Anal. Chem. 1985, 57, 2005-2007 (6) Tlttarelll, P.; Zerlla, T.; Ferrarl, G. Appl. Spectfosc. 1964, 3 8 , 715-720. (7) Lundberg, E.; Johansson, 0.Anal. Chem. 1976, 48, 1922-1926. (8) Harnly, J. M.; O’Haver, T. C.; Golden, 8.; Wolf, W. R. Anal. Chem. 1979, 57, 2007-2014. (9) Marshall, J.; LittleJohn,D.; Ottaway. J. M.; Harnly, J. M.; Mlller-Ihli, N. J.; O’Haver, T. C. Analyst (London) 1963, 108, 178-188. (IO) Horllck, G. Appl. Spectrosc. 1976,30, 113-123. (11) Parsons, M. L.; McElfresh, P. M. Flame Spectroscopy: Atlas of Spectral Llnes”. 1st ed.; IfVPlenum: New York, 1971. (12) Harrlson, G. R. “Wavelength Tables”; M.I.T. Press: Cambridge, MA, 1969. (13) Alder, J. F.; Samuel, A. J.; Snook, R. D. Spectfochlm. Acta, Pari S 1976, 316, 509-514. (14) Ottaway, J. M.; Shaw, F. Appl. Spectrosc. 1977, 3 1 , 12-17. (15) Pearse, R. W. B.; Gaydon, A. G. “The Identiflcatlon of Molecular
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Spectra”, 3rd ed.;Chapman 8. Hall: London, 1963.
Paolo Tittarelli* Rosa Lancia Tiziana Zerlia Stazione Sperimentale per i Combustibili Viale A. De Gasperi 3 20097 San Donato Milanese, Italy
RECEIVED for review December 26,1984. Accepted April 11, 1.985. This work was presented in part at the 11th FACSS Meeting, Philadelphia, PA, 16-21 Sept 1984 (p 100).
Differential Scanning Calorimetry for Rapid Exothermic Transitions Sir: In a recent paper by Eckhoff and Bagley (I), the freezing of distilled water was examined by differential scanning calorimetry (DSC), using the Perkin-Elmer Models DSC-1B and DSC-2. It was reported that the instruments gave incorrect energy values for the freezing exotherm, and the authors suggested that these instruments are unable to measure rapid thermal phenomena in a sample. In fact, DSC can be used for accurate measurement of any kind of thermal behavior, whether endothermic or exothermic, rapid or slow, provided that the instantaneous heat flow rate into or out of the sample does not exceed the maximum rating of the differential power measuring system. From the data presented by Eckhoff and Bagley, it is clear that this rating was exceeded in every experiment reported; the differential power measuring circuit was saturated for part of the exothermic transition, and therefore the transition energy was not measured accurately. That is, the experiments were carried out in such a way that peaks went off scale on the least sensitive range of the instrument. The purpose of this paper is to describe a procedure for measurements on highly supercooled samples and to report the results obtained with a 20-mg sample of distilled water, exhibiting 18.5 degrees of supercooling. The measurement was made on a Model DSC-2, and the measured exothermic transition energy agreed with the endothermic transition energy, as predicted. The paper will also include a discussion of the thermal equivalent circuit of the DSC sample holder. There are two mechanisms for providing the energy required by a sample material: through the thermal resistance from the sample holder to the surrounding enclosure at ambient temperature, and through a virtual thermal resistance created by the temperature control system. With this virtual resistance very small with respect to the other thermal resistances affecting heat transfer, heat exchange between the sample and the enclosure is made negligible. THERMAL EQUIVALENT CIRCUIT Figure 1shows the equivalent thermal circuit for the sample holder, using the nomenclature of the 1964 paper in which the theory of DSC was first presented (2).In this model, RT is the thermal resistance from the sample holders, at temperatures THSand THR, to the ambient temperature TA; a typical value for RT at ambient temperature is 600 deg s cal-l. Tp and Rp represent the temperature source and the thermal source resistance created by the closed-loop temperature control system. A typical value for Rp is 2 deg s cal-’. Ro is the resistance between the sample and the sample holder; a typical value for Ro is 200 deg s cal-’.
