ASTM E2070-13(2018)
(Test Method)Standard Test Methods for Kinetic Parameters by Differential Scanning Calorimetry Using Isothermal Methods
Standard Test Methods for Kinetic Parameters by Differential Scanning Calorimetry Using Isothermal Methods
SIGNIFICANCE AND USE
6.1 These test methods are useful for research and development, quality assurance, regulatory compliance, and specification acceptance purposes.
6.2 The determination of the order of a chemical reaction or transformation at specific temperatures or time conditions is beyond the scope of these test methods.
6.3 The activation energy results obtained by these test methods may be compared with those obtained from Test Method E698 for nth order and accelerating reactions. Activation energy, pre-exponential factor, and reaction order results by these test methods may be compared to those for Test Method E2041 for nth order reactions.
SCOPE
1.1 Test Methods A, B, and C determine kinetic parameters for activation energy, pre-exponential factor and reaction order using differential scanning calorimetry from a series of isothermal experiments over a small ( ≈10 K) temperature range. Test Method A is applicable to low nth order reactions. Test Methods B and C are applicable to accelerating reactions such as thermoset curing or pyrotechnic reactions and crystallization transformations in the temperature range from 300 to 900 K (nominally 30 to 630°C). These test methods are applicable only to these types of exothermic reactions when the thermal curves do not exhibit shoulders, double peaks, discontinuities or shifts in baseline.
1.2 Test Methods D and E also determines the activation energy of a set of time-to-event and isothermal temperature data generated by this or other procedures
1.3 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.4 These test methods are similar but not equivalent to ISO DIS 11357, Part 5, and provides more information than the ISO standard.
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use. Specific precautionary statements are given in Section 8.
1.6 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
General Information
Relations
Buy Standard
Standards Content (Sample)
This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E2070 − 13 (Reapproved 2018)
Standard Test Methods for
Kinetic Parameters by Differential Scanning Calorimetry
Using Isothermal Methods
This standard is issued under the fixed designation E2070; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
1.1 Test MethodsA, B, and C determine kinetic parameters 2.1 ASTM Standards:
for activation energy, pre-exponential factor and reaction order D3350Specification for Polyethylene Plastics Pipe and Fit-
usingdifferentialscanningcalorimetryfromaseriesofisother- tings Materials
mal experiments over a small (≈10 K) temperature range.Test D3895Test Method for Oxidative-Induction Time of Poly-
Method A is applicable to low nth order reactions. Test olefins by Differential Scanning Calorimetry
Methods B and C are applicable to accelerating reactions such D4565Test Methods for Physical and Environmental Per-
asthermosetcuringorpyrotechnicreactionsandcrystallization formance Properties of Insulations and Jackets for Tele-
transformations in the temperature range from 300 to 900 K communications Wire and Cable
(nominally 30 to 630°C). These test methods are applicable D5483Test Method for Oxidation Induction Time of Lubri-
only to these types of exothermic reactions when the thermal catingGreasesbyPressureDifferentialScanningCalorim-
curves do not exhibit shoulders, double peaks, discontinuities etry
or shifts in baseline. D6186Test Method for Oxidation Induction Time of Lubri-
cating Oils by Pressure Differential Scanning Calorimetry
1.2 Test Methods D and E also determines the activation
(PDSC)
energy of a set of time-to-event and isothermal temperature
E473Terminology Relating to Thermal Analysis and Rhe-
data generated by this or other procedures
ology
1.3 The values stated in SI units are to be regarded as
E537Test Method for The Thermal Stability of Chemicals
standard. No other units of measurement are included in this
by Differential Scanning Calorimetry
standard.
E698Test Method for Kinetic Parameters for Thermally
Unstable Materials Using Differential Scanning Calorim-
1.4 These test methods are similar but not equivalent to
ISODIS11357,Part5,andprovidesmoreinformationthanthe etry and the Flynn/Wall/Ozawa Method
E967Test Method for Temperature Calibration of Differen-
ISO standard.
tial Scanning Calorimeters and Differential ThermalAna-
1.5 This standard does not purport to address all of the
lyzers
safety concerns, if any, associated with its use. It is the
E968Practice for Heat Flow Calibration of Differential
responsibility of the user of this standard to establish appro-
Scanning Calorimeters
priate safety, health, and environmental practices and deter-
E1142Terminology Relating to Thermophysical Properties
mine the applicability of regulatory limitations prior to use.
E1445Terminology Relating to Hazard Potential of Chemi-
Specific precautionary statements are given in Section 8.
cals
1.6 This international standard was developed in accor-
E1858Test Methods for Determining Oxidation Induction
dance with internationally recognized principles on standard-
Time of Hydrocarbons by Differential Scanning Calorim-
ization established in the Decision on Principles for the
etry
Development of International Standards, Guides and Recom-
E1860Test Method for Elapsed Time Calibration of Ther-
mendations issued by the World Trade Organization Technical
mal Analyzers
Barriers to Trade (TBT) Committee.
