ozawat thermochim acta 253 (1995) 183-188

8/6/2019 OzawaT Thermochim Acta 253 (1995) 183-188

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thermochimicaactaThermochimica Acta 253 ( 1995) 183- 188

Linearityand non-linearityn DSC:l " ' i IA critique on modulatedDSC*Takeo Ozawa u'*. Katsuhiko Kanari b

" Daicel Chemical Industires, Ltd., Tsukuba ResearchCenter, Tsukuba 305,Japanb Electrotechnical Laboratorv. Tsukuba i05. JaoanReceived8 August 1994;accepted 3 August 1994

AbstractRecently a new interesting technique, modulated differential scanning calorimetry(MDSC), was published.This technique s basedon Fourier transformation,and its sound

applicabilitydepends n linearity in thermalprocesses,ecausehe Fourier transformation sbased on linearity or the superposition principle. If a processunder observation is anon,linear process, he physicalmeaningof the Fourier transformation should be closelyexamined.The DSC output signalsof someprocesses,uch as melting and reaction, havebeen revealed o be non-linear.Therefore,differentiationof linearity from nonJinearity inDSC thermalresponsess very important for modulatedDSC, and they arediscussedn thispaper. In this discussion,methods o analyzeoscillatingoutput signalsare alsoproposed.Keywords: econvolution; SC; DynamicDSC; Linearity;MDSC

IntroductionRecently a new technique of heat-flux differential scanning calorimetry, hf-DSC,was set forth [,2]. In this new technique, emperatureoscillation is added to linearheating, and oscillating thermal response n the form of the temperature differenceis recorded.The thermal response s analyzed by Fourier transformation, and the* Corresponding author.* Presented at the Intemational and III Sino-Japanese Symposium on Thermal Measurements, Xi'an,4-6 June 1994.

0040-6031/95/$09.501995 ElsevierScience .V. All rights reserveds,sD/0040-603 94)02088-4


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184 T. Ozawa. K. KanarilThermochimica Acta 253 (199t 183-188responsen phasewith the temperatureoscillation s thought to be causedbyreversible hangeor equilibriumchange n the sample,while the response ut ofphasewith the temperature scillation s postulated o be non-reversible hange[,2]. Becausehe Fouriertransformations applied o the response,t is implicitlyassumedhat lirtearity,or the superposition rinciple,holds n the response.However, his is not alwaysclear,and non-linear esponsesave beenobservedin DSC results[3]. In order to elucidate he applicability of this very interestingtechnique,t is essentialo differentiateinear from non-linear esponsen hf-DSC.What canbe done and what cannotbe doneby this technique? he answero thisquestion s strictly connectedwith the linearity and non-linearity of the process.norder to obtain, he correct answer, hermal responses, uch as the baseline,peakarea and peakshape,are considered heoreticallyand their linearitiesare examinedby experimental acts. It is elucidatedhat linearity holds for the baselineand peakarea,but thermal response y sample ransformation s a non-linearprocess, o thephysicalmeaningof observationof the transformation behaviorby this techniqueis not clear. However, the thermal responseby electrical heating, insteadof thesample transformation, is a linear process.These theoretical considerationsaredescribed n detail, and methods to analyze oscillating thermal response areproposedn this paper.

2. LinearityFirst of all, the mathematicaland physicalmeaningof linearity and nonJinearityshould be made clear in relation to DSC. Linearity can be expressedby theequations

f(x +y):f\x) +f(y) (1)f(ax) : q71*1 Q)

For DSC, x andy express uantities o be measuredand the function expressestsoutput signal. Eq. (l ) is the basis of the superposition rinciple and Eq. (2)indicatesproportionality for quantitativemeasurement.Hitherto,two contradictorypostulations avebeenmade.Tateno 4] postulatedthe applicabilityof a linear theory of measurement, nd he deconvolutedDTAcurves o reaction ate curvesby usinga transfer unction of the DTA apparatus.The transfer unction correspondso f(u) for a pulsiveu in Eq. (l), and thedeconvolutions a procedure or obtainingx + y +. . ., which occursconsecu-tively, from the observed(x + y + ) by using the transfer function. Thisprocedure f deconvolutionwasmadeon the basisof linearityor the superpositionprinciple.However,O'Neill [5] analyzed he heat transfer n powercompensationDSC (pc-DSC)and postulatedhat transformation f the sample, uchasmelting,proceeds y a heat-transfer-controllingechanism.Thesewo contradictory ostulates re expressedchematicallyn Fig. l. Accord-ing to the linear theory of measurement,he output signal is deformed by


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T. Ozawa, K. Kanarif Thermochimica Acta 253 (jgg5) lE3_lgg(a l

Quantity to be measured

timeFig._ l. Schematic expressions of (a) linear theory of measurementmechanism. Reproduced lrom Ref. t3l with permission.

(b)

F

ooo

a

oo

o

tri.irrl$, ti&i'ttime

and (b) heat-transfer-controlling

the transfer function as shown in Fig. la, and the peak height is proportional to thequantity to be measured.For "^u-pl., the DSC peak hefiht oimetting of a puresample should be proportional to the sample mass, f [r*?ritv-t "rds in a meltingprocess.However, o'Neill's postulate [5] can be expressedas in Fig. lb. The heat istransferred by temperature gradient and the sampre temperature is kept constantduring the transformation, while the temperatureof the heat source s controlled toincrease inearly. Therefore, the temperaturegradient arso ncreasesinearly and therate of transformation is controlled by this linearly increasing emperaturegradient,through which the heat is suppried. Thus the uioitio.rut "i.rgy'input due to thetransformation is increased Iinearry in pc-DSC, and the aoaltonat temperaturedifference s similarly increased inearly in nr-nsc [3]. The ,"r* p."r. shapecan beobtained by both types of DSC, because he heat transfer mechanism s the same[3]' Non-linearity is implicitly postulated by this heat transfer mechanrsmdue to theconstant sample temperature during the transformation.


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186 T. Ozawa, K. Kanarif ThermochimicaActa 253 (1995) 183,1883. Experimental facts and theoretical considerations

Experimental verification was made by one of the present authors [3]. With apc-DSC, indium melting peakswere observedby changing the samplemass,and theresults are reproduced n Fig. 2 . Similar results were also obtained using DTA of amicro-sample. From these results we can conclude that the above heat-transfer-controlled melting proceeds n both types of DSC, and it can be concluded that thesample temperature is kept approximately constant during the melting. Thus, thetemperature distribution within the sample can be neglected during the melting.However, a linear process was also observed with an hf-DSC [3]. Pulsiveelectrical heating was made by inserting an electrical heater into the sample cell, andsome peaks were obtained by changing the current and duration. They werecompared in relation with the current, and the peak height was proportional to thecurrent. A peak of long duration was compared with a peak synthesized bysuperposition of successivepeaks of short duration, and excellent agreement wasobtained between them. In other words, by adding f(x) and "f(y) we can getf(x + y), and this synthesized"f(x + y) is in excellent agreement with the f(x + y)observed for x * y. Thus it was concluded that linearity holds in the case ofelectrical heating.

Similar results were obtained by using peaks obtained by infrared irradiation ofthe sample pan of a pc-DSC [3]. Thus linearity holds in both types of DSC, whent )

3 0 s

157.00c

Fig. 2. DSC curves of melting for high-purity indiumlrom Ref. [3 ] with permission.

timeThe broken line is for remelting.Reproduced

*)>\kL


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T. Ozav,a,K. KanarilTherntochinticaActa 253 (1995) 183,188heat evolves or absorbs independently of the sample temperature. In this case heevolved or absorbed heat is consumed by the sample temperature change.For the baseline, t is clear that linearity holds, because inearity has been used nheat capacity measurementsby using the baseline shift. From a theoretical view-point, the temperature distribution in an hf-DSC apparatus was analyzed n detailfor sample cells of cyl indrical symmetry [6] and for sample cells of general geometry[7]. In the dynamic steadystate at a constant heating rate, the temperature of everypart around the sample and the reference material is increased linearly, so that thetemperature gradient at any point is proportional to the heating rate and it is alsoproportional to the heat capacity of the part to which the heat is supplied from theparticular point.

Similarly, linearity holds in the peak area, becauseproportionality has been usedin measurements of transformation heats. The linearity in this case was alsotheoretically considered for sample cells of cylindrical symmetry [6] and for samplecells of general geometry [7]. The following fundamental equation for heat conduc-tion holds for the additional temperature, Z', due to a sample transformation in anhf-DSC apparatus

d7'' dLH)"Y' : rp a, * a, (3)whereV, tr, c, p, t andLH are respectively Laplacianoperator, he thermalconductivity, the heat capacity, the density, the time and the transformation heat.By integration from the beginning of the peak to its end, we get the followingequation for any point [7]

f (4).) r ' a t tnwhere the thermophysical properties of the sample, i.e., c and p, are assumed to beconstant during the transformation, and the integral on the left side corresponds othe peak area. Thus the integral equation for the peak area is mathematicallyequivalent to the equation for temperature distribution in the steady state, andlinearity holds for the peak area, as it holds for the baseline.

4. Concluding remarksAs seen n the above discussion, inearity holds for a baseline and peak area.Therefore, modulated DSC can be applied to heat capacity measurements,butapplicability of this technique to melting behavior observation is not certainlyproved. For this application, further considerationsusing a theoretical approachare necessary.For theoretical consideration of the baseline, an analytical approach can beapplied. In this approach, it can be assumed that the temperature distribution

within the sample is negligibly small, though linearity holds for the case ofdetectable emperature distribution in the sample. Therefore, a fundamental equa-

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T. Ozawa. K. KanariiThermochimica Acta 25 3 (199, 183-188

tion for DTA derived by Vold [8] can be used and the temperature oscillation isintroduced into the equation. Thus, the relation of a baseline with experimentalparameters,such as heat transfer coefficients,heating rate, samplemass,etc., can bederived as an analytical solution.For examination of modulated DSC results on sampletransformation, the aboveanalytical approach cannot be applied, but computer simulation seems to be apowerful tool. For this approach another assumption can be introduced, i.e.,constant temperature of the sample during the transformation.These approacheswill give us valuable suggestions,and we will report them inthe near future.

ReferencesI M. Reading,. Ell iott ndV.L.Hil l,J.Therm. nal..401993)49.[2] P.S.Gill, S.R.SauerbrunnndM. Reading,. Therm. nal.,40 1993) 31.[3] T. Ozawa,NetsuSokutei, (19^17)5..[4] J. Tateno, rans. araday oc., 2(1966) 885.t5l M.J.O'Neill,Anal.Chem., 6 1964) 238.[6] T. Ozawa, ull.Chem. oc. pn., 9(1966) 071.[7] T. Ozawa nd K. Kanari,Thermochim cta, o be submitted.[8] M.J.Vold,Anal.Chem., 1(1949) 83.

ozawat thermochim acta 253 (1995) 183-188

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