In the absence of thermal behavior in the sample, there is a steady heat flow rate from Tp to TA in each sample holder, through RP and R p This heat flow rate is a function of the total temperature gradient and the value of RT. As shown in the equivalent circuit, this heat flow rate bypasses the sample. The equivalent circuit shows the incremental heat flow rates associated with an assumed sample heat flow rate of W s cal s-l. Linear network theory predicts that this heat flow rate will be divided between Rp and RT in accordance with their resistance ratio. Thus the fraction of W , that flows through Rp, and is recorded by the calorimeter, is given by WM = (1 - p)wS where
Since RP is very much smaller than RT, the calorimetric calibration is essentially invariant with respect to RT; the ordinate calibration of a DSC varies by less than 2% over its operating temperature range, although RT varies by almost an order of magnitude. This is the fundamental difference between true power-compensated DSC and “heat flux” DSC. It should be understood that the measured heat flow rate WM is algebraically added to the existing heat flow rate through Rp. In the case of an exothermic transition, the heat flow rate out of the sample is accommodated by reducing the power supplied to the sample heater by the same amount. The heat flow rate through RT to the enclosure at ambient temperature is essentially unchanged. The thermal equivalent circuit also shows that the measurement of sample energy exchange will be unaffected by variations in Rotthe coupling resistance. It is impossible to avoid small variations in this parameter as different samples are introduced, but these variations influence only the rate of heat flow, not the integrated heat flow rate associated with a transition. In general, it is preferable to minimize Rot for improved temperature resolution and S I N ratio, but it is occasionally necessary to increase Ro, as will be discussed below. Finally, the equivalent circuit shows that phenomena on the reference side of a DSC sample holder are analyzed independently of those on the sample side. In a Model DSC-lB, the thermal coupling resistance between the two sample holders is of the same order of magnitude as RT, and the response is thus dominated by the much smaller resistance Rp; in the Model DSC-2, the sample holders are in separate cavities, and the thermal coupling vanishes. The orthogonality of the two temperature-control and power-measurement systems in DSC was analyzed in a study
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 9, AUGUST 1985
signal and Ro dependence of temperature calibration.
T
EXOTHERMIC TRANSITIONS IN SUPERCOOLED SAMPLES When a supercooled sample starts to freeze, its temperature rises rapidly. For small samples, the temperature rise is proportional to the transition energy and inversely proportional to the total heat capacity of the sample and the sample pan. For a certain sample size, the temperature just reaches the transition temperature TF before the exotherm is complete; for any sample larger than this, the temperature remains at TF until all of the transition energy has been transferred to the sample holder. Under these conditions the temperature differential across Ro is given by
ATo = AT,, TA
Flgure 1. Sample holder equivalent thermal circuit.
+ Tpt
where ATSC is the amount of supercooling and Tp is the program rate. The largest acceptable temperature differential ATo is defined by ATo(max) = RoW,,,
b TA Flgure 2. Eckhoff-Bagiey equivalent thermal circuit.
of exothermic measurements by DSC (3). In the same paper, it was demonstrated that the ordinate calibration is independent of the value of Ro. Eckhoff and Bagley (I) presented two heat-balance equations in their paper, from which a thermal equivalent circuit may be derived; this circuit is shown in Figure 2, using the same notation as that in Figure 1. In this model of the sample holder, the thermal coupling resistance Ro appears in series with RT, the thermal resistance to ambient temperature, and the sample is located at the junction between these two circuit elements. An additional element R3 represents thermal coupling between the two sample holders. Eckhoff and Bagley included R3 for completeness, but recognized in the following discussion that R3 may safely be neglected in the analysis of sample behavior. In this model, the sample heat flow rate must divide between Ro and RT. This means that the ordinate calibration will be temperature dependent; RT is actually smaller than Ro for temperatures above 200 OC, with the result that more than half of the sample heat flow rate would go unrecorded at temperatures higher than this. It will also be apparent that the sample temperature Ts is defined by the ratio between Ro and RT, leading to unacceptable dependence of temperature calibration on these parameters. It should be clear from the earlier discussion that Figure 2 is not a valid thermal equivalent circuit for a DSC sample holder. An instrument designed in this way would exhibit Ro dependence and temperature dependence of the ordinate
where W,, is the rated differential power of the instrument. In a Model DSC-2, the specified differential power rating is 20 mcal s-l, and a typical value for Ro is 200 deg s cal-l, giving a AT0 of 4 degrees. In the case of pure water, which exhibits supercooling of 15 degrees or more, the differential power measuring system will be overloaded for samples weighing more than approximately 0.5 mg. For a Model DSC-lB, capable of measuring 32 mcal s-l, the corresponding sample weight would be approximately 0.8 mg. The recommended solution for this problem is to increase Ro artifically and thus to reduce the heat flow rate into the sample holder for a given temperature gradient. This may be done most conveniently by placing a thin disk of thermal insulating material between the sample pan and the sample holder. As emphasized above, the measurement of transition energies is not affected in any way by this procedure.
CORRECTION FOR SPECIFIC HEAT DISCONTINUITY In an analysis of DSC peak shapes (4),Gray has shown that the transition energy is delivered to the sample in the temperature interval ATF defined by the leading edge of the recorded peak, while the area under the trailing edge represents the heat absorbed by the sample while returning to the program temperature after the transition is complete. The indicated energies in these two sections of the peak are given by A", = A",- (Co iCs)AT, and
A",= (C,+ Cs + AC~)ATF where Co is the thermal capacity of the sample pan, Cs is the thermal capacity of the sample prior to the transition, and ACs is the increase in the sample thermal capacity at the transition temperature. When ACs is very small, the terms proportional to AT, are nearly equal, and the peak area AHM is essentially the same as the transition energy AHF. In the case of water, there is a large increase in the specific heat at the melting point, and a correction should be made to the indicated transition energy; the true transition energy is given by A H F = A H M - ACSATF Thus there will be a negative correction to the endothermic peak area and a positive correction to the exothermic peak area. If there is supercooling behavior, the supercooling in-
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ACs = (0.02009)(0.504) = 0.0101 cal K-’
Temperature (K) T,
270
250
260
RESULTS AND DISCUSSION
,
I
FREEZING
I I
lOmcal sec-1
FUSION
Figure 3 shows the endothermic and exothermic transitions as recorded by the DSC-2. The endothermic transition width was found to be 8.75 K. This temperature interval multiplied by AC, gives a correction of 0.088 cal, which is subtracted from the measured peak area of 1.691 cal to give the heat of fusion AHF = 1.691 - (8.75)(0.0101) = 1.603 cal ~
1
+
/
,- - - - - - - - - - - -
-3-
I
1 .
