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ISSN 0012-5016, Doklady Physical Chemistry, 2007, Vol. 416, Part 2, pp. 265–270. © Pleiades Publishing, Ltd., 2007.Original Russian Text © A.A. Koptelov, Yu.M. Milekhin, S.A. Gusev, 2007, published in Doklady Akademii Nauk, 2007, Vol. 416, No. 4, pp. 496–501.
265
Studies of the formal kinetics of thermal decompo-sition of energetic materials in the condensed phase arenecessary for estimating the reaction rates under theconditions of their synthesis, storage, processing, anduse and also for solving a number of basic problems ofchemical physics [1]. Although experimental equip-ment is continuously being improved, the publisheddata on the kinetic parameters of thermal decomposi-tion of energetic materials show very wide scatters. Inour opinion, these scatters are largely determined bysubjective factors, first of all, insufficiently detaileddesign of experiments. Among methods for studyingthe kinetics of transformation of energetic materials onheating, the most informative are thermal analysis (TA)methods: differential scanning calorimetry (DSC), dif-ferential thermal analysis (DTA), and dynamic and iso-thermal thermogravimetry (TG). The purpose of thiswork is to analyze the sources of errors in determiningthe kinetic parameters of thermal decomposition ofenergetic materials from thermal analysis data and tofind the experimental conditions under which theseerrors are minimal. The quantitative data presentedbelow on decomposition kinetics mainly describe octo-gen (HMX); however, the conclusions drawn are alsoequally applicable to other types of energetic materials.
The scatter of experimental values of the kineticconstants for thermal decomposition of HMX in thecondensed phase can be illustrated by the monomolec-ular reaction data (Table 1):
(1)
where
η
is the degree of decomposition,
τ
is time,
E
isthe activation energy, and
Z
is the preexponential factor.
In our calculations, along with the data in Table 1,we also used other estimates for the first-order reaction,which were taken from Burnham and Weese’s work [4].
dηdτ------ Z
ERT-------–⎝ ⎠
⎛ ⎞ 1 η–( ),exp=
By DSC, DTA, and TG methods, they obtained 38 dif-ferent values of
E
and
Z
, which range from 125 to216 kJ/mol, depending on
η
and
T
.
Using these kinetic constants, let us estimate thecritical temperatures
T
cr
for HMX samples shaped asflat plates with a half-thickness
r
of
2.5
×
10
–3
to1.0 mm. In an isothermal experiment,
T
cr
is found fromthe equation
(2)δ*E
RTcr----------⎝ ⎠
⎛ ⎞exp r2ρQZE
λRTcr2
--------------------= .
Application of Thermal Analysis Methods to Studyingthe Kinetics of Thermal Decomposition of Energetic Materials
A. A. Koptelov, Yu. M. Milekhin, and S. A. Gusev
Presented by Academician Yu.D. Tret’yakov May 21, 2007
Received May 21, 2007
DOI:
10.1134/S0012501607100016
Soyuz Federal Center for Dual-Use Technologies,ul. Akademika Zhukova 42, Dzerzhinskii,Moscow oblast, 140090 Russia
PHYSICALCHEMISTRY
Table 1.
Kinetic parameters of thermal decomposition ofHMX in the condensed phase (monomolecular reaction) [2
−
4]
No. Method
T
i
– ,K
E
,kJ/mol
Z
,s
–1
1 DSC 504–547 220.6 5.00
×
10
19
2 DSC, DTA, TG 473–533 165.0 2.00
×
10
13
3 IR spectroscopy 439–467 204.3 6.31
×
10
19
4 DTA 502–542 187.1 7.94
×
10
15
5 Chromatography 503–543 221.4 2.51
×
10
18
6 TG (O
2
) 502–542 208.0 1.26
×
10
17
7 TG (air) 502–542 205.1 1.00
×
10
17
8 DTA (N
2
, 3.5 MPa) 502–542 108.4 3.16
×
10
8
9 TG (N
2
, 10
5
MPa) 502–542 146.9 3.16
×
10
11
10 DSC 544–558 214.6 6.31
×
10
18
11 Barometry 403–453 158.6 1.58
×
10
9
12 Barometry 534–549 205.1 6.31
×
10
17
13 DTA 433–523 113.0 1.00
×
10
10
14 DTA 458–533 172.5 2.00
×
10
14
15 Barometry 453–458 171.6 6.31
×
10
12
16 Barometry 423–443 150.7 6.31
×
10
10
17 Barometry 449–503 158.6 1.58
×
10
11
* Initial and final values of the investigation temperature range.
T f*
266
DOKLADY PHYSICAL CHEMISTRY
Vol. 416
Part 2
2007
KOPTELOV et al.
