Thermal Analysis of Polymeric Materials Part 15 ppsx

Appendix 10–Extreme DTA and DSC __________________________________________________________________ 826 or a liquid, such as silicon oil. Reference and sample are placed around their respective thermocouples inside the high-pressure container. The thermocouple output is recorded for the measurement of temperature and temperature difference. Special safety precautions must be observed when using high-pressure DTA. Particularly, special enclosures must be in place to contain the DTA in case of failure if the pressure-transmitting agent is gaseous or can easily evaporate. High-speed thermal analysis is possible, as shown for example in Fig. 3.95 (see also Sect. 6.2). Many industrial processes are very fast. Spinning of fibers, for example, may be done at rates of 100 to 10,000 m min 1 . A temperature change of 60 K over a length of 1 m in the thread-line, then, causes rates of temperature change of 6,000–600,000 K min 1 (100–10,000 K s 1 ). Such heating rates must be compared to times for large-amplitude molecular motion, such as rearrangements of polymer chains by conformational adjustments, described in Sects. 1.3.5–8 and 5.3.4. The conformational motion may have a picosecond timescale (10 12 s), i.e., in the time the fiber goes through the above temperature gradient, each backbone bond may rearrange billions of times, sufficient to cause intricate changes to various useful structures, particularly when, in addition, strains are imparted on the fiber. Most DTA and DSC equipment can be adjusted to measure at rates from about 0.1 K min 1 to perhaps 100 K min 1 . With some modification, changes of sample size and altering of heater size, etc., this can be brought to a range of 0.01 K min 1 to 1,000 K min 1 . One can cover in this way five orders of magnitude in heating rates. Even faster DTA needs special considerations. Permitting a temperature gradient of ±0.5 K in a disc-like sample, the equation in Fig. A.10.1 can be used to calculate the maximum sample dimensions for given cooling or heating rates (see also Figs. 4.65 and 4.66). Obviously the limit of fast-heating DTA has not been reached. Just dipping samples in cooling baths or heating baths can produce rates up to 10,000 K min 1 with reasonable control [5]. A unique solution to fast DTA is the foil calorimeter, shown schematically in Fig. A.10.4. A copper-foil is folded in such a way that two sheets of the sample (also very thin, so that the mass remains small) can be placed between them. The copper foil is used as the carrier of electrical current for fast heating. Between the inner portion of the stack of copper foil and sample, a thin copper–constantan thermocouple is placed. Only three folds of the stack are shown. In reality, many more folds make up the stack so that there are no heat losses from the interior and measurements can be made under adiabatic conditions. Heating rates of up to 30,000 K min 1 (500 K s 1 ) have been accomplished. Measured is temperature, time, and the-rate-of- change of temperature for a given heat input. With such fast heating rates it becomes possible to study unstable compounds by measuring faster than the decomposition kinetics of the compound. This super-fast calorimeter has seen little application, likely because it requires a new calorimeter for each sample. A more recent step to speed up DSC was taken in connection with a commercial power-compensation DSC (High Performance DSC) [6] and is now available as HyperDSC ® from the Perkin-Elmer Inc. It is claimed to reach 500 K min 1 . For a pan of a diameter of 5 mm, the heating rates calculated in Fig. A.10.1 corresponds to sample masses of 20, 2, and 0.2 mg, showing that it is the heating and cooling capacity of the DSC that limits fast calorimetry, not the properties of the sample. Appendix 10–Extreme DTA and DSC __________________________________________________________________ 827 Fig. A.10.4 To accomplish and exceed the fast calorimetry just suggested, one can turn to integrated circuit calorimetry. The measuring methods may be modulated calorime- try, such as AC calorimetry or the 3 7 -method [7,8], thin-film calorimetry as shown in Fig. A.10.4, or may involve standard heating or cooling curves as well as DSC configurations as illustrated in Chap. 4. Modern