A N I M P R O V E D M E T H O D OF A N A L Y Z I N G C U R V E S IN DIFFERENTIAL SCANNING CALORIMETRY WILLIAM P. BRENNAN, BERNARD MILLER, AND JOHN C. WHITWELL Textile Research Institute and Department of Chemical Engineering, Princeton University,Princeton, N . J . 08540
A new technique for analyzing data which appears to improve the use of the traces obtained by differential scanning calorimetry is based on a procedure for obtaining the true heat capacity base line for a system undergoing a thermal event, allowing enthalpy change to be isolated from heat capacity contributions. Experimental data on cotton and poly(methy1 methacrylate) illustrate the resulting improvement in determination of enthalpy changes, peak temperatures, and kinetic data.
A D I F F E R E N T I A L scanning calorimeter-e.g.,
the PerkinElmer DSC-is capable of determining inherent thermal properties such as heat capacity and of measuring enthalpic changes for thermal events-i.e., transitions or reactions (Watson et al., 1964). I n a n y application involving an enthalpic event, a quantitative result depends on the determination of the area generated on the recording trace. This area must be determined with reference to a base line, which corresponds to the output of the differential readout when no thermal event is occurring. If this base line is the same before and after the event, it would seem a simple matter to interpolate i t during the event. A problem arises when it becomes apparent that the base line has been appreciably shifted at the end of the event. Many examples of this shift have been observed in our laboratories or have appeared in the literature. One report (Rogers and Morris, 1966) describes the extent of this for a decomposition reaction, and includes the observation that “if the contour of the base line during the reaction cannot be determined . . . it is impossible to integrate the curve with any confidence.” I n most reported work the base line has been approximated either by a straight line drawn between assumed initial and final points of the event or by extending both pre-event and postevent base lines until they connect with a line dropped vertically froni the peak temperature point. These two alternatives are shown in Figure 1. Figure 1, A , implies that the effective heat capacity changes linearly with temperature during the course of the event, while Figure 1, B , implies that it changes abruptly a t the peak temperature from the heat capacity value of the initial sample to that of the final product. Obviously, neither is likely to be true for any actual experiment. A method for the determination of the true base line is here proposed and implications with regard to the measurement of quantities of heat are discussed.
Heat Capacity
The well-documented general method for determining heat capacities with the DSC is reviewed here as a convenient starting point (Brenner and O’Neill, 1965). I n the determination of heat capacity a reference base line is established on the recorder chart, using empty sample pans. This line has three parts: the first under isothermal conditions, the second over the desired programmed temperature range, and the last again isothermal with the system maintained a t the upper temperature limit. This three-part sequence is then repeated with the sample placed in one of the two pans. Characteristic resultant recorder traces are illustrated, superimposed, in Figure 2. The amplitude of the deflection of the sample output from that with empty reference pans, measured in units of dq/dt (enthalpy per unit time), is equal to mC,T. I n subsequent remarks ?n,Cp,l is replaced by tp,,, the total heat capacity for the i t h substance. During isothermal operation the position of the pen on the chart is independent of the heat capacity of the system and is dependent only on the radiation characteristics of that system. Thus, the isothermal base lines a t the beginning and end of the scan are not identical if the radiation characteristics are different a t these two temperatures. Finally the characteristics discussed are under the restriction of negligible conductive effects. The development is also under this restriction; if conductive effects are appreciable, they may be included with minor alterations of the methods described, introducing considerable complexity in the mathematical model.
T
I
A
B 1
tFigure 1. 314
I h E C
Standard base line interpolations
FUNDAMENTALS
Figure 2.
Heat capacity determination using DSC
event-for
example, a t point iM-by f
Figure 3.
Constructions for calculation of true base line
1. Empty sample pans 2. Sample in one pan
3. Recycled scan o f product A. Initiation of event as evidenced b y departure of curve 2 from line A€ (extrapolated curve for reactant i f no thermal event occurred) 8 . Termination of event as evidenced b y intersection o f curve 2 with that o f curve 3 (recycled product)
CD.