The names and numerical values of the parameters ofEq. (2) that were used in our calculations are presentedin Table 2. We can write
δ
* =
ϕ
(Bi)
δ
0
[5], where, for theplanar geometry, we have
δ
0
= 0.88; the function
ϕ
(Bi)
for arbitrary Biot numbers can be calculated as
The results of calculating
T
cr
are shown in Fig. 1.Table 3 presents the most probable values of the criticaltemperatures and the root-mean-square deviations
σ
,which were calculated under the assumption that thedistribution is Gaussian. The found
T
cr
values corre-spond best to the values
E
0
= 165 kJ/mol and
Z
0
=2.0
×
10
13
s
–1
, which were recommended [6] for use in“global kinetics” models (Fig. 1, curve
2
). These
E
and
Z
data were further used as calculation reference val-ues. The data found from barometric experiments(Fig. 1, curves
11
,
15
–
17
) and by IR spectroscopy(Fig. 1, curve
3
) are farther than
±
2
σ
from the val-ues (the exception is barometric curve
12
). Somecauses of the inaccuracy of determining the kinetic con-stants by barometric methods have been considered inthe literature [1, 7].
The experimental data indicate that the HMX sam-ples decompose with acceleration (are ignited) either
ϕ Bi( ) = Bi2----- Bi2 4+ Bi–( ) Bi2 4+ Bi 2––
Bi----------------------------------------- .exp
Tcr
Tcr
directly at the onset of melting or after partial melting[4, 8]. An abrupt increase in the decomposition reactionrate in the formation of the liquid phase is also charac-teristic of other energetic materials. Therefore, the cal-culated Tcr values exceeding the melting point Tm(for HMX, ~550 K) are conditional. Previously deter-mined [8] extremely high (1100–8800 kJ/mol) E valuesare likely to be related to attempts to describe thedecomposition curves in the melting zone.
Let us consider the possibilities of the DSC methodin studying the kinetics of decomposition of energeticmaterials in the solid phase. DSC devices record thepower q (W) of heat release (absorption) on heating atest sample. According to a mathematical model ofDSC [9], the set of equations describing the thermaldecomposition of energetic materials in the dynamicheating mode has the form
(3)
(4)
ΨΣCsdqdτ------ q– Cs Cr–( )
dT p
dτ---------+=
– m0QZE
RTs
---------–⎝ ⎠⎛ ⎞ f η( ),exp
dηdτ------ Z
ERTs
---------–⎝ ⎠⎛ ⎞ f η( ).exp=
Table 2. Names and numerical values of the parameters of Eqs. (2) and (3)
Characteristic Notation Dimension Numerical value
Sample half-thickness r m from 2.5 × 10–6 to 1.0 × 10–3
Sample density ρ kg/m3 1340*
Thermal conductivity of HMX λ W/(m K) 0.29 [15]
Specific heat of HMX decomposition Q J/kg 2.0 × 106 [15]
Sample surface area S m2 1.6 × 10–5
Specific heat of HMX C J/(kg K) 391 + 2.09T [8]
Initial weight of sample m0 kg from 1.0 × 10–7 to 5.0 × 10–5
Specific heat of aluminum container (crucible) Ct J/(kg K) 785 + 0.42T
Container weight mt kg 6.5 × 10–5
Biot number Bi** – from 7.0 × 10–3 to 3.0
Initial temperature of sample T0 K 293
* ρ = kρ0, where ρ0 = 1810 kg/m3 [15] and k = 0.74 (closest packing of identical spheres).
** Bi = .r
ΨSλ-----------
Table 3. Most probable values of the critical temperatures at various characteristic sizes of samples of an energetic materialwith characteristics identical to those of HMX
r, mm 2.48 × 10–3 6.74 × 10–3 1.83 × 10–2 4.98 × 10–2 0.135 0.368 1.00
, K 614 596 584 564 547 529 509
σ, K 19 11 21 20 14 6 9
Tcr
DOKLADY PHYSICAL CHEMISTRY Vol. 416 Part 2 2007
APPLICATION OF THERMAL ANALYSIS METHODS TO STUDYING THE KINETICS 267
Here, Cs = Cm + Ctmt, Cr = Ctmt, dTp/dτ = b is the pro-grammed heating rate, Tp is the heating temperature,and m ≈ m0(1 – η) is the current weight of the sample ofenergetic material; the names and numerical values ofthe other quantities are presented in Table 2. The valid-ity of using Eq. (3) is confirmed by available experi-mental data. The heat resistance ΨΣ in Eq. (3) is gener-ally a complex function of heat-transfer conditions andtemperature differences in the sample–container–tem-perature sensor–heater system. We restrict our consid-eration to the case where ΨΣ = Ψ, where Ψ is limited bythe device; then, in Eqs. (3) and (4), Ts = Tp – qΨ =T0 + bτ – qΨ. The minimal value Ψ = Ψ0 for “classical”DSC devices (Perkin-Elmer) that was found in studyingthe melting of metals was estimated at 75 K/W [10].The position of the baseline for q in the model consid-ered is determined by solving the set of Eqs. (3) and (4)at m0 = 0, and in a real experiment, by the method ofsuccessive approximations [11]. In measurements inthe mode b = const, it is most convenient to comparequantities of the same scale, namely, the “true” rate ofchange in η with an increase in temperature
(5)
and the “normalized” DSC signal
(6)
