Thermal Analysis of Polymeric Materials
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weighed. The heat that flowed from the sample to the measuring ice is thus measured as in the experiment of Black. At the bottom of Fig. 4.27, some of the experimental data are listed that were obtained by Lavoisier. This calorimeter was also used to determine the thermal effects of living animals such as guinea pigs.
4.2.2 Isothermal and Isoperibol Calorimeters
The name calorimeter is used for the combination of sample and measuring system, kept in well-defined surroundings, the thermostat or furnace. To describe the next layer of equipment, which may be the housing, or even the laboratory room, one uses the term environment. For precision calorimetry the environment should always be kept from fluctuating. The temperature should be controlled to ±0.5 K and the room should be free of drafts and sources of radiating heat.
Calorimeters can be of two types, 1. isothermal and isoperibol, or 2. adiabatic. Isothermal calorimeters have both calorimeter and surroundings at constant To. If only the surroundings are isothermal, the mode of operation is isoperibol (Gk. , equal,
(-,, surround). In isoperibol calorimeters the temperature changes with time, governed by the thermal resistance between the calorimeter and surroundings. In adiabatic calorimeters, the exchange of heat between a calorimeter and surroundings is kept close to zero by making the temperature difference small and the thermal resistance large.
To better assess heat losses, twin calorimeters have been developed that permit measurement in a differential mode. A continuous, usually linear, temperature change of calorimeter or surroundings is used in the scanning mode. The calorimetry, described in Sect. 4.3 is scanning, isoperibol twin-calorimetry, usually less precisely called differential scanning calorimetry (DSC).
Perhaps the best-known isothermal phase-change calorimeter is the Bunsen1 icecalorimeter, invented in 1870 [5]. This calorimeter is strictly isothermal and has, thus, practically no heat-loss problem. The drawing in Fig. 4.28 shows the schematics. The measuring principle is that ice, when melting, contracts, and this volume contraction is measured by weighing the corresponding amount of mercury drawn into the calorimeter. The unknown sample is dropped into the calorimeter, and any heat exchange is equated to the heat exchanged with the measuring ice. Heat-loss or -gain of the calorimeter is eliminated by insulation with a jacket of crushed ice and water, contained in a Dewar vessel. An ice-calorimeter is particularly well suited to measure very slow reactions because of its stability over long periods of times. The obvious disadvantage is that all measurements must be carried out at 273.15 K, the melting temperature of ice. For a more modern version see [6].
Figure 4.29 represents an isoperibol drop calorimeter [7]. The surroundings are at almost constant temperature and are linked to the sample via a heat leak. The recipient is a solid block of metal making it an aneroid calorimeter (Fr. anéroïde, not using a liquid, derived from the Gk.). The solid block eliminates losses due to
1 Robert Wilhelm Bunsen, 1911–1899. Professor of Chemistry at the University of Heidelberg, Germany. He observed in 1859 that each element emits light of a characteristic wavelength. These studies of spectral analysis led Bunsen to his discovery of cesium and rubidium.
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Fig. 4.28
Fig. 4.29
evaporation and stirring that occur in a liquid calorimeter, as shown in Fig. 4.30. But its drawbacks are less uniformity of temperature relative to the liquid calorimeter and the need of a longer time to reach steady state.
For measurement, the sample is heated to a constant temperature in a thermostat above the calorimeter (not shown), and then it is dropped into the calorimeter, where heat is exchanged with the block. The small change in temperature of the block is
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used to calculate the average heat capacity. Heats of reaction or mixing can also be measured by dropping one of the components into the calorimeter or breaking an ampule of one of the reactants to initiate the process. More advanced versions of this type of calorimeter involve compensation of heat flow, as shown in Sect. 4.2.5.
