................................................. 551
14.2 Typical Stress–Strain Behavior...................................................................... 552
14.3 Stress–Strain Relationships............................................................................ 554
14.4 Specific Physical Tests.................................................................................... 556
14.4.1 Tensile Strength................................................................................ 556
14.4.2 Tensile Strength of Inorganic and Metallic Fibers and Whiskers.... 557
14.4.3 Compressive Strength....................................................................... 558
14.4.4 Impact Strength................................................................................ 559
14.4.5Hardness........................................................................................... 560
14.4.6 Brinell Hardness............................................................................... 561
14.4.7 Rockwell Hardness........................................................................... 561
14.4.8 Shear Strength.................................................................................. 562
14.4.9 Abrasion Resistance......................................................................... 562
14.4.10Failure............................................................................................... 563
14.5Summary........................................................................................................ 563
Glossary.....................................................................................................................564
Exercises.................................................................................................................... 565
Answers..................................................................................................................... 566
Additional Reading................................................................................................... 566
Chapter 15 Additives.................................................................................................................... 569
15.1Plasticizers...................................................................................................... 569
15.2Antioxidants................................................................................................... 573
15.3 Heat Stabilizers.............................................................................................. 575

15.4 Ultraviolet Stabilizers..................................................................................... 575
15.5 Flame Retardants............................................................................................ 576
15.6Colorants......................................................................................................... 577
15.7 Curing Agents................................................................................................. 578
15.8 Antistatic Agents: Antistats............................................................................ 578
15.9 Chemical Blowing Agents.............................................................................. 579
15.10Compatibilizers.............................................................................................. 579
15.11 Impact Modifiers............................................................................................ 580
15.12 Processing AIDS............................................................................................ 580
15.13Lubricants....................................................................................................... 580
15.14 Microorganism Inhibitors............................................................................... 580
15.15Summary........................................................................................................ 580
Glossary.............................................................................................educed viscosity

Viscosity number

ηsp/c

ηred or ηsp/c

Inherent viscosity

Logarithmic viscosity number

(ln ηr)/c

Intrinsic viscosity

Limiting viscosity number

Limit (ηsp/c)c→0 or limit
(ln ηr)/cc→0

ηinh or (ln ηr)/c
LVN

and K values to make a plot of log LVN versus log M since the log of Equation 3.31, that is, Equation
3.32, is a straight line relationship, where the slope is a and intercept K. In reality, a is determined
from the slope but K is determined by simply selecting a known LVN–M couple and using the determined a value to calculate the K value:
log LVN = a log M + log K





(3.32)

The intrinsic viscosity or limiting viscosity number, like melt viscosity, is temperature dependent
and decreases as the temperature increases as shown in Equation 3.33:
LVN = A e E /RT



(3.33)



However, if the original temperature is below the theta temperature, the viscosity will increase
when the mixture of polymer and solvent is heated to a temperature slightly above the theta
temperature.

Viscosity measurements of dilute polymer solutions are carried out using a viscometer, such as
either of those pictured in Figure 3.20b. The viscometer is placed in a constant temperature bath and
the time taken to flow through a space is measured.

nsp/c or (ln nr)/c

(a)

(b)

nsp/c

(ln nr)/c

Concentration, g/dL

FIGURE 3.20  (a) Reduced and inherent viscosity–concentration lines for a dilute polymer solution, (b)
drawings of two common solution viscometers.


89

Molecular Weight of Polymers

TABLE 3.8
Typical K Values for the Mark–Houwink Equation
Polymer
Low-density polyethylene
High-density polyethylene
i-Polypropylene

Polystyrene
Poly(vinyl chloride)
Poly(vinyl acetate)
Poly(methyl acrylate)
Polyacrylonitrile
Poly(methyl methacrylate)
Poly(ethylene terephthalate)
Nylon-66

Solvent

Temperature (K)

K × 105 dL/g

Decalin
Decalin
Decalin
Decalin
Chlorobenzene
Acetone
Acetone
Dimethylformamide
Acetone
m-Cresol
90% Aqueous formic acid