versions of such fast heating calorimeters are based on silicon-chip technology for measuring heat flow using thermopiles integrated in the chip and resistors for heating. Increasingly more numerous chips have become available based on free-standing membranes of SiN x , produced by etching the center of a properly coated Si-chip, as illustrated in Figs. A.10.5 and A.10.6, which contain each a schematic of the calorimeter. Figure A.10.5 illustrates the performance of several integrated circuit thermopiles, ICT, by Xensor Integration. The calibration without sample shows a constant output voltage of the sensor, decreasing at higher frequency, depending on the membrane used. The phase-angle response shows an analogous change at higher frequency. A linearity-check revealed that dynamic heat capacity measurements should be possible over the frequency range of 1 mHz to 100 Hz, compared to the 0.1 Hz limit typical for TMDSC of Sect. 4.4. This allows to analyze nanogram samples deposited from solution on the position shown in Fig. A.10.5. Similar chips for the measurements of heat capacity of samples of below 1 mg, contained in small aluminum pans, showed time resolutions well below 1 s when using heat-pulses [9]. Finally, Fig. A.10.6 illustrates the change in the crystallization rate when going from slow cooling, measured by standard DSC, to fast standard DSC, and to, ultimately, a chip calorimeter with 5,000 K s 1 (300,000 K min 1 ) [10]. The cooling rates were varied over five orders of magnitude by using different calorimeters. The cooling rates in the figure from right to left are: for 4 mg in a standard DSC, 0.02, 0.03, 0.08, 0.2, and 0.3 K s 1 ; for 0.4 mg in a fast DSC: 0.3, 0.8, 3, and 8 K s 1 ; for Appendix 10–Extreme DTA and DSC __________________________________________________________________ 828 Fig. A.10.5 Fig. A.10.6 0.1  g in an IC calorimeter as shown in the sketch: 0.45, 0.9, 2.4, and 5 kK s 1 . In this chip-calorimeter, the SiN x film, supported by the Si-chip frame was about 500 nm thick. The cooling was achieved by overcompensating the heater power with a cooled purge gas. The heating experiments could also be used to study the reorganization, as discussed in Sect. 6.2. Appendix 10–Extreme DTA and DSC __________________________________________________________________ 829 Fig. A.10.7 The principle of microcalorimetry is illustrated with Fig. A.10.7 (  TA™ of TA Instruments, Inc.). The tip of an atomic force microscope, AFM, is replaced by a Pt- wire that can be heated and modulated, as is illustrated in detail with Fig. 3.96. A typical resolution is about 1.0  m with heating rates up to 1,000 K min 1 . A temperature precision of ±3 K and a modulation frequency up to 100 kHz has been reached. The figure shows the control circuit for localized thermal analysis. In this case the probe contacts the surface at a fixed location with a programmed force, controlled by the piezoelectric feedback of the AFM. A reference probe is attached next to the sample probe with its tip not contacting the sample, allowing for differential measurements. Equal dc currents are applied to both probes and are controlled by the temperature-feedback circuit to achieve a constant heating rate. The temperature of the sample, T s , can be determined, after calibration from the resistance of the probe. Since the resistance of platinum has an almost linear dependence on temperature between 300 and 600 K, R s , can be expressed as shown in Fig. A.10.7, where R s o and  are the resistance and the temperature coefficient of the platinum wire at T o . The resistance of all lead wires and the heavy Wollaston-portion of the sensor are neglected. Their resistance is small when compared to the V-shaped platinum tip. In addition to the dc current, an ac current can be superimposed on the tip to obtain a temperature-modulation. The dc difference of power between the sample and reference probes is determined by measuring the dc voltage difference between two probes after low-pass filtering (LPF) and the ac voltage difference is measured by the lock-in amplifier (LIA). This arrangement permits an easy deconvolution of the underlying, dc and the reversing ac effects (see also Sect. 4.4). An example which illustrates a qualitative local thermal analysis with a micro- calorimeter is given in Fig. 3.97, the limits were probed in [12]. Appendix 10–Extreme DTA and DSC __________________________________________________________________ 830 References for Appendix 10 1. Wunderlich B (2000) Temperature-modulated Calorimetry in the 21 st Century. Thermochim Acta 355: 43–57. 2. White GK (1979) Experimental Techniques in Low Temperature Physics. Clarendon, Oxford, 3 rd edn. 3. Gmelin E (1987) Low-temperature Calorimetry: A particular Branch of Thermal Analysis. Thermochim Acta 110: 183–208. 4. Lounasma OV (1974) Experimental Principles and Methods below 1 K. Academic Press, London; see also Bailey CA, ed (1971) Advanced Cryogenics. Plenum Press, New York. 5. Wu ZQ, Dann VL, Cheng SZD, Wunderlich B (1988) Fast DSC Applied to the Crystallization of Polypropylene. J Thermal Anal 34: 105–114. 6. Pijpers TFJ, Mathot VBF, Goderis B, Scherrenberg RL, van der Vegte EW (2002) High- speed Calorimetry for the Study of the Kinetics of (De)vitrification, Crystallization, and Melting of Macromolecules. Macromolecules 35: 3601–3613. 7. Jeong Y-H (2001) Modern Calorimetry: Going Beyond Tradition. Thermochim. Acta 377: 1–7. 8. Jung DH, Moon IK, Jeong Y-H (2003) Differential 3 7 Calorimeter. Thermochim. Acta 403: 83–88. 9. Winter W, Höhne WH (2003) Chip-calorimeter for Small Samples. Thermochim. Acta 403: 43–53. 10. Adamovsky SA, Minakov AA, Schick C (1003) Thermochim. Acta 403: 55–63. Figure modified from a presentation at the 8 th Lähnwitz Seminar of 2004, courtesy of Dr. C. Schick. 11. Moon I, Androsch R, Chen W, Wunderlich B (2000) The Principles of Micro-thermal Analysis and its Application to the Study of Macromolecules. J Thermal Anal and Calorimetry 59: 187–203. 12. Buzin AI, Kamasa P, Pyda M, Wunderlich B (2002) Application of Wollaston Wire Probe for Quantitative Thermal Analysis, Thermochim Acta, 381, 9–18. Appendix 11 __________________________________________________________________ 831 Fig. A.11.1 Description of an Online Correction of the Heat-flow Rate A basic analysis of the heat-flow rate in a DSC was derived in Sect. 4.3.6 with Figs. 4.69 and 4.70. To use these equations, special baseline calibrations are necessary, as illustrated in Fig. 2.29. With the development of data analysis by computer, it has become possible to make a number of corrections online, i.e., during the run. The Tzero™ method is the first of such programs [1]. Figure A.11.1 illustrates a conventional DSC baseline over an extensive temperature range of 450 K. Considerable improvement arises from the correction with the Tzero™ method before recording the data. The main correction concerns the asymmetry of the instrumenta- tion which is not likely to change from run to run. An initial calibration sets the instrument parameters and leads to such a much improved performance. In this appendix the Tzero™ method of TA Instruments will be summarized [2]. To analyze the instrument performance, the DSC shown in Fig. 4.54 is used. Note that in the literature cited to this appendix the used quantities are represented by different symbols than used here [2,3]. The furnace or block temperature T b is called often T o (hence Tzero™ method). The heat-flow rate called dQ/dt = 0 is called q $ in [2], and should not to be confused with the here used rate of temperature change dT/dt = q. The temperature difference used in Chap. 4 and maintained here is written as  T = T r  T s , a positive quantity on heating of a sample run versus an empty pan, giving a positive differential heat flow into the sample. (In [2,3], in contrast,  T is set equal to T s  T r , making 0 proportional to  T). Next, the true sample and reference temperatures which determine the actual heat- flow rates into the sample and reference pans are computed by modeling the DSC of Fig. 4.54 by an electric circuit which behaves analogously, as described in [4], for Appendix 11–Online Correction of the Heat-flow Rate __________________________________________________________________ 832 Fig. A.11.2 (2) example. The thermal resistances are equated to electrical resistances and the heat capacities with capacitors. The heat-flow rates are then given by electrical currents calculated from Ohms law and lead to similar answers as in Figs. 4.64–72. In the sketch A of Fig. A.11.2 the equivalent electrical circuit for a conventional DSC