Thermal Analysis of Polymeric Materials
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2.1 Heat, Temperature, and Thermal Analysis |
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Fig. 2.14
Fig. 2.15
These first thermal analysis simulations open the view how future macroscopic experiments may be directly connected to molecular structure and motion. Much progress is naturally needed to establish proper simulation parameters and to expand the low-temperature analyses to quantum-mechanical calculations.
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With neural networks, as described in Appendix 4, the prediction of properties of unmeasured substances on the basis of data of known substances became possible (see introduction to Sect. 3.4). For thermal analysis, it has been possible to extend measured heat capacities at high temperatures into the region of low temperature where measurement is more difficult, as well as predict the theta-temperatures needed for the description of the vibrational heat capacities [4,5].
2.2 The Laws of Thermodynamics
Only a brief introduction to thermodynamics is offered in this Section. It should serve as a refresher of prior knowledge and a summary of the important aspects of the material needed frequently for thermal analysis. It is a small glimpse at what must be securely learned by the professional thermal analyst. For an in-depth study, some of the textbooks listed at the end of the chapter should be used as a continual reference. This does not mean that without a detailed knowledge of thermodynamics one cannot begin to make thermal analysis experiments, but it does mean that for increasing understanding and better interpretation of the results, a progressive study of thermodynamics is necessary.
2.2.1 Description of Systems
To start with a thermodynamic description of an object of interest, it is best to define it as a system as illustrated schematically in Fig. 2.16. The system must be well delineated from its surroundings by real or imaginary boundaries. These surroundings are in the widest sense the rest of the universe. This makes the surroundings ill defined. One simply does not know the size, content, and behavior of the universe. For this reason, it is convenient to work with the system alone and separately assess and account for the effects of the surroundings.
To be sure what ensues within the system, one monitors carefully its state and its boundaries. All transport across the boundaries or creation which increases an extensive quantity within the system is counted as positive (+). All that is lost across the boundaries or within the system is counted as negative ( ). This assignment of plus and minus is characteristic from the point of view of the scientists. They consider themselves as the spokespersons for the system. Engineers often have the opposite view since their responsibility is to make use of the products of a system. They will count quantities lost by the system as a gain. This double set of definitions causes confusion, but by now cannot be eliminated and requires care to avoid mistakes.
A term describing the transport across the boundaries is called flux, and the creation of any extensive quantity within the system is called production. Immediately, one recognizes that the conservation laws of mass and energy, which is discussed in Sect. 2.1.2, forbid their production. A final set of new terms is also shown in Fig. 2.16. It is a classification of the types of systems. The first type is an open system, represented by an open container, for example, as used for the sample placement in thermogravimetry. In such a system mass and energy flux may occur
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Fig. 2.16
across the boundaries of the system to be analyzed. Across the boundaries of the system flows heat from the furnace, and mass can be lost by evaporation, or gained by interaction with the atmosphere. All changes in mass are recorded by a balance, as described in Sect. 4.6.
The second type of system, shown in Fig. 2.16 is the closed system. It does not permit exchange of matter, but energy may still be exchanged. A closed vessel can still be heated or cooled from the outside. Calorimetry is an example for a closed system in thermal analysis. The inner surface of the closed sample pan is the boundary of the system across which only heat flows. This heat is being measured, so that a controlled experiment is performed as discussed in Sects. 4.2–4.4.
The third type of system sketched in Fig. 2.16 is an isolated system. Neither energy nor matter, may pass across its system boundaries. Obviously, it is challenging to use an isolated system for measurement by thermal analysis.
From this initial discourse into the basics of thermal analysis, it can be learned that thermogravimetry should, for full system description, be coupled with calorimetry, so that both mass and energy changes can be followed. Calorimetry, in turn, is restricted to closed systems, since changes in the amount of matter within the system would distort the analysis. A simple rule for calorimetry is to check the sample weight after completion of each measurement to make sure the system was truly closed.
