Designation: E2041 − 13´1Standard Test Method forEstimating Kinetic Parameters by Differential ScanningCalorimeter Using the Borchardt and Daniels Method1This standard is issued under the fixed designation E2041; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.ε1NOTE—Warning statements were editorially corrected throughout in September 2013.1. Scope1.1 This test method describes the determination of thekinetic parameters of activation energy, Arrhenius pre-exponential factor, and reaction order using the Borchardt andDaniels2treatment of data obtained by differential scanningcalorimetry. This test method is applicable to the temperaturerange from 170 to 870 K (−100 to 600°C).1.2 This treatment is applicable only to smooth exothermicreactions with no shoulders, discontinuous changes, or shifts inbaseline. It is applicable only to reactions with reaction ordern ≤ 2. It is not applicable to acceleratory reactions and,therefore, is not applicable to the determination of kineticparameters for most thermoset curing reactions or to crystalli-zation reactions.1.3 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.4 This test method is similar, but not equivalent to,ISO 11357, Part 5, that contains provisions for additionalinformation not supplied by this test method.1.5 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:3E473 Terminology Relating to Thermal Analysis and Rhe-ologyE537 Test Method for The Thermal Stability of Chemicalsby Differential Scanning CalorimetryE698 Test Method for Arrhenius Kinetic Constants forThermally Unstable Materials Using Differential Scan-ning Calorimetry and the Flynn/Wall/Ozawa MethodE967 Test Method for Temperature Calibration of Differen-tial Scanning Calorimeters and Differential Thermal Ana-lyzersE968 Practice for Heat Flow Calibration of DifferentialScanning CalorimetersE1142 Terminology Relating to Thermophysical PropertiesE1445 Terminology Relating to Hazard Potential of Chemi-calsE1641 Test Method for Decomposition Kinetics by Thermo-gravimetry Using the Ozawa/Flynn/Wall MethodE1970 Practice for Statistical Treatment of ThermoanalyticalData2.2 ISO Standards:4ISO 11357 Part 5: Determination of Temperature and/orTime of Reaction and Reaction Kinetics3. Terminology3.1 Definitions—Specific technical terms used in this testmethod are defined in Terminologies E473, E1142, and E1445,including calibration, calorimeter, differential scanningcalorimetry, enthalpy, peak, reaction, repeatability,reproducibility, and slope.4. Summary of Test Method4.1 Atest specimen is heated at a linear rate in a differentialscanning calorimeter or other suitable calorimeter through aregion of exothermic reaction behavior. The rate of heatevolution, developed by a chemical reaction, is proportional tothe rate of reaction. Integration of the heat flow as a function oftime yields the total heat of a reaction.1This test method is under the jurisdiction ofASTM Committee E37 on ThermalMeasurements and the direct responsibility of Subcommittee E37.01 on Calorimetryand Mass Loss.Current edition approved Sept. 15, 2013. Published September 2013. Originallyapproved in 1999. Last previous edition approved in 2008 as E2041 – 08ε1. DOI:10.1520/E2041-13E01.2Borchardt, H.J., Daniels, F., Journal of the American Chemical Society, Vol 79,1957, pp. 41–46.3For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.4Available from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor, New York, NY 10036, http://www.ansi.org.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States14.2 The Borchardt and Daniels2data treatment is used toderive the kinetic parameters of activation energy, Arrheniuspre-exponential factor, and reaction order from the heat flowand total heat of reaction information obtained in 4.1 (seeSection 5).5. Basis of Methodology5.1 Kinetic reactions may be modeled with a number ofsuitable equations. The Borchardt and Daniels2method makesuse of the rate equation to describe the dependence of the rateof reaction on the amount of material present.dα/dt 5 k~T!~1 2 α!n(1)where:dα/dt = reaction rate (min−1)α = fraction reacted (dimensionless),k(T) = rate constant at temperature T (min−1), andn = reaction order (dimensionless).5.2 For a reaction conducted at temperature (T), the rateequation of Eq 1, may be cast in its logarithmic form:

[email protected]α/dt# 5

[email protected]~T!#

[email protected] 2 α# (2)This equation has the form of a straight line, y = mx + b,where a plot of the logarithm of the reaction rate (ln[dα/dt])versus the logarithm of the fraction remaining ln[1−α] yieldsa straight line, the slope of which is equal to n and the interceptis equal to ln[k(T)].5.3 The Borchardt and Daniels2model also makes use of theArrhenius equation to describe how the reaction rate changesas a function of temperature:k~T! 5 Ze·E/RT(3)where:Z = Arrhenius pre-exponential factor (min−1),E = Activation energy (J mol−1),T = Absolute temperature (K), andR = Gas constant (= 8.314 J mol−1K−1).5.4 The Arrhenius equation Eq 3 also may be cast in itslogarithmic form:

[email protected]~T!# 5

[email protected]# 2 E/RT (4)The equation has the form of a straight line, y = mx + b,(where y ≡ ln[k(T)], m ≡ E/R, x ≡1/T and b ≡ ln[Z]) where a plotof the logarithm of the reaction rate constant (ln[k(T)]) versusthe reciprocal of absolute temperature (l/T) produces a straightline, the slope of which is equal to −E/R and the intercept ofwhich is ln[Z].5.5 As an alternate to Eq 2 and 4, the rate and Arrheniusequations may be combined and cast in its logarithmic form:

[email protected]α/dt# 5

[email protected]#

[email protected] 2 α# 2 E/RT (5)The resultant equation has the form z = a + bx + cy (wherez ≡ ln[dα/dt], ln[Z] ≡ a, b ≡ n, x ≡ ln[1−α], c ≡ E/R, and y ≡l/T) and may be solved using multiple linear regression datatreatment.5.6 The values for dα/dt,(1−α) and T needed to solve Eq2, Eq 4 and Eq 5, are experimental parameters obtained froma single linear heating rate DSC experiment scanning throughthe temperature region of the reaction exotherm as shown inFig. 1.5.7 Kinetic results obtained by this test method may becompared with those obtained by Test Method E698.6. Significance and Use6.1 This test method is useful in research and development.6.2 The determination of the appropriate model for a chemi-cal reaction or transformation and the values associated with itskinetic parameters may be used in the estimation of reactionperformance at temperatures or time conditions not easilytested. This use, however, is not described in this test method.7. Interferences7.1 Because of its simplicity and ease of use, the Borchardtand Daniels2method is often the method of choice forcharacterization of the kinetic parameters of a reaction system.The Borchardt and Daniels method, like all tools used toevaluate kinetic parameters, is not applicable to all cases. Theuser of this test method is expressly advised to use this testmethod and its results with caution.7.2 Tabulated below are some guidelines for the use of theBorchardt and Daniels2method.7.2.1 The approach is applicable only to exothermic reac-tions.NOTE 1—Endothermic reactions are controlled by the kinetics of theheat transfer of the apparatus and not by the kinetics of the reaction.7.2.2 The reaction under investigation must have a constantmechanism throughout the whole reaction process. In practice,this means that the reaction exotherm upon heating must besmooth, well shaped (as in Fig. 1) with no shoulders, multiplepeaks or discontinuous steps.7.2.3 The reaction must be nth order. Confirmation of an nthorder reaction may be made by an isothermal experiment suchas that described in Appendix X1.7.2.4 Typical reactions which are not nth order and to whichBorchardt and Daniels2kinetic may not be applied for predic-tive purposes include many thermoset curing reactions andcrystallization transformations.7.2.5 The nth order kinetic reactions anticipate that thevalue of n will be small, non-zero integers, such as 1 or 2.Values of n greater than 2 or that are not simple fractions, suchas1⁄2 = 0.5, are highly unlikely and shall be viewed withcaution.7.2.6 The Borchardt and Daniels2method assumes tempera-ture equilibrium throughout the whole test specimen. Thismeans that low heating rates, (that is, 10 K/min), smallspecimen sizes (5 mg) and highly conductive sealed specimencontainers, for example, aluminum, gold, platinum, etc., shouldbe used.7.3 Since milligram quantities of specimen are used, it isessential that the specimen be homogeneous and representativeof the test sample from which they are taken.7.4 Toxic or corrosive effluents, or both, may be releasedwhen heating the test specimen and may be harmful toE2041 − 13´12personnel or to the apparatus. Operating with a venting orexhaust system is recommended.8. Apparatus8.1 Differential Scanning Calorimeter (DSC)—The instru-mentation required to provide the minimum differential scan-ning calorimetric capability for this method includes thefollowing:8.1.1 DSC Test Chamber, composed of the following:8.1.1.1 Furnace(s), to provide uniform controlled heating ofa specimen and reference to a constant temperature at aconstant rate within the applicable temperature range of thistest method.8.1.1.2 Temperature Sensor, to provide an indication of thespecimen/furnace temperature to 60.01 K.8.1.1.3 Differential Sensor, to detect heat flow differencebetween the specimen and reference equivalent to 1 µW.8.1.1.4 A means of sustaining a test chamber environmentof purge gas at a rate of 10 to 50 mL/min.NOTE 2—Typically, 99.9+ % pure nitrogen, helium, or argon isemployed. Use of dry purge gas is recommended and is essential