Figure 4.30 illustrates the liquid calorimeter. It also operates in an isoperibol manner. The cross-section represents a simple bomb or reaction calorimeter, as is ordinarily used for the determination of heats of combustion [8]. The reaction is carried out in a steel bomb, filled with oxygen and the unknown sample. The reaction is started by electrically burning the calibrated ignition wire. The heat evolved during
Fig. 4.30
the ensuing combustion of the sample is then dissipated in the known amount of water that surrounds the bomb, contained in the calorimeter pail. From the rise in temperature and the known water value, W, of the whole setup, the heat of reaction can be determined:
H = W T
where H is the heat of reaction and T, the measured increase in temperature. The water value is equivalent to the heat capacity of a quantity of water which equals that of the whole measuring system.
Much of the accuracy in bomb calorimetry depends upon the care taken in the construction of the auxiliary equipment of the calorimeter, also called the addenda. It must be designed such that the heat flux into or out of the measuring water is at a minimum, and the remaining flux must be amenable to a calibration. In particular, the loss due to evaporation of water must be kept to a minimum, and the energy input from the stirrer must be constant throughout the experiment. With an apparatus such as shown in Fig. 4.30 anyone can reach, with some care, a precision of ±1%, but it is possible by most careful bomb calorimetry to reach an accuracy of ±0.01%.
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In both the aneroid and liquid calorimeters, a compromise in the block and pail construction has to be taken. The metal or liquid must be sufficient to surround the unknown, but it must not be too much, so that its temperature-rise permits sufficient accuracy in T measurement. The calibration of isothermal calorimeters is best done with an electric heater in place of the sample, matching the measured effect as closely as possible. To improve the simple calculation of the output of the aneroid and liquid calorimeters, a loss calculation must be carried out as described in the next section.
4.2.3 Loss Calculation
To improve the measured T, in isoperibol calorimetry, the heat losses must naturally be corrected for. Loss calculations are carried out using Newton's law of cooling, written as Eq. (1) in Fig. 4.31 (see also Fig. 4.9). The change in temperature with
Fig. 4.31
time, dT/dt, is equal to some constant, K, multiplied by To, the thermal head (i.e., the constant temperature of the surroundings), minus T, the measured temperature of the calorimeter. In addition, the effect of the stirrer, which has a constant heat input with time, w, must be considered in the liquid calorimeter in Eq. (2). The same term w also corrects any heat loss due to evaporation.
The graph in Fig. 4.32 and the data in Table 4.2 show a typical example of calorimetry with a liquid calorimeter. The experiment is started at time t1 and temperature T1. The initial rate of heat loss is determined in the drift measurement. If the thermal head of the calorimeter, To, is not far from the calorimeter temperature, a small, linear drift should be experienced. The measurement is started at t2, T2. This process may be combustion, mixing of two liquids, or just dropping a hot or cold sample into the calorimeter. A strong temperature change is noted between t2 and t3.
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This is followed by the period of final Table 4.2 drift, between t3 and t4. The experi
ment is completed at time t4 and temperature T4.
The detailed analysis of the curve is continued in Fig. 4.32. The temperatures and times, T2, T3 and t2, t3, respectively, are established as the points where the linear drifts of the initial and final periods are lost or gained. The equations for the initial and final drifts, Ri and Rf, are given by Eq. (3) and Eq. (4). They are used to evaluate Newton’s law constant K, as shown by Eq. (5), and the stirrer correction w, as shown by Eq. (6). With these characteristic constants evaluated, the actual jump in temperature T, can be corrected. Now, the
value of Tcorrected is equal to the uncorrected temperature difference
minus the integral over the rate of temperature-change given by New-
ton’s law (i.e., taken as if there had been no reaction). The integral goes from the time t2 to t3, as shown in Eq. (7). If one now defines an average temperature, T, as expressed by Eq.(8), Tcorrected can be written as is shown in Fig. 4.32 as Eq. (9).
Fig. 4.32
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Graphically, T can be found by assuming that the heat losses change proportionally to the changes in the amplitude of the curve—i.e., T is fixed at a value that makes the two shaded areas of the figure equal in size (note that the thermal head, To, drops out of the calculation).