E1970PracticeforStatisticalTreatmentofThermoanalytical
Data
These test methods are under the jurisdiction of ASTM Committee E37 on
Thermal Measurements and is the direct responsibility of Subcommittee E37.01 on
Calorimetry and Mass Loss. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved April 1, 2018. Published May 2018. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 2000. Last previous edition approved in 2013 as E2070–13. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/E2070-13R18. the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2070 − 13 (2018)
E2041Test Method for Estimating Kinetic Parameters by where:
Differential Scanning Calorimeter Using the Borchardt
n, ₥, and p = partial reaction order terms.
and Daniels Method
NOTE 1—There are a large number of conversion function expressions
for[f(α)]. Thosedescribedherearethemostcommonbutarenottheonly
E2046TestMethodforReactionInductionTimebyThermal
functionssuitableforthesetestmethodsEq1isknownasthegeneralrate
Analysis
5,6
3 equation while Eq 3 is the accelerating (or Sestak-Berggren) equation.
2.2 ISO Standard:
Eq 4 is the accelerating Avrami equation. Eq 2 is used for nth order
ISODIS11357Part5:DeterminationofTemperatureand/or
reactions while Eq 3 or Eq 4 are used for accelerating reaction, such as
Time of Reaction and Reaction Kinetics thermoset cure and crystallization transformations.
5.3 For a reaction conducted at temperature (T), the accel-
3. Terminology
eratingrateEq3andtherateequationEq1maybecastintheir
3.1 Specific technical terms used in these test methods are
logarithmic form.
defined in Terminologies E473, E1142, and E1445, including
₥ n
dα/dt 5 k T α 1 2α (5)
~ ! ~ !
the terms calorimeter, Celsius, crystallization, differential
ln@dα/dt# 5 ln@k~T!#1₥ ln@α#1n ln@1 2α# (6)
scanning calorimetry, general rate law, isothermal, peak, and
reaction.
Thisequationhastheformz=a+bx+cyandmaybesolved
using multiple linear regression analysis where x = ln[α], y =
4. Summary of Test Method
ln[1 – α], z = ln[dα/dt], a = ln[k(T)], b = ₥ and c = n.
4.1 A test specimen is held at a constant temperature in a
NOTE 2—The rate equation (Eq 3) reduces to the simpler general rate
differential scanning calorimeter throughout an exothermic
equation (Eq 2) when the value of reaction order parameter₥ equals zero
reaction.The rate of heat evolution, developed by the reaction,
thereby reducing the number of kinetic parameters to be determined.
is proportional to the rate of reaction. Integration of the heat
5.4 For reactions conducted at temperature (T), the acceler-
flow as a function of time yields the total heat of reaction.
ating rate equation of Eq 4 may be cast as:
4.2 An accelerating (Sestak-Berggren or Avrami models),
4,5,6 ln 2 ln 1 2 α 5 p ln k T 1p ln t (7)
@ ~ !# @ ~ !# @ #
nthorderdata,ormodelfreetreatment isusedtoderivethe
kinetic parameters of activation energy, pre-exponential factor
This equation has the form of y = mx + b and may be solved
and reaction order from the heat flow and total heat of reaction
by linear regression where x = ln[t], y = ln[-ln(1 – α)], with p
information obtained in 4.1. (See Basis for Methodology,
= m, b = p ln[k(T)], and t = time.
Section 5.)
5.5 TheArrhenius equation describes how the reaction rate
changes as a function of temperature:
5. Basis of Methodology
2E/RT
k~T! 5Ze (8)
5.1 Reactions of practical consideration are exothermic in
nature; that is, they give off heat as the reaction progresses.
where:
Furthermore, the rate of heat evolution is proportional to the
–1
Z = pre-exponential factor (s ),
rateofthereaction.Differentialscanningcalorimetrymeasures –1
E = activation energy (J mol ),
heat flow as a dependent experimental parameter as a function
T = absolute temperature (K),
–1 –1
of time under isothermal experimental conditions. DSC is
R = gas constant = (8.314 J mol K ), and
useful for the measurement of the total heat of a reaction and
e = natural logarithm base = 2.7182818.
the rate of the reaction as a function of time and temperature.
5.6 Eq 8 cast in its logarithmic form is:
5.2 Reactions may be modeled with a number of suitable
ln k T 5 ln Z 2 E/RT (9)
@ ~ !# @ #
equations of the form of:
Eq 9 has the form of a straight line, y = mx + b, where a plot
dα/dt 5 k~T! f~α! (1)
of the logarithm of the reaction rate constant (ln[k(T)]) versus
where:
the reciprocal of absolute temperature (l/T) is linear with the
–1
dα/dt = reaction rate (s ),
slope equal to –E/R and an intercept equal to ln[Z].
α = fraction reacted (dimensionless),
5.7 As an alternative to Eq 6 and Eq 7, the rate and
–1
k (T) = specific rate constant at temperature T (s ),
Arrhenius equations combined and cast in logarithmic form is:
f(α) = conversion function. Commonly used functions
ln@dα/dt# 5 ln@Z# 2 E/RT1m ln@α#1n ln@1 2α# (10)
include:
n
f α 5 1 2α (2)
~ ! ~ ! Eq10hastheform, z= a+ bx+ cy+ dw,andmaybesolved
₥ n
using multiple linear regression analysis.
f α 5α 1 2α (3)
~ ! ~ !
~p 2 1!⁄p
f ~α! 5 p~1 2 α!@2 1 n ~1 2 α!# (4) where:
z = ln[dα/dt]
3 a = ln[Z]
Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
b =-E/R
4th Floor, New York, NY 10036, http://www.ansi.org.
