Thermal Analysis of Polymeric Materials

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Fig. 4.59

Fig. 4.60

The use of the upper, differential amplifier loop in Fig. 4.59 closes the T loop. The temperature-difference signal is directly amplified and a corresponding power is used to compensate the imbalance between the sample and reference calorimeter temperatures so that only a small temperature difference remains, as indicated in Fig. 4.60. This differential-power signal is also sent to the computer as WD, the

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differential heat-flow rate. The remaining temperature difference is in this way proportional to the differential power applied.

The mathematical description for the closed T loop is based on the equations of Fig. 4.60 which are rewritten as Eqs. (1) and (2) in Fig. 4.61. The amplifier gain, X, is introduced in Eq. (3). Combining Eqs. (1) and (2) yields Eq. (4), which one can reorganize into Eqs. (5) and (6) for T and W, respectively, the true differential

Fig. 4.61

temperature and power necessary for compensation calorimetry. The measured signalTmeas is made small by choosing a large X, as shown in Eq. (8), which can be derived by combining the equations indicated. It, however, cannot become zero, as required for true compensation calorimetry, since for zero Tmeas, W would also be zero. The “closed-loop” lines in Fig. 4.60 indicate that the measured caloric quantity of the power compensation DSC is also a temperature difference as in a heat-flux DSC.

One additional point needs to be considered. The commercial DSC is constructed in a slightly different fashion. Instead of letting the differential amplifier correct only the temperature of the sample calorimeter by adding power, only half is added, and an equal amount of power is subtracted from the reference calorimeter. This is accomplished by properly phasing the power input of the two amplifiers. A check of the derivations shows that the result does not change with this modification.

4.3.4 Calibration

Differential scanning calorimetry is not an absolute measuring technique, calibrations are thus of prime importance. Calibrations are necessary for the measurement of temperature, T (in K); amplitude, expressed as temperature difference, T (in K) or as heat-flow rate, dQ/dt (in J s 1 or W); peak area H (in J); and time, t (in s or min). Figure 4.62 shows the analysis of a typical first-order transition, a melting transition.

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Fig. 4.62

The curve is characterized by its baseline and the peak of the (endothermic) process. Characteristic temperatures are, the beginning of melting, Tb, the extrapolated onset of melting, Tm, the peak temperature, Tp, and the point where the baseline is finally recovered, the end of melting, Te. The beginning of melting is not a very reproducible point. It depends on the sensitivity of the equipment, the purity of the sample, and the degree to which equilibrium was reached when the sample crystallized. The extrapolated onset of melting is most reproducible if there is only a small temperature gradient within the sample and if the analyzed material melts sharply. The DSC curve should, under these conditions, increase practically linearly from an amplitude of about 10% of its deviation from the baseline up to the peak and show the heating rate as its slope when plotted against time or reference temperature. The extrapolation back to the baseline then gives an accurate measure of the equilibrium melting temperature, Tm.

If there is a broad melting range, it is better characterized by the melting peak temperature, Tp. It represents the temperature of the largest melting rate. The temperature of the recovery of the baseline Te is a function of the design of the calorimeter. In Fig. 4.63 several reference materials are listed that may be used for temperature calibration to an accuracy of ±0.1 K or better and for establishment of a link to the International Temperature Scale (ITS, see Appendix 8).

The next calibration concerns the area of the DSC trace or the amplitude at any one temperature. The peak area below the baseline in Fig. 4.62 can be compared with the melting peaks of standard materials such as the benzoic acid, urea, indium, or anthracene, listed at the bottom of the figure. The amplitudes measured from the baseline established in the heat-capacity mode of measurement are usually compared with the heat capacity of standard aluminum oxide in the form of sapphire. The heat capacity of sapphire is free of transitions over a wide temperature range and has been

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Fig. 4.63

measured accurately by adiabatic calorimetry and reviewed by Archer in the reference listed at the bottom of Fig. 4.63. Although both calibration methods should give identical results, it is better to select the method that matches the application. For highest precision it is recommended to match the calibration areas or amplitudes with the measurement at a similar temperature.

A final calibration concerns time. Heating rates or modulations may not be constant over a wide temperature range, because of, for example, the effect of nonlinearity of the thermocouples (see Sect. 4.1). It is thus useful to occasionally check heating rate and modulation over the regions of temperature of interest. A good stopwatch is usually sufficient for this check.

Measurements at different heating rates may lead to different amounts of instrument-lag, i.e., the temperature marked on the DSC trace can only be compared to a calibration of equal heating rate and baseline deflection. A simple lag correction makes use of the slope of the indium melting peak when plotted vs. sample temperature as a correction to vertical lines on the temperature axis. In some commercial DSCs this lag correction is included in the analysis program. It must be considered, however, that different samples have different thermal conductivities and thermal resistances so that different lags are produced as shown, for example, in Fig. 4.94, for an analysis with TMDSC.

4.3.5 Mathematical Treatment

The mathematical treatment of calorimetry is complicated since there are no prefect insulators for heat. Even a vacuum is no barrier since radiation can transport heat, particularly at elevated temperatures. For this reason one uses differential techniques where reference and sample calorimeters are placed in symmetrical environments with

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similar heat losses and gains. The differential measurement should then need only a small asymmetry correction. The heat-flux paths, however, are hard to assess mathematically, as is illustrated in Appendix 11 when describing a first attempt of making online corrections for heat-flow rates. Different DSCs require, in addition, different mathematical models. Finally, it must be observed that for the assessment of the time-dependence the thermal diffusivity must be known (see Fig. 4.65, below).

A schematic of a single DSC cell is given in Fig. 4.64. It is chosen to provide basic insight, and the results can be transferred to more involved calorimeters with appropriate changes in geometry, thermal resistances, and calibration constants. The

Fig. 4.64

cylindrical geometry in the center of the cell, chosen as an example, leads to a onedimensional solution for the heat flow along the radius, r. The end-effects which occur for a length of one to two radii from the top and bottom are neglected. The radius Ri to the edge of the sample is assumed for all numerical calculations to be initially 4.0 mm. At Ro, the surface of the metal block is reached, with no further change in temperature to the heater. With a negligible temperature gradient within the sample, the heat capacity of the sample will be shown in Sect, 4.3.6 to determine the heat-flow rate at steady state, i.e., under conditions when the temperature of the calorimeter changes with the same rate dT/dt at all points. Only then can this problem be solved easily. First, however, the mathematical problem with a temperature gradient within the sample will be treated and then simplified.

Six conditions are set for the description. In case they are not fulfilled, further corrections need to be included in the presented calculations. (1) The thermocouples or resistance thermometers are assumed to be inert, meaning that they themselves do not contribute to the differential heat flow. (2) All material interfaces must be without thermal resistance. The main contacts are between the sample and cell, and from the

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cell to the block. No effects due to these interfaces are included in the calculations.

(3) There should be no or only negligible packing effects within the sample. (4) The heat capacity and thermal conductivity are assumed to be constant with temperature, or change so gradually that they do not create significant transients of their own. Major changes caused by thermal transitions within the sample are treated separately as distinct transients. (5) Only heat transfer by conduction or radiation is permitted. This assumption is necessary because one cannot assess heat transfer by the always irreproducible convection. (6) Sample and reference must be thermally independent of each other, so that there is no cross-flow of heat. Such cross-flow occurs for larger temperature differences between reference and sample placed too closely. In the calorimeter chosen for Fig. 4.64 there is no inert atmosphere. Most commercial DSCs work with purge gases which are prone to convection and cause then difficult to control heat losses or gains. For the typical commercial DSC this means that the gas flow should be as free of turbulence as possible (laminar flow, low flow-rate). The temperature of the gas should be that of the block, and its flow must be controlled and measured at the inlet and exit of the DSC. Inadvertent leaks, or too high flow rates, can cause major systematic errors.

The first step in the calculation is to describe the steady state which is reached when all parts of the calorimeter change temperature with the same rate q in K min 1. The block temperature at the contact surface with the sample cell, T(Ro), is governed by the controller as shown in Eq. (1) of Fig. 4.64. For convenience one sets T(Ro) at the beginning of the experiment equal to some constant temperature To and assumes that it rises linearly with the heating rate q. To calculate the heat flow into the sample at distance r to follow the steady-state heating rate q, one can set up Eq. (2) as the integral over dr from 0 to r of the surface of the cylinder, A, of length (2 r = A). The heat exchanged per unit volume of the sample is cs sq for the change in temperature, dT, per time-change, dt. With constant cs, s, q, and , the integration over the whole sample is easily completed. The right part of Eq. (2) illustrates the simple result of this integration.