measurement is drawn. The heat-flow rate into the sample calorimeter (pan + sample) is represented by 0 s , and 0 r is the heat-flow rate into the empty pan which is the reference calorimeter. The heat-flow rate into the sample itself should then be 0 = 0 s  0 r and matches Eq. (3) of Fig. 4.69 when assuming the thermal resistances from the furnace to the measured temperatures are equal (R spl = R rpl = R): Including also the heat capacities of the sample and reference platforms, C spl and C rpl , into the model leads to the bottom sketch in Fig. A.11.2. The heat-flow rate into the sample needs now a four-term heat-flow-equation making use of a second tempera- ture difference  T b = T s  T b besides  T = T r  T s : The second and third terms express the imbalance of the thermal resistances and heat capacities outside the calorimeters. The fourth term, the effect of different heating rates between reference and sample calorimeter. This last term is of importance when a transition occurs in the sample and does not follow the assumptions made for Figs. 4.71 and 4.72 [3]. (Note the differences to [2] from changed symbols and signs, 0 = q $ ,  T =  T,  T b =  T o , q s = dT s /dt, as well as from expanded subscripts). (1) Appendix 11–Online Correction of the Heat-flow Rate __________________________________________________________________ 833 Fig. A.11.3 The use of the corrections B in Fig. A.11.2 needs two calibration runs of the DSC of Fig. 4.54. The heat capacities of the calorimeter platforms, C spl and C rpl , and the resistances to the constantan body, R spl and R rpl , must be evaluated as a function of temperature. First, the DSC is run without the calorimeters, next a run is done with sapphire disks on the sample and reference platforms without calorimeter pans. From the empty run one sets a zero heat-flow rate for 0 s and 0 r . This allows to calculate the temperature-dependent time constants of the DSC, written as - s = C spl R spl and - r = C rpl R rpl , and calculated from the equations in the lower part of Fig. A.11.2. For the second run, the heat-flow rates are those into the sapphire disks, known to be mc p q, as suggested in Figs. 4.54 and 4.70. The heat-capacity-correction terms are zero in this second calibration because no pans were used. From these four equations, all four platform constants can be evaluated and the DSC calibrated. As the next step, the effect of the pans must be considered. The differences between sample and reference pan cause similar heat-flow problems as the platforms and can be assessed by the diagram of Fig. A.11.3. The dash-dotted lines indicate the sample and reference platforms from which 0 s and 0 r , contained in Eq. (3), enter into the calorimeters. The heat-flow rate into the sample, 0 sample , however, is modified by the heat resistance between pan and platform, R pn , and the heat capacity of the pan C spn . The actual sample temperature inside the calorimeter pan, in turn, is not T s , but the temperature T spn . For the reference side, one assumes the same thermal resistance R pn , but because of the possibly different mass of the reference pan, a different C rpn . Assuming further that the reference calorimeter is empty, as is the usual operation procedure, there is no 0 reference . Using the two measured temperature differences  T and  T b listed in Fig. A.11.3 and inserting them into the heat-flow rate expressions of sketch B of Fig. A.11.2 yields the two heat-flow rates of Fig. A.11.3 and their difference, expressed in Appendix 11–Online Correction of the Heat-flow Rate __________________________________________________________________ 834 Eq. (2), above. To obtain the actual heat-flow rate into the sample, 0 sample , one has to compute the heat-flow rates into the two pans. Since both pans are out of the same material C spn = m spn c pan and C rpn = m rpn c pan , where c pan is the specific heat capacity of the pan material and m represents the corresponding mass. Since all 0 r goes into the empty pan, one can use its value to compute 0 sample : Insertion of the measured values for 0 s and 0 r , contained in Fig. A.11.3, leads to an overall equation for 0 sample . The needed sample and reference pan temperatures in Eq. (3) can be calculated with some simple assumptions about the contact resistance