For a thermodynamic description of a system, there is one important set of additional requirements: The system must be in mechanical and thermal equilibrium. Equilibrium requires that the intensive variables, such as concentration, pressure, and temperature, are constant throughout the system. If the system is not in equilibrium, however, not all is lost. Analysis is still possible, but with a greater effort. One must then recognize that the overall system can be considered as being composed of many
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subsystems as illustrated in Fig. 2.80, below. Each subsystem is a chosen portion of the system which in itself is in equilibrium, or at least sufficiently close to equilibrium that the error introduced by assuming it is, is negligible. Although one can now describe the nonequilibrium system in terms of the many subsystems, each subsystem must be analyzed separately for the description of the system as a whole. This is often a difficult task and may even become impossible for a large number of subsystems.
The extensive properties of the overall system that is not in equilibrium, such as volume or energy, are simply the sums of the (almost) equilibrium properties of the subsystems. This simple division of a sample into its subsystems is the type of treatment needed for the description of irreversible processes, as are discussed in Sect. 2.4. Furthermore, there is a natural limit to the subdivision of a system. It is reached when the subsystems are so small that the inhomogeneity caused by the molecular structure becomes of concern. Naturally, for such small subsystems any macroscopic description breaks down, and one must turn to a microscopic description as is used, for example, in the molecular dynamics simulations. For macromolecules, particularly of the flexible class, one frequently finds that a single macromolecule may be part of more than one subsystem. Partially crystallized, linear macromolecules often traverse several crystals and surrounding liquid regions, causing difficulties in the description of the macromolecular properties, as is discussed in Sect. 2.5 when nanophases are described. The phases become interdependent in this case, and care must be taken so that a thermodynamic description based on separate subsystems is still valid.
2.2.2 The First Law of Thermodynamics
A system in a given state can be described by a set of independent variables that must be known or established by experiment. The functions of state can then be represented by a mathematical relationship between this set of independent variables. Typical functions of state of importance for thermal analysis, such as total energy, U, temperature, T, volume, V, pressure, p, number of moles, n, and mass, M, are described in Sect. 2.1.2. More details about the first law are also given in Sect. 2.1.5 with Fig. 2.10. For example, the total energy of one mole of an ideal gas is a function of state fixed by only one variable, the temperature, T, since for an ideal monatomic gas the kinetic energy is equal to the total energy (U = 3RT/2, as seen in Fig. 2.9). A more general description of U is given by the differential function dU, listed as Eq. (1) in Fig. 2.10. For one mole of an ideal gas dn and ( U/ V) are zero, reducing the independent variables to T [( U/ T)V,n = 3R/2 and dU = (3/2)RdT]. Expanding this discussion to a general system of one component, three independent variables are expected. For systems of multiple components, additional concentration terms increase the number of variables.
Thermodynamics provides the framework of functions of state. All our experimental experience can be distilled into the three laws of thermodynamics (or four, if one counts the zeroth law, mentioned in the discussion of thermometry in Sect. 4.1). Not so precisely, these three laws have been characterized as follows [6]: In the heat-to-work-conversion game the first law says you cannot win, the best you
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can do is break even. The second law says you can break even only at absolute zero of temperature, and the third law, finally, says you can never reach absolute zero. Indeed, one finds that it is difficult to win in thermodynamics. Somewhat more precisely, as discussed in Sect. 2.1.5, the first law is based on the fact that energy is conserved in experiments that concern thermal analysis. One can transfer energy from one system to another, but one cannot create or destroy it. Many thermal analysis experiments are based on the first law alone.
2.2.3 The Second Law of Thermodynamics
The second law of thermodynamics restricts the first law: dU = Q + w (see Sect. 2.1.5). From experience, it is not possible to arbitrarily subdivide dU among heat,Q, and work, w. Through the ages man has tried to make engines which convert more heat into work, either by violating the first law, or by dividing Q and w in favor of more work (perpetuum mobiles of the first and second kind, respectively). The frustration with the unattainable perpetuum mobile of the second kind has been gathered in the last century into the formulation of the second law. The formulation by Thomson (Lord Kelvin) is listed in Fig. 2.17.
