Designation: E2890 − 12´1Standard Test Method forKinetic Parameters for Thermally Unstable Materials byDifferential Scanning Calorimetry Using the KissingerMethod1This standard is issued under the fixed designation E2890; 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—Research report information was editorially added to 14.1 in September 2015.1. Scope1.1 This test method describes the determination of thekinetic parameters of Arrhenius activation energy and pre-exponential factor using the Kissinger variable heating rateiso-conversion method (1, 2)2and activation energy andreaction order by the Farjas method (3) for thermally unstablematerials. The test method is applicable to the temperaturerange from 300 to 900 K (27 to 627°C).1.2 Both nth order and accelerating reactions are addressedby this method over the range of 0.5 n 4and1p 4where n is the nth order reaction order and p is the Avramireaction order (4). Reaction orders n and p are determined bythe Farjas method (3).1.3 This test method uses the same experimental conditionsas Test Method E698. The Flynn/Wall/Ozawa data treatment ofTest Method E698 may be simultaneously applied to theseexperimental results.1.4 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.5 There is no ISO equivalent to this standard.1.6 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 CalorimetryE691 Practice for Conducting an Interlaboratory Study toDetermine the Precision of a Test MethodE698 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 PropertiesE1231 Practice for Calculation of Hazard Potential Figures-of-Merit for Thermally Unstable MaterialsE1860 Test Method for Elapsed Time Calibration of Ther-mal AnalyzersE1970 Practice for Statistical Treatment of ThermoanalyticalDataE2041 Test Method for Estimating Kinetic Parameters byDifferential Scanning Calorimeter Using the Borchardtand Daniels Method1This test method is under the jurisdiction ofASTM Committee E37 on ThermalMeasurements and is the direct responsibility of Subcommittee E37.01 on Calo-rimetry and Mass Loss.Current edition approved Sept. 1, 2012. Published October 2012. DOI: 10.1520/E2890-12E01.2The boldface numbers in parentheses refer to the list of references at the end ofthis standard.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.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1E2161 Terminology Relating to Performance Validation inThermal Analysis and Rheology3. Terminology3.1 Technical terms used in this test method are defined inTerminologies E473, E1142, and E2161. Referenced termsinclude Arrhenius equation, baseline, calibration, Celsius, dif-ferential scanning calorimeter, endotherm, enthalpy, figure-of-merit, first-deviation-from baseline, full-width-at-half-maximum, Kelvin, onset point, peak, peak value, relativestandard deviation, standard deviation, thermal analysis andthermal curve.4. Summary of Test Method4.1 A series of test specimens are heated at a minimum offour different linear rates in a differential scanning calorimeterthrough a region of exothermic reaction behavior. The rate ofheat evolution, created by a chemical reaction, is proportionalto the rate of reaction and is measured as a function oftemperature and time.4.2 The temperature corresponding to the maximum rate ofreaction (measured at the heat flow maximum of the exother-mic reaction peak) is recorded at each linear heating rate. Thisobserved temperature is corrected for instrument thermalresistance. Activation energy and pre-exponential factor arederived from the linear regression of the natural logarithm ofthe heating rate, normalized to the square of the absolutetemperature, versus the reciprocal absolute temperature of heatflow at the peak maximum. The approach is known as theKissinger method (1, 2).4.3 A reaction type is determined for the specimen from theshape of the reaction exotherm under isothermal temperatureconditions.4.4 Once a reaction type is determined kinetic parameters oforder (either n or p) are determined using the shape of thereaction exotherm measured by the time at full-width-at-half-maximum (tFWHM). This approach is known at the Farjasmethod (3). The activation energy and reaction order arederived from the linear regression of the natural logarithm ofthe time at full-width-at-half-maximum versus the reciprocal ofabsolute temperature at maximum reaction rate (heat flow).5. Basis of Methodology5.1 For reactions that are exothermic in nature, the rate ofheat evolution is proportional to the rate of the reaction.Differential scanning calorimetry measures the heat flow as thedependent experimental parameter versus temperature (ortime) as the independent parameter.5.2 Reactions may be modeled with a number of suitableequations of the form:da⁄dt 5 k~T! f~α! (1)where:da/dt = reaction rate (s-1),α = fraction reacted or conversion (dimensionless),k(T) = specific rate constant at temperature T, andf(α) = conversion function (dimensionless).Commonly used functions include:f1~α! 5 ~1 2 α!n(2)f2~α! 5 p~1 2 α