Generated trace for thermal behavior of empty pans
AD. Initial approximation to a line representing sensible heat contribution o f reactant during thermal event. At any point along this line f) ep,reaot f, where f equals fracdistance to CD should be .qual to ( 1 tion reocted CB. Initial approximation to a line representing sensible heat contribution of product during thermal event. At any point along this line distance to CD should equal fep.vrod f AB. Initial approximation to a line representing sum o f sensible heat contributions from reactant and product. Any point along this line is equal to sum of distances from line CD to lines AD and CB, obviously a t one specific temperature
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the relationship
area I area 11
= area I
+
[Some investigators recommend accounting for thermal lag between instrument and sample by not dropping a vertical line from point JI, but instead using one that slopes to the left a t an angle determined from a standard run with a pure metal. For reasons similar to those presented in the Appendix, we do not believe this is necessary for work with polymeric materials.] 3. Using these values off, calculate the sensible heat contributions of reactant and product at these points. (ep,prod T a y be determined a t a n y point during the event, while Cp,reaotmust be determined from the extrapolated line A E through the zone of the thermal event.) 4. Add the sensible heat contribution from step 3 to give a revised base line during the course of the event. 5 . Gsing the new event base line, recalculate f's (step 2 ) and sensible heat contributions (step 3 ) . 6. Iterate until the calculated base line does not change. -4single iteration appears to be adequate in any of the experiments described in the discussion of experimental results. An event with no appreciable true base line shift is merely a special case of the general procedure and a straight line may be drawn through the initial and final points with no appreciable error. Resolution of Curves
The true heat of the event is now proportional to the area enclosed between curve A X B of Figure 3 and a newly
1
PROGRAMMED
ISQTHERMAL
I
Base line for Reaction or Transition
One use for the DSC is the determination of the enthalpy change accompanying a thermal event. This operation also precedes any kinetic analysis. I n the recorder trace of such an event the base line must, of necessity, be different before and after the event has taken place, because of change of phase, mass, or composition of the sample. The traces in Figure 1 are typical examples, drawn to illustrate a pronounced base line shift. T o establish a better approximation to the true base line, three runs are required (Figure 3 ) . Each follows the temperature sequence described for the heat capacity determination: an initial isothermal portion to establish a base line before the event, a second portion to scan the temperature range of interest, and, finally, another isotherinal portion to locate the final base line and to facilitate the matching of the three curves. I n Figure 3 the three curves are identified, each containing all three portions superimposed for ease of comparison. The assignment of the end of the event to point B is in disagreement with published views (Perkin-Elmer Thermal Analysis .Vewsletter, 1966). Argument for this choice is presented in the Appendix. A good approximation to the correct line connecting A and B may be established by an iterative procedure:
1. Assume A B is the indicated straight line between initial point A and final point B of the thermal event. 2. Calculate trial values f at various points during the
Figure 4.
Figure 5.