343
408
408

373
303
298
298
298
298
298
298

39
68
11
16
71
11
6
17
10
1
110

Flory, Debye, and Kirkwood showed that [η] is directly proportional to the effective hydrodynamic volume of the polymer in solution and inversely proportional to the molecular weight M.
The effective hydrodynamic volume is the cube of the RMS end-to-end distance, (r2)3/2. (It is of
interest to note that volume fraction and mole fraction are generally proportional to one another.)
This proportionality constant, N, in the Flory equation for hydrodynamic volume, Equation 3.34,
has been considered to be a universal constant independent of solvent, polymer, temperature, and
molecular weight:

[η ] =

N (r 2 )3 / 2
M


(3.34)

The actual average end-to-end distance r is related to the nonsolvent expanded average end-toend distance ro using the Flory expansion factor a as follows:


r = a ro

(3.35)

Substitution of Equation 3.35 into Equation 3.34 and rearrangement give


[η]M = N (ro2 )3/ 2a2

(3.36)

The values for α vary from 0.5 for the Flory theta solvents to about 3 for polymers in good
solvents.
In Equation 3.31, α values for random coils range from 0.5 for theta solvents to 0.8 for good
solvents, 0 for hard spheres, about 1 for semicoils, and 2 for rigid rods.
The theta temperature corresponds to the Boyle point in an imperfect gas and is the range in
which the virial coefficient B in the expanded gas law becomes zero. This same concept applies to
the modification of the gas law (PV = nRT) used to determine the osmotic pressure of a polymer
solution and is simply the van’t Hoff equation that reduces to Equation 3.37 when B = 0.

π=


RTC
Mn

(3.37)


90

Carraher’s Polymer Chemistry

For linear polymers at their theta temperature, that is, the temperature where the chain attains
unperturbed dimensions, the Flory equation resembles the Mark–Houwink equation, where α is
equal to 1.0 as shown below:


[η ] = K M 1/ 2 α3 = K ′ M 1/ 2

(3.38)

The intrinsic viscosity of a solution, like the melt viscosity, is temperature dependent and
decreases as temperature increases:


[η] = Ae E /RT

(3.39)

While the description of viscosity is complex, the relative viscosity is directly related to the flowthrough times using the same viscometer as shown in Equation 3.34, where t and t0 are the flow
times for the polymer solution and the solvent, respectively, and the density of the solution (ρ) and
solvent (ρ0) is related as in Equation 3.40:



η
ρt
=
= ηr
η0
ρ0t0


(3.40)

Since the densities of the dilute solution and solvent are almost the same, they are normally canceled giving Equation 3.41:



η
t
=
= ηr
η0
t0


(3.41)

Thus, the relative viscosity is simply a ratio of flow times for the polymer solution and solvent.
Reduced viscosity is related to the LVN by a virial equation 3.42:



ηsp
=[η ] + k1[η ]2c + k ′[η ]3c 2 +
c



(3.42)

For most solutions, Equation 3.42 reduces to the Huggins viscosity relationship, Equation 3.38,



ηsp
=[η ] + k1[η ]2c
c


(3.43)

which allows [η] to be determined from the intercept of the plot of ηsp/c versus c and is the basis for
the top plot given in Figure 3.20.
Another relationship often used in determining the [η] is called the inherent viscosity equation
and is given in Equation 3.44.

(ln ηr )
= [η] − k2 [η]2 c
c


(3.44)


91

Molecular Weight of Polymers

Again, a plot of (ln ηr)/c versus c gives a straight line with the intercept [η] (or LVN) after
extrapolation to zero polymer concentration. This is the basis of the lower plot in Figure 3.20. While
k1 and k2 are mathematically such that


k1 + k2 = 0.5

(3.45)

many systems appear not to follow this relationship.
We will now turn our attention from the viscosity of dilute solutions and look at the viscosity
of melted polymers. The viscosity of melted polymers is important in transferring resins and in
polymer processing such as determining the correct conditions to have a specific flow rate for injection processing and in determining the optimum conditions to obtain the necessary dimensions of
extruded shapes. Fillers, plasticizers, temperature, solvents, and molecular weight are just some of
the variables that influence the viscosity of polymer melts. Here, we will look at the dependence
of melt viscosity on polymer molecular weight. Polymer melts have viscosities on the order of
10,000 MPa (1 centipoise is equal to 0.001 Pa/s).

For largely linear polymers, such as PP, where there are not present particularly bulky side
chains, the viscosity or flow is mainly dependent on the chain length. In most polymers, the melt
viscosity–chain length relationship has two distinct regions where the region division occurs when
the chain length reaches some length called the critical entanglement chain length, Z (or simply
critical chain length) where intermolecular entanglement occurs. This intermolecular entanglement
causes the individual chains in the melt to act as being much more massive because of the entanglement. Thus, the resistance to flow is a combination of the friction and entanglement between chains
as they slide past one another. Below the critical entanglement length, where only the friction part is
important, the melt viscosity η is related to the weight-average molecular weight by


η = K l M w1.0

(3.46)

Above the critical chain length, where both the friction and entanglement are important, the
relationship is


η = K h M w3.4

(3.47)

where Kl is a constant for the precritical entanglement chain length and K h is for the situation above
Z and where both K values are temperature dependent. The first power dependence is due to the
simple increase in molecular weight as chain length increases, but the 3.4 power relationship is due
to a complex relationship between chain movement as related to entanglement and diffusion and
chain length.
The critical chain length is often the onset of “strength”-related properties and is generally considered the lower end for useful mechanical properties. The Z value for polymers varies but is typically between about 200 and 1000 units in length. For instance, the Z value for PS is about 700; for
polyisobutylene, about 600; for poly(decamethylene sebacate), about 300; for poly(methyl methacrylate), about 200; and for poly(dimethyl siloxane), about 1000.
A number of techniques have been developed to measure melt viscosity. Some of these are

listed in Table 3.9. Rotational viscometers are of varied structures. The Couette cup and bob
viscometer consist of a stationary inner cylinder, the bob, and an outer cylinder, cup, that is
rotated. Shear stress is measured in terms of the required torque needed to achieve a fixed rotation rate for a specific radius differential between the radius of the bob and cup. The Brookfield
viscometer is a bob and cup viscometer. The Mooney viscometer, often used in the rubber


92

Carraher’s Polymer Chemistry

TABLE 3.9
Viscosity Measuring Techniques and Their
Usual Range
Technique

Typical Range (Poise)

Capillary pipette
Falling sphere
Parallel plate
Falling coaxial cylinder
Stress relaxation
Rotating cylinder
Tensile creep

0.01–1000
1–100,000
10,000–109
100,000–1011
1000–1010

1–1012
100,000–greater than 1012

industry, measures the torque needed to revolve a rotor at a specified rate. In the cone and plate
assemblies, the melt is sheared between a flat plat and a broad cone whose apex contacts the
plate containing the melt.
A number of capillary viscometers or rheometers have been employed to measure melt viscosity.
In some sense, these operate on a principle similar to the simple observation of a trapped bubble
moving from the bottom of a shampoo bottle when it is turned upside down. The more viscous the
shampoo, the longer it takes for the bubble to move through the shampoo.

3.10 SUMMARY


1.Some naturally occurring polymers such as certain proteins and nucleic acids consist of
molecules with a specific molecular weight and are called monodisperse. However, many
other natural polymers, such as cellulose and natural rubber, and most synthetic polymers consist of molecules with different molecular weights and are called polydisperse.
Many properties of polymers are dependent on their chain length. Since the melt viscosity
increases exponentially with chain length, the high energy costs of processing high-molecular-weight polymers are not often justified.
2. The distribution of chain lengths in a polydisperse system may be represented on a typical
probability-like curve. M n is the smallest in magnitude of the typically obtained molecular weights and is a simple arithmetic mean that can be determined using any technique
based on colligative properties, such as osmotic pressure, boiling point elevation, freezing
point depression, and end group determination. M w is larger than M n and is referred to
as the second power relationship for disperse polymer chains. This value is most often
determined by light-scattering photometry. Light-scattering photometry and the colligative related values are referred to as absolute molecular weight values because there is
a direct mathematical connection between molecular weight and the particular property
used to determine the molecular weight.
3.For monodisperse samples, M n = M w . For polydisperse samples, the ratio of M w /M n is
a measure of the polydisparity and is given the name polydispersity index. The viscosity molecular weight must be calibrated using samples whose molecular weight has been
determined using an absolute molecular weight determination technique; thus, it is not an