foroperation at subambient temperatures.8.1.2 Temperature Controller, capable of executing a spe-cific temperature program by operating the furnace(s) betweenselected temperature limits, that is, 170 to 870 K, at a rate oftemperature change of up to 10 K/min constant to 60.1 K/min.8.1.3 Data Collection Device, to provide a means ofacquiring, storing, and displaying measured or calculatedsignals, or both. The minimum output signals required for DSCare heat flow, temperature, and time.8.2 Containers (pans, crucibles, vials, etc.), that are inert tothe specimen and reference materials, and which are of suitablestructural shape and integrity to contain the specimen andreference in accordance with the specific requirements of thistest method.8.3 While not required, the user will find useful calculator orcomputer and data analysis software to perform the necessaryleast squares best fit or multiple linear regression data treat-ments required by this test method.8.4 Balance—to weigh specimens, or containers, or both, to610 µg with a capacity of at least 100 mg.9. Calibration9.1 Perform any calibration procedures recommended bythe apparatus manufacturer in the instrument operator’smanual.9.2 Calibrate the DSC temperature signal over the range ofthe reaction using Test Method E967.9.3 Calibrate the DSC heat flow signal using Practice E968.FIG. 1 Idealized DSC CurveE2041 − 13´1310. Procedure10.1 Weigh 1 to 10 mg of test specimen to a precision of610 µg into a sample container and hermetically seal thecontainer. Weigh the specimen and container to 610 µg. Loadthe test specimen into the apparatus using an equivalent emptyspecimen container as the reference. Close the DSC samplechamber and prepare the apparatus for an experimental run.NOTE 3—This test method is based upon a “non-self heating” assump-tion. Combinations of specimen size and reaction kinetics that produceheat flow greater than 8 mW fail this assumption and produce erroneousresults. Small specimen sizes may be used to obtain this critical non-selfheating assumption.10.2 Equilibrate the specimen at a temperature 40 K belowthe first exothermic behavior.NOTE 4—This temperature may be determined from a previouslyrecorded exploratory run using Test Method E537.10.3 Heat the test specimen at a rate of 5 K/min to atemperature 10 K higher than the completion of the exothermicreaction as indicated by the return to baseline. Record the heatflow and sample temperature throughout this region.NOTE 5—Other heating rates (10 K/min) may be used but shall beindicated in the report.Agreement of results undertaken at several heatingrates will provide confidence in the method and efficacy of the results.10.4 Cool the specimen container to ambient temperatureand reweigh. Record and report any change in mass from thatobserved in 10.1 prior to the test.10.5 Calculate reaction order (n), activation energy (E), andArrhenius pre-exponential factor (Z) according to the proce-dures in Section 11.11. Calculation11.1 Construct a linear baseline from a point on the baselinebefore the reaction exotherm to a point on the baseline after thereaction.11.2 Construct a perpendicular line from the baseline to thepeak of the thermal curve and record this value in mW. Onlyresults for which the maximum heat flow (as expressed by thisline) are less than 8 mW shall be used in these calculations. Ifthe heat flow at the peak maximum is greater than 8 mW,reduce the specimen size or heating rate and rerun theexperiment (see Note 3).11.3 Integrate the total peak area bounded by the peak itselfand the constructed baseline to obtain the heat of the reaction(∆H)inmJ.11.4 Identify the temperatures that correspond approxi-mately to 10 and 90 % of the peak area obtained in 11.3.11.5 Select a temperature interval which provides a mini-mum of ten equally-spaced values between the temperaturelimits determined in 11.4.11.6 At each of the ten temperatures identified in 11.5,record the rate of reaction (dH/dt ) in mW, temperature (T)inK and heat of reaction remaining (∆HT) in mJ as illustrated inFig. 1.NOTE 6—It is convenient to prepare a table of these values.11.7 For each of the fractional areas obtained in 11.6,determine the fraction remaining (1 − α) and the fractional rateof reaction (dα/dt) using the following equation:~1 2 α! 5 ∆HT/∆H (6)dα/dt 5 ~dH/dt!/∆H (7)NOTE 7—In this and all subsequent calculations, retain all availablesignificant figures rounding only the final result to the number ofsignificant figures described in Section 13.NOTE 8—The values for (1 − α) should range between 0.9 and 0.1depending upon the value