between pan and platform R pn [2]: T spn = T s  0 s R pn and T rpn = T r  0 r R pn . The commercial software includes typical values derived from data given in [5], considering geometry, purge gas, and pan construction. To summarize, the correction for asymmetry of the measuring platforms in Fig. 4.54 due to their thermal resistances and their heat capacities with Eq. (3) eliminates the main effect of the baseline curvature with temperature, as is shown in Fig. A.11.1. Further corrections include the mass differences of the pans and the thermal resistance between calorimeters and measuring platforms. Remaining problems are the temperature gradients within the sample and their changes on heating or cooling, deformations of the sample pan and the accompanying changes in contact resistances which occur when samples expand or shrink. Also, the position of the calorimeters should be fixed and must not be altered during measurement, as can happen by vibrations or mechanical shocks. Changes of heat transfer due to radiation can be caused by fingerprints on calorimeters, platforms, and furnace cavity since fingerprints are easily converted at high temperatures to carbon specks with higher emissivity and absorptivity. Once cleaned, it is helpful to use dust-free and clean nylon gloves and to touch the calorimeters with tweezers only. The purge gas must not change in flux and pressure from the calibration and must not develop turbulent flow or convection currents. Finally, the room temperature must not fluctuate and set up changing temperature gradients within the DSC. Avoidance of all these potential errors is basic to good calorimetry. References to Appendix 11: 1. Waguespack L, Blaine RL (2001) Design of A New DSC Cell with Tzero™ Technology. Proc 29 th NATAS Conf in St. Louis, MO, Sept 24–26, Kociba KJ, Kociba BJ, eds 29: 721–727. 2. Danley RL (2003) New Heat-flux DSC Measurement Technique. Thermochim Acta 395:201–208. 3. Höhne G, Hemminger W, Flammersheim HJ (2003) Differential Scanning Calorimetry, 2 nd edn. Springer, Berlin. 4. Holman JP (1976) Heat Transfer, 4 th edn. McGraw-Hill, New York, pp 97–102. 5. Madhusudana CV (1996) Thermal Contact Conductance. Springer, New York. (3) Appendix 12 __________________________________________________________________ 835 Fig. A.12.1 Derivation of the Heat-flow-rate Equations The heat flow across any surface area, A, is given in Fig. A.12.1 by the heat-flow rate per unit area dQ/(Adt) in Eq. (1), a vector quantity in J m 2 s 1 . It is equal to the negative of the thermal conductivity,  in J m 1 s 1 K 1 , multiplied by the temperature gradient (dT/dr). Equation (2) represents the differential heat flow into the volume V and can be derived from the definition of the heat capacity dQ/dT = mc p . The symbols have the standard meanings; ' is the density and c p , the specific heat capacity per unit mass, so that m = V ' . Standard techniques of vector analysis allow to equate the heat flow into the volume V to the heat flow across its surface. This operation leads to the linear and homogeneous Fourier differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity in m 2 s 1 , which is equal to the thermal conductivity  divided by the density and specific heat capacity. The Laplacian operator is / 2 = 0 2 / 0 x 2 + 0 2 / 0 y 2 + 0 2 / 0 z 2 , where x, y, and z are the space coordinates. In the present cylindrical symmetry, the Laplacian, operating on temperature T, can be represented as d 2 T/dr 2 reducing the equation to one dimension. Equations (1)  (3) form the basis for the further mathematical treatment of differential thermal analysis, as is given, for example by Ozawa T (1966) Bull. Chem. Soc. Japan 39: 2071. The superposition principle is obeyed by the solutions of Eq. (3), i.e., the combined effect of a number of causes acting together is the sum of the effects of the causes acting separately. This allows the description of phase transitions by evaluating several separate solutions and adding them, as suggested in Fig. A.12.2. For a modeling of the glass transition in a DSC with R i = 4 mm, see Fig. A.12.3. [...]... for the further mathematical treatment of differential thermal analysis, as is given, for example by Ozawa T (1966) Bull Chem Soc Japan 39: 2071 The superposition principle is obeyed by the solutions of Eq (3), i.e., the combined effect of a number of causes acting together is the sum of the effects of the causes acting separately This allows the description of phase transitions by evaluating several... refers to the thermal diffusivity of the glass, and ks' to that of the liquid Fig A.12.3 837 Appendix 13 Description of Sawtooth-modulation Responses In this Appendix a number of applications of sawtooth modulations are described with modeling and actual results, starting with the sawtooth modulation by utilizing a standard DSC and analysis without Fourier analysis, followed... subscripts) Fig A.11.2 Appendix 11–Online Correction of the Heat-flow Rate 833 The use of the corrections B in Fig A.11.2 needs two calibration runs of the DSC of Fig 4.54 The heat capacities of the calorimeter platforms, Cspl and Crpl, and the resistances to the constantan body, Rspl and Rrpl, must be evaluated as a function of temperature First, the DSC is run without the... shown in the lower graph of Fig A.13.2 During the heating segment, the lightly, vertically dotted area, is part of the positive heat-flow rate HFh During the cooling segment, the underlying portion of the lightly diagonally dotted area is opposite in direction and must be added to HFc to define the raised pseudo-isothermal baseline level (Ps) Next, a constant, irreversible thermal process with a latent... smaller fluctuations The imbalance in Cp (x) is calculated from Eq (3) It is a measure of the kinetics of the cold crystallization by assessing the evolution of the latent heat It speeds up until about 12 min into the quasi-isothermal experiment, and then slows to completion at about 60 min An example of modulation cycles of PET at 450.8 K (before major melting) and 512.2 K (at the melting peak) is displayed... a modeling of the glass transition in a DSC with Ri = 4 mm, see Fig A.12.3 836 Appendix 12–Derivation of the Heat-flow-rate Equations Fig A.12.2 Initially one finds the steady state of the glass with Eq (4) of Fig 6.65 At time t' the heat flux into the glass stops and it approaches the constant temperature T = qt' [= T1(Ri) at t'] with the transient T2 of Eq (6),... on cold crystallization of PET (see Figs 4.74 and 4.136–139) A latent heat does not change the temperatures of Fig A.13.1, so that the heat-flow rates need to be modified, as is shown in the upper graph of Fig A.13.2 The constant latent heat is indicated by the vertically shaded blocks and is chosen to compensate the effect of the underlying heating rate, so that the level of Ps is moved to zero The... heat-flow rate expressions of sketch B of Fig A.11.2 yields the two heat-flow rates of Fig A.11.3 and their difference, expressed in 834 Appendix 11–Online Correction of the Heat-flow Rate Eq (2), above To obtain the actual heat-flow rate into the sample, 0sample, one has to compute the heat-flow rates into the two pans Since both pans are out of the same material Cspn... cycle, but melting continues from (a) to (b) despite of the beginning of cooling Comparing the heat-flow rates to the data taken at 450.8 K allows an approximate separation of the melting, as marked by the shadings Comparing the heating and cooling segments suggests an approximate equivalence of (b) and (c), so that the 840 Appendix 13–Description of Sawtooth-modulation Response ... evaluation of bo, equal to the total heat-flow rate, , and is completed with determination of a1 =2 and the value of b1 = 2, the first harmonic terms of the Fourier series described in Sect 4.4.3 A sinusoidal curve has no further terms in Eq (4) Higher harmonics need to be considered if the sliding average bo is not constant over the modulation Appendix 13–Description of Sawtooth-modulation . Seminar of 2004, courtesy of Dr. C. Schick. 11. Moon I, Androsch R, Chen W, Wunderlich B (2000) The Principles of Micro -thermal Analysis and its Application to the Study of Macromolecules. J Thermal. Spinning of fibers, for example, may be done at rates of 100 to 10,000 m min 1 . A temperature change of 60 K over a length of 1 m in the thread-line, then, causes rates of temperature change of 6,000–600,000. Eq. (3), i.e., the combined effect of a number of causes acting together is the sum of the effects of the causes acting separately. This allows the description of phase transitions by evaluating