Solving the set of Eqs. (3) and (4) under the initialconditions T(0) = 0 and η(0) = 0 at given initial E0 andZ0 values gave (Ti) values. The kinetic constants Eand Z at f(η) = 1 – η were determined as the parametersat which the data are best approximated by thefunction W(T) found from the integral form of Eq. (5).For f (η) of more complex form, we used data process-ing methods proposed before [4, 5]. The criterion ofquality of thermograms obtained in the numericalexperiment was the relative deviations ε of the calcu-lated constants E and Z from E0 and Z0. The characterof the influence of b, m0Q, and Ψ on the ratio betweenW* and W becomes clearer after minor transformationsof Eq. (3) using the inequality |qΨ| Tp, which is validfor small samples:
(7)
Equation (7) shows that the representation of thekinetic curve (W* → W) by the DSC signal is the moreaccurate the better the conditions m0QbΨ → 0 andΨCsb → 0 are met. An increase in b, m0Q, and Ψ causesa progressive increase in the deviation of the curves
WdηdT------≡ Z
b--- E
RT-------–⎝ ⎠
⎛ ⎞ f η( )exp=
W*q
Qm0b--------------.–=
Wi*
Wi*
W* W W*Qm0bΨE
RT p2
-----------------------⎩ ⎭⎨ ⎬⎧ ⎫
exp≈
– ΨCsbdW*dT
----------- CmQm0-----------.–
W*(T) from the curves W(T), which describe the “pure”kinetics, and, correspondingly, in the errors in deter-mining the kinetic constants. In the example given inFig. 2, the error of determining E at Ψ = 750 K/W, m0 =10 mg, and b = 0.015 K/s is ε = (E/E0 – 1) × 100% ≈ 16%.
The maximum admissible heating rate in studyingthe decomposition of energetic materials in the solidphase should be taken to be that at which, before theonset of melting, the degree of decomposition is η ≈ 1.From this condition, solving the set of Eqs. (3) and (4),we obtain that the maximal b value in studying the ther-mal decomposition of HMX in the solid phase shouldnot exceed 0.015 K/s (~1.0 K/min). The minimaladmissible heating rate can be estimated from the con-dition that the signal-to-noise ratio N is suitable, whichcan be written as
(8)
Here, qmax is the maximal heat release power in thesample, q* is the device sensitivity, and Nmin is the min-imum admissible N value. Taking Nmin = 100 and qmax =1 µW, we obtain q* = 0.1 mW. For base characteristicsof the sample, these values correspond to bmin =
qmax q*Nmin,≥
550
–6
Tcr, K
lnr[mm]
700
–2
650
600
500–4 0
1234567891011121314151617
Fig. 1. Critical ignition temperature versus logarithm of thehalf-thickness of an HMX sample shaped as a flat plate atvarious kinetic parameters of the monomolecular reactionof thermal decomposition. The numbering of curves in thefigure corresponds to the numbering of methods in Table 1.
268
DOKLADY PHYSICAL CHEMISTRY Vol. 416 Part 2 2007
KOPTELOV et al.
0.0017 K/s (~0.1 K/min). These estimates of the admis-sible range of b for HMX are in good agreement withexperimental data [4, 5].
Figure 3 shows the admissible region of studyingthe thermal decomposition of HMX in the solid phasein m0–b coordinates. The upper boundary for m0 wasfound from the condition that the value ε ≈ 5% for thecalculated E and/or lnZ values should not be exceeded.Samples of relatively large weight may be unsuitablebecause of the possible creation of conditions of adynamic mode of thermal self-ignition. There exists acritical heating rate above which the kinetic factor can-not prevent explosion [12]. The absence of thermalexplosion is guaranteed by satisfying the condition
(9)
where Ω is the dimensionless heating rate and Ω* is itscritical value. For a first-order reaction in the quasi-sta-
tionary approximation, we have Ω* = , where
γ = .
The calculations were performed for the most prob-able Tcr found above for HMX and their correspondingsample thicknesses 2r; the other parameters were takenfrom Table 2. The maximum admissible weights (m0 =
Ω CbE
RTcr----------⎝ ⎠
⎛ ⎞ / QZ( )exp Ω*,≤=
e2γ1 γe–--------------
CRTcr2
QE---------------
2ρrS) of samples of energetic materials that correspondto Ω = Ω* and the thicknesses of such samples at vari-ous b are presented below.
It is seen that the conditions under which there is nothermal explosion in the mode b = const in DSC devicesare reachable; therefore, the factor that limits the heat-ing rate in studying energetic materials by DSC is pri-marily the distortion of the position and shape of thedevice output signal (with respect to the curve describ-ing the “true” decomposition kinetics) because of finiteheat resistance. In DSC study of solid-phase energeticmaterials, powder samples are typically used. In thiscase, not only can Ψ significantly exceed 75 K/W, butit can also vary, depending on the fractional composi-tion of the sample, the packing density in the container,T, and b, which, in real experiments, can lead to verylarge errors and scatters in estimating the kinetic con-stants.
In combustion of most individual energetic materi-als, a melt film (more precisely, a two-phase region)forms on their surface [13]. The totality of physico-chemical processes in this region along with the heattransfer from the gas phase determines the combustionrