416
Anal. Chem. 1986, 58,416-421
Equivalent-Circuit Modeling of a Heat-Flux Differential Scanning Calorimetry Cell. Analysis of Thermal Resistance Factors and Comparison with Experimental Data Guang-Way Jang and Krishnan Rajeshwar* Department of Chemistry, The University of Texas a t Arlington, Arlington, Texas 76019
An equivalent-clrcult model Is presented for a commerclal heat-flux dlfferentlal scanning calorlmetry (DSC) cell. Thls model Is developed In a form that permlts dlrect comparlson wlth experlmental data. The varlous thermal resistance factors In the cell are computed vla this model. Comparlsons of model predlctlons wlth experlmental data are presented for the melting endotherms of In, Sn, and Zn. I n partlcular, the Influence of varlables, such as the heatlng rate and the thermal characterlstlcs of sample and the purge gas atmosphere, Is crltlcally examlned. Thls study demonstrates for the first time, both vla model as well as by experiments, the Importance of heat exchange between the sample and reference channels and heat leakage through the purge gas.
Differential scanning calorimetry (DSC) is rapidly becoming a powerful tool in the arsenal of techniques that is currently available to the analytical chemist ( I ) . However, theoretical developments have lagged somewhat behind the many innovations that have characterized the practical application of this technique. For example, equivalent-circuit models have been presented by many authors for the description of DSC systems, both of the heat-flux type (2-6) as well as of the power-compensation design (7,8). However, in the majority of cases, the model implications were developed within a framework that rendered direct testing via experimentation a difficult task. The main contribution of this paper, thenefore, is the development and experimental testing of an equivalent-circuit model for a commercial DSC cell of the heat-flux type. We focus in particular on the various resistance factors in this cell and their sensitivity to variables such as heating rate, sample characteristics, and purge gas atmosphere.
EXPERIMENTAL SECTION A Du Pont 1090Thermal Analysis System fitted with the Model 910 DSC accessory module, was used in this study. The software supplied by the manufacturer was used for the most part for the analyses of DSC thermograms. All fusion endotherms were recorded only after one or two initial "conditioning" heat-cool cycles through the transition. mmmercial samples of In, Sn, and Zn (99.999% purity or better) were used as received. The purge gas was flushed through the DSC cell at the rate of ca. 80 mL/min. Sealed sample pans were used in all the cases. The sample sensor thermal resistance was varied by inserting an A1 disk in the sample and reference pan bottoms. These disks were fashioned out of the Al pans supplied by the manufacturer. Programmed heating rates were nominally in the range from 1 'C/min to 40 'C/min, for the results presented herein. In selected instances, heating rates down to 0.2 "C/min were utilized to check thermal lag effects. Nominal sample mass was ca. 10 mg.
THEORY The starting point for our model development utilizes, as in previous studies (2,9,10) two basic relationships, namely one describing the conservation of energy and the other arising from the linear dependence of heat-flow on temperature 0003-2700/86/0358-0416$01.50/0
differential (Newton's law). Figure 1 presents this simple equivalent-circuit model. The key elements are the sample at temperature Ts, the sample holder at temperature TsH,and the heater (source at temperature Tp). The heat-flow paths have thermal resistance RD, which is the thermal resistance between heater and sample-pan holder, and Rs, which is the thermal resistance between sensor and sample (note that this is a composite resistance). The sample and its container have a mass heat capacity, Cs. For the reference channel of the DSC cell, an analogous situation exists, as illustrated in Figure 1. The heat-flow rates (thermal currents) through the two channels are, respectively, dqsldt and dqRldt. The assumptions underlying the present model are the following: (a) The various resistances and capacitances are temperature independent. (b) The cell is completely symmetric; for example, resistances RD on both sides (cf. Figure 1) exactly match each other. (c) The rate of utilization of thermal energy, dhldt, in an endothermic thermal event is positive. Similarly, the heat flow from the source to sample is taken as positive. Conservation of energy in the DSC cell (2,9, IO) requires that the net heat flow to the sample is partitioned between that required to raise the sample temperature and the energy needed for the thermal event (e.g., melting)
In terms of Newton's law, the heat-flow rate is governed by the temperature differential and the particular resistance element (i.e., the thermal analogue of Ohm's law)
Also
(3)
+
In eq 2, RT = RD Rs (cf. Figure 1). For the reference channel, expressions similar to eq 1-3 apply except that now dhldt = 0. It is then easily shown that the heat flow rate inthe sample channel and the DSC ordinate signal, which is the differential heat flow, are respectively given by
-dqs - - = dqR
dt
dt
dTp d h (cs-cR)+-R dt dt
d2ss dt2
C - (5)
These master equations have been derived previously by Baxter (2) and Gray (9). Equation 5 shows that the recorded output, dqs/dt - dqR/dt,is the sum of three terms (9,lO): (a) a base line displacement that is dependent on the heat ca0 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986 417
Flgure 3. Heat leakage and exchange paths in a DSC cell involving the purge gas and the sample and reference channels, respectively. Refer to the text for definition of symbols.