absolute molecular weight determining technique, but it requires simple equipment, and is
easy to measure. The Mark–Houwink relationship, LVN = KM a , is used to relate molecular weight and viscosity.
4.The number-average molecular weight is dependent on the number of polymer chains
while the weight-average molecular weight is dependent on the size of the chains. Thus,


Molecular Weight of Polymers

93

there is a correlation between the way the molecular weight is obtained and the type of
molecular weight obtained.
5. Molecular weight distribution is most often measured using some form of chromatography.
In GPC, crosslinked polymers are used in a column and act as a sieve allowing the larger
molecules to elute first. After calibration, the molecular weight of the various fractions of
the polymer can be determined. Combinations such as chromatography coupled with lightscattering photometry are used to obtain the molecular weight of the various fractions in a
continuous manner.
6.While some techniques such as membrane osmometry and light-scattering photometry
give absolute molecular weight, other techniques such as viscometry give only relative
molecular weights unless calibrated employing a technique that gives the absolute molecular weight. After calibration between viscometry values and chain length through some
absolute molecular weight method, viscometry is a fast, inexpensive, and simple method to
monitor the molecular weight.
7.In general, polymers are soluble in less solvents and to a lower concentration than similar
smaller molecules. This is because entropy is the driving force for solubility and smaller
molecules have larger entropy values when solubility is achieved in comparison to polymers. Polymers also take longer to dissolve since it takes time for the solvent molecules to
penetrate the polymer matrix.
8.Flory and Huggins developed an interaction parameter that may be used as a measure of
the solvent power of solvents for amorphous polymers. Flory and Krigbaum introduced the
idea of a theta temperature, which is the temperature at which an infinitely long polymer
chain exists as a statistical coil in a solvent.

9.Hildebrand developed solubility parameters to predict the solubility of nonpolar polymers
in nonpolar solvents. The solubility parameter is the square root of the CED. For polar solvents, special solvent–polymer interactions can be incorporated into the solubility parameter approach.

GLOSSARY
Affinity chromatography: Chromatography in which the resin is designed to contain moieties that
interact with particular molecules and/or units within a polymer chain.
Bingham plastic: Plastic that does not flow until the external stress exceeds a critical threshold
value.
Brownian motion: Movement of larger molecules in a liquid that results from a bombardment of
smaller molecules.
Buoyancy factor: In ultracentrifugation experiments, it determines the direction of polymer transport under the effect of centrifugal forces in the cell.
Chromatography: Family of separation techniques based on the use of a medium that shows selective absorption.
Colligative properties: Properties of a solution that are dependent on the number of solute molecules present.
Cloud point: Temperature at which a polymer starts to precipitate when the temperature is lowered.
Cohesive energy density (CED): Heat of vaporization per unit volume.
Commercial polymer range: Molecular weight range high enough to have good physical properties but not too high for economical processing.
Cryometry: Measurement of number-average molecular weight from freezing point depression.
Ebulliometry: Measurement of number-average molecular weight from boiling point elevation.
Effective hydrodynamic volume: Cube of the RMS end-to-end distance of a polymer chain.
Electrophoresis: Form of chromatography that uses an electric field to separate molecules.


94

Carraher’s Polymer Chemistry

End-group analysis: Determination of number-average molecular weight by the determination of
end groups.
Flory–Huggins theory: Theory used to predict the equilibrium behavior between liquid phases
containing polymer.

Fractional precipitation: Fractionation of polydisperse systems by the addition of small amounts
of nonsolvent to a solution of polymer.
Fractionation of polymers: Separation of a polydisperse polymer into fractions of similar molecular weight.
Gel permeation chromatography: Type of liquid–solid elution chromatography, which separates
solutions of polydisperse polymer solutions into fractions containing more homogeneous
chain sizes by means of a sieving action of a swollen crosslinked polymeric gel. Also
called size exclusion chromatography.
High-performance liquid chromatography (HPLC): Chromatography in which pressure is
applied that causes the solution to pass more rapidly through the column.
Hildebrand (H): Unit used for solubility parameter values.
Ion exchange chromatography: Chromatography that separates molecules on the basis of their
electrical charge employing the polyanionic or polycationic resins.
Kauri-butanol values: Measure of the aromaticity of a solvent.
Low-angle laser light-scattering photometry (LALLS): Light scattering that employs low-angle
measurements, minimizing the effect of polymer shape on the scattering.
Mark–Houwink equation: Relates limiting viscosity number to molecular weight;
Mn < Mv < M w < Mz .
Matrix-assisted laser desorption/ionization mass spectrometry (MALDI MS): Mass spectrometry in which the sample is placed in a matrix that contains a strong UV absorber chosen
to match the UV absorption of the laser, which allows the molecules to become volatilized
with minimal fragmentation.
Melt index: Measure of the flow related inversely to melt viscosity.
Monodisperse: System containing molecules of only one chain length.
Multiangle low-angle laser light-scattering photometry (MALS): Similar to LALLS except
where the necessary angle ratios are made together; employs low-angle measurements,
minimizing the effect of polymer shape on the scattered light.
Number-average molecular weight: Arithmetical mean value obtained by dividing the sum of the
molecular weights by the number of molecules.
Oligomer: Polymers with 2–10 repeat units. Oligos means “few.”
Osmometry: Gives the number-average molecular weight from osmotic pressure measurements.
Polydisperse: Mixture of polymer chains of different lengths.

Raoult’s law: States that the vapor pressure of a solvent in equilibrium with a solution is equal to the
product of the mole fraction of the solvent and the vapor pressure of the pure solvent. This
relationship is used in obtaining the number-average molecular weights.
SEC-MALS and SEC-LALLS: Coupled chromatography and light-scattering photometry
that allows the determination of a number of important values along with chain length
distribution.
Sedimentation equilibrium experiment: Ultracentrifugation technique that allows chain length
information to be determined.
Semipermeable membrane: Membrane that permits the diffusion of solvent molecules but not
large molecules.
Size exclusion chromatography (SEC): Chromatography in which separation is by molecular
size or differences in hydrodynamic volume; also called gel permeation chromatography
(GPC); can use the universal calibration approach to obtain molecular weight.
Solubility parameter: A numerical value equal to the square root of the CED, which is used to
predict polymer solubility.


Molecular Weight of Polymers

95

Theta solvent: Solvent in which the polymer chain exists as a statistical coil.
Theta temperature: Temperature at which a polymer of infinite molecular weight begins to
precipitate.
Ultracentrifuge: Centrifuge that increases the force of gravity by as much as 100,000 times, causing a distribution of materials in a solution to separate in accordance with chain length.
Vapor pressure osmometry: Technique for determining the number-average molecular weight by
measuring the relative heats of evaporation of a solvent from a solution and pure solvent.
Viscosity: Resistance to flow.
Intrinsic viscosity: The limiting viscosity number obtained by extrapolation of the reduced viscosity to zero concentration.
Reduced viscosity: Specific viscosity divided by the polymer concentration.

Relative viscosity: Ratio of the viscosities of a solution and its solvent.
Specific viscosity: Difference between the relative viscosity and 1.
Weight-average molecular weight: Second power average of molecular weight; dependent on the
size of the particular chains.
Zimm plot: Type of double extrapolation used to determine the weight-average molecular weight
in light-scattering photometry.