Typical DSC traces
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established curved base line replacing the initial straight dashed line connecting AB. The curvature of the base line indicates that the shape of the original trace, A M B , may well be changed when the new base line is used. If the curvature of the new base line is pronounced, even the peak temperature may be displaced. The resolution of curve AMB, Figure 3, to place i t on a base from which heat capacity effects have been removed, is
illustrated in Figure 4, where curves 1, 2, and 3 represent empty pans, reaction, and products, respectively. As the first step, curves 2 and 3 are redrawn wit'h reference to a straight line representing (dp/dt)l = 0, starting beyond the point where lag of the instrument has been overcome. Distances Xi and Yi of Figure 4, replotted on this new straight , horizontal base, produce the curves of Figure 5 with (dg/dt)z and (dq/dt)s having absolute values, whereas in Figure 4 they were only relative t,o the thermal behavior of the sample pans. After this first step, curve 1 has beconle (dp/dt)l = 0, as planned, curve 3 represents (dg/dt)3 = Cp,prodT. Curve 2 represents, by a he;at balance on the system, = fCp,prodT (1 - f)Cp,reactT X (Aoe-EIRT)vtrLwhere X = unit enthalpy change, A,g-EIRT = k , the rate, m = remaining mass of reactant, and n = t'he apparent order of t.he event. Any other appropriate rate expression can obviously be substituted. To resolve curve 2 further, so that its final form may represent the heat contribution of the event alone, the sensible heat cont,ributions must be removed. As a first step, the true base line for the zone of the thermal event is obtained according to the procedure outlined above. I n Figure 6, curve 2 is repeated with base line corrected t o account for heat capacity effects. If heights XI, Xp, . . . , Xi, from this curve are plotted against the corresponding Ti,the curve of Figure 7 results. The shape of the event curve may thus have been changed. This last curve is free of effects and is a t,rue picture of the thermal event. The area under the curve now shows the heat, contribution of the thermal event only and the equation for the corrected line is dg/dt = X (Aoe-E'RT)mn. IF7he11 reactant is totally dest'royed during the event, curve 3 is identical to curve I . The previous discussion applies, as this t,ype of reaction is only a special case of the general situation. Effect of Scan Rate, h i increase in scan rate will magnify any ba5e line shift because the $ y / d t displacements for any given C, are proportional to T, the programmed temperature scan rate. A change in scan rate produces no problem in heat capacity determinations. When a thermal event is to be studied, however, a straight-
+
dq dt
Figure 6.
Thermal event trace with true base line
+
e,
Figure 7.
Heat contribution of thermal event only
45r
M
Figure 8. 316
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FUNDAMENTALS
Analysis of curve for cotton pyrolysis
line approximation between the initial and final points of the event can, at a n y one scan rate, cause appreciable error. Any variable effect of scan rate can be eliminated by using the true base line procedure described. Examples of Experimental Results
Cotton undergoes a n endothermic pyrolysis when heated in nitrogen, resulting in partial volatilization. Because of this and the formation of solid pyrolysis products, there is a substantial base line shift. The original curves showing the thermal behavior (Figure 8) have been redrawn, with curve 1 corresponding to dq/dt = 0. The difference between areas I and 11, which would be the error resulting from using a straight-line approxiniatioii, is less than 0.5yo of the tot,al peak area. On the other hand, particularly in the initial stages of t8hereaction, the shape of curve A’M’B’ (corrected t o eliminate heat capacity contributions) is noticeably modified, which could have significant consequences in the use of these dat,a in kinetic analysis.
A more extreme case of base line shift due to the total volatilization of a reactant is illustrated in Figure 9, for poly (methyl methacrylate). The curves, plotted as in the previous example, show three effects resulting from base line shift. The difference between areas I and I1 represents a potential error in area measurement of approximately 9%, the shape of the corrected curve A’M’B’ has been altered, and the peak temperature has been shifted. Curve 3 is identical with curve 1 because of total volatilization. Effect of Scan Speed
The effect of scan speed can be demonstrated by comparing the curves of Figure 9 with those of Figure 10. These scans were obtained on identical samples of poly (methyl methacrylate) a t 20’ and 10’ C.per minute, respectively. Base line shift is noticeably less a t the slower scan speed; consequently the effects noted in Figure 9 are diminished. Area measurement error and peak temperature shifts are negligible, but there is still some noticeable influence on the shape of the curve.
Figure 9. Analysis of curve for PMMA decomposition
Figure 10.
Effect of scan speed on PMMA decomposition curve VOL.
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Conclusions
Differential scanning calorimetry may lead to erroneous results if the effect of base line shift is ignored when investigating thermal events. The true base line, representing the sensible heat contributions of the system in transformation, is not either of the straight-line approximations commonly used. The use of either approximation can result in appreciable errors in area measurements, peak temperature, and curve shape. Inconsistencies in data produced a t different scan rates can be explained by the neglect of the consequences of base line shift. Although the analysis and experimental results reported are based on the use of the Perkin-Elmer DSC, a similar treatment should be applicable to any DTA apparatus used for quantitative calorimetry Appendix
The assignment of the end of the event to point B is acknowledged to be in disagreement with a published view that the fusion process is complete when the peak of the curve is reached (Perkin-Elmer Thermal Analysis Newsletter, 1966). This concept originates from a basic analysis of scanning calorimetry (O’Neill, 1964), in which the melting of a pure substance was considered to proceed with a constant interfacial area between liquid and solid. This approach included the prescription that, at conventional scanning rates, the temperature of the liquefied portion of the sample did not rise during fusion. I n a recent extension of this concept (Gray, 1968), the conditions for fusion were assumed specifically as ‘ I (1) the sample temperature is uniform and equal to that of the container, (2) the thermal phenomenon is only temperature and not time dependent, and (3) dT,/dt = 0 during melting.” Gray stated that the area generated after the peak maximum represented the sensible heat contribution required by the molten sample as it caught up with the programmed temperature. Based on visual observations and analysis of experimental curves for pure metals we believe the above view is oversimplified. Contrary to the picture presented by O’h’eill, melting of ordinary solid samples does not proceed with constant area of contact between liquid and solid phases. A previously melted sample will be in the rough shape of a half sphere, resting on its major circular cross section. As melting progresses, the cross section of the phase boundary becomes smaller, a t first very slowly but, near the end of the process, very rapidly. Consequently, the last portion of the fusion will occur a t a very slow rate, dp/dt will be reduced, and that part of the process will correspond to the downward portion of the curve. Only a small sensible heat will be required at the end of the process, since there will have been a temperature drop through the liquid at all times to provide the driving force for fusion. I n addition, the maximum enthalpic change required to bring the liquid to the programmed temperature will be negligible over the small temperature range required for the process in comparison to the
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FUNDAMENTALS
enthalpy change for fusion. The same argument holds for a granular solid. As fusion progresses, the individual particles decrease in size and surface area, so that the rate of heat transfer at the end of the process is low and the instrument trace is falling during the last portion of the fusion. Beyond the question of the correct picture of temperatureprogrammed fusion of a pure substance there is a n additional argument that, for impure substances or materials with inherent heterogeneity-Le. , polymers-fusion or any other thermal event requiring heat does not occur a t a single temperature. There is appreciable time for the earliest transformed portion to increase in temperature before the transition is completed, and the transition rate again will be reduced by the diminishing surface area of the solid. As in fusion of pure substances, catch-up sensible heats will be small compared to enthalpies of fusion, reaction, etc. These arguments strongly imply that the end of a n endothermic event is not a t the peak temperature but, in terms of the DSC output, very close to the point where the trace is coincident,al with the heat capacity contribution of the product. Nomenclature (in Consistent Units)
A0
Cp
= pre-exponential factor = heat capacity per unit mass
C, = total heat capacity dq/dt = enthalpy per unit time E = activation energy f = fraction reacted k = rate constant m = mass of sample n = apparent order of event R = gas constant t = time T = temperature T = programmed temperature rate X , Y = displacements SUBSCRIPTS prod = product react = reactant 1, 2, 3 = curve designations GREEKLETTER X
= unit enthalpy change
literature Cited
Brenner, N., O’Neill, 11.1. J., Instrument News (Perkin-Elmer Corp.), 16, No. 2 (1965). Gray, A. P., in “Analytical Calorimetry,”p. 209, R. S. Porter and J. F. Johnson. eds.. Plenum Press. New York.’ 1968. O’Neill, M. J., AnaL‘Chem. 36, 1235 (1964). Perkin-Elmer Thermal Analysis Newsletter, No. 6 (1966). Rogers, R. N., Morris, E. D., Jr., Anal. Chem. 38, 410 (1966). Watson, E. S., O’Neill, M. J., Justin, J., Brenner, N., Anal. Chem. 36, 1233 (1964). RECEIVEDfor review November 18, 1968 ACCEPTED February 26, 1969