Several approximations are possible for the evaluation of T. Often, it is sufficient to use the graphical integration just suggested, or to count the corresponding squares of the curve drawn on millimeter paper. If the heat addition is electrical, the temperature rise is close to linear, and the average time is the location of the average temperature. For heats of combustion, the rise in temperature is frequently exponential and can be integrated in closed form.
With all these corrections discussed, it may be useful to practice an actual loss calculation, using the data in Table 4.2 (answer: T = 1.3710 K). If the temperature is measured with a calorimetric mercury-in-glass thermometer, an emergent stem correction becomes necessary if the thermometer extended out of the bath liquid. This emergent stem correction can be made as the last correction for the calculated T, using the equation derived in Fig. 4.4.
4.2.4 Adiabatic Calorimetry
Figure 4.33 shows a sketch of an adiabatic calorimeter. With an adiabatic calorimeter an attempt is made to follow the temperature increase of an internally heated calorimeter raising the temperature of the surroundings so that there is no net heat flux between the calorimeter and surroundings. The electrically measured heat input into the calorimeter, coupled with the measurement of the sample temperature, gives the information needed to compute the heat capacity of calorimeter plus sample. If truly adiabatic conditions could be maintained, the heat input, Hmeas, divided by the temperature change, T, would already be the heat capacity. Subtracting the heat absorbed by the empty calorimeter, its water value, Co, completes the data analysis.
Adiabatic calorimeters have only become possible with advanced designs for electrical temperature measurement and the availability of regulated electrical heating. The first adiabatic calorimeter of this type was described by Nernst in 1911 [9]. Special equipment is needed for low-temperature calorimetry, below about 10 K as are described in Sect. 4.3. Modern calorimeters [10–13] are more automated than the adiabatic calorimeter shown in Fig. 4.33, but the principle has not changed from the original design by Nernst.
The shown calorimeter has an accuracy of ±0.1% and a temperature range of 170 to 600 K. A sample of 100–300 g is placed in two sets of silver trays, one outside and one inside a cylindrical heater. In the middle of the sample, the tip of the platinum resistance thermometer can be seen. Sample trays, thermometer, and heater are enclosed in a rounded steel shell, which for ease of temperature equilibration is filled with helium of less than one pascal pressure. The shell is covered with a thin silver sheet on the outside, gold-plated to reduce radiation losses. The calorimeter is then hung in the middle of the adiabatic jacket, drawn in heavy black. This adiabatic jacket is heated by electrical heaters and cooled by a cold gas flow, as indicated by the dials of the instruments pictured on the left at the bottom. The whole assembly, calorimeter and adiabatic jacket, is placed in a sufficient vacuum to avoid convection.
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Fig. 4.33
To measure the deviation from the ideal, adiabatic condition, the temperature difference between the calorimeter and the jacket, the adiabatic deviation, is continuously monitored between the points A and B (= TA TB). The heat losses as a function of the adiabatic deviation are measured and calibrated for each temperature (bi and bf). The sample temperature is then determined with the platinum resistance thermometer, using a precision galvanometer and a Wheatstone bridge, indicated on
the right in Fig. 4.33. The heat input into the sample, Hmeas, is measured by the watthour meter, indicated on the bottom right (unit Ws J). A typical experiment involves
an increase in temperature in steps of between 1 and 20 K.
The loss and heat capacity calculations are indicated in Fig. 4.34. The curve represents the adiabatic deviation, TA TB. During the initial isothermal period (up to Ti), the drift Ri of the sample is followed with the platinum resistance thermometer as a function of time. The temperature difference between A and B is monitored continuously throughout the measurement. During the heating cycle (heater on) the temperature changes quickly, so that it cannot be determined. Temperature measurement is started again at Tf, as soon as a final, linear drift, Rf, is obtained.
This discussion of the measurement of heat capacity by adiabatic calorimetry gives insight into the difficulties and the tedium involved. Today, computers handle the many control problems, as well as the data treatment. The rather involved experimentation is the main reason why adiabatic calorimetry is not used as often as is required by the importance of heat capacity for the thermodynamic description of matter.
The main advantage of adiabatic calorimetry is the high precision. The cost for such precision is a high investment in time. For the measurement of heat capacities of linear macromolecules, care must be taken that the sample is reproducible enough to warrant such high precision. Both chemical purity and the metastable initial state must be defined so that useful data can be recorded.
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Fig. 4.34
4.2.5 Compensating Calorimeters
Compensating calorimeters are constructed so that it is possible to compensate the heat effect with an external, calibrated heat source or sink. These calorimeters often operate close to isothermal conditions between the calorimeter and surroundings. Functionally the simplest of the compensating calorimeters is that designed by Tian and Calvet [14]. The schematic of Fig. 4.35 shows that the sample is contained in the central calorimeter tube that fits snugly into a silver socket, across which the heat exchange occurs. The heat generated or absorbed by the sample is compensated by the Peltier effect of hundreds of thermocouples. A short summary of the three thermoelectric effects, of which one is the Peltier effect, is given in Fig. 4.36 (see also Fig. 4.7 for the application of the Seebeck effect). If heat is generated in the calorimeter, the Peltier thermocouples shown in Fig. 3.35 are cooled. If heat is absorbed, the Peltier thermocouples are heated by reversing the current. These thermocouples are arranged in series, with one junction at the silver socket (calorimeter), and the other junction at the thermostated metal block (surroundings). The whole silver socket is covered evenly with the thermocouple junctions, as shown on the lefthand side of the drawing. In addition to the Peltier thermocouples, there is a set of measuring thermocouples, considerably smaller in number (approximately ten to twenty). These measuring thermocouples are interspersed between the Peltier thermocouples and are used to measure the temperature difference T between the silver socket and the metal block. In the figure, the measuring thermocouples are drawn separately on the right-hand side. The heat, Q, generated or absorbed by the Peltier effect is given by Eq. (1) where the Peltier coefficient is represented by -, i is the positive or negative current, and the time interval is t. The Peltier coefficient for
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Fig. 4.35
Fig. 4.36
a single thermocouple is of the order of magnitude 10 3 to 10 4 J C 1. With this amount of reversible heat, the major heat effect in the calorimeter is compensated and thus measured directly.
The heat losses, ., need to be discussed next on hand of Fig. 4.37. Obviously, the main heat loss should be through heat conducted by the thermocouples. If the
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Fig. 4.37
temperature difference between the calorimeter and the surroundings is not exactly equal to zero, this heat loss is given by c T (loss a). Another loss comes from the convection of air between calorimeter and surroundings. Again, one can assume that this convection loss is, at least approximately, proportional to the temperature difference T (loss b). Finally, a fraction of the losses must go through the areas which are not covered by thermocouples—for example, losses by radiation. These losses, as a catch-all, can be assumed to be a certain fraction of the total loss, . (loss c) as summarized in the figure.
All these heat losses can now be evaluated from the emf of the measuring thermocouples. The thermocouple emf is given in Eq. (2) of Fig. 4.37. It is equal to the thermocouple constant, o, multiplied by the sum of the temperature differences of all the measuring couples. Equation (3) is the final expression for the heat loss, ., expressed in terms of the emf. It is arrived at by elimination of T in (a) and (b), with help of Eq. (2), and the addition of (a), (b), and (c). All constants—c, , ando—have to be evaluated by calibration.
The overall calorimeter equation of the Calvet calorimeter is finally given by Eq. (4). The overall heat effect, H, is equal to the time integral over the Peltier compensation, the major effect to be measured, corrected for two factors: the timeintegral over the just-discussed losses, ., and, if the temperature does not stay exactly constant during the experiment, a correction term which involves the heat capacity of the calorimeter and the sample, C. All three terms can be evaluated by the measurement of the Peltier current i, the measurement of the emf of the measuring thermocouples, and a measurement of the change of the emf with time. The last term is needed for the calculation of the heat capacity correction which is written in Eq. (4). The last two terms in Eq. (4) are relatively small as long as the operation is close to isothermal.