EXERCISES

















1.Which of the following is polydisperse with respect to chain length: (a) casein, (b) commercial polystyrene, (c) paraffin wax, (d) cellulose, or (e) Hevea brasiliensis?
2. If the number-average molecular weight for LDPE is 1.4 million, what is the corresponding
average chain length?
3.What are the number and weight-average molecular weights for a mixture of five molecules, each having the following molecular weights: 1.25 × 106; 1.35 × 106; 1.5 × 106;
1.75 × 106; and 2.00 × 106?

4.What is the most probable value for the polydispersity index for (a) a monodisperse polymer and (b) a polydisperse polymer synthesized by a condensation technique?
5.List in increasing order of values: LVN = KM a , M z , M n , and M w .
6. Which of the following provides an absolute measure of the molecular weight of polymers:
(a) viscometry, (b) cryometry, (c) osmometry, (d) light-scattering photometry, and (e) GPC?
7.What is the relationship between the intrinsic viscosity or limiting viscosity number and
average molecular weight?
8.What molecular weight determination techniques can be used to fractionate polydisperse
polymers?
9. Which of the following techniques yields a number-average molecular weight: (a) viscometry, (b) light-scattering photometry, (c) ultracentrifugation, (e) osmometry, (e) ebulliometry,
and (f) cryometry?
10.What kind of molecular weight do you generally get from light-scattering photometry?
11.What is the value of the exponent a in the Mark–Houwink equation for polymers in theta
solvents?
12. How many amino groups are present in each molecule of nylon-66 made from an excess of
hexamethylenediamine?
13.What is the value of the exponent in the Mark–Houwink equation for a rigid rod?
14. If the values of K and a in the Mark–Houwink equation are 1 × 10−2 cm3/g and 0.5, respectively, what is the average molecular weight of a polymer whose solution has an intrinsic
viscosity of 150 cc/g?
15.Which polymer of ethylene will have the highest-molecular weight: (a) a trimer, (b) an
oligomer, or (c) UHMWPE?
16.What is a Zimm plot?


96

Carraher’s Polymer Chemistry

17.What type of molecular weight average, M v or M w, is based on colligative properties?
18.What principle is used in the determination of molecular weight by vapor pressure
osmometry?

19.Why does the melt viscosity increase faster with molecular weight increase than other
properties such as tensile strength?
20. In spite of the high cost of processing, ultrahigh-molecular-weight polyethylene is used for
making trash cans and other durable goods. Why?
21.Under what conditions are the weight- and number-average molecular weight the same?
22.What is the driving force for polymer solubility?
23.What are the colligative methods for measuring the molecular weight and what kind of
molecular weight do you get?
24.What is the advantage of using viscometry to measure molecular weight?
25.Which will yield the higher apparent molecular weight values in the light-scattering
method: (a) a dust-free system or (b) one in which dust particles are present?
26.Does HPLC need to be calibrated before it can give absolute molecular weights?
27.Which of the following does modern LC allow the calculation of: (a) weight-average
molecular weight, (b) radius of gyration, (c) number-average molecular weight, (d) molecular weight distribution, and (e) polydispersity index?
28. What is the significance of the virial constant B in osmometry and light-scattering equations?
29.According to Hildebrand, what is a regular solvent?
30. Which of the two steps that occur in the solution process, (a) swelling and (b) dispersion of
the polymer particles, can be accelerated by agitation?
31.Define CED.
32.For solution to occur, the change in Gibbs free energy must be (a) 0, (b) <0, or (c) >0.
33.Will a polymer swollen by a solvent have higher or lower entropy than the solid polymer?
34.Define the change in entropy in the Gibbs free energy equation.
35.Is a liquid that has a value of 0.3 for its interaction parameter a good or a poor solvent?
36.What is the value of the Gibbs free energy change at the theta temperature?
37.What term is used to describe the temperature at which a polymer of infinite molecular
weight precipitates from a dilute solution?
38.At which temperature will the polymer coil be larger in a poor solvent: (a) at the theta
temperature, (b) below the theta temperature, or (c) above the theta temperature?
39.If the solubility parameter for water is 23.4 H, what is the CED for water?
40.What is the heat of mixing of two solvents having identical solubility parameters?

