Designation: E2958 − 14Standard Test Methods forKinetic Parameters by Factor Jump/ModulatedThermogravimetry1This standard is issued under the fixed designation E2958; 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 These test methods describe the model-free determina-tion of Arrhenius activation energy by thermogravimetry usingthe factor jump (1)2(Method A) or modulated thermogravim-etry (2) (Method B) techniques. With the assumption of afirst-order kinetic model, the pre-exponential factor is addition-ally determined.1.2 These test methods are applicable to materials withwell-defined decomposition profiles, namely, a smooth, con-tinuous mass change.1.3 These test methods are applicable to decompositionoccurring in the range from 400 K to 1200 K (nominally 100°Cto 900°C). The temperature range may be extended dependingon the instrumentation and material used.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 standard similar 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-ologyE1142 Terminology Relating to Thermophysical PropertiesE1582 Practice for Calibration of Temperature Scale forThermogravimetryE1641 Test Method for Decomposition Kinetics by Thermo-gravimetry Using the Ozawa/Flynn/Wall MethodE1877 Practice for Calculating Thermal Endurance of Ma-terials from Thermogravimetric Decomposition DataE1970 Practice for Statistical Treatment of ThermoanalyticalDataE2040 Test Method for Mass Scale Calibration of Thermo-gravimetric AnalyzersE2550 Test Method for Thermal Stability by Thermogravi-metry3. Terminology3.1 Definitions—Technical terms used in this test methoddefined in Terminologies E473 and E1142 include Arrheniusequation, activation energy, Celsius, failure criterion, pre-exponential factor, reaction order, and thermogravimetricanalysis.4. Summary of Test Method4.1 These test methods consist of heating a test specimenweighing a few milligrams at a heating rate of about 1 K/minwith a superimposed step-and-hold (factor jump) or sinusoidal(modulated) temperature program through the decompositiontemperature region. The specimen mass rate-of-change iscontinuously calculated and recorded as a function of tempera-ture. The activation energy is then determined from the massrate-of-change at two (or more) closely spaced temperatureregions. The activation energy thus determined is based on noassumed reaction model or mechanism and thus is model free.4.2 Assuming a first-order reaction model (n = 1), theadditional reaction parameter logarithm-of-the-pre-exponential-factor (ln[Z]) is additionally determined.4.3 Activation energy and logarithm-of-the-pre-exponential-factor may be displayed as a function of averagetemperature or conversion to provide additional informationabout the constancy of the decomposition reaction relative tothese experimental parameters.5. Significance and Use5.1 The activation energy may be used to calculate thermalendurance and an estimate of the lifetime of the material atspecified temperatures using Test Method E1877.1These test methods are under the jurisdiction of ASTM Committee E37 onThermal Measurements and is the direct responsibility of Subcommittee E37.01 onCalorimetry and Mass Loss.Current edition approved April 1, 2014. Published June 2014. DOI: 10.1520/E2958-14.2The boldface numbers in parentheses refer to a 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 States15.2 The kinetic parameters determine by this test methodmay be used in quality assurance, research and development.5.3 The kinetic parameters of activation energy and loga-rithm of the pre-exponential factor determined by this methodhave little intrinsic value in themselves. Most practical appli-cations of this information, such as lifetime estimation (seeTest Method E1877), also require an estimation of the preci-sion of the respective values. Determination of that precisionby replicated determination is a non-manditory part of thisstandard.6. Apparatus6.1 The essential equipment required to provide minimumthermogravimetric analytical capability of this test methodinclude:6.1.1 A thermobalance, composed of (a)afurnace toprovide uniform controlled heating of a specimen at a constantrate up to 100 K/min within the temperature range fromambient to 1200 K; (b)atemperature sensor to provide anindication of the specimen/furnace temperature to within 60.1K; (c)anelectrobalance to continuously measure the specimenmass with a minimum capacity of 20 mg and a sensitivity of650 µg; and (d) a means of sustaining the specimen/containerunder atmospheric control of an inert or reactive purge gas of99.99 % purity at a rate of 20 mL/min to 50 mL/min 6 5mL/min.6.1.2 A temperature