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Anal. Chem. 1984, 56, 2868-2870
Differential Scanning Calorimetry for Rapid Exothermic Transitions Steven R. Eckhoff* Agricultural Engineering Department, Kansas State University, Manhattan, Kansas 66506
Edward B. Bagley Northern Regional Research Center, Agricultural Research Service, US. Department of Agriculture, 1815 North University, Peoria, Illinois 61604
A theory of differential scanning calorimetry (DSC) is developed without the usual assumption that the sample and reference cells are at the same temperature. The results show that the heat generated by the sample cannot fully be accounted for by the DSC during fast exothermic processes, such as during phase transition after supercooling or during exothermic reactions. Experimental results with distilled water verified the theory. Up to a 44% difference between endothermic and exothermic peak areas was observed. Sample sire was found to greatly affect the degree of difference between endothermic and exothermic peaks.
0.625 to 10.0 "C/min. Exothermic and endothermic peak areas were determined by duplicating,cutting, and weighing each peak in quadruplicate. Base lines were drawn by using a procedure similar to that of Guttman and Flynn (3). The DSC was calibrated with a 3.62-mg sample of indium metal as reference. Various materials which do not appreciably supercool were analyzed on the DSC to show linearity of the DSC over the temperature range -40-156 "C. Samples of indium (156 "C), methyl stearate (38 "C), cyclohexane (6.5 "C), and mercury (-40 " C ) were used with sample sizes of 6.63, 5.71, 3.62, and 2.01 mg for the indium, 3.13 mg for the methyl stearate, 5.98 mg for the cyclohexane, and 40.04 mg for the mercury. Several runs also made on a Perkin-Elmer DSC-I1 calorimeter to demonstrate that the phenomenon was not limited to the particular instrument used.
A differential scanning calorimeter (DSC) is a valuable research tool for determining many of the thermodynamic properties of materials. The electrical output of a DSC is assumed to be directly related to the heat flow into or out of the sample being studied, thus providing a quantitative method for determining the phase transitions, glass transitions, and specific heat of a material. However, anomalous behavior has been reported in the study of highly exothermic materials. Cantor (1) in a study on salt hydrates attempted to use distilled water to calibrate a power-compensated DSC. He found that the apparent exotherm could be less than the apparent endotherm by up to 25%. Addition of AgI to inhibit supercooling resulted in exotherms of the same size as the endotherms. Cantor suggested that the instrument's differential power circuit could be saturated during the exothermic transition. In a paper on DSC analysis of highly exothermic reactions (2),two incidents were noted where sample size affected the amount of heat measured. This effect was attributed to the self-heating of the sample. Self-heating occurs when the generation of heat within the sample, due to chemical reaction or phase change, is more rapid than the rate of heat dissipation from the sample. The result is heat accumulation by the sample, thus increasing its temperature. Although this study was with nonsupercooled materials, the instrument behavior was similar to that observed by Cantor. It is the purpose of this study to show how the theory of differential scanning calorimetry can be expended to predict and account for the anomalous behavior that occurs during highly exothermic reactions. This theory will be verified experimentally using distilled water,
THEORY
EXPERIMENTAL SECTION Various-sized samples of distilled water ranging from 1.24 to 19.30 mg were cycled on a Perkin-Elmer DSC 1B calorimeter. Distilled water was chosen because it has a high latent heat (80 cal/g) and because it supercools significantly (approximately 20 "C). The samples were cycled at various scan speeds ranging from
A schematic of the DSC sample cell is shown in Figure 1. The major components of the Perkin-Elmer DSC-1B sample holders are the stainless steel cups and supports that encase a platinum wire temperature sensor and an etched Nichrome heater. The platinum wire sensor is located between the heater and the bottom of the sample holder. The holder is in contact with the Nichrome heater in order to maintain the sensor at the preprogrammed temperature. The difference between the amount of electrical power required to maintain the sample holder and the reference holder a t the desired temperature is the output of the instrument. Previous derivations for the governing DSC equations (4-6) assumed that the major limiting heat resistance is between the heater (or sensor) and the sample. They assumed that the heat lost to the environment was the same for both the sample and the reference cell. In the following derivation these assumptions are not made. Heat balances for the sample and the reference cell yield the following two relationships
where T s is the sample temperature, TR is the reference temperature, TAis the heat sink temperature, and Tp is the programmed temperature or the temperature at the sensor. CS and C R are the specific heats of the sample plus sample container and the reference container, respectively. R1 is the thermal resistance between the temperature sensing location and the sample, R2 is the thermal resistance between the sample and the heat sink, and R3 is the thermal resistance between the sample cell and the reference cell. The rate of
0003-2700/84/0356-286S$01.50/00 1984 Amerlcan Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
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Table I. Effect of Scan Speed on Peak Area (13.17-mg Sample) scan speed, OC/min
endotherm, cal/g
0.625 1.230 2.500 5.000 10.000
76.6 75.9 77.3 78.4 74.9 76.6 f 1.3 av
exotherm, cal/g
70 difference
53.7 52.1 50.9 50.7 48.1 51.1 i 2.1" av
30 31 34 35 36
"The range noted here is not entirely random. Table 11. Effect of Sample Weight in Peak Areas at a Scan Speed of 2.5 "C/min
Figure 1. Schematic of a DSC cell.
