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1992 | Book

Heat Capacity Read chapter Read first chapter

Thermophysical Properties of Polymers

Author: Prof. Dr. Yuli K. Godovsky

Publisher: Springer Berlin Heidelberg

Included in: Professional Book Archive

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About this book

Among various branches of polymer physics an important position is occupied by that vast area, which deals with the thermal behav­ ior and thermal properties of polymers and which is normally called the thermal physics of polymers. Historically it began when the un­ usual thermo-mechanical behavior of natural rubber under stretch­ ing, which had been discovered by Gough at the very beginning of the last century, was studied 50 years later experimentally by Joule and theoretically by Lord Kelvin. This made it possible even at that time to distinguish polymers from other subjects of physical investigations. These investigation laid down the basic principles of solving the key problem of polymer physics - rubberlike elasticity - which was solved in the middle of our century by means of the statistical thermodynamics applied to chain molecules. At approx­ imately the same time it was demonstrated, by using the methods of solid state physics, that the low temperature dependence of heat capacity and thermal expansivity of linear polymers should fol­ low dependencies different from that characteristic of nonpolymeric solids. Finally, new ideas about the structure and morphology of polymers arised at the end of the 1950s stimulated the development of new thermal methods (differential scanning calorimetry, defor­ mation calorimetry), which have become very powerful instruments for studying the nature of various states of polymers and the struc­ tural heterogeneity.

Table of Contents

Frontmatter

Thermal Properties of Polymers

Frontmatter
1. Heat Capacity
Abstract
The heat capacity of any solid body is determined by the normal modes of vibrations available to its structure. If the spectrum of vibrational states, ρ(ω)dω, which gives the number of modes of vibration whose frequencies lie between ω and ω + dω, is established for a solid body the heat capacity C of a unit volume of the body can be immediately determined as
$$ C = {k_B}\int {p(\omega )} {\left( {\frac{{\hbar \omega }}{{{k_B}T}}} \right)^2}\frac{{\exp \left( {\hbar \omega /{k_B}T} \right)}}{{{{\left[ {\exp (\hbar \omega /{k_B}T) - 1} \right]}^2}}}$$
(1.1)
.
Yuli K. Godovsky
2. Thermal Conductivity
Abstract
Most polymers are insulating systems, therefore, any electronic effects are absent in them and heat conduction occurs as a result of lattice vibrations. Theoretical consideration of the thermal conductivity of the crystalline dielectrics where the lattice vibrations can be resolved into normal modes which can then be treated as phonons leads to the Debye equation [1]
$$x = \frac{1}{3}C(T)\bar c\bar l$$
(2.1)
where C is the heat capacity per unit volume, c is the average phonon velocity and l0303 is the phonon mean free path. The temperature dependence of the thermal conductivity of crystalline solids (Fig. 2.1) can be understood according to this equation as follows. At high temperatures where most of the phonons are excited χ ~ T-1 since C constant and l0305 is proportional to T-1. With decreasing temperature the number of interacting phonons decreases exponentially and correspondingly both χ and l0305 should increase exponentially. The further decrease of temperature must lead to the situation when l0303 will be compared with the dimension of the sample and therefore its temperature dependence should disappear. The phonons then will be scattered only by the boundaries of the sample which characterized the mean free path.
Yuli K. Godovsky
3. Thermal Expansion
Abstract
Thermal expansion, or more widely thermal deformation, is characterized by the changes of the dimensions of a body resulting from the temperature changes. Similar to the thermal conductivity, the thermal expansivity occurs due to anharmonicity of various modes of lattice vibrations [1, 2]. Any formulations concerning the thermal expansivity of solids are closely related to the main ideas of an equation of state for solids. According to the original Gruneisen assumption [1] the internal energy of a solid can be divided into a static and athermal portion. This assumption leads to an equation of state for the pressure which contains two terms: one of the terms corresponds to the static interaction (internal pressure) and the other represents the thermal pressure due to the expansivity of the lattice vibrations. Thus, the most widely used form of the equation of state for solids is
$$P = {P_i} + \gamma {P_T}$$
(3.1)
which corresponds to the usual Mie-Gruneisen approximation. In this equation Pi = -dUL/dV is the internal pressure, γ is the Gruneisen parameter, PT = Ut/V is the thermal pressure, and UT is the thermal energy. Differentiation of Eq. (3.1) with respect to temperature at constant volume yields
$${\left( {\frac{{\partial P}}{{\partial T}}} \right)_v} = \alpha {K_T} = \gamma \frac{{{C_v}}}{V}$$
(3.2)
where α is the thermal expansion coefficient, and KT is the isothermal bulk modulus.
Yuli K. Godovsky
4. Experimental Methods and Instrumentation
Abstract
Two methods are usually used for the determination of the heat capacity of polymers over a wide temperature range: adiabatic calorimetry and differential scanning calorimetry (DSC).
Yuli K. Godovsky

Thermal Behavior of Polymers under Mechanical Deformation and Fracture

Frontmatter
5. Thermomechanics of Glassy and Crystalline Polymers
Abstract
Conventionally, the deformation of solids is treated within the framework of elasticity theory in terms of stresses and strains, i.e. in purely mechanical terms. Although experimental determination of stress-strain relationship provides important information concerning the deformation process, it is quite obvious that this conjugated variable pair is only one of several pairs which can be used to describe the response of a material on deformation. The necessity of using the thermodynamic instead of mechanical approach became evident when investigators began to consider deformation in terms of thermodynamic potentials, such as internal energy, free energy etc., rather than in terms of potential energy. The first law of thermodynamics shows that energy is conserved in all deformation processess either reversible or irreversible. Therefore, the mechanical response of any material reflects exactly that amount of energy which accompanied the deformation process as heat and/or changes of internal energy. What this means to an experimentalist is that the temperature or thermal variations brought about by adiabatic or isothermal processes have to be measured simultaneously with stresses and strains. The thermal effects and temperature variations accompanying deformation of solids are usually rather small and it is their correct measurement which constitutes the major difficulty in passing from mechanical to a thermodynamic approach.
Yuli K. Godovsky
6. Thermomechanics of Molecular Networks and Rubberlike Materials
Abstract
On simple elongation, rubberlike materials are capable of undergoing very large reversible elastic deformations. The modulus of elasticity, which unlike solids is strongly dependent on deformation, is some order of magnitude lower than the bulk modulus. Unlike the solids, the modulus of elasticity of the deformed networks and rubberlike materials is proportional to the absolute temperature (excluding very initial deformations). A very striking feature of the thermomechanical behavior of elastomers is a strong dependence of the linear thermal expansivity on deformation: the initial positive thermal expansion decreases drastically with deformation and in the vicinity of 0–10% deformation the expansion becomes negative and at moderate deformations it reaches the value typical for gases. All these facts demonstrate that the thermomechanical behavior of rubberlike materials differs in principle from that of solids. High elastic deformations are characteristic only for those deformation modes which are connected with the elasticity of the form. Since the volume compressibility of rubberlike materials is very small (the same order as for liquids) the thermodynamics of their uniform (volume) deformation is similar to the uniform deformation of solids and liquids.
Yuli K. Godovsky
7. Thermodynamic Behavior of Solid Polymers in Plastic Deformation and Cold Drawing
Abstract
The mechanical work spent on the irreversible deformation of solids is always at least partly dissipated. Therefore, independent of the sign of the heat effect resulting from the elastic deformation, plastic deformation is always accompanied by an exothermic effect. A schematic diagram of thermo-mechanical behavior of an ordinary solid and a solid polymer is shown in Fig. 7.1. After a small amount of initial cooling, resulting from elastic deformation, evolution of heat accompanying the beginning of plastic deformation occurs. The appearence of plastic deformation is accompanied by heat evolution independent of whether it is localized (necking) or distributed along the sample uniformly. If the plastic deformation is accompanied with a neck formation, which is typical of the cold drawing of the majority of glassy and crystalline polymers well below glass transition or melting point, then the heat generated locally may lead to a considerable local temperature rise. This temperature rise may strongly influence the cold drawing of the sample. At low drawing rates the neck propagates uniformly along the sample. Under certain conditions, however, especially at a high rate of extension for some polymers, instabilities of neck propagation can be observed (self-oscillation phenomenon), which is also closely related with the local thermal effects.
Yuli K. Godovsky
8. Thermal Behavior of Solid Polymers During Fracture
Abstract
The fracture of polymers is inevitably accompanied by thermal events. The sources of the thermal events may be both the deformational processes and the rupture of macromolecules. Pure elastic (brittle) fracture of solids is more the exception than the rule [1]. Normally the appearence of the initial sources of fracture — cracks — results in the local plastic deformation. A considerable part of the work, corresponding to the plastic deformation is transformed into heat and this local heat build-up leads to the local rises in temperature during the crack propagation in solids. Unlike the typical low molecular solids in which the plastic deformation at the tip of the cracks is the main mechanism of the local temperature rises under rupture, in polymers one more source of the rise in temperature is possible which is a direct consequence of the chain structure of macromolecules. A part of a macromolecule stressed almost to its load-bearing capacity represents an extremely powerful source of the elastically stored energy. The scission of any single bond in these adequately long segments of the highly stressed macromolecules is inevitably accompanied with dissipation of all the energy stored in all the other bonds. Therefore, this process can also be a very powerful source of the local rise in temperature.
Yuli K. Godovsky
9. Experimental Methods and Instrumentation
Abstract
Three types of measurements are usually used for the study of the thermal behavior of polymers under deformation: the temperature changes resulting from the deformation of the sample, the temperature dependence of the stress or force and direct calorimetric measurements of heat effects in various deformation modes.
Yuli K. Godovsky
Backmatter
Metadata
Title
Thermophysical Properties of Polymers
Author
Prof. Dr. Yuli K. Godovsky
Copyright Year
1992
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-51670-2
Print ISBN
978-3-642-51672-6
DOI
https://doi.org/10.1007/978-3-642-51670-2