controller, capable of executing aspecific temperature program by operating the furnace betweenselected temperature limits at a rate of temperature change of1 K/min to 100 K/min constant to within 61 % or anisothermal temperature which is maintained constant to within60.05 K.6.1.3 A data collection device, to provide a means ofacquiring, storing, and displaying measured or calculatedsignals, or both. The minimum output signals required for thistest method are mass, mass rate-of-change, temperature, andtime.6.1.4 Auxiliary instrumentation or data analysis capabilityconsidered useful in conducting this method includes:6.1.4.1 For Method B, the ability to apply a sinusoidaltemperature program of a 100 s to 300 s period and 60Kto6K amplitude upon the underlying linear temperature programor isothermal conditions.6.1.4.2 For Method B, the capability to continuously calcu-late activation energy and logarithm of the pre-exponentialfactor.NOTE 1—Alternative capabilities are described in Refs (3-7).6.2 Containers (pans, crucibles, and so forth) that are inertto the specimen and that will remain dimensionally stable overthe temperature range from ambient to 1200 K.6.3 High-Purity (99.99 %) Nitrogen Supply, for purge gas.NOTE 2—Other atmospheres may be used but shall be reported.6.4 Cryogenic Mill capable of grinding up to 4 mg ofmaterial at a temperature less than 173 K (–100°C).7. Sampling, Test Specimens, and Test Units7.1 Since milligram quantities of specimens are used, it isessential that the specimens be representative of the samplesfrom which they are taken.All specimens should be thoroughlymixed prior to sampling and should be sampled by removingportions form various parts of the sample. These portionsshould in turn be combined and mixed well to ensure arepresentative specimen for the determination.7.2 Powdered or granular specimens that have a highsurface-to-volume ratio, are preferred, although films, fibers,and fabric may be used providing that care is taken to ensurethat all specimens are uniform in size and shape. Where thesample is a part or is in the form of pellets, the specimen maybe prepared by filling, rasping or cryogenic milling.NOTE 3—The specimen size and surface-to-volume ratio are known toaffect the results of this test. A narrow range of specimen sizes should beused as noted in 10.1 and 12.1. Uniformity in particle size can beachieved, without the loss of volatiles, by using a cryoscopy (liquidnitrogen) mill to grind the sample to a powder. To prevent the condensa-tion of moisture, the mill should be opened only after returning to ambienttemperature, or the operation should be performed in a glove box filledwith dry gas.7.3 In the absence of other information, the samples areassumed to be analyze as-received except for the mechanicaltreatment noted in 7.2. If some heat treatment, such as drying,is applied to the sample prior to analysis, this treatment and anyresulting mass loss shall be reported.7.4 Some materials may require more sophisticatedconditioning, such as maintaining the sample in a specifiedtemperature and relative humidity for an extended period oftimes. Such conditioning may be conducted, but proceduraldetails shall be included in the report.8. Preparation of Apparatus and ExperimentalConditions8.1 Prepare the thermogravimetric analyzer using the pro-cedures described in the manufacturer’s operations manual.8.2 Identify the weight loss to be used as the failurecriterion. Report this value.NOTE 4—The value of 5 % mass loss of the specific decomposition stepis commonly used in thermogravimetry and accelerated lifetime testing asthe failure criteria (see Test Method E1641).9. Calibration and Standardization9.1 Calibrate the temperature scale of the thermogravimetricanalyzer at 1 K/min using Practice E1582.9.2 Calibrate the mass loss scale of the thermogravimetricanalyzer using Test Method E2040.METHOD AFACTOR JUMP METHOD10. Procedure10.1 Place 2 mg to 4 mg of the specimen into a clean, taredinstrument specimen container.NOTE 5—Other specimen size may be used but shall be reported.NOTE 6—Powdered or granular specimens should be distributed evenlyE2958 − 142over the specimen holder so as to maximize the exposed surface.10.2 Equilibrate the specimen for 1 minute at a temperature20 K below the known decomposition onset temperature.Establish the mass scale at the conversion fraction of zero (α =0.0 %).NOTE 7—The decomposition onset temperature may be obtained froma scouting experiment using Test Method E2550 at 5 K/min.10.3 Begin recording experimental data. Initiate an isother-mal temperature program for 150 s. At the end of thisisothermal period, measure and