Sbirrazzuoli, N., Brunel, D., and Elegant, L., Journal of Thermal Analysis,Vol
x =1/T
38, 1992, pp. 1509–1524.
c = ₥
Sestak, J., and Berggren, G., Thermochimica Acta, Vol 3, 1971, p. 1.
y = ln[1 – α]
Gorbachiev, V.M., Journal of Thermal Analysis, Vol 18, 1980, pp. 193–197.
E2070 − 13 (2018)
6. Significance and Use
d = n, and
w = ln[1 – α].
6.1 These test methods are useful for research and
development, quality assurance, regulatory compliance, and
5.8 If activation energy values only are of interest, Eq 11
specification acceptance purposes.
may be solved under conditions of constant conversion to
yield:
6.2 Thedeterminationoftheorderofachemicalreactionor
transformation at specific temperatures or time conditions is
ln@∆t# 5 E/RT1b (11)
beyond the scope of these test methods.
where:
6.3 The activation energy results obtained by these test
∆t = lapsedtime(s),atconstantconversionandatisothermal
methods may be compared with those obtained from Test
temperature, T, and
Method E698 for nth order and accelerating reactions.Activa-
b = constant.
tion energy, pre-exponential factor, and reaction order results
Eq11hastheformofastraightline,y=mx+b,whereaplot
by these test methods may be compared to those for Test
of the logarithm of the lapsed time under a series of differing
Method E2041 for nth order reactions.
isothermal conditions versus the reciprocal of absolute tem-
perature (l/T) is linear with a slope equal to E/R.
7. Interferences
5.9 If activation energy values only are of interest, Eq 11
7.1 Theapproachisapplicableonlytoexothermicreactions.
maybesolvedunderconditionsofconstantconversionandthe
NOTE 3—Endothermic reactions are controlled by the rate of the heat
equality dα/dt = dH/dt/(H) to yield:
transfer of the apparatus and not by the kinetics of the reaction and may
ln dH/dt 52E/RT1b 5 m/T1b (12)
@ # not be evaluated by these test methods.
7.2 These test methods are intended for a reaction mecha-
where:
nism that does not change during the transition. These test
H = total heat of reaction (mJ),
methods assume a single reaction mechanism when the shape
dH/dt = instantaneous heat flow (mW),
ofthethermalcurveissmooth(asinFig.2andFig.3)anddoes
b = constant, and
not exhibit shoulders, multiple peaks, or discontinuation steps.
m = slope (K)
7.3 Test method precision is enhanced with the selection of
Eq12hastheformofastraightliney=mx+b,whereaplot
of the logarithm of the heat flow (ln[dH/dt]) at the peak of the the appropriate conversion function [f(α)] that minimizes the
number of experimental parameters determined. The shape of
exotherm under a series of differing isothermal temperature
conditions versus the reciprocal of the absolute temperature the thermal curve, as described in Section 11, may confirm the
selection of the nth order or accelerating models.
(1/T) is linear with a slope equal to E/R.
5.10 Aseries of isothermal experiments by Test MethodA, 7.4 Typical nth order reactions include those in which all
but one of the participating species are in excess.
B, and C described in Section 11 at four or more temperatures,
determines the kinetic parameters of activation energy, pre-
7.5 Typical accelerating reactions include thermoset cure,
exponentialfactorandreactionorder.Alternatively,thetimeto
crystallization and pyrotechnic reactions.
a condition of constant conversion for a series of experiments
7.6 For nth order kinetic reactions, these test methods
at four or more temperatures obtained by this or alternative
anticipate that the value of n is small, non-zero integers, such
Test Method D, described in Section 12, may be used to
as 1 or 2. These test methods should be used carefully when
determine activation energy only.
values of n are greater than 2 or are not a simple fraction, such
5.11 A series of not less than four isothermal DSC
as ⁄2 = 0.5.
experiments, covering a temperature range of approximately
7.7 Acceleratingkineticreactionsanticipatethatmandnare
10Kandatimelessthan100min(suchasthoseshowninFig.
fractionsbetween0and2andthattheirsum(m+n)islessthan
1) provides values for dα/dt,α,(1–α) and T to solve Eq 6, Eq
3.
7, Eq 9, and Eq 10.
7.8 Accelerating kinetic reactions anticipate that p is an
5.12 A series of not less than four isothermal DSC experi-
integer often with a value of ≤4.
mentscoveringatemperaturerangeofapproximately10Kand
a time less than 100 min provides dH/dt and T to solve Eq 12 7.9 Since these test methods use milligram quantities, it is
essential that the test specimens are homogeneous and repre-
5.13 A variety of time-to-event experiments such as oxida-
sentative of the larger samples from which they are taken.
tion induction time methods (Specification D3350 and Test
Methods D3895, D4565, D5483, D6186, and E1858) and 7.10 Test specimens may release toxic and corrosive efflu-
reaction induction time methods (Test Method E2046) provide ents that may be harmful to personnel or apparatus. Operation
with a venting or exhaust system is recommended.
values for ∆t and T to solve equation Eq 11.
E2070 − 13 (2018)
NOTE 1—This figure is for a crystallization application in which the reaction rate increases with decreasing temperature. Chemical reactions show an
increase in reaction rate with increasing temperature.
FIG. 1 Heat Flow Curves at a Series of Isothermal Temperatures
NOTE 4—Typically inert purge gases that inhibit sample oxidation are
8. Hazards
99.9+% pure nitrogen, helium or argon. Dry gases are rec
...
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: E2070 − 13 (Reapproved 2018)
Standard Test Methods for
Kinetic Parameters by Differential Scanning Calorimetry
Using Isothermal Methods
This standard is issued under the fixed designation E2070; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
1.1 Test Methods A, B, and C determine kinetic parameters 2.1 ASTM Standards:
for activation energy, pre-exponential factor and reaction order D3350 Specification for Polyethylene Plastics Pipe and Fit-
using differential scanning calorimetry from a series of isother- tings Materials
mal experiments over a small ( ≈10 K) temperature range. Test D3895 Test Method for Oxidative-Induction Time of Poly-
Method A is applicable to low nth order reactions. Test olefins by Differential Scanning Calorimetry
Methods B and C are applicable to accelerating reactions such D4565 Test Methods for Physical and Environmental Per-
as thermoset curing or pyrotechnic reactions and crystallization formance Properties of Insulations and Jackets for Tele-
transformations in the temperature range from 300 to 900 K communications Wire and Cable
(nominally 30 to 630°C). These test methods are applicable D5483 Test Method for Oxidation Induction Time of Lubri-
only to these types of exothermic reactions when the thermal cating Greases by Pressure Differential Scanning Calorim-
curves do not exhibit shoulders, double peaks, discontinuities etry
or shifts in baseline. D6186 Test Method for Oxidation Induction Time of Lubri-
cating Oils by Pressure Differential Scanning Calorimetry
1.2 Test Methods D and E also determines the activation
(PDSC)
energy of a set of time-to-event and isothermal temperature
E473 Terminology Relating to Thermal Analysis and Rhe-
data generated by this or other procedures
ology
1.3 The values stated in SI units are to be regarded as
E537 Test Method for The Thermal Stability of Chemicals
standard. No other units of measurement are included in this
by Differential Scanning Calorimetry
standard.
E698 Test Method for Kinetic Parameters for Thermally
1.4 These test methods are similar but not equivalent to Unstable Materials Using Differential Scanning Calorim-
etry and the Flynn/Wall/Ozawa Method
ISO DIS 11357, Part 5, and provides more information than the
ISO standard. E967 Test Method for Temperature Calibration of Differen-
tial Scanning Calorimeters and Differential Thermal Ana-
1.5 This standard does not purport to address all of the
lyzers
safety concerns, if any, associated with its use. It is the
E968 Practice for Heat Flow Calibration of Differential
responsibility of the user of this standard to establish appro-
Scanning Calorimeters
priate safety, health, and environmental practices and deter-
E1142 Terminology Relating to Thermophysical Properties
mine the applicability of regulatory limitations prior to use.
E1445 Terminology Relating to Hazard Potential of Chemi-
Specific precautionary statements are given in Section 8.
cals
1.6 This international standard was developed in accor-
E1858 Test Methods for Determining Oxidation Induction
dance with internationally recognized principles on standard-
Time of Hydrocarbons by Differential Scanning Calorim-
ization established in the Decision on Principles for the
etry
Development of International Standards, Guides and Recom-
E1860 Test Method for Elapsed Time Calibration of Ther-
mendations issued by the World Trade Organization Technical
mal Analyzers
Barriers to Trade (TBT) Committee.
E1970 Practice for Statistical Treatment of Thermoanalytical
Data
These test methods are under the jurisdiction of ASTM Committee E37 on
Thermal Measurements and is the direct responsibility of Subcommittee E37.01 on
Calorimetry and Mass Loss. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved April 1, 2018. Published May 2018. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 2000. Last previous edition approved in 2013 as E2070 – 13. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/E2070-13R18. the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2070 − 13 (2018)
E2041 Test Method for Estimating Kinetic Parameters by where:
Differential Scanning Calorimeter Using the Borchardt
n, ₥, and p = partial reaction order terms.
and Daniels Method
NOTE 1—There are a large number of conversion function expressions
E2046 Test Method for Reaction Induction Time by Thermal for [f(α)]. Those described here are the most common but are not the only
functions suitable for these test methods Eq 1 is known as the general rate
Analysis
5,6
equation while Eq 3 is the accelerating (or Sestak-Berggren) equation.
2.2 ISO Standard:
Eq 4 is the accelerating Avrami equation. Eq 2 is used for nth order
ISO DIS 11357 Part 5: Determination of Temperature and/or
reactions while Eq 3 or Eq 4 are used for accelerating reaction, such as
Time of Reaction and Reaction Kinetics
thermoset cure and crystallization transformations.
5.3 For a reaction conducted at temperature (T), the accel-
3. Terminology
erating rate Eq 3 and the rate equation Eq 1 may be cast in their
3.1 Specific technical terms used in these test methods are
logarithmic form.
defined in Terminologies E473, E1142, and E1445, including
₥ n
dα/dt5 k~T! α ~1 2 α! (5)
the terms calorimeter, Celsius, crystallization, differential
ln dα/dt 5 ln k T 1₥ ln α 1n ln 1 2 α (6)
@ # @ ~ !# @ # @ #
scanning calorimetry, general rate law, isothermal, peak, and
reaction.
This equation has the form z = a + bx + cy and may be solved
using multiple linear regression analysis where x = ln[α], y =
4. Summary of Test Method
ln[1 – α], z = ln[dα/dt], a = ln[k(T)], b = ₥ and c = n.
4.1 A test specimen is held at a constant temperature in a
NOTE 2—The rate equation (Eq 3) reduces to the simpler general rate
differential scanning calorimeter throughout an exothermic
equation (Eq 2) when the value of reaction order parameter ₥ equals zero
reaction. The rate of heat evolution, developed by the reaction,
thereby reducing the number of kinetic parameters to be determined.
is proportional to the rate of reaction. Integration of the heat
5.4 For reactions conducted at temperature (T), the acceler-
flow as a function of time yields the total heat of reaction.
ating rate equation of Eq 4 may be cast as:
4.2 An accelerating (Sestak-Berggren or Avrami models),
4,5,6 ln@2 ln 1 2 α # 5 p ln@k T #1p ln@t# (7)
~ ! ~ !
nth order data, or model free treatment is used to derive the
kinetic parameters of activation energy, pre-exponential factor
This equation has the form of y = mx + b and may be solved
and reaction order from the heat flow and total heat of reaction
by linear regression where x = ln[t], y = ln[-ln(1 – α)], with p
information obtained in 4.1. (See Basis for Methodology,
= m, b = p ln[k(T)], and t = time.
Section 5.)
5.5 The Arrhenius equation describes how the reaction rate
changes as a function of temperature:
5. Basis of Methodology
2E/RT
k T 5 Z e (8)
~ !
5.1 Reactions of practical consideration are exothermic in
nature; that is, they give off heat as the reaction progresses.
where:
Furthermore, the rate of heat evolution is proportional to the
–1
Z = pre-exponential factor (s ),
rate of the reaction. Differential scanning calorimetry measures –1
E = activation energy (J mol ),
heat flow as a dependent experimental parameter as a function
T = absolute temperature (K),
–1 –1
of time under isothermal experimental conditions. DSC is
R = gas constant = (8.314 J mol K ), and
useful for the measurement of the total heat of a reaction and
e = natural logarithm base = 2.7182818.
the rate of the reaction as a function of time and temperature.
5.6 Eq 8 cast in its logarithmic form is:
5.2 Reactions may be modeled with a number of suitable
ln@k~T!# 5 ln@Z# 2 E/RT (9)
equations of the form of:
Eq 9 has the form of a straight line, y = mx + b, where a plot
dα/dt5 k T f α (1)
~ ! ~ !
of the logarithm of the reaction rate constant (ln[k(T)]) versus
where:
the reciprocal of absolute temperature (l/T) is linear with the
–1
dα/dt = reaction rate (s ), slope equal to –E/R and an intercept equal to ln[Z].
α = fraction reacted (dimensionless),
5.7 As an alternative to Eq 6 and Eq 7, the rate and
–1
k (T) = specific rate constant at temperature T (s ),
Arrhenius equations combined and cast in logarithmic form is:
f (α) = conversion function. Commonly used functions
ln dα/dt 5 ln Z 2 E/RT1m ln α 1n ln 1 2 α (10)
include: @ # @ # @ # @ #
n
f ~α! 5 ~1 2 α! (2)
Eq 10 has the form, z = a + bx + cy + dw, and may be solved
₥ n using multiple linear regression analysis.
f ~α! 5 α ~1 2 α! (3)
p 2 1 ⁄p
~ !
f α 5 p 1 2 α 2 1 n 1 2 α (4)
~ ! ~ !@ ~ !# where:
z = ln[dα/dt]
3 a = ln[Z]
Available from American National Standards Institute (ANSI), 25 W. 43rd St.,
b = -E/R
4th Floor, New York, NY 10036, http://www.ansi.org.
Sbirrazzuoli, N., Brunel, D., and Elegant, L., Journal of Thermal Analysis, Vol x = 1/T
38, 1992, pp. 1509–1524.
c = ₥
Sestak, J., and Berggren, G., Thermochimica Acta, Vol 3, 1971, p. 1.
y = ln[1 – α]
Gorbachiev, V.M., Journal of Thermal Analysis, Vol 18, 1980, pp. 193–197.
E2070 − 13 (2018)
6. Significance and Use
d = n, and
w = ln[1 – α].
6.1 These test methods are useful for research and
development, quality assurance, regulatory compliance, and
5.8 If activation energy values only are of interest, Eq 11
specification acceptance purposes.
may be solved under conditions of constant conversion to
yield:
6.2 The determination of the order of a chemical reaction or
transformation at specific temperatures or time conditions is
ln Δt 5 E/RT1b (11)
@ #
beyond the scope of these test methods.
where:
6.3 The activation energy results obtained by these test
Δt = lapsed time (s), at constant conversion and at isothermal
methods may be compared with those obtained from Test
temperature, T, and
Method E698 for nth order and accelerating reactions. Activa-
b = constant.
tion energy, pre-exponential factor, and reaction order results
Eq 11 has the form of a straight line, y = mx + b, where a plot
by these test methods may be compared to those for Test
of the logarithm of the lapsed time under a series of differing
Method E2041 for nth order reactions.
isothermal conditions versus the reciprocal of absolute tem-
perature (l/T) is linear with a slope equal to E/R.