Figure 1. Simple equivalent circuit for the sample and reference
channel of a heat-flux DSC cell. Refer to the text for definition of symbols.
(c) After Fusion. At the instant when the sample is completely molten, dhldt = 0. The exponential decay model has been used (9, 10, 13, 14) to describe the thermogram after fusion, i.e.
0
From eq 10
MODEL DEVELOPMENT
Figure 2. Analysis parameters and components of a hypothetical DSC thermogram.
pacity difference between sample and reference and the heating rate, (b) a term that is dependent on the rate of enthalpy change in the sample, and (c) the thermal lag in the sample channel. This lag is given by the product of the time constant, RTCS (11)and the slope of the thermogram d2qs/dt2. Consider a melting process. The thermogram corresponding to this thermal event may be decomposed into three successive stages: (a) Before Fusion. dh/dt = 0. Since Cs, dTp/dt, and RT are constant, d2qs/dt2is also zero (cf. eq 4). Thus from eq 5
Note that this expression forms the basis for specific-heat determinations via DSC (12). ( b ) During Fusion. Utilizing the assumption that fusion is truly an isothermal process, the source temperature Tp may be represented by
(7) In eq 7, T, is the melting point of the sample, and the time t is measured from the onset of fusion. The following general expressions thus may be derived for the ordinate output (eq B), the time to peak maximum from the onset of melting, T,, (eq 91, and the peak amplitude, (dqs/dt)m, (eq 10):
We shall utilize the preceding background material as the springboard for our model development. Equation 6 describes the steady-state condition in the DSC cell in the absence of a thermal transition (cf. Figure 2). Again, considering the specific example of fusion, the additional heat absorption in the sample channel perturbs this steady-state; the net heat flow in this case being driven by the temperature differential, TSH - T e When the thermal lag is small (i.e., for low heating rates and thermally conductive samples), the approximation, = DSC peak temperature, cf. Figure 2), will TsH T, (T,, be good (see below). Additionally, since Ts = TIM, Newton's law yields the following expression: Tmax -
A plot of T,, vs. (dq/dt),, (generated for example from a family of DSC thermograms at varying heating rate), should yield a straight line whose slope and intercept should permit computation of Rs and T,, respectively. Equation 9 for the usual case when the product RTCs is small reduces to eq 14
Substitution of eq 14 in eq 12 yields eq 15
(2)max (m)gdt 2AHRT
=
T,
These and other relevant parameters in a DSC thermogram are illustrated in Figure 2.
TM
Rs
'I2 1 dTp
(
I
=
2 AH(d Tp /dt) RT
) 2
(15)
should be also proportional to (dTp/dt)1/2via eq 13. In all of the above, the influence of heat exchange between the sample and reference channels and of the thermal characteristics of the ambient gas was not explicitly considered. Figure 3 shows a schematic illustrating the inclusion of these resistive elements, and Figure 4 is the corresponding (more detailed) version of the equivalent circuit (cf. Figure 1). As Figure 3 illustrates, the additional resistances are RG, the gas thermal resistance between the heating block to sample or reference, RG', the gas thermal resistance between sample and reference, and RD', the disk thermal resistance between sample and reference pan holders. The analogues of eq 13 for this
418
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986
.-(
1 - 1-K
1 +
&)
For the limiting case when there is negligible heat exchange between sample and reference (i.e., RD’, RG, and RG’ are very high), we have dTR/dTsH = 1/K, and the following simplified counterparts of eq 20-22:
and d2q = 1- K dTp -
+-
+-
dt2
Figure 4. Equivalent circuit of a heat-flux DSC system including the heat leakage and exchange paths illustrated In Figure 3.