41.If the density of a polymer is 0.85 g/cc and the molar volume is 1,176,470 cc, what is the
molecular weight?
42.Name some steps that occur when a polymer is dissolved.
43.Why is it important to determine polymer chain length?
44.Why do δ values decrease as the molecular weight increases in a homologous series of
aliphatic polar solvents?
45.Which would be a better solvent for polystyrene: (a) n-pentane, (b) benzene, or (c)
acetonitrile?
46. Which will have the higher or greater slope when its reduced viscosity or viscosity number
is plotted against concentration: a solution of polystyrene (a) in benzene or (b) in n-octane?
47.What are general typical values for a in the viscosity relationship to molecular weight?
48.When is the Flory equation similar to the Mark–Houwink equation?
49.What is the term used for the cube root of the hydrodynamic volume?
50.Explain why the viscosity of a polymer solution decreases as the temperature increases.
51.Is MALDI MS restricted to use for natural polymers such as proteins and nucleic acids?
52. Which of the following would you expect to be the most soluble in water–ethanol, hexane,
or benzene? Why?


Molecular Weight of Polymers

53.If entropy is the driving force for mixing and solubility, why is there such a focus on
enthalpy through the various approaches with solubility parameters and other similar
values?
54. If MALDI MS is of such great use to biopolymer chemists, why is it not more widely used
by synthetic polymer chemists?

ANSWERS
1.b, c, d, e.
2.50,000.

6
6
3.
M n  = 1.57 × 10 ; M w = 1.62 × 10 .
4.(a) 1, (b) 2.
5.
Mn < Mv < M w < Mz .
6.b,c,d.
7.LVN = KMa.
8.GPC, ultracentrifugation.
9.d, e, f.
10.Weight-average molecular weight.
11.0.5.
12.2 (end groups).
13.2.0.
14.2.25 × 108.
15.c.
16.A type of double extrapolation for the determination of the weight-average molecular
weight for high polymers in which both the concentration and the angle of the incident
beam are extrapolated to zero.
17.Number-average molecular weight.
18.Owing to higher vapor pressure, the solvent evaporates faster from a pure solvent than
from a solution; an application of Raoult’s law.
19.The melt viscosity is proportional to the molecular weight to the 3.4 power.
20.The extremely high-molecular-weight polymer is much tougher.
21.For a monodisperse system.
22.Entropy.
23.Many such as membrane osmometry, end-group analysis. Number-average molecular
weight.
24.Inexpensive equipment and relatively fast.

25.b.
26.Yes.
27.All of them.
28.B is a constant related to the interaction of the solvent and polymer.
29.One in which London forces are the predominant intermolecular attractions.
30.b.
31. Cohesive energy density is equal to the strength of the intermolecular forces between molecules, which is equal to the molar energy of vaporization per unit volume.
32.b.
33.Higher.
34.ΔS = (ΔH − ΔG/T) at constant T.
35.A good solvent.
36.0.
37.Theta temperature.

97


98

Carraher’s Polymer Chemistry

38.b.
39.548 cal/cc.
40.0.
41. 1,000,000 g.
42.Wetting or polymer coming into contact with the solvent, invasion of the polymer matrix
by the solvent (swelling), and finally going into solution.
43.Since there is a relationship between chain length and property.
44. The contribution of the polar group becomes less significant as the alkyl portion increases.
45.The answer depends on what “better” means. From a toxicity standpoint, it is probably
n-pentane since it is the least toxic. From a solubility standpoint, it is b.
46.(a) Benzene is a better solvent, and therefore the value of a is greater.
47.About 0.5–0.8.
48.When α = 1.
49.Root-mean-square end-to-end distance.
50. As shown in the Arrhenius equation, log viscosity = log A + E/2·3RT, viscosity is inversely
related to T.
51.No, it can be used with both synthetic and natural polymers.
52.Ethanol. Because it has the OH group that can hydrogen bond to the water and because it
is the only polar molecule.
53.It is an effort to try to find solvents that will minimize the enthalpy term.
54.MALDI MS requires that the polymer be soluble is a volatile solvent such as water and

most polymers are not soluble in such solvents.