7. Interferences
5.9 If activation energy values only are of interest, Eq 11
7.1 The approach is applicable only to exothermic reactions.
may be solved under conditions of constant conversion and the
NOTE 3—Endothermic reactions are controlled by the rate of the heat
equality dα/dt = dH/dt / (H) to yield:
transfer of the apparatus and not by the kinetics of the reaction and may
ln@dH/dt# 5 2E/RT1b 5 m/T1b (12)
not be evaluated by these test methods.
7.2 These test methods are intended for a reaction mecha-
where:
nism that does not change during the transition. These test
H = total heat of reaction (mJ),
methods assume a single reaction mechanism when the shape
dH/dt = instantaneous heat flow (mW),
of the thermal curve is smooth (as in Fig. 2 and Fig. 3) and does
b = constant, and
not exhibit shoulders, multiple peaks, or discontinuation steps.
m = slope (K)
Eq 12 has the form of a straight line y = mx + b, where a plot 7.3 Test method precision is enhanced with the selection of
the appropriate conversion function [f(α)] that minimizes the
of the logarithm of the heat flow (ln[dH/dt]) at the peak of the
exotherm under a series of differing isothermal temperature number of experimental parameters determined. The shape of
the thermal curve, as described in Section 11, may confirm the
conditions versus the reciprocal of the absolute temperature
(1/T) is linear with a slope equal to E/R. selection of the nth order or accelerating models.
7.4 Typical nth order reactions include those in which all
5.10 A series of isothermal experiments by Test Method A,
but one of the participating species are in excess.
B, and C described in Section 11 at four or more temperatures,
determines the kinetic parameters of activation energy, pre-
7.5 Typical accelerating reactions include thermoset cure,
exponential factor and reaction order. Alternatively, the time to
crystallization and pyrotechnic reactions.
a condition of constant conversion for a series of experiments
7.6 For nth order kinetic reactions, these test methods
at four or more temperatures obtained by this or alternative
anticipate that the value of n is small, non-zero integers, such
Test Method D, described in Section 12, may be used to
as 1 or 2. These test methods should be used carefully when
determine activation energy only.
values of n are greater than 2 or are not a simple fraction, such
5.11 A series of not less than four isothermal DSC
as ⁄2 = 0.5.
experiments, covering a temperature range of approximately
7.7 Accelerating kinetic reactions anticipate that m and n are
10 K and a time less than 100 min (such as those shown in Fig.
fractions between 0 and 2 and that their sum (m + n) is less than
1) provides values for dα/dt, α, (1 – α) and T to solve Eq 6, Eq
3.
7, Eq 9, and Eq 10.
7.8 Accelerating kinetic reactions anticipate that p is an
5.12 A series of not less than four isothermal DSC experi-
integer often with a value of ≤4.
ments covering a temperature range of approximately 10 K and
a time less than 100 min provides dH/dt and T to solve Eq 12 7.9 Since these test methods use milligram quantities, it is
essential that the test specimens are homogeneous and repre-
5.13 A variety of time-to-event experiments such as oxida-
sentative of the larger samples from which they are taken.
tion induction time methods (Specification D3350 and Test
Methods D3895, D4565, D5483, D6186, and E1858) and 7.10 Test specimens may release toxic and corrosive efflu-
reaction induction time methods (Test Method E2046) provide ents that may be harmful to personnel or apparatus. Operation
values for Δt and T to solve equation Eq 11. with a venting or exhaust system is recommended.
E2070 − 13 (2018)
NOTE 1—This figure is for a crystallization application in which the reaction rate increases with decreasing temperature. Chemical reactions show an
increase in reaction rate with increasing temperature.
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E2070 − 13 E2070 − 13 (Reapproved 2018)
Standard Test MethodMethods for
Kinetic Parameters by Differential Scanning Calorimetry
Using Isothermal Methods
This standard is issued under the fixed designation E2070; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 Test Methods A, B, and C determine kinetic parameters for activation energy, pre-exponential factor and reaction order using
differential scanning calorimetry from a series of isothermal experiments over a small ( ≈10 K) temperature range. Test Method
A is applicable to low nth order reactions. Test Methods B and C are applicable to accelerating reactions such as thermoset curing
or pyrotechnic reactions and crystallization transformations in the temperature range from 300 to 900 K (nominally 30 to 630°C).
ThisThese test method ismethods are applicable only to these types of exothermic reactions when the thermal curves do not exhibit
shoulders, double peaks, discontinuities or shifts in baseline.
1.2 Test Methods D and E also determines the activation energy of a set of time-to-event and isothermal temperature data
generated by this or other procedures
1.3 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.4 ThisThese test method ismethods are similar but not equivalent to ISO DIS 11357, Part 5, and provides more information
than the ISO standard.
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety safety, health, and healthenvironmental practices and determine the
applicability of regulatory limitations prior to use. Specific precautionary statements are given in Section 8.