[email protected] ln ~1 2 α!#~p 2 1!⁄p(3)where:n = nth reaction order (dimensionless), andp = Avrami reaction order (dimensionless).NOTE 1—There are a large number of conversion function expressionsfor f(a) (5). Those described here are the more common ones but are notthe only functions suitable for this method. Eq 2 is known as the Law ofMass Action (6) while Eq 3 is the Avrami equation (4).5.3 The Arrhenius equation (7) describes how the reactionrate changes as a function of temperature:k~T! 5 Ze2E⁄RT(4)where:Z = pre-exponential factor (s-1),E = activation energy (J mol-1),T = absolute temperature (K),R = gas constant (8.314 J mol-1K-1), ande = natural logarithm base (2.7182818).5.4 Eq 1 and Eq 4 may be combined to yield the general rateequation:da⁄dt 5 f~α!Ze2E⁄RT(5)5.5 As the temperature increases, the rate of reaction willincrease until a maximum is reached and then the rate declinesback to “zero” as the reactant is consumed. When the rate ofreaction is displayed as a function of increasing temperature,the shape of this response is called a “peak”. The mathematicalderivative of the reaction rate at the peak maximum equalszero. Taking the derivative of Eq 5 over time at the maximumpoint for the heating with constant rate β, then casting inlogarithmic form and assuming that

[email protected] ~f ~α!! ⁄ dt#50, leadsto Eq

[email protected]β ⁄ Tm2# 5

[email protected]⁄ E# 2 E⁄RTm(6)where:β = heating rate (K s-1), andTm= temperature a peak maximum (K).NOTE 2—The assumption of

[email protected] ~f ~α!! ⁄ dt#50 holds strictly onlyfor 1st order reaction but is considered a “reasonable” approximation forother nth order or Avrami reactions.5.6 Eq 6 is of the form Y5mX1b.Ifln[β/Tm2] is set equal toYand 1/Tmis set equal to X, then a display ofYversus X yieldsa slope (mK) equal to –EK/R and an intercept (bK) equal toln[ZR/EK] where Z and EKare the pre-exponential factor andthe activation energy, respectively, determined by the Kissingermethod.5.7 The shape of the reaction exothermic peak may becharacterized by the time at full-width-at-half-maximum(tfwhm) (3)