Fig. 2.17
To express the second law mathematically, one defines a new function of state, the entropy, S. The differential form of S is written in the figure. The reasoning behind this equation is simple. For all different reversible cycles between given states the heats exchanged must be identical and naturally, the same must be true for the different amounts of work. Figure 2.18 shows a schematic of two reversible cycles between a and b in the space of states. It is obvious that QI must be equal to QII and wI equal to wII. If this were not so, one could run the two cycles in opposite
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Fig. 2.18
directions, converting each time QII QI heat into wI wII work. This means, one would do what the Thomson statement says cannot be done—namely, have an engine which does nothing but extract heat from a reservoir and perform an equal amount of work. Since experience tells us this cannot be so for reversible processes, heat and work must be functions of state and can be written as total differentials dQ and dw,
as shown in Fig. 2.17. The ratio dQreversible/T defines the new, second-law function entropy, S, and wreversible is, naturally, also a function of state, called the Helmholtz free energy, F. The total differentials, dS and dF, can be written as for the other
functions of state, as also shown in Fig. 2.17. The first law of thermodynamics can, finally, be written as a second-law expression which holds only for reversible processes and at constant temperature, as shown by the boxed equation in the figure. The second law assigns thus limiting values to the amounts of heat and work that can be exchanged in a reversible process. Much of the present form of presentation of thermodynamics was derived towards the end of the last century by Gibbs1 [7].
The just derived functions are awkward to use for thermal analysis as is the total energy in Fig. 2.10. The introduction of the enthalpy function was the solution. Figure 2.19 shows the use of H instead of U in connection with the second-law functions. It leads to the analogous expressions for the free enthalpy, G, replacing the free energy F (also called Gibbs free energy or function). The advantage of G lies, as with H, in its easy measurement at constant pressure. Since all functions used for the definition of H are functions of state, one can easily write the proper total differentials. The transformations needed for this change of variables are known as
1 J. Willard Gibbs, 1839–1903. Theoretical physicist and chemist, introduced the theory of thermodynamics as key element into physical chemistry. He was appointed Professor of Mathematical Physics at Yale University in 1871.
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Fig. 2.19
Legendre transformations and given in Appendix 5. Together with the Maxwell relations, the correlations at the bottom of Fig. 2.19 are derived.
To connect the verbal statement of the second law to a mathematical expression, one can use the Thomson engine, shown schematically in Fig. 2.20. The reversible engine is an isothermal, closed system; the system including the two dotted boxes, is isolated since there is no further contact with the surroundings. Heat, Q, is taken
Fig. 2.20
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from the heat reservoir and converted via the reversible engine into work. The engine regains the initial functions of state after one revolution in either direction. In the forward direction, as shown in the figure, the engine has after one cycle taken-in heat Q and produced (lost) work w. Such process is not possible according to the Thomson statement. The reversible engine has, by definition not changed its functions of state, including the entropy. The heat reservoir, however, lost Q and, thus, the entropy of the isolated system has decreased. The Thomson statement forbids such change ( S < 0). The Thomson statement permits, however, the reverse process of conversion of work into heat ( S > 0). The expression S 0 is, thus, also a statement of the second law. A process in an isolated system with a negative entropy change is forbidden (has never been observed). A positive change in S in an isolated system points to a permitted, irreversible process, and if there is no change in S, the process is reversible ( S = 0).
By use of the boxed equation in Fig. 2.20, one gets a measure of irreversibility of a process in both, isolated and closed systems which will form the basis of the irreversible thermodynamics discussed in Sect. 2.4. Since U under the given conditions is zero independent of the changes in the surroundings, the following must hold: F < 0. If, in turn, F = 0, the system is in equilibrium, and an increase in F,F > 0, is forbidden by the second law.
2.2.4 The Third Law of Thermodynamics
The third law is particularly easy to understand if one combines the macroscopic entropy definition with its statistical, microscopic interpretation through the Boltzmann equation1 S = k ln W, as shown in Fig. 2.21. The symbol k is the Boltzmann constant, the gas constant R divided by Avogadro’s number NA, and W is the thermodynamic probability, representing the number of ways a system can be arranged on a microscopic level. Figure A.5.3 in Appendix 5 gives some additional information on the Boltzmann equation. One can state the third law then, as proposed by Nernst2, formulated by Lewis and Randall [8], and reprinted in Fig. 2.21. This statement sets a zero of entropy from which to count. For measurement by calorimetry, one can assume that a perfect crystal at absolute zero has zero entropy, since W for perfect order must be one, making the logarithm zero. Measuring the heat capacity, C, lets one determine the absolute entropy of such crystal at any other temperature by adding increments of C/T, as indicated in the second boxed equation in Fig. 2.21, and derived for heat capacities at constant volume and pressure in Appendix 5, Fig. A.5.4.
1 Ludwig E. Boltzmannn, 1844–1906. Physicist whose greatest achievement was in the development of statistical mechanics. After receiving a Ph.D. from the University of Vienna (1866) he held professorships at Vienna, Graz, Munich, and Leipzig. The equation S = k ln W is carved above Boltzmann’s name on his tombstone in the Zentralfriedhof of Vienna.
2 Walter H. Nernst, 1864–1941. Professor of Physics in Göttingen and Berlin. He was one of the founders of modern physical chemistry. His formulation of the third law of thermodynamics gained him the 1920 Nobel Prize for Chemistry.
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Fig. 2.21
The three laws of thermodynamics just summarized give the basis for all thermal analysis. Figure 2.22 illustrates the thermodynamic functions as they relate to the heat capacity and will be discussed in Sect. 2.3. Equation (1) is based on the first law alone, Eqs. (2) and (3) need the second law. With the third law, the entropy at zero kelvin is fixed, so that Eq. (2) leads to an absolute value for the entropy. The enthalpy is not known as an absolute value. By convention the enthalpy of all
Fig. 2.22
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elements in equilibrium at 298.15 K (25oC) is set to be zero. By thermochemistry the enthalpy of all compounds can then be determined, as discussed in Sect. 4.2.7.
Figure 2.23 displays the change of enthalpy and free enthalpy of polyethylene from zero kelvin, based on heat capacity data, described in Sect. 2.3.7, and heat of fusion measurements (see, for example, Sect. 4.3.7). There is no jump in free enthalpy at the melting temperature, only a change in the slope. The double arrow marks TS as the difference between H and G as shown in Fig. 2.22, Eq. (3).
Fig. 2.23
2.2.5 Multi-component Systems
Multi-component systems have more than one type of molecule mixed within the system. For the description of the system one needs, thus, one more independent variable for each additional component. Gases always mix, i.e., an unlimited number of multi-component systems can be generated. Liquids still dissolve a variety of gases, other liquids, or even solids, but molecules with weaker interactions than in their respective pure states, may only mix to a limited degree. Common knowledge is that polar and non-polar molecules, like oil and water are immiscible. Although crystals sometimes also dissolve other solids, liquids, or gases, they do so to a lesser degree than the liquid state. Not only must the interaction between the different molecules be favorable for mixing, but the molecular geometry must also be proper to assemble mixed crystals. The mixing equations discussed next are for ideal solutions, i.e., systems that have colligative properties (see Fig. 1.52). Solutions that deviate slightly from the ideal case are called regular solutions. They have a small heat of mixing, but can still be approximated by an ideal entropy of mixing. Real solutions have a substantial heat of mixing and a nonideal entropy of mixing.