Thermal Analysis of Polymeric Materials Part 18 ppt

Appendix 12–Derivation of the Heat-flow-rate Equations __________________________________________________________________ 836 Fig. A.12.3 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' [= T 1 (R i ) at t'] with the transient T 2 of Eq. (6), but at the same time t' there also begins a new transient, T 3, for the liquid which reaches the new steady state at time  , as described by Eq. (5) with t = t*. Note, that k s refers to the thermal diffusivity of the glass, and k s ' to that of the liquid. Appendix 13 __________________________________________________________________ 837 Fig. A.13.1 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 by the analysis of the sawtooth-modulation data after fitting to a Fourier series. Such temperature modulation can be done with any standard DSC which can be programed for a series of consecutive heating and cooling steps. In the example given, a Perkin-Elmer Pyris-1™ DSC is used with a calibration as described in Sect. 4.3 [1]. The modulation is illustrated in Fig. A.13.1. The heavy line represents the change in sample temperature, the dashed lines represent the underlying temperature increase by 1.0 K min 1 and the reversing change by ±3 K min 1 . Within each cycle, the upper and lower limits of the heat-flow rates, HF h and HF c (proportional to the temperature differences  T), are read from the measured temperature-difference response, shown in the lower graph of Fig. A.13.2 for an idealized, instantaneously reacting calorimeter and a sample without latent heat contributions. The response HF(t) is represented by the heavy line, jumping from zero to the heating response in heat-flow rate, HF h , to the cooling response HF c . If the response were not instantaneous, one would have had to wait until the steady states were reached and then extrapolate the steady-state response back to the beginnings of the heating and cooling segments to produce a result similar to the one shown in Fig. A.13.2. For a heat capacity that changes linearly with temperature, the analysis shown next is still possible for the slow response, but a nonlinear change in heat capacity would destroy the stationarity. The reversible heat-flow rate can be Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 838 Fig. A.13.2 deconvoluted by subtracting <HF(t)> from HF(t) as 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 HF h . During the cooling segment, the underlying portion of the lightly diagonally dotted area is opposite in direction and must be added to HF c to define the raised pseudo-isothermal baseline level (Ps). Next, a constant, irreversible thermal process with a latent heat is added to the modulation cycles, as is found 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 reversing specific heat capacity is given by: mc p.rev = (HF h  HF c ) / (q h  q c )(1) Note that under the linearity condition of thermal response, HF is positive (endother- mic) on heating and negative (exothermic) on cooling. Analogously, the rates of temperature change q h and q c change their sign on going from heating to cooling. In TMDSC the nonreversing contribution can only be assessed indirectly by subtracting the reversing C p from the total C p , or by an analysis in the time domain. Any error in the reversing or total heat capacity will be transferred to the non- reversing heat capacity. The analysis using the standard DSC method, however, allows a direct measurement of the nonreversing effect by determining the difference in the heat capacities on heating and cooling at steady state. This quantity is called the imbalance in heat capacity, and can be written as: mc p.imbalance = (HF h /q h )  (HF c /q c )(2) Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 839 Fig. A.13.3 The irreversible heat-flow rate is calculated from Eq. (2) by separating HF h and HF c into their reversible, underlying, and irreversible parts for heating and cooling. Assuming linear response among the heat-flow rates, caused by the true heat capacity and a constant, irreversible heat-flow rate, one can derive the following equation: HF irreversible = mc p.imbalance /[(1/q h )  (1/q c )] (3) The cold crystallization of PET was analyzed by running the series of quasi- isothermal experiments at 388 K and evaluated as shown in Fig. A.13.3. The reversing C p (  ) decreases, as expected from the lower C p of the crystallized sample. The same decrease can be calculated from the integral of the total heat-flow rate over time, the latent heat, and used to estimate the heat capacity as sum of contributions of amorphous and crystalline fractions (+). As expected, both data sets agree, but the reversing heat capacity has smaller fluctuations. The imbalance in C p ( 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 in Fig. A.13.4. The measurements at 450.8 K reach steady state on both, heating and cooling, the ones at 512.2 K, only on cooling. On heating at 512.2 K, the heating is interrupted at 0.5 min by the cooling 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 Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 840 Fig. A.13.4 (4) marked HF h accounts for practically all irreversible melting (a + b) and the reversible heat capacity. The cooling segment when represented by HF c , contains almost no latent heat, and similarly, the heat-flow rate HF h ' is a measure of the heat-flow rate on heating without latent heat. Turning to the analyses of periodic functions, such as the modulated heat-flow rate HF(t), using Fourier series, one can find in any mathematics text the derivation of the following recursion formula [2]: where  is a running integer starting from 1, and the constant b o and the maximum amplitudes a  and b  are given by: The data deconvolution for  = 1 starts with the evaluation of b o , equal to the total heat-flow rate, <HF(t)>, and is completed with determination of a 1 =2<HF sin (t)> and the value of b 1 = 2<HF cos (t)>, 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 b o is not constant over the modulation (6) (5) Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 841 Fig. A.13.5 Fig. A.13.6 period (loss of stationarity) or if the frequency is not constant. The Fourier series for a sawtooth which is symmetric about <q>t is given in Fig. A.13.5. Figure A.16.6 illustrates the attainment of steady state after the beginning of saw- tooth modulation and on changing from a heating to a cooling segment. Based on these equations, the following curves are computed for Figs. A.13.7–10 by using the same conditions as in the equations shown in Figs. 4.67 and 4.68. Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 842 Fig. A.13.8 Fig. A.13.7 Figure A.13.7 illustrates the temperature modulation and resulting heat-flow rate,  T, for a case that reaches steady state before changing the direction of the change in temperature, similar to the case of Fig. A.13.4 at 450.8 K. Figure A.13.8 illustrates the case that steady state is not reached. Figures A.13.9 and A.13.10 show enlarged graphs of  T and their various harmonic components. Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 843 Fig. A.13.9 Fig. A.13.10 The data calculated from Figs. A.13.7 and A.13.9 by the standard DSC method given in Fig. 4.54, using the response after steady state has been reached at the end of each cycle give the proper heat capacity, as listed in Table A.13.1, below. The data from Figs. A.13.8 and A.13.10 cannot be analyzed in this way since in this case, steady state is never reached. For Figs. A.13.8 and A.13.10 the evaluation methods illustrated with Figs. 4.54 and 4.92 for the frequency of the first harmonic must be Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 844 Fig. A.13.11 considered. Doing so, the last column in the table shows that, again, the proper answer is obtained for the first harmonic as long as stationarity is preserved. Every higher harmonic will under the chosen conditions also give the correct results. Table A.13.1 Results of the Calculation of Heat Capacity Figures: C s /K  C r /K Standard DSC Deviation (%) TMDSC A.13.7 and 9 6  1 = 5 4.997  0.06 5.00 A.13.8 and 10 60  10 = 50 14.916  70.2 50.00 Footnote to the table: The heat capacities C s and C r , when divided by the Newton’s Law constant K, are given in seconds, s; the linear heating and cooling rates of the sawtooth q are 0.04 K s  1 ; the temperature amplitude set for the sawtooth, A, is ±1.0 K; the period p is 100 s; the frequency 7 = 2%/p = 0.062832 rad s  1 ; at the time t = 0: T s = T r = T o ; C p (standard DSC)/K = T/q. The calculation for the first harmonic gives: C p (TMDSC)/K = [A  /(A7)] × [1 + (C r 7/K) 2 ] ½ ; the total length of analysis is 400 s, one point is calculated every second; for the standard DSC, T is taken at the maximum or at the steady state of T/K; for the quasi-isothermal TMDSC the last 100 points of the first harmonic of the maximum modulation amplitude were averaged (the 4 th modulation cycle in Figs. A.13.7 and 8). The next five figures show experimental results on the melting of pentacontane, C 50 H 102 , a paraffin which melts practically reversibly [3]. Figure A.13.11 illustrates the comparison of a standard DSC trace and a quasi-isothermal analysis with a Appendix 13–Description of Sawtooth-modulation Response __________________________________________________________________ 845 Fig. A.13.12 ±0.05 K sinusoidal modulation amplitude. The main part of the melting of C 50 H 102 is clearly reversible. The standard DSC shows the expected broadening in the melting peak due to the lag of the instrument. The extrapolated onset of the melting temperature, as defined for the standard DSC in Fig. 4.62, agrees with point D of the quasi-isothermal TMDSC. Lissajous figures of the heat-flow rate plotted versus the sample temperature in Fig. A.13.12 for points A to D indicates that the final heat of fusion is too large for complete melting in one modulation cycle, as was also seen for the reversibly melting indium in Fig. 4.109, 4.134 and 4.135. At point A in Figs. A.13.11 and A.13.12 the reversing heat-flow rate is already considerably larger than expected for the crystals as can be judged from the Lissajous figure for the melt at the highest temperature indicated. The deviation from a perfect ellipse indicates further, that the response is not symmetric in the heating and cooling cycle. The melting and crystallization have either different rates over the modulation cycle, or the mechanisms are more complicated than just melting and crystallization, they may involve, for example, growth of metastable crystals on cooling, followed by annealing to stabler crystals before reaching their melting temperature. The missing area of the explained by the major melting which is not completed within the modulation cycle, as seen at temperatures C and D. Even if there were enough time within the modulation cycle to complete melting and crystallization, the simulated response of a single reversible cycle of melting followed by crystallization in Fig. A.13.13 shows that a straight-forward analysis of the heat-flow rate HF(t), as outlined in Sect. 4.4.3, would not lead to a good representation of the melting and crystallization process. The main reason for the deviations is the nonstationary response. The sawtooth modulation shown in Fig. A.13.14 allows an easy interpretation of the actual data for a similar pentacontane sample as in Figs. A.13.11 and A.13.12. [...]... comparison of a standard DSC trace and a quasi-isothermal analysis with a Fig A.13.11 Appendix 13–Description of Sawtooth-modulation Response 845 ±0.05 K sinusoidal modulation amplitude The main part of the melting of C50H102 is clearly reversible The standard DSC shows the expected broadening in the melting peak due to the lag of the instrument The extrapolated onset of the... the total length of analysis is 400 s, one point is calculated every second; for the standard DSC, T is taken at the maximum or at the steady state of T/K; for the quasi-isothermal TMDSC the last 100 points of the first harmonic of the maximum modulation amplitude were averaged (the 4th modulation cycle in Figs A.13.7 and 8) The next five figures show experimental results on the melting of pentacontane,... using standard DSC baselines are shown in the integral analysis of the heat-flux rates in Fig A.13.15 This analysis bypasses the nonstationarity problems of the Fourier analysis Fig A.13.14 Appendix 13–Description of Sawtooth-modulation Response 847 Fig A.13.15 References to Appendix 13 1 Details about the data analysis without Fourier transformation for sawtooth-type... (2001) Data Analysis Without Fourier Transformation for Sawtooth-type Temperature-modulated DSC J Thermal Anal Calorim 66: 677–697 2 A collection of Fourier series for different curves can be seen, for example in: Lide DR, ed (2002/3) Handbook of Chemistry and Physics, 83rd ed CRC Press, Boca Raton 3 The TMDSC with Fourier analysis of the melting pentacontane and the calculations using saw-tooth analysis. .. paraffin, comparison of melting temperatures with poly(1-alkene)s 754, 755, 755, 756 cooperative melting, described with the Ising model 554, 555, 555 even, heat capacities of C4H10 to C18H38 327, 327 lower limit, of chain-folding 491 of molecular nucleation 254, 255 melting temperatures 192, 193 molecular dynamics calculations for, parameters 44 partially fluorinated, condis crystal of 562, 563, 563 see... see: poly(ethylene-co-octene-1) melting 611–623, 612, 613, 616–623 by AFM 269, 270 data 544 kinetics 256, 611, 612 of equilibrium crystals 298, 299, 611, 612, 719–723, 720–722 of fractions 615– 618, 616– 618 of lamellae 155, 612, 614 of the fold surface 623, 623 peak, dependence on Tc 613 range of 614 superheating 611, 612, 612 temperature, extrapolation 154, 155, 192, 193 mesophase 490, 510, 569–571, 569–571,... A.17.2 The Moiré lines are related to the spacings of the basis lines and their angle of rotation Adding a “defect,” as is illustrated in Fig A.17.4 to the lower layer, the pattern of Fig 5.93 results, based on the distortion of the line lattice caused by the edge dislocation An analogous interpretation is possible for the double layer of foldedchain crystals of polyethylene lamellae, shown in Fig 5.94 Fig... 542, 543 thermochemistry of combustion 322, 322 ethyl alcohol, see: ethanol ethyl benzoate, diluent for polyethylene 715 ethylene, melting parameters 542 reaction type 212 F fluorine 4, 5 atomization, heat of 324 bonding 3 fullerene 3, 172, 483, 484, 793 crystals with solvent 484 heat capacity of 325, 325, 326 thermal analysis and NMR 172–174, 173 thermodynamic functions, plot of H, TS, G 326, 326 TMDSC... transition analysis 142 gold, density 5 melting parameters 538 graphite 25, 172 atomization, heat of 324 bond energy 324 frequency spectrum 115 -temperature 111, 115 heat capacity of 325, 325, 326 thermochemical data 323 thermochemistry of combustion 322, 322 gutta percha, see: 1,4-poly(2-methyl butadiene), trans gypsum, see: calcium sulfate dihydrate H helium, atomic diameter 3 class of molecule... atomization, heat of 324 bond energy 324 bromide, melting parameters 539 chloride, melting parameters 539 cyanide, melting parameters 542 iodide, melting parameters 539 melting parameters 539 phosphide, melting parameters 539 sulfide, class of molecule 8 melting parameters 539 thermochemistry of the combustion of 320–322, 321 thermochemical data 323 I ice, see: water indium, heat of fusion 339 Lissajous . main part of the melting of C 50 H 102 is clearly reversible. The standard DSC shows the expected broadening in the melting peak due to the lag of the instrument. The extrapolated onset of the. followed by partial recrystallization (6) and final melting (7). Quantitative analyses of the various stages of this process using standard DSC baselines are shown in the integral analysis of the. graph of Fig. A.13.2. During the heating segment, the lightly, vertically dotted area, is part of the positive heat-flow rate HF h . During the cooling segment, the underlying portion of the