[email protected]# 5 EF⁄

[email protected] ⁄Z# (7)where:tfwhm= the full-width-at-half-maximum time (s), andt’ = an arbitrary function (s-1).E2890 − 12´125.8 Eq 7 is of the form Y5mX1b.Ifln[tFWHM] is set equalto Y and 1/Tmis set equal to X, then a display of Y versus Xyields a slope (mF) equal to EF/R and an intercept (bF) equal toln[t’/Z] where EFis the activation energy determined by theFarjas method.5.9 The reaction order, n or p, is determined through anempirical relationship based on t’.6. Significance and Use6.1 This test method is useful for research and development,quality assurance, regulatory compliance and specification-based acceptance.6.2 The kinetic parameters determined by this method maybe used to calculate thermal hazard figures-of-merit accordingto Practice E1231.7. Interferences7.1 This test method assumes a single reaction mechanismconstant over the reaction conversion temperature range of thematerial under evaluation. Some overall reactions of interestare known to include a series of competing reaction mecha-nisms that lead to changes in reaction order with conversion(8). This method addresses the reaction only at a singleconversion value at the maximum reaction rate—often about0.7.7.2 Method precision is enhanced with the selection of theappropriate conversion function [f(α)]. The shape of the ther-mal curve, as described in 11.2, may confirm the selection ofthe nth order or accelerating reaction models.7.2.1 Typically nth reactions include many (but not all)decomposition reactions or those where one of the participatingspecies is in excess.7.2.2 Typical accelerating (Avrami) reactions include ther-moset cure, crystallization, and some pyrotechnic reactions.7.3 Since this method uses milligram quantities of material,it is essential for the test specimens to be homogeneous andrepresentative of the larger sample from which they are taken.7.4 Acritical literature evaluation of kinetic methods reportsthat the Kissinger method is the most accurate method fordetermining activation energy in many cases (9).8. Apparatus8.1 Differential Scanning Calorimeter (DSC)—The essentialinstrumentation required to provide the minimum differentialscanning calorimetric capability for this method includes (a) afurnace(s) to provide uniform controlled heating or cooling ofa specimen and reference to a constant temperature or at aconstant rate over the range of 300 K to 900 K, (b) atemperature sensor to provide a measurement of the specimentemperature to 6 0.01 K, (c) differential sensors to detect aheat flow difference between the specimen and reference witha range of 100 mW readable to 6 1µW, (d) a means ofsustaining a test chamber environment of inert purge gas at apurge rate of 10 to 100 mL/min within 6 5 mL/min, (e) atemperature controller, capable of executing a specific tem-perature program by operating the furnace(s) between selectedtemperature limits over the range of ambient to 900 K (627 °C)at a rate of 0.1 to 20 K/min constant to 1 % or at an isothermaltemperature constant to 0.1 K, (f) a data collection device,toprovide a means of acquiring, storing, and displaying measuredor calculated signals or both. The minimum output signalsrequired are heat flow, temperature, and time.8.2 Containers (pans, crucibles, vials, lids, closures, seals,etc.) that are inert to the specimen and reference materials (ifany) and that are of suitable structural shape and integrity tocontain the specimen (even under internal pressure developedduring the reaction) and reference in accordance with thespecific requirements of this test method.NOTE 3—Many users find glass, gold or gold coated hermetically sealedcontainers of low headspace volume advantageous for testing with highenergy materials. The selected container shall meet the necessary internalpressure rating to withstand internal pressure buildup.8.3 A means, tool or device to close, encapsulate or seal thecontainer of choice.8.4 Analytical Balance with a capacity of at least 100 mg toweigh specimens or containers, or both to 6 10 µg.8.5 Auxiliary instrumentation considered useful but notessential for conducting this method would include coolingcapability to hasten cooling to ambient temperature conditionsat the end of the test.9. Hazards9.1 This test method is used to determine the properties ofthermally reactive materials. The user of this test method shalluse the smallest quantity of material (typically a few milli-grams) needed to obtain the desired analytical results.9.2 Special precautions shall be taken to protect personneland equipment when the apparatus in use requires the insertionof specimens into a heated furnace. Typical special precautionsinclude adequate shielding, ventilation of equipment and faceand hand protection for users. A safety analysis prior to testingis recommended.10. Calibration and Standardization10.1 Perform any calibration procedures recommended bythe manufacturer as described in the operator’s manual toensure that the apparatus is calibrated at each heating rate used.10.2 Calibrate the heat flow signal using 99.99+% indium,Practice E968, and the same type of specimen container to beused in the subsequent test for kinetic parameters.10.3 Calibrate the temperature signal using 99.99+%indium, Practice E967, and the same type of specimen con-tainer and heating rates to be used in the subsequent test forkinetic parameters.10.4 Calibrate the elapsed time signal using Test MethodE1860.10.5 Determine the thermal resistance (φ) from the leadingedge slope ~S 5 ∆ q ⁄ ∆ T! in (mW/K) of the indium meltingendotherm as shown in Fig. 1 and 12.1.11. Procedure11.1 Scouting Experiment:E2890 − 12´1311.1.1 Usinga1to5mgtest specimen, weighed to aprecision of 6 0.1 mg, perform a scouting experiment usingTest Method E537 to determine the temperature of first-deviation-from baseline (To) and the heat of reaction (∆H).11.2 Determination of Reaction Type:11.2.1 Weigh into a specimen container 1 to 5 mg of the testspecimen, with a precision of 60.1 mg, and hermetically sealthe container. DO NOT load the test specimen into theapparatus. Load an empty specimen container into the refer-ence chamber. Close the DSC chamber and prepare theapparatus for an experimental run.11.2.2 Select an isothermal test temperature correspondingto 10 % peak area (∆H) from the scouting experiment per-formed in 11.1. Equilibrate the apparatus for 1 min at this testtemperature.11.2.3 Initiate the experiment, recording heat flow as afunction of time.11.2.4 Open the DSC sample chamber and quickly load thetest specimen container into the apparatus. Immediately closethe sample chamber. Record the thermal curve for 20 min orun