The temperature gradient at any radius r, written as grad T = dT1/dr, is connected with the heat-flow rate as given by Eq. (3) of Fig. 4.65. It derives from the boxed equation in Fig. 4.65, the one-dimensional form of the Fourier equation of heat flow. More details about the Fourier equation and its application are given in Appendix 13. Combining Eqs. (2) and (3) and integration from r to radius Ri, one gets the temperature difference between the outside of the sample at Ri and the position at distance r from the center, as given by Eq. (4). The temperature T1(Ri) lags behind the block temperature by an amount given by a similar calculation using the geometry and thermal diffusivity of the cell. Under steady-state conditions, naturally, it also increases linearly with q.

Some typical parameters for DSC measurements are listed at the top of Fig. 4.66. The largest sample holder is assumed to have a radius of 0.4 cm. Such a sample holder would need on the order of magnitude of 1 2 g of materials. From Eq. (4) one can easily find the different temperature distributions that result from sample holders of different diameters. In order to keep the temperature gradient small, one may want to keep the heating rate between 1 and 5 K min 1. With a smaller sample holder of a 0.04 cm radius, a size requiring 1 2 mg of sample, the ordinate must be multiplied by

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4.3.6 DSC Without Temperature Gradient

The steady-state calculations have set the stage for computation of the DSC performance without, or better with negligible, temperature gradient within the sample. A temperature gradient within the sample pan should not exceed 2–3 kelvins for typical standard DSC experiments. The first two equations of Fig. 4.67 express the heat-flow rates into the sample and reference making use of Newton’s law of cooling (see Fig. 4.9). The heat-flow rate, dQ/dt, is strictly proportional to the difference between block temperature, Tb, and sample temperature, Ts. The proportionality

Fig. 4.67

constant, K, is dependent on the material of construction of the cell (or the heat-leak disk or cylinder), but under the chosen conditions of negligible temperature gradient within the sample, it is independent of the thermal diffusivity of the sample. Inserting the temperature of the block, Tb, and the total heat flow into the sample, Qs, into the equation for heat-flow rate into the sample leads to the equations of Fig. 4.67. Similar equations can be written for the heat-flow rate into the reference. Choosing the initial conditions at time t = 0 Tb, Ts, and To are all zero, the solution of the heat-flow-rate equations is given in Fig. 4.68. Instead of expressing the solution in terms of the heat Q, one can write an expression for the sample temperature as a function of time, as indicated, and drawn in the figure as an example.

The plot at the bottom of Fig. 4.68 shows the increase in block temperature, qt, and the change in sample temperature, Ts, in analogy to Fig. 4.55. Initially, at time zero, block and sample temperatures are identical. Then, as the experiment begins, the sample temperature lags increasingly behind Tb until steady state is reached. The difference between the block and sample temperatures at steady state is qCp/K, a quantity which is strictly proportional to the heating rate and the heat capacity of the

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Fig. 4.68

sample. The restriction of the sample to a small size, so that there is no temperature gradient within it, has led to a relatively simple mathematical description of the changes in temperature with time.

For the measurement of a constant heat capacity, the sketches of Fig. 4.55 and 4.68 show that only measurement of Tb and Ts is necessary. The value of K can be obtained by calibration. With this discussion, the principle of heat capacity measurement is already clarified. The further mathematical treatment of the DSC will show how to handle the calibration by using a differential set-up that is designed, as mentioned above, to minimize the effort to evaluate heat losses and gains.

The next stage in the mathematical treatment is summarized in Fig. 4.69. It models the measurement of a heat capacity that changes with temperature, but sufficiently slowly that a once achieved steady state is maintained. The changing heat capacity causes, however, different heating rates for sample and reference, so that the previous calculation cannot be applied. The two top equations express the temperature difference between the block and sample, and block and reference, respectively. They are derived simply from Newton’s law expressed in temperature as seen in Fig. 4.68 after steady state has been achieved. The differences written show that further simplification is possible if the overall heat capacity of the sample calorimeter (pan and sample) is made to be equal to Cp' + mcp, where Cp' is the heat capacity of the empty calorimeter (its water value), m is the sample mass, and cp is the specific heat capacity of the sample in J K 1 g 1. A further simplification is possible if one makes Cp' also the heat capacity of the reference calorimeter by using an empty pan of the same weight as the sample pan. This leads to the boxed Eq. (3). The equality of the heat capacities of the sample and reference holders are best adjusted experimentally (±0.1 mg), otherwise, a minor complication in the mathematics is necessary, requiring knowledge of the different masses and the heat capacity of aluminum or other