TF
+
The calculated exothermic transition energy is within 1% of ;\L the endothermic energy, the agreement being well within the
-‘AT,-;
270
The exothermic transition exhibited a peak heat flow rate of 15.6 mcal s-l, and a transition width of 1.5 K, followed by an exponential decay with a width of 2.2 K. The supercooling interval was 18.5 K, and the measured peak area was 1.413 cal. Thus the heat of crystallization is given by AHc = 1.413 (18.5 1.5)(0.0101) = 1.615 cal
290
280
Temperature (K)
Flgure 3. Fusion and freezing of distilled water.
terval AT,, is included in the correction. When the specific heat correction is necessary, the base line under the peak is constructed by extrapolating the pretransition and posttransition base lines and joining them by a perpendicular line through the point of maximum heat flow.
EXPERIMENTAL SECTION The analysis was performed with a Perkin-Elmer Model DSC-2, using a 0.25 in. diameter,0.004 in. mica disk between the sample and the sample holder. The sample was 20.09 mg of distilled water, encapsulated in a gold pan. The temperature program rate was 1.25 K min-’ for both heating and cooling studies. Rowas found to have a value of 1340 K s cal-’, and the measured transition temperatures were corrected in the usual manner for thermal lag. The ordinate calibration of the instrument was established on the basis of the fusion of the water sample, corrected as described above. The heat of fusion and the associated change in specific heat were taken as 79.8 cal g-’ and 0.504 cal g-’ K-’, respectively. For this sample, the heat of fusion and the thermal capacity change are given by A& = (0.02009)(79.8) = 1.603 cal
normal experimental error for measurements of this kind. These results confirm that there is no difference between endothermic and exothermic measurements in power-compensated DSC, even when large temperature gradients are generated, provided that the instrument is operated within its linear range. The DSC-2 was chosen to demonstrate this fact; however, similar results would be obtained with the DSC-1B and other power-compensated DSC instrumentation. In conclusion,it is recommended that a DSC, like any other analytical instrument, should be operated within its specified linear range. Users should recognize that an off-scale deflection of a strip-chart recorder indicates saturation and that area measurements made under these conditions are inaccurate. The solution to overloading caused by supercooling of a sample material is either to reduce the sample size or to introduce enough additional thermal resistance to keep the indicated sample heat flow rate on scale.
LITERATURE CITED (1) (2) (3) (4)
Eckhoff, S. R.; Bagley, E. B. Anal. Cbern. 1984, 56, 2868-2870. O’Neill, M. J. Anal. Chem. 1984, 36, 1238-1245. O’Neill, M. J. Anal. Chem. 1875, 4 7 , 630-637. Gray, A. P. “Analytical Calorimetry”; Porter, R. S., Johnson, J. F., Eds.; Plenum Press: New York, 1968; pp 209-218.
M. J. O’Neill Perkin-Elmer Corporation Norwalk, Connecticut 06859-0093
RECEIVED for review March 6,1985. Accepted April 29,1985.
and
High-Sensitivity Laser Fluorometer Sir: It is well-known that laser-induced fluorescence is a very sensitive technique for ultratrace analysis. Various combinations of laser sources and detection systems have been used in the past to achieve increased sensitivity (1-4). Hirschfeld et al. reported on the detection of one molecule of polyethyleneiminebound to y-globulin, tagged with 80-100 fluorescein isothiocyanate molecules (I). This was accomplished by illuminating the sample a t light intensities high enough to produce photochemical bleaching during the observation period, thus producing a short fluorescence pulse which contained the maximum signal the fluorescent tags could produce. More recently, Dovichi et al. achieved a de0003-2700/85/0357-2007$01.50/0
tection limit of 8.9 X M for Rhodamine 6G (R6G) in the liquid phase using a flow cytometer system with a probe volume of 11pL (2). This is the equivalent of 22000 molecules of R6G flowing through the probe volume during a 1-s integration time. In this work, we describe a relatively simple and highly sensitive laser fluorometric system which has been used to achieve a detection limit of approximately 8000 molecules of R6G. The R6G was adsorbed onto the surface of small (10 wm diameter) silica spheres which were viewed individually with a fluorescence microscope. Advantages of this technique are (i) there is no solvent fluorescence or Raman scatter and 0 1985 American Chemlcal Society