record the mass rate-of-change(dαv/dt), temperature (Tv) and conversion (αv).10.4 Initiate a temperature step-and-hold sequence by in-creasing the temperature by 10.0 K 6 0.2 K and holding thattemperature to within 0.05 K for 150 s. At the end of thisisothermal period, record the mass rate-of change (dαp/dt),temperature (Tp) and conversion (αp).NOTE 8—The time required to establish temperature equilibrium and tomake an accurate mass rate-of-change measurement may vary by instru-ment and temperature. These conditions are thought to embrace thoseachievable by all instrument designs. Other temperature steps and isother-mal hold periods may be used but shall be reported (see Appendix X1).10.5 Using the data obtained in 10.3 and 10.4, determine theactivation energy, logarithm of the pre-exponential factor andtemperature (T) using Eq 1, Eq 2, and Eq 3. Record thesevalues along with the conversion (α) at the end of the secondisothermal region.10.6 Initiate a second step-and-hold cycle by decreasing thetemperature by 5.0 K 6 0.1 K and holding that temperature towithin 0.05 K for 150 s. At the end of the isothermal region,record the mass-rate-of change (dαv/dt), temperature (Tv) andconversion (αv).10.7 Using the data obtained in 10.4 and 10.6, determine theactivation energy, logarithm of the pre-exponential factor, andtemperature using Eq 1, Eq 2, and Eq 3. Record these valuesalong with the conversion (α) at the end of the secondisothermal region.10.8 Repeat 10.4 – 10.7 until the decomposition weight lossis complete or until the upper temperature limit of theapparatus is reached.10.9 Create a table of activation energy and logarithm of thepre-exponential factor versus conversion. Select the activationenergy and logarithm of the pre-exponential factor nearest thefailure criterion conversion level from 8.2.NOTE 9—Most uses of activation energy and logarithm of the pre-exponential factor required an estimation of their precision. Mean valuesand standard deviations for both values may be obtained from a minimumof three replicate determinations (see Practice E1970).10.10 Report the mean activation energy (E) and its percentrelative standard deviation (σE/E) and the mean logarithm ofthe pre-exponential factor (ln[Z]) and its percent relativestandard deviation (σln[Z]/ln[Z]) at the temperature closest tothe failure criterion of 8.211. Calculations11.1 Calculations are as follows:E 5 $RTpTvln @~dαp⁄ dt!⁄~dαv⁄ dt!#%⁄~Tp2 Tv! (1)

[email protected] , min21# 5

[email protected]~dαv⁄ dt!⁄~1 2 α ⁄ 100 %!#1E⁄RT (2)T 5 ~Tp1 Tv!⁄2 (3)where:E = Activation energy, J/mol,R = Gas constant (= 8.31451 J/(mol K)),Tp= Temperature at the end of the higher temperatureisothermal plateau, K,Tv= Temperature at the end of the lower temperatureisothermal plateau, K,T = Average temperature between Tpand Tv,K,dαp/dt = Mass rate-of-change at the end of the higher tem-perature isothermal plateau, % / min,dαv/dt = Mass rate-of-change at the end of the lower tem-perature isothermal plateau, % / min,ln = Natural logarithm to the Napier base e,α = Fraction reacted or conversion, %,Z = Pre-exponential factor, min-1, anddα/dt = Mean mass rate-of-change for two adjacent step-and-hold segments = (dαvdt + dαp/dt)/2.NOTE 10—The logarithm of the pre-exponential factor (ln[Z]) calcu-lated in Eq 2 is determined assuming a first-order kinetics reaction.METHOD BMODULATED THERMOGRAVIMETRY METHOD12. Procedure12.1 Place 2 mg to 4 mg of the specimen into a clean, taredinstrument specimen container.NOTE 11—Other specimen size may be used but shall be reported.NOTE 12—Powdered or granular specimens should be distributedevenly over the specimen holder so as to maximize the exposed surface.12.2 Equilibrate the specimen for 1 minute at a temperature20 K below the known decomposition onset temperature.Establish the percent mass loss scale at 100 % (α = 0 %).NOTE 13—The decomposition onset temperature may be obtained froma scouting experiment using Test Method E2550 at 5 K/min.12.3 Begin recording experimental data including averagetemperature, average mass, conversion, activation energy andlogarithm of the pre-exponential factor. Initiate a modulatedtemperature program with amplitude of 64.9 K – 5.1 K (thatis, 9.8 K to 10.2 K peak-to-peak) and a period of 300 s.NOTE 14—The time required to establish dynamic equilibrium mayvary with instrument and temperature. These conditions are thought toembrace the dynamic equilibrium achievable by all instrument designs.Other periods and amplitudes may be used but shall be reported (seeAppendix X1).NOTE 15—The recording of other signals such as the mass rate-of-change, modulated temperature and modulated mass