ADDITIONAL READING
Albertsson, A. 2008. Chromatography for Sustainable Polymeric Materials: Renewable, Degradable, and
Recyclable, Springer, New York.
Brown, W. 1996. Light Scattering Principles and Development, Verlag, New York.
Carraher, C., Molloy, H. M., Tiernan, T. O., Taylor, M. L., Schroeder, J. 1982. Identification of Thermal
Degradation Products Using Coupled Thermogravimetric Analysis-Mass Spectroscopy: Development
and Instrumentation: Advances in Organometallic Polymers, Chapter 7, Dekker, New York.
Carraher, C., Sabir, T., Carraher, C. L. 2009. Inorganic and Organometallic Macromolecules, Springer, New
York.
Chabra, R., Richardson, J. 2008. Non-Newtonian Flow and Applied Rheology: Engineering Applications,
Elsevier, New York.
Debye, P. J. 1944. Light scattering analysis, J. Appl. Phys., 15:338.
Debye, P. J., Bueche, A. M. 1948. Intrinsic viscosity, diffusion, and sedimentation rates of polymers in solution, J. Chem. Phys., 16:573.
Einstein, A. 1910. Theory of the opalescence of homogeneous liquids and liquid mixtures in the neighborhood
of the critical state, Ann. Physik., 33:1275.
Hansen, C. 2007. Hansen Solubility Parameters: A User’s Handbook, CRC Press, Boca Raton, FL.
Huggins, M. L. 1942. The viscosity of dilute solutions of long-chain molecules. IV. Dependence on concentration, J. Amer. Chem. Soc., 64:2716.
Jenekhe, S. A., Kiserow, D. 2004. Chromogenic Phenomena in Polymers, Oxford University Press, New York.
Krigbaum, W. R., Flory, P. J. 1952. Treatment of osmotic pressure data, J. Polym. Sci., 9:503.
Kulichke, W. 2004. Viscometry of Polymers and Polyelectrolytes, Springer, New York.
Mac, W., Borger, L. 2006. Analytical Ultracentrifugation of Polymers and Nanoparticles, Springer, New
York.
Mark, H., Whitby, G. S. 1940. Collected Papers of Wallace Hume Carothers on High Polymeric Substances,
Interscience, New York.
Montaudo, G., Lattimer, R. 2002. Mass Spectrometry of Polymers, CRC Press, Boca Raton, FL.
Oliver, R. 1998. HPLC of Macromolecules, 2nd Ed., Oxford University Press, Cary, NY.
Pasch, H. 2003. MALDI-TOF Mass Spectrometry of Synthetic Polymers, Springer, New York.

Molecular Weight of Polymers

99

Pasch, H., Schrepp, W. 2003. MALDI-TOF of Polymers, Springer-Verlag, Berlin.
Pasch, H., Trathnigg, B. 1999. HPLC of Polymers, Springer-Verlag, New York.
Podzimek, S. 2011. Light Scattering, Size Exclusion Chromatography and Asymmetric Flow Field Flow
Fractionation, Wiley, Hoboken, NJ.
Rayleigh, J. W. S. (Lord) 1871. On the light from the sky, its polarization and color, Phil. Mag., 41:107.
Schartl, W. 2007. Light Scattering from Polymer Solutions and Nanoparticle Dispersions, Springer, New
York.
Staudinger, H. 1928. Uber diekonstitution der Hochpolymeren, Ber. Bunsenges Phys. Chem., 61:2427.
Svedberg, T., Pederson, K. O. 1940. The Ultracentrifuge, Clarendon, Oxford, UK.
Yau, W. 2009. Size-Exclusion Chromatography, Wiley, Hoboken, NJ.
Zimm, B. H. 1948. The scattering of light and the radical distribution function of high polymer solutions, J.
Chem. Phys., 16:1093 and Apparatus and methods for measurement and interpretation of the angular
variation of light scattering: Preliminary results on polystyrene solutions, J. Chem. Phys., 37:19.