1.6 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
D3350 Specification for Polyethylene Plastics Pipe and Fittings Materials
D3895 Test Method for Oxidative-Induction Time of Polyolefins by Differential Scanning Calorimetry
D4565 Test Methods for Physical and Environmental Performance Properties of Insulations and Jackets for Telecommunications
Wire and Cable
D5483 Test Method for Oxidation Induction Time of Lubricating Greases by Pressure Differential Scanning Calorimetry
D6186 Test Method for Oxidation Induction Time of Lubricating Oils by Pressure Differential Scanning Calorimetry (PDSC)
E473 Terminology Relating to Thermal Analysis and Rheology
E537 Test Method for The Thermal Stability of Chemicals by Differential Scanning Calorimetry
E698 Test Method for Kinetic Parameters for Thermally Unstable Materials Using Differential Scanning Calorimetry and the
Flynn/Wall/Ozawa Method
E967 Test Method for Temperature Calibration of Differential Scanning Calorimeters and Differential Thermal Analyzers
E968 Practice for Heat Flow Calibration of Differential Scanning Calorimeters
E1142 Terminology Relating to Thermophysical Properties
E1445 Terminology Relating to Hazard Potential of Chemicals
E1858 Test Methods for Determining Oxidation Induction Time of Hydrocarbons by Differential Scanning Calorimetry
ThisThese test method ismethods are under the jurisdiction of ASTM Committee E37 on Thermal Measurements and is the direct responsibility of Subcommittee E37.01
on Calorimetry and Mass Loss.
Current edition approved Sept. 15, 2013April 1, 2018. Published October 2013May 2018. Originally approved in 2000. Last previous edition approved in 20082013 as
E2070 – 08.E2070 – 13. DOI: 10.1520/E2070-13.10.1520/E2070-13R18.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2070 − 13 (2018)
E1860 Test Method for Elapsed Time Calibration of Thermal Analyzers
E1970 Practice for Statistical Treatment of Thermoanalytical Data
E2041 Test Method for Estimating Kinetic Parameters by Differential Scanning Calorimeter Using the Borchardt and Daniels
Method
E2046 Test Method for Reaction Induction Time by Thermal Analysis
2.2 ISO Standard:
ISO DIS 11357 Part 5: Determination of Temperature and/or Time of Reaction and Reaction Kinetics
3. Terminology
3.1 Specific technical terms used in thisthese test methodmethods are defined in Terminologies E473, E1142, and E1445,
including the terms calorimeter, Celsius, crystallization, differential scanning calorimetry, general rate law, isothermal, peak, and
reaction.
4. Summary of Test Method
4.1 A test specimen is held at a constant temperature in a differential scanning calorimeter throughout an exothermic reaction.
The rate of heat evolution, developed by the reaction, is proportional to the rate of reaction. Integration of the heat flow as a
function of time yields the total heat of reaction.
4,5,6
4.2 An accelerating (Sestak-Berggren or Avrami models), nth order data, or model free treatment is used to derive the kinetic
parameters of activation energy, pre-exponential factor and reaction order from the heat flow and total heat of reaction information
obtained in 4.1. (See Basis for Methodology, Section 5.)
5. Basis of Methodology
5.1 Reactions of practical consideration are exothermic in nature; that is, they give off heat as the reaction progresses.
Furthermore, the rate of heat evolution is proportional to the rate of the reaction. Differential scanning calorimetry measures heat
flow as a dependent experimental parameter as a function of time under isothermal experimental conditions. DSC is useful for the
measurement of the total heat of a reaction and the rate of the reaction as a function of time and temperature.
5.2 Reactions may be modeled with a number of suitable equations of the form of:
dα/dt 5 k T f α (1)
~ ! ~ !
where:
–1
dα/dt = reaction rate (s ),
α = fraction reacted (dimensionless),
–1
k (T) = specific rate constant at temperature T (s ),
f (α) = conversion function. Commonly used functions include:
n
f α 5 12 α (2)
~ ! ~ !
₥ n
f α 5 α 12 α (3)
~ ! ~ !
~p 2 1!⁄p
f α 5 p 1 2 α @2 1 n 1 2 α # (4)
~ ! ~ ! ~ !
where:
n, ₥, and p = partial reaction order terms.
NOTE 1—There are a large number of conversion function expressions for [f(α)]. Those described here are the most common but are not the only
functions suitable for thisthese test method.methods Eq 1 is known as the general rate equation while Eq 3 is the accelerating (or Sestak-Berggren)
5,6
equation. Eq 4 is the accelerating Avrami equation. Eq 2 is used for nth order reactions while Eq 3 or Eq 4 are used for accelerating reaction, such
as thermoset cure and crystallization transformations.
5.3 For a reaction conducted at temperature (T), the accelerating rate Eq 3 and the rate equation Eq 1 may be cast in their
logarithmic form.
₥ n
dα/dt 5 k~T! α ~12 α! (5)
ln dα/dt 5 ln k T 1₥ ln α 1n ln 12 α (6)
@ # @ ~ !# @ # @ #
This equation has the form z = a + bx + cy and may be solved using 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 rate equation (Eq 2) when the value of reaction order parameter ₥ equals zero thereby
Available from American National Standards Institute (ANSI), 25 W. 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org.
Sbirrazzuoli, N., Brunel, D., and Elegant, L., Journal of Thermal Analysis, Vol 38, 1992, pp. 1509–1524.
Sestak, J., and Berggren, G., Thermochimica Acta, Vol 3, 1971, p. 1.
Gorbachiev, V.M., Journal of Thermal Analysis, Vol 18, 1980, pp. 193–197.
