Designation: E2070 − 13Standard Test Method forKinetic Parameters by Differential Scanning CalorimetryUsing Isothermal Methods1This standard is issued under the fixed designation E2070; 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.1. Scope1.1 Test Methods A, B, and C determine kinetic parametersfor activation energy, pre-exponential factor and reaction orderusing differential scanning calorimetry from a series of isother-mal experiments over a small ( ≈10 K) temperature range. TestMethod A is applicable to low nth order reactions. TestMethods B and C are applicable to accelerating reactions suchas thermoset curing or pyrotechnic reactions and crystallizationtransformations in the temperature range from 300 to 900 K(nominally 30 to 630°C). This test method is applicable only tothese types of exothermic reactions when the thermal curves donot exhibit shoulders, double peaks, discontinuities or shifts inbaseline.1.2 Test Methods D and E also determines the activationenergy of a set of time-to-event and isothermal temperaturedata generated by this or other procedures1.3 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 toISO DIS 11357, Part 5, and provides more information than theISO standard.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. Specific precau-tionary statements are given in Section 8.2. Referenced Documents2.1 ASTM Standards:2D3350 Specification for Polyethylene Plastics Pipe and Fit-tings MaterialsD3895 Test Method for Oxidative-Induction Time of Poly-olefins by Differential Scanning CalorimetryD4565 Test Methods for Physical and Environmental Per-formance Properties of Insulations and Jackets for Tele-communications Wire and CableD5483 Test Method for Oxidation Induction Time of Lubri-cating Greases by Pressure Differential Scanning Calorim-etryD6186 Test Method for Oxidation Induction Time of Lubri-cating Oils by Pressure Differential Scanning Calorimetry(PDSC)E473 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-calsE1858 Test Method for Determining Oxidation InductionTime of Hydrocarbons by Differential Scanning Calorim-etryE1860 Test Method for Elapsed Time Calibration of Ther-mal AnalyzersE1970 Practice for Statistical Treatment of ThermoanalyticalData1This 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. 15, 2013. Published October 2013. Originallyapproved in 2000. Last previous edition approved in 2008 as E2070 – 08. DOI:10.1520/E2070-13.2For 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 States1E2041 Test Method for Estimating Kinetic Parameters byDifferential Scanning Calorimeter Using the Borchardtand Daniels MethodE2046 Test Method for Reaction Induction Time by ThermalAnalysis2.2 ISO Standard:3ISO DIS 11357 Part 5: Determination of Temperature and/orTime of Reaction and Reaction Kinetics3. Terminology3.1 Specific technical terms used in this test method aredefined in Terminologies E473, E1142, and E1445, includingthe terms calorimeter, Celsius, crystallization, differentialscanning calorimetry, general rate law, isothermal, peak, andreaction.4. Summary of Test Method4.1 A test specimen is held at a constant temperature in adifferential scanning calorimeter throughout an exothermicreaction. The rate of heat evolution, developed by the reaction,is proportional to the rate of reaction. Integration of the heatflow as a function of time yields the total heat of reaction.4.2 An accelerating (Sestak-Berggren or Avrami models),nth order data, or model free treatment4,5,6is used to derive thekinetic parameters of activation energy, pre-exponential factorand reaction order from the heat flow and total heat of reactioninformation obtained in 4.1 (See Basis for Methodology,Section 5.)5. Basis of Methodology5.1 Reactions of practical consideration are exothermic innature; that is, they give off heat as the reaction progresses.Furthermore, the rate of heat evolution is proportional to therate of the reaction. Differential scanning calorimetry measuresheat flow as a dependent experimental parameter as a functionof time under isothermal experimental conditions. DSC isuseful for the measurement of the total heat of a reaction andthe rate of the reaction as a function of time and temperature.5.2 Reactions may be modeled with a number of suitableequations of the form of:dα/dt 5 k~T! f~α! (1)where:dα/dt = reaction rate (s–1),α = fraction reacted (dimensionless),k (T) = specific rate constant at temperature T (s–1),f(α) = conversion function. Commonly used functionsinclude:f1~α! 5 ~1 2 α!n(2)f2~α! 5 α₥~1 2 α!n(3)f3~α! 5 p~1 2 α