case can be shown to be eq 16 and 17 using simple equivalent circuit analyses (cf. Figure 4)
-dqS - - = dqR dt dt
TSH-
RS
TS
+ TRH-
TSH - T P -
TRH -
RD‘
RD dt
Also, eq 23 in this limiting case reduces to eq 23a
1 =1 -K1 Rs K RD As we shall attempt to illustrate in the next section, eq 13, 15, 20-22, and 20a-22a (and the implications thereof) are amenable to direct verification via experiment. Additionally, comparisons of model predictions with experimental data can be made to permit an assessment of the importance of RG, RG’,and RD‘.
RESULTS AND DISCUSSION
K will in general range from 0 to 1. Note that the factor K comprises contributions from the sample characteristics (e.g., thermal conductivity) as well as instrument operational parameters such as the heating rate. For the reference channel, we make the following reasonable assumption:
We can now write down the following expressions for the temperature derivative (eq 20) and time derivatives (eq 21 and 22) of the ordinate DSC signal, dq/dt = dqs/dt - dqR/dt
d (dq / dt)
-
dTSH
By equating the slopes in eq 21 and 22, we obtained an expreeaion for Rs in terms of K and the disk resistance factors, i.e.
The experimental variables that were utilized to test the model predictions were the following: the sample itself, the purge gas, and the sensor/sample resistance. The latter was varied by inserting AI disks in the sample and reference pans (see Experimental Section). The melting endotherms for selected metals (In, Sn, and Zn) were chosen as the test reactions for analyses. Figure 5a illustrates a typical experimental test of the (dTp/dt)1/2dependence of (dqsldt),,, as predicted by eq 15. The data are shown for In with the purge gas and sensor/sample resistance as parameters. In all the cases, the correlation was good ( r 2 0.990). In eq 15, since Rs and TMwill be constant for a given system, T,, should also scale with (dTp/dt)1/2(see above). Figure 5b bears out this expectation, again for experimental data analogous to those in Figure 5a. The “limiting” ordinate intercept of T,,, vs. (dTp/dt)1/2 plots (i.e., for the case, dTp/dt = 0), should correspond to the sample melting point, TM: Figure 5b also illustrates that the requirement that a value for TMthus obtained be insensitive to experimental variables such as thermal resistance factors, is more or less met. The one exception is the He case, where facile heat leakage through RG and RG’ (cf. Figure 4) can be shown as the culprit (see below). As in the case of the (dq/dt),,, parameter above (cf. Figure 5a), the linearity of the T, vs. (dTp/dt)1/2response also deteriorates at heating rates >-15 OC/min. Figure 6 contains an experimental test of eq 13, again for In. The linear response predicted by this equation is obviously met in this case except at heating rates greater than ca. 15 “C/min (not shown) where the thermal lag starts to exert on influence. Correlation coefficients 20.990 were also observed for all other systems examined in this study. The intercept again yields a value for T M in each case (cf. eq 13). Table I presents values of TM for In obtained from plots such as those shown in Figures 5b and 6. The upper limit for the heating rate, which was defined previously as ca. 15 OC/min, would obviously depend on the magnitude of RT (Le., a higher RT would translate to a lower
ANALYTICAL CHEMISTRY, VOL. 48,NO. 2, FEBRUARY 1986
25
419
In
162 3
dq
20
161
15
160
'iE')rrax imwi
Tmax I"C1
159
10
158
5
In
157
0
0
1
2
3
4
5
155 5
0
/3
162
J/
10
15
20
25
/2
161
Flgure 8. Plot of T,, vs. (dqldt),, (cf. eq 13) for In. Notation for data is the same as in Figure 5. The lines are the least-squares fit to the data points.
Table I. Comparison of Melting-Point (T,) Determinations for In Using Three Different Analysis Strategies of a DSC Thermogram
160
*ma* I"l 159
purge gas
no. of data points"
Ar
23
NZ
10
air
6
He
9
158
157
156
155
method lb
TM,"C method 2c
method 3d
156.70 (0.26)e 156.60 (0.28) 156.68 (0.16) 155.83 (0.22)
156.00 (0.29) 156.29 (0.29) 156.35 (0.14) 155.70 (0.25)
156.40 (0.19) 156.47 (0.30) 156.51 (0.14) 155.78 (0.25)
"The data are pooled from those obtained under conditions of varying sample/sensor resistance and sample size. *The values are the onset temperature, To (cf. Figure 2), obtained from DSC runs at 1 OC/min. cValues obtained from plots of T, vs. (dqldt),, (dq/dt),, (cf. eq 13). dValues obtained from plots of T,,, vs. (dTp/dt)'/2. e Standard deviation. Table 11. Thermal Resistance Factors for Test Samples in Air
Flgure 5. Plots (dqldt),,, vs. (dTpldt)"2 (a) and T,, vs. (dTpldt)"2 (b) for In. The numbers, 0-3 refer to the number of Ai disks inserted between pan and sample. The lines are the least-squares fit to the data points.
cutoff value for the heating rate beyond which nonlinearity would set in). The pitfalls associated with the identification of TMwith the peak onset temperature To (cf. Figure 2) are clear from an examination of Figure 7. The linear dependence of Toon dTp/dt confirms the finding of a previous author (15)for the same system (In). Only in the regime of an infinitesmdy small heating rate, do the two parameters To and TMmerge with each other. Values of Toobtained at dTp/dt = 1 OC/rnin are also presented in Table I for comparison. For eq 13 and 15, a higher slope is predicted with higher Rs and RT, respectively. This expectation is fulfiied in Figures
sample Zn In Sn
thermal resistance factor, lo2 "C/mW method 1" method 2b method 3c
lit. value: cm "C/W
10.03 27.12 46.49
1.007 1.372 1.681
7.87 11.07 17.73
4.92 5.76 11.63
"Computed from slopes of T,, vs. (dq/dt),,, plots (cf. eq 13). bComputed from slopes of (dqldt),, vs. (dTp/dt)1/2plots (cf. eq 15). "Computed from slopes of d2q/dt2vs. dTp/dt after dividing through by K (cf. eq 21a). dThese values are the reciprocals of the thermal conductivities as reported in Ho, C. Y.;Powell, R. W.; and Lilev. P. E. J. Phvs. Chem. Ref. Data 1972.1. 351. 406. 416. 5a and 6 (recall that the thermal conductivity of the various purge gases are ordered thus: He >> N2, Air > Ar). Furthermore, a value for RT can be computed from the slopes of
420
ANALYTICAL CHEMISTRY, VOL. 58, NO. 2, FEBRUARY 1986 159
I
t
158
400 300
*
d2q
h
0
e
I
157
200
(E)
I-O
100 156
0 0
155 0
5
10
30
40
dTp/dt I"C/rnin]
20
15
20
10
Figure 9. Plots of d291dt2vs. dT,ldtfor Zn, In, and Sn in air. Solid lines are the least-squares fit to the data points.
UP, (OC/rnin) dt
Figure 7. Variation of the endotherm onset temperature. T,, with df,,ldt for In in various purge gas atmospheres. The lines are simply drawn through the data points.
600 -
/
d20
Time, (min)
Figure 10. Plots of T,, vs. time and d291dt2vs. time for In in air. Refer to the text for definition of symbols. Table 111. Demonstration of the Importance of the Heat Leakage Effect through Purge Gas 0
10
20
30
40
dTp/dt I"C/rninl Flgure 8. Plots of d291dt2 vs. dT,ldt (cf. eq 21 and 22) for In. Notation is the same as in Figure 5. The lines are the least-squares fit to the data points.
plots of (dqs/dt),, vs. (dTp/dt)1/2if the LWS are known (cf. eq 15). Assembled in Table I1 are values of RT thus obtained. Also presented in this compilation are values of Rs obtained from the slopes in Figure 5a (cf. eq 13) and literature values for the sample resistivity. Important here is the relative trend in Rs and RT for the various samples rather than their absolute magnitudes. (Comparison of absolute magnitudes between Rs, RT, and literature data is precluded at the present time by the lack of knowledge on contact area and RD.) However, it is satisfactory to note that Rs and RT scale accurately with the known sample characteristics. Figure 8 illustrates typical plots of d2q/dt2vs. dTp/dt (cf. eq 21 and 22) for In, with purge gas and sample/sensor thermal resistance again as parametric variables. Figure 9 presents similar plots for the three metal speciments in air. The trends in Figures 8 and 9 generally mimic those discussed in the preceding paragraphs, namely that the slopes increase with increasing (a) thermal conductivity of the purge gas, (b) sample/sensor thermal conductivity, and (c) thermal conductivity of the sample itself. Data pertinent to point (c) above are also included in Table 11. Furthermore, the rate of change of slope becomes smaller with increasing sample/sensor thermal resistance (cf. Figure 8). To examine why this happens, consider eqn 21. The slopes of d2q/dt2 vs.
atmosphere (purge gas pair)
d2q/dt2ratioa (1- K ) ration % differenceb
Ar, N2 Ar, air air, Nz He, Nz He, Ar He, air
0.858 0.850 1.009 1.253 1.461 1.242
0.837 0.844 0.992 2.074 2.477 2.089
2.4 0.8 1.6 65.5 69.5 68.0
OHeating rate: 10 OC/min. * % difference = [[(l- K) ratio dzq/dt2ratio]/(l - K ) ratio) X 100. dTp/dt plot are equal to [KIRs + (1 - K)/RD' + 1/RG + ~ / R G 'according ] to eq 21. The limiting values of the slopes (Le., at high sample/sensor resistance levels) would then tend toward the value (1 - K)/RD' 1/RG + 2/RG',which will be constant for a given purge gas and sample holder system. We shall now take up the question of heat leakage through the purge gas in greater detail. Consider eq 22. If we assume negligible heat leakage through the purge gas (i.e., high RG and RG'),the last two terms of the slope [ ( l- K ) / R D+ 2(1 - K)/RD' 1/RG 2/RG')drop out yielding a modified slope of (1 - K ) ( l / R D+ l/RD'). Now, ratios of d2q/dt2values can be generated for pairs of purge gases (N2,air, etc.) and a given heating rate, wherein the factors RD and RD' remain sensibly constant. Thus, the magnitude of these ratios should equal the term (1 - K,)/(l- Kb), which can also be measured independently from DSC data such as those depicted in Figure 10. (Here K, and Kb are the thermal lag factors corresponding to the two purge gases that are chosen.) Such a comparison
+
+
+
Anal. Chem. 1906, 58,421-427
Table IV. Heat Exchange Effect between Sample and Reference Channels in DSC Data for In in A r Atmosphere thermal resistance factor, 10' OC/mW from eq 21a % difference"
no. of A1 disks
from eq 20a
0 1 2
8.38
7.42
11.05
12.72 20.70
18.1
3
26.18
10.83 16.95 19.08
column in Table IV shows, the degree of heat exchange is dependent on the magnitude of the sample/sensor thermal resistance.
ACKNOWLEDGMENT The authors thank the Du Pont Co. for instrumental support.