heat generation by the sample is denoted as dhldt. By subtracting (2) from (1) we get
dh _ dt
Differentiating Newton's law of cooling
where dqldt is the energy flux to the sample. Upon rearrangement this becomes d2q dTp dTs
--
--Rip
+-
dt dt2 dt Substituting (5) into (3) yields the relationship
Making the assumption that T p = most conditions (7)
TR,
which is valid under
Since dqldt = (Tp- Ts)/R1,which is Newton's law of cooling dh _ -
dt
When Ts = TR = Tp eq 8 becomes the classically derived DSC equation (4-6).
In instances when Ts # Tp (TR= Tp), as in the case of highly exothermic materials which self-heat, and for rapid exothermic reactions, the last two terms
2-
Ts - T P R3
and
Ts- Tp R2
or
Ts - T R ~
R2
(since Tp =
TR)
(11)
sample wt, mg
endotherm, cavg
exotherm, calig
difference
1.24 2.49 3.98 9.71 13.17 19.30
73.1 73.7 73.3 77.7 78.4 76.6
60.0 59.4 61.7 59.4 52.5 42.8
11 19 16 24 33 44
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affect the DSC thermogram. Exothermic materials will self-heat when the exothermic process is very rapid and heat is liberated at a rate faster than it can be dissipated. Both terms are opposite in sign to the d2q/dt2and dq/dt terms, indicating that when Ts # T Rthe exothermic peak will be smaller than expected. Of the two terms, (Ts- T p ) / R 3is the smaller due to the larger resistance of R3 (heat must pass through two pans, two thermal boundary layers, and the pan separation distance) and probably can be neglected. However, in the term ( T s- T R ) / R Pan, increase in sample temperature due to self-heating results in an increased flux to the environment, which will not be accounted for by the calorimeter. On most DSC instruments there is usually an indicator light that shuts off if the temperature at the sensor is not the same as the programmed temperature. One can rightly ask why this does not turn off when the sample self-heats. The answer is that the sensor does not measure the sample temperature but the temperature a t a point between the heater and the sample. During self-heating the heat flux from the sample is increased, yet due to the distance from the sample to sensor, the sensor temperature is not increased. Although heat is flowing from the sample to the sensor, the liquid nitrogen heat sink is large enough that additional heat is required to flow from the heater to the sensor in order to maintain the sensor a t the programmed temperature.
RESULTS AND DISCUSSION Tables I and I1 illustrate the effect of scan speed and sample weight, respectively, on the endotherms and exotherms of the distilled water samples. The percent difference used in the tables is defined as (endotherm area - exotherm area)/endotherm area X 100. The tables show that while there may be a slight relationship between scan speed and percent difference, there is a significant relationship between sample weight and percent difference. Within experimental error, sample weight did not affect the magnitude of the measured endotherm but had a large effect on the magnitude of the measured exotherms. The percent difference between endothermic and exothermic peak areas decreased as the sample weight decreased from a level of 44% for a 19.30-mg sample to 11% for a 1.24-mg sample. The phase-transition temperatures varied less than 2 "C over the range of sample weights and scan speeds.
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
To further demonstrate that this relationship between percent difference and sample weight is not limited to the particular Perkin-Elmer DSC-IB instrument used, a 16.14-mg sample of distilled water was cycled on a Perkin-Elmer DSC-I1 calorimeter. The results showed that the average endothermic peak area was 35% larger than the exothermic peak area. Linear regression of the datum in Table I1 and the data point taken with the DSC-I1 had a best fit line with the equation % difference = 1.68 (sample weight) 10.50. The coefficient of determination was 0.96. The reasons for the linear behavior are unknown at this time. The approximately 10% difference observed at very small sample sizes can be explained on thermodynamic grounds. The dependence of AHf (heat of fusion) on temperature, given by Kirchhoff's law, is
+
where CL and Cs are the specific heats of the liquid and solid, respectively, and, in general, will be functions of temperature. For water, CL - Cs can be taken as 0.5 cal/g "C over the temperature range 0 to -20 "C, so that for supercooling to -17.5 "C the change in AHf would be 8.75 cal/g. This would amount to a 10.9% decrease in the peak area for an exotherm. This would explain the offset on the ordinate axis described by the linear equation % difference = 1.68 (sample weight) + 10.50. The change in percent difference observed by increasing scan speed is not as appreciable as observed for sample weight. However, Table I does show that an increase in scan speed from 0.625 to 10.0 "C/min resulted in a decrease in apparent exothermic values from 53.7 to 48.1 cal/g. Based on the theory presented herein, these results are to be expected. As scan speed is increased, the programmed temperature is decreasing faster away from the temperature of the self-heating sample, thus producing a larger differential between Ts and Tp. It takes longer for the sample temperature to dissipate its heat and catch up to the reference temperature. The result is that more heat is not measured by the DSC as scan speed increases. Results from the indium, methyl stearate, cyclohexane, and mercury samples did not exhibit the same difference between exothermic and endothermic peak areas as was demonstrated by the distilled water samples. Endothermic and exothermic peak areas agreed within 5% for all samples. Except for the mercury, which had an average exothermic peak area 1.06% smaller than the average endothermic peak area, the discrepancy in the peak areas was such that the exotherm was larger than the endotherm. ONeill(8) studied the supercooling phenomena associated with indium and concluded that the same latent heat values could be obtained from endothermic and exothermic runs. They concluded that apparent discrepancies observed between endothermic and exothermic values are due to either improper experimental techniques, errors in base-line interpolation, or improper interpretation of the results. The data presented in this article dispute their conclusion for high exothermic materials that self-heat. Although O'Neill studied supercooling, indium cannot be considered a good material for studying supercooling or self-heating because of its low level
of supercooling (less than 1.6 OC) and its low latent heat (6.79 cal 9-l). As discussed previously, the difference observed between endothermic and exothermic values is a function of the temperature difference between the sample and the reference cell. This means that the degree of supercooling can dictate this temperature differential which causes the discrepancy between endothermic and exothermic values. The sample will never self-heat above the melting point temperature, and, thus, for indium the maximum temperature difference that can be observed between the sample and the reference cell is 1.6 "C. Water, on the other hand, supercools more than 20 "C and thus can have a temperature difference of about 20 "C. Indium also has a low latent heat compared to water (6.79 vs. 80 cal g 3 . Since the self-heating phenomenon is a problem associated with the dissipation of heat, samples with larger latent heats will exhibit the problem to a greater extent. In general, O'Neill's conclusions need to be restricted to samples which only marginally supercool and which have a low latent heat. In such cases, the uncertainty in drawing the base line may be of a greater magnitude than the loss of heat due to self-heating. In fact, our results with indium showed a slightly larger exotherm than endotherm. Uncertainty in drawing the base line may account for this difference.
CONCLUSIONS Care must be taken in analyzing exothermic data taken from a power compensated DSC. If the sample size is too large, and the sample undergoes a rapid highly exothermic transition, self-heating of the sample will occur and heat will be lost from the sample. This heat cannot be accounted for by the instrument. Practically, the sample size should be minimized, scan speed should be slow, and the sample should be dispersed throughout the sample pan as much as possible for reliable results. Endothermic values are to be considered more reliable for phase-change values in samples that supercool. ACKNOWLEDGMENT This is Contribution No. 84-435-5 from the Kansas Agricultural Experiment Station. We thank J. W. Hagemann of the Northern Regional Research Center, Peoria, IL, for his technical assistance and advice. LITERATURE CITED Cantor, S . Thermochim. Acta 1979, 3 3 , 69-86. Duswalt, A. A. "Analytlcal Calorimetry"; Porter, R. S., Johnson, J. F., Eds.; Plenum Press: New York, 1968; pp 313-317. Guttman, C. M.; Fiynn, J. H. Anal. Chem. 1973, 45, 408-410. Dumas, J. P.; Clausse, D.; Broto, F. Thermochim. Acta 1075, 13, 261-275. Gray, A. P. "Analytical Calorimetty"; Porter, R. S.,Johnson, J. F., Eds.; Plenum Press: New York, 1968; pp 209-218. Wendlandt, W. W. "Thermal Methods of Analysis", 2nd ed.; Wiley: New York, 1965. Brennan, W. P.; Miller, B.; Whitewell, J. C. Thermochim. Acta 1971, 2, 354-356. O'Neill, M. J. Anal. Chem. 1975, 4 7 , 630-637.
RECEIVED for review May 29,1984. Accepted August 6,1984. The mention of firm names or trade products does not imply that they are endorsed or recommended by the U S . Department of Agriculture over other firms or similar products not mentioned.
