Thermophysics of Polymers I: Theory

352 Pages·1999·10.001 MB·English

by  PD Dr. rer. nat. Herbert Baur (auth.)

Herbert Baur
Thermophysics of Polymers I
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Herbert Baur
Thermophysics
of Polymers I
Theory
With 79 Figures and 2 Tables
i
Springer
PD Dr. rer. nat. Herbert Baur
SonnenwendstraBe 41
D-67098 Bad Diirkheim
ISBN-13:978-3-642-64216-6
Library of Congress Cataloging-in-Publication Data
Baur, Herbert.
Thermophysics of polymers I : theory / Herbert Baur.
p. cm.
Includes bibliographical references and index.
ISBN-J3:978-3-642-64216-6 e-ISBN-J3:978-3-642-59999-6
DOl: 10.1007/978-3-642-59999-6
I. Polymers--Thermal properties. J. Title.
QCI73.4.P65B38 1999
620.1'92--dc21 98-53819
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Preface
"Physics is physics" a physicist may think and be surprised that there is suppos
ed to be a "polymer physics". However, physics lives to a large extent from ab
stractions. The Newtonian laws of mechanics could only be formulated and
start their triumphant march after one had learned to disregard friction (the
dissipative effects which occur during motion). The physicist likes to regard
particles, e. g. molecules, as mass points or rigid spheres. Many "laws" are based
on this abstraction. However, if one builds a polyethylene molecule using the
Stuart-Brigleb models, one obtains a flexible chain with a diameter of about
5 cm (2 inches), which can attain a length of 2 km (1.24 miles). Even on the true
microscopic scale, such a molecule is certainly not a mass point, nor is it a
structureless rigid sphere. The flexibility and the enormous length of the
threadlike molecules induce properties in polymers which do not usually occur
in physics, or which are considered to be negligible. In this respect, it is, there
fore, certainly possible to speak of a polymer-specific physics, a "polymer
physics". The physical peculiarities, which are caused by the very long flexible
chain molecules, are mainly of an entropic nature. One can even attribute an
entropy to an individual polymer molecule. These molecules usually respond
dissipatively to an external perturbation, i. e. with a production of entropy. For
this reason, thermodynamics is the basis of the polymer-specific physics.
"Polymer physics" is at its roots a thermophysics of polymers, as indicated by
the title of this book.
Experimental aspects will be treated by B. Wunderlich in Vol. II of this book.
The objective of the present volume is to provide a simple description of the
theoretical framework which constitutes the basis of polymer physics.
According to the statements above, it is clear that this framework must be
established with the help of thermodynamics. Unfortunately, a complete theory
does not exist to date.
The problems of polymer physics can be formulated with an almost arbitrary
complexity. But this is not our objective. In order to illustrate the theoretical
skeleton, we have only treated simple, easily comprehensible problems of poly
mer physics, but, in detail and with all the necessary information on the assump
tions used. The description of dissipative processes within the framework of
thermodynamics shows clearly that these are generally non-linear phenomena.
Nevertheless, we will also confine ourselves here mainly to the simpler linearized
cases. The so-called glass transition, for example, is found to be a typical dissipa
tive process which cannot be linearized.
VI Preface
At this point I would like to thank once again: Mrs. C. Paciello (Bad
Diirkheim) who very carefully translated the originally German text into
English, Prof. Dr. B. Wunderlich (Knoxville, TN) who critically perused the en
tire text, Prof. Dr. G. Kanig (Ludwigshafen/Rhein) who kindly gave me the orig
inal of the Fig. 8.3, and Mr. F. Maron (BASF AG, Ludwigshafen/Rhein) who pre
pared most of the figures. I also wish to thank Dr. M. Hertel (Springer-Verlag,
Heidelberg), whose unremitting patience I greatly appreciate.
Bad Diirkheim, May 1998 H.Baur
Contents
1 Theoretical Aspects of Polymer Thermodynamics .................. 1
2 Equilibrium Thermodynamics of Simple Fluids . . . . . . . . . . . . . . . . . . . . . 5
2.1 Fundamental Equations and Equations of State . . . . . . . . . . . . . . . . . 5
2.2 Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
2.3 Response Functions ....................................... 15
2.4 Equilibrium and Stability Conditions ......................... 20
3 Homogeneous Mixtures. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
3.1 Mass Variables. . . ... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
3.2 Fundamental Equations, Subsidiary
and Stability Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30
3.3. Activities, Standard and Mixing Terms . . . . . . . . . . . . . . . . . . . . . . .. 35
3.4 Classes of Mixtures, Excess Quantities ........................ 39
3.5 Simple Non-ideal Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43
4 Heterogeneous Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
4.1 Equilibrium and Stability with Respect to the Macroscopic Degrees
of Freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
4.2 Gibbs' Fundamental Equation, Gibbs' Phase Rule,
Response Functions ....................................... 57
5 Homogeneous Elastic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
5.1 Fundamental Equations and Equations of State. . . . . . . . . . . . . . . .. 61
5.2 Response Functions ....................................... 64
5.3 Energy and Entropy Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
6 Molecular Aspects of Some Equilibrium Properties of Polymers . . . . . . .. 77
6.1 Linkage of Thermodynamics with Statistical Mechanics ......... 77
6.2 The Polymer Coil ......................................... 87
6.3 The Two-level System ...................................... 104
6.4 The Extended Polymer Chain ............................... 116
6.4.1 The Linear Chain Composed of Equal Masses .. . . . . . . . . . . . . . . .. 117
6.4.2 The Linear Chain Composed of Two Different, Alternating Masses 136
6.4.3 The Bending Modes of the Linear Chain ...................... 145
6.4.4 The Planar Zigzag Chain ................................... 152
VIII Contents
6.5 The Ideal Polymer Crystal .................................. 156
6.5.1 Born-Von Karman (Harmonic) Approximation ................ 156
6.5.2 Continuum Approximations ................................ 166
6.5.3 The Stockmayer-Hecht Model ............................... 173
6.5.4 Anharmonic Effects ....................................... 188
7 Systems with Macroscopically Relevant Internal Degrees
of Freedom ............................................... 197
7.1 Preconditions, Fundamental Equations and Equations of State .... 198
7.2 Dynamic Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204
7.3 Dynamic Response Functions ............................... 210
7.4 Relaxation Phenomena ..................................... 220
7.5 Linear Response to Periodic Perturbation ..................... 234
7.6 Connection with the Linear Theory of Aftereffects .............. 243
7.7 Systems with Several Intercoupled Internal Degrees of Freedom ... 256
7.8 The Glass Transition ....................................... 270
8 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 297
8.1 Phenomenological Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 298
8.2 Melting and Crystallization of Polymers. . . . . . . . . . . . . . . . . . . . . .. 303
8.3 Condis Crystals ........................................... 321
References ................................................ 337
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343
CHAPTER 1
Theoretical Aspects of Polymer Thermodynamics
Theoretical macroscopic-phenomenological physics is usually divided into
three areas: continuum mechanics (hydrodynamics, theory of elasticity), elec
trodynamics and thermodynamics. Thermodynamics deals with thermal pro
perties of materials, the most prominent representative of which is the heat
capacity.
A strict tripartition, however, can only be made with certain idealizations.
There are classes of matter in which the other areas are decisively affected by
thermodynamics. A pure mechanics of gases, for example, cannot be formulat
ed. The mechanical properties of a gas are essentially dependent upon its
entropy. Their explanation is, therefore, mainly a thermodynamic problem.
Entropy can also have a major influence on the mechanical properties of
polymers. The best-known example is the rubber elasticity found in cross-link
ed or long-chain, entangled polymers. In the case of polymers, however, yet
another reason blurs the division into three classical areas. Polymers usually
behave dissipatively when responding to an external perturbation, i. e., the
external perturbation is linked to a non-negligible entropy production. For ex
ample, when one measures Young's modulus E'(w) or the dielectric constant
E' (w) of a polymer as a function of frequency w of an external sinusoidal
disturbance, these mechanical or electrical quantities are accompanied by
equally measurable, so-called loss quantities E" (w) and E" (w) which are of
thermodynamic origin and character. The loss quantities are a measure of the
entropy produced per half period of the perturbation. According to the
Kramers-Kronig relations, however, the reactive quantities E' (w) and E' (w)
cannot be separated from the dissipative quantities E"(w) and E"(W) (see Sect.
7.5). Rather, the functions E'(w) and E'(W) can be determined unequivocally
from the functions E" (w) and E"(W) (and vice versa). In this case also, the sup
posedly mechanical or electrical problem proves to be a thermodynamic
problem.
Thus, the thermodynamics of polymers has to do more than simply explain
the purely thermal behaviour of this class of substances. It assumes the role of
a comprehensive thermophysics of polymers. In the following, however - as is
common in classical equilibrium thermodynamics - we will confine ourselves
to homogeneous or heterogeneous systems at rest that are not exposed to a
locally variable external force field. Transport phenomena, such as heat con
duction, viscosity, diffusion, and electrical conductivity, will be ignored or only
mentioned in passing when it seems appropriate. We will consider the thermo-
H. Baur, Thermophysics of Polymers I
© Springer-Verlag Berlin Heidelberg 1999
2 1 Theoretical Aspects of Polymer Thermodynamics
mechanical phenomena of polymers in detail, but will forgo consideration of
the theory of thermoelectric and thermomagnetic behaviour, which can be
developed in analogy to thermomechanics.
Equilibrium thermodynamics is certainly the basis and at the same time an
important component of polymer thermophysics. Its basic principles can be
most simply described by means of a homogeneous, fluid, one-component sys
tem (Chap. 2). In practice, however, equilibrium thermodynamics can only be
applied in this simple form to non-cross-linked polymers with a low molecular
mass and very narrow molecular mass distribution. As the polymerization pro
cess leads to different lengths in the chain molecules, a hundred percent frac
tionation is impossible with longer chain molecules, and there are often low
molecular-mass impurities or deliberate admixtures present, polymers almost
always constitute homogeneous mixtures (Chap. 3). It is a typical feature of
polymers that they are incapable of forming perfect mixtures. The perfect mix
ture, used as a reference system for low-molecular-mass substances, is replaced
by an ideal-athermal mixture in the case of polymers (Sect. 3.4). Polymers in a
segregated or partially crystalline state, as well as filled or reinforced polymers,
constitute heterogeneous systems (Chap. 4). Furthermore - in contrast to fluid
media - polymers in a partially crystalline or crystalline state, in a glass- or
rubber-like state, as well as melts of high-molecular-mass polymers in certain
non-equilibrium states, resist not only a change in volume, but also a change
in shape. Polymers are, thus, often elastic systems (Chap. 5). In this connection,
it is important to differentiate between energetic and entropic elasticity,
which, among other things, explains the polymer-specificity of rubber elasticity
(Sect. 5.3).
The specific properties of polymers are mainly due to the fact that the long
chain-like molecules of which they are composed are capable of producing a
multitude of conformational isomers. The thermal excitation of these isomers
results in coiled polymer molecules which constitute an equilibrium state with
maximal entropy (Sect. 6.2). The simplest model for a polymeric liquid is the
two-level system (Sect. 6.3). Despite the absence of the characteristics due to
thermally excitated conformational isomerism in an ideal polymer crystal con
sisting of extended chain molecules, there are still polymer-specific effects.
These are caused by the pronounced hybrid interaction between the lattice
units. Strong covalent bonds, that cannot be treated as central forces, act in the
chain direction. In contrast to this, predominantly weak van-der-Waals- or
dipol-forces act perpendicular to the chain direction (Sect. 6.4 and 6.5).
Equilibrium thermodynamics requires that all changes of state occur so
slowly that the microscopic internal degrees of freedom of the system under
consideration are capable of adjusting their equilibrium in response. Due to
their conformational degrees of freedom, the polymers have a class of internal
degrees of freedom which attain equilibrium relatively slowly. This severely
restricts the applicability of equilibrium thermodynamics to the problems of
polymer physics. If the rate of external perturbation of a thermodynamic
system is of the same order of magnitude as the rate with which an internal
degree of freedom of the system tries to adjust to equilibrium, equilibrium
thermodynamics is no longer valid. The perturbation process becomes an irre-