E2070 − 13 (2018)
reducing the number of kinetic parameters to be determined.
5.4 For reactions conducted at temperature (T), the accelerating rate equation of Eq 4 may be cast as:
ln@2 ln ~1 2 α!# 5 p ln@k ~T!#1p ln@t# (7)
This equation has the form of y = mx + b and may be solved by 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 rate changes as a function of temperature:
2E/RT
k~T! 5 Z e (8)
where:
–1
Z = pre-exponential factor (s ),
–1
E = activation energy (J mol ),
T = absolute temperature (K),
–1 –1
R = gas constant = (8.314 J mol K ), and
e = natural logarithm base = 2.7182818.
5.6 Eq 8 cast in its logarithmic form is:
ln@k~T!# 5 ln@Z# 2 E/RT (9)
Eq 9 has the form of a straight line, y = mx + b, where a plot of the logarithm of the reaction rate constant (ln[k(T)]) versus the
reciprocal of absolute temperature (l/T) is linear with the slope equal to –E/R and an intercept equal to ln[Z].
5.7 As an alternative to Eq 6 and Eq 7, the rate and Arrhenius equations combined and cast in logarithmic form is:
ln dα/dt 5 ln Z 2 E/RT1m ln α 1n ln 12 α (10)
@ # @ # @ # @ #
Eq 10 has the form, z = a + bx + cy + dw, and may be solved using multiple linear regression analysis.
where:
z = ln[dα/dt]
a = ln[Z]
b = -E/R
x = 1/T
c = ₥
y = ln[1 – α]
d = n, and
w = ln[1 – α].
5.8 If activation energy values only are of interest, Eq 11 may be solved under conditions of constant conversion to yield:
ln@Δt# 5 E/RT1b (11)
where:
Δt = lapsed time (s), at constant conversion and at isothermal temperature, T, and
b = constant.
Eq 11 has the form of a straight line, y = mx + b, where a plot of the logarithm of the lapsed time under a series of differing
isothermal conditions versus the reciprocal of absolute temperature (l/T) is linear with a slope equal to E/R.
5.9 If activation energy values only are of interest, Eq 11 may be solved under conditions of constant conversion and the
equality dα/dt = dH/dt / (H) to yield:
ln dH/dt 52E/RT1b 5 m/T1b (12)
@ #
where:
H = total heat of reaction (mJ),
dH/dt = instantaneous heat flow (mW),
b = constant, and
m = slope (K)
Eq 12 has the form of a straight line y = mx + b, where a plot of the logarithm of the heat flow (ln[dH/dt]) at the peak of the
exotherm under a series of differing isothermal temperature conditions 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 to a
E2070 − 13 (2018)
condition of constant conversion for a series of experiments at four or more temperatures obtained by this or alternative Test
Method D, described in Section 12, may be used to determine activation energy only.
5.11 A series of not less than four isothermal DSC experiments, covering a temperature range of approximately 10 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, Eq 7, Eq 9, and Eq
10.
5.12 A series of not less than four isothermal DSC experiments covering a temperature range of approximately 10 K and a time
less than 100 min provides dH/dt and T to solve Eq 12
5.13 A variety of time-to-event experiments such as oxidation induction time methods (Practice(Specification D3350 and Test
Methods D3895, D4565, D5483, D6186, and E1858) and reaction induction time methods (Test Method E2046) provide values
for Δt and T to solve equation Eq 11.
6. Significance and Use
6.1 ThisThese test method ismethods are useful for research and development, quality assurance, regulatory compliance, and
specification acceptance purposes.
6.2 The determination of the order of a chemical reaction or transformation at specific temperatures or time conditions is beyond
the scope of thisthese test method.methods.
6.3 The activation energy results obtained by thisthese test methodmethods may be compared with those obtained from Test
Method E698 for nth order and accelerating reactions. Activation energy, pre-exponential factor, and reaction order results by
thisthese test methodmethods may be compared to those for Test Method E2041 for nth order reactions.
7. Interferences
7.1 The approach is applicable only to exothermic reactions.
NOTE 3—Endothermic reactions are controlled by the rate of the heat transfer of the apparatus and not by the kinetics of the reaction and may not be
evaluated by thisthese test method.methods.
NOTE 1—This figure is for a crystallization application in which the reaction rate increases with decreasing temperature. Chemical reactions show an
increase in reaction rate with increasing temperature.
FIG. 1 Heat Flow Curves at a Series of Isothermal Temperatures
E2070 − 13 (2018)
7.2 ThisThese test method ismethods are intended for a reaction mechanism that does not change during the transition.
ThisThese test method assumesmethods assume a single reaction mechanism when the shape of the thermal curve is smooth (as
in Fig. 2 and Fig. 3) and does not exhibit shoulders, multiple peaks, or discontinuation steps.
7.3 Test method precision is enhanced with the selection of the appropriate conversion function [f(α)] that minimizes the number
of experimental parameters determined. The shape of the thermal curve, as described in Section 11, may confirm the selection of
the nth order or accelerating models.
7.4 Typical nth order reactions include those in which all but 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, thisthese test method anticipatesmethods anticipate that the value of n is small, non-zero
integers, such as 1 or 2. ThisThese test methodmethods should be used carefully when values of n are greater than 2 or are not
...
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