[email protected] 1 n ~1 2 α!#~p 2 1!⁄p(4)where:n, ₥, and p = partial reaction order terms.NOTE 1—There are a large number of conversion function expressionsfor [f(α)].4Those described here are the most common but are not the onlyfunctions suitable for this test method. Eq 1 is known as the general rateequation while Eq 3 is the accelerating (or Sestak-Berggren) equation.5,6Eq 4 is the accelerating Avrami equation. Eq 2 is used for nth orderreactions while Eq 3 or Eq 4 are used for accelerating reaction, such asthermoset cure and crystallization transformations.5.3 For a reaction conducted at temperature (T), the accel-erating rate Eq 3 and the rate equation Eq 1 may be cast in theirlogarithmic form.dα/dt 5 k~T! α₥~1 2 α!n(5)

[email protected]α/dt# 5

[email protected]~T!#1₥

[email protected]α#1n

[email protected] 2 α# (6)This equation has the form z = a + bx + cy and may be solvedusing multiple linear regression analysis where x = ln[α], y =ln[1 – α], z = ln[dα/dt], a = ln[k(T)], b = ₥ and c = n.NOTE 2—The rate equation (Eq 3) reduces to the simpler general rateequation (Eq 2) when the value of reaction order parameter ₥ equals zerothereby reducing the number of kinetic parameters to be determined.5.4 For reactions conducted at temperature (T), the acceler-ating rate equation of Eq 4 may be cast as:

[email protected] ln ~1 2 α!# 5 p

[email protected] ~T!#1p

[email protected]# (7)This equation has the form of y = mx + b and may be solvedby linear regression where x = ln[t], y = ln[-ln(1 – α)], with p= m, b = p ln[k(T)], and t = time.5.5 The Arrhenius equation describes how the reaction ratechanges as a function of temperature:k~T! 5 Ze2E/RT(8)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 = 2.7182818.5.6 Eq 8 cast in its logarithmic form is:

[email protected]~T!# 5

[email protected]# 2 E/RT (9)Eq 9 has the form of a straight line, y = mx + b, where a plotof the logarithm of the reaction rate constant (ln[k(T)]) versusthe reciprocal of absolute temperature (l/T) is linear with theslope equal to –E/R and an intercept equal to ln[Z].5.7 As an alternative to Eq 6 and Eq 7, the rate andArrhenius equations combined and cast in logarithmic form is:

[email protected]α/dt# 5

[email protected]# 2 E/RT1m

[email protected]α#1n

[email protected] 2 α# (10)Eq 10 has the form, z = a + bx + cy + dw, and may be solvedusing multiple linear regression analysis.where:z = ln[dα/dt]a = ln[Z]b =-E/Rx =1/Tc = ₥y = ln[1 – α]3Available from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor, New York, NY 10036, http://www.ansi.org.4Sbirrazzuoli, N., Brunel, D., and Elegant, L., Journal of Thermal Analysis,Vol38, 1992, pp. 1509–1524.5Sestak, J., and Berggren, G., Thermochimica Acta, Vol 3, 1971, p. 1.6Gorbachiev, V.M., Journal of Thermal Analysis, Vol 18, 1980, pp. 193–197.E2070 − 132d = n, andw = ln[1 – α].5.8 If activation energy values only are of interest, Eq 11may be solved under conditions of constant conversion toyield:

[email protected]∆t# 5 E/RT1b (11)where:∆t = lapsed time (s), at constant conversion and at isothermaltemperature, T, andb = constant.Eq 11 has the form of a straight line, y = mx + b, where a plotof the logarithm of the lapsed time under a series of differingisothermal conditions versus the reciprocal of absolute tem-perature (l/T) is linear with a slope equal to E/R.5.9 If activation energy values only are of interest, Eq 11may be solved under conditions of constant conversion and theequality dα/dt = dH/dt /(H) to yield:

[email protected]/dt# 52E/RT1b 5 m/T1b (12)where:H = total heat of reaction (mJ),dH/dt = instantaneous heat flow (mW),b = constant, andm = slope (K)Eq 12 has the form of a straight line y =mx + b, where a plotof the logarithm of the heat flow (ln[dH/dt]) at the peak of theexotherm under a series of differing isothermal temperatureconditions versus the reciprocal of the absolute temperature(1/T) is linear with a slope equal to E/R.5.10 A series of isothermal experiments by Test Method A,B, and C described in Section 11 at four or more temperatures,determines the kinetic parameters of activation energy, pre-exponential factor and reaction order.Alternatively, the time toa condition of constant conversion for a series of experimentsat four or more temperatures obtained by this or alternativeTest Method D, described in Section 12, may be used todetermine activation energy only.5.11 A series of not less than four isothermal DSCexperiments, covering a temperature range of approximately10 K and a time less than 100 min (such as those shown in Fig.1) provides values for dα/dt, α,(1–α) and T to solve Eq 6, Eq7, Eq 9, and Eq 10.NOTE 1—This figure is for a crystallization application in which the reaction rate increases with decreasing temperature. Chemical reactions show anincrease in reaction rate with increasing temperature.FIG. 1 Heat Flow Curves at a Series of Isothermal TemperaturesE2070 − 1335.12 A series of not less than four isothermal DSC experi-ments covering a temperature range of approximately 10 K anda time less than 100 min provides dH/dt and T to solve Eq 125.13 A variety of time-to-event experiments such as oxida-tion induction time methods (Practice D3350 and Test MethodsD3895, D4565, D5483, D6186, and E1858) and reactioninduction time methods (Test Method E2046) provide valuesfor ∆t and T to solve equation Eq 11.6. Significance and Use6.1 This test method is useful for research and development,quality assurance, regulatory compliance and specificationacceptance purposes.6.2 The determination of the order of a chemical reaction ortransformation at specific temperatures or time conditions isbeyond the scope of this test method.6.3 The activation energy results obtained by this testmethod may be compared with those obtained from TestMethod E698 for nth order and accelerating reactions. Activa-tion energy, pre-exponential factor, and reaction order resultsby this test method may be compared to those for Test MethodE2041 for nth order reactions.7. Interferences7.1 The approach is applicable only to exothermic reactions.NOTE 3—Endothermic reactions are controlled by the rate of the heattransfer of the apparatus and not by the kinetics of the reaction and maynot be evaluated by this test method.7.2 This test method is intended for a reaction mechanismthat does not change during the transition. This test methodassumes a single reaction mechanism when the shape of thethermal curve is smooth (as in Fig. 2 and Fig. 3) and does notexhibit shoulders, multiple peaks or discontinuation steps.7.3 Test method precision is enhanced with the selection ofthe appropriate conversion function [f(α)] that minimizes thenumber of experimental parameters determined. The shape ofthe thermal curve, as described in Section 11, may confirm theselection of the nth order or accelerating models.7.4 Typical nth order reactions include those in which allbut one of the participating species are in excess.7.5 Typical accelerating reactions include thermoset cure,crystallization and pyrotechnic reactions.7.6 For nth order kinetic reactions, this test method antici-pates that the value of n is small, non-zero integers, such as 1or 2. This test method should be used carefully when values ofn are greater than 2 or are not a simple fraction, such as1⁄2 =0.5.7.7 Accelerating kinetic reactions anticipate that m and n arefractions between 0 and 2 and that their sum (m + n) is less than3.FIG. 2 Heat Flow Curve for an nth Order ReactionE2070 − 1347.8 Accelerating kinetic reactions anticipate that p is aninteger often with a value of ≤4.7.9 Since this test method uses milligram quantities it isessential that the test specimens are homogeneous and repre-sentative of the larger samples from which they are taken.7.10 Test specimens may release toxic and corrosive efflu-ents that may be harmful to personnel or apparatus. Operationwith a venting or exhaust system is recommended.8. Hazards8.1 Special precautions shall be taken to protect personneland equipment when the apparatus in use requires the insertionof specimens into a heated furnace. These special precautionsinclude adequate shielding and ventilation of equipment andface and hand protections for users (see Note 6).9. Apparatus9.1 A differential scanning calorimeter (DSC) that providesthe minimum calorimetric capability for this test methodincludes:9.1.1 A DSC Test Chamber, composed of:9.1.1.1 A Furnace(s), that provides uniform controlled heat-ing of a specimen and reference to constant temperature at aconstant rate between 300 and 900 K.9.1.1.2 A Temperature Sensor, that indicates the specimen/furnace temperature to 60.01 K.9.1.1.3 A Differential Sensor, that detects heat flow differ-ences between the specimen and reference equivalent to 1 µW.9.1.1.4 Ameans of sustaining a purge gas rate of 10 to 50 65 mL/minute in the test chamber.NOTE 4—Typically inert purge gases that inhibit sample oxidation are99.9+ % pure nitrogen, helium or argon. Dry gases