14.9
17.1
% difference = [(value in column 2 - value in column 3)/value in column 21 X 100. (I
is shown in Table 111. It is seen that the percent difference between the two sets of values (which is a measure of the importance of the purge gas heat leakage contribution) is small in all the cases except when He is involved as the purge gas. This is perhaps not too surprising in view of the facile thermal conduction path that is available through this gas. Finally, the data in Figure 8 permit an assessment of the degree of heat exchange between the sample and reference channels (i.e., through RD', cf. Figure 4). Consider eq 20a and 21a for this purpose. If the influence of R D ' was truly negligible, the slope of d2q/dt2vs. dTp/dt plots divided by K (cf. eq 21a) should equal the right-hand side of eq 20a. Table IV presents a set of data for In in Ar atmosphere. The disparate nature of the two sets of thermal resistance factors clearly points to the importance of RD'. Furthermore as the last
42 1
LITERATURE CITED Wendlandt, W. W. Anal. Chem. 1984, 56, 250R-261R. Baxter, R. A. I n "Thermal Analysis"; Schwenker, R. F., Jr., Garn, P. D., Eds.; Academic Press: New York, 1969; Vol. 1, p 65. Claudy, P.; Commercion, J. C.; Letoffe, J. M. Thermochlm. Acta 1983, 68, 305-316. Claudy, P.; Commercion, J. C.; Letoffe, J. M. Thermochim. Acta 1983, 68, 317-327. Schonborn, K. H. Thermochlm. Acta 1983, 69, 103-114. Van der Plaats, G. Thermochim, Acta 1984, 72, 77-82. O'Nelll, M. J. Anal. Chem. 1964, 3 6 , 1238-1245. Eckhoff, S. R.; Bagley, E. R. Anal. Chem. 1984, 56, 2868-2870. Gray, A. P. I n "Analytical Calorimetry"; Porter, R. S., Johnson, J. F., Eds.; Plenum Press: New York, 1968; Vol. 1, p 209. Brennan, W. P. Ph.D. Thesis, Princeton University, 1971. Flynn, J. H. I n "Analytical Calorlmetry": Porter, R. S., Johnson, J. F., Eds.; Plenum Press: New York, 1972; Vol. 3, p 17. O'Neill, M. J. Anal. Chem. 1966, 38, 1331-1336. Brennan, W. P.; Mlller, B.; Whltwell, J. C. I n "Analytical Calorimetry"; Porter, R. S., Ed.; Plenum Press: New York, 1970; Vol. 2, p 441. Flynn, J. H. NBS Spec. Publ. (US.)1970, no. 338, 119. Flynn, J. H. I n "Thermal Analysis: Proceedings of the 3rd Internatlonal Conference on Thermal Analysis"; Weidemann, H. G., Ed.; Birkhauser Verlag: Basel, 1972; Vol. 1, p 127.
RECEIVED for review June 27,1985. Accepted September 12, 1985.
Digital Filters for Noise Reduction in Optical Kinetic Experiments Seshadri Jagannathan* and Ramesh C. Pate1
Chemistry Department, Clarkson University, Potsdam, New York 13676
Nolse is a slgnlflcant problem In hlgh-sensltlvlty techniques such as chemlcal relaxation kinetlcs. I t Is important to separate the signlflcant data-contalnlng slgnals from the nolse. The use of high-speed A/D converters makes It Convenient to acquire and store signals in dlgltlzed form. Hence, software implementatlon of slgnal enhancement techniques Is convenient and versatile. The theory, design, and use of recursive digital filters for recovering kinetic signals from experimental relaxation kinetic data are presented. The frequency approach is used to develop the theory and design of these fllters. I t Is also demonstrated that these filters can be used effectlvely on data wlth slgnal-to-nolse ratios of unlty. Flnaliy, the filters are used on data from temperature-jump and electric fleld light scatterlng experiments (both of which require high sensltlvlty) to demonstrate their effectlveness.
Relaxation methods provide some of the most versatile techniques for studying chemical reactions ( I ) . Since the concentration changes involved in these reactions are relatively small, highly sensitive methods me required for detection. Due to this stringent requirement, reliable techniques are required for separating the inherent signal corresponding to the chemical reaction from the underlying noise. Since computer-controlled data acquisition systems are convenient and
popular, digital filters are appropriate for this purpose (2-5). These filters can be part of the hardware of the data acquisition unit or implemented as a computer program (6). Software implementation of digital filters offers greater diversity and flexibility over the hardware counterpart, and hence, only the former will be dealt with in this paper. The theory and design of digital filters using the frequency approach (6-8) (as contrasted with the classical polynomial approach (9-14)) will be discussed, along with their application to relaxation kinetic data. The general equation for the relaxation curve obtained from a kinetic experiment is given by (15)
X, =
i=O
Aie-klt
(1)
where n is frequently 12. For illustrative purposes, the more commonly encountered form of the equation ( n = 1) will be used.
X , = A.
+ Ale-klt
(2) The parameters of interest to the kineticist, Al and k l , are usually obtained from a linear least-squaresfit of the logarithm of the first derivative of the data. Numerical derivatives tend to amplify small errors in the data (16,17). In order to obtain accurate estimates of the parameters, the data must be sampled correctly (to prevent aliasing) and should be free from noise. Figure 1 is a numerically simulated relaxation curve
0003-2700/86/0358-0421$01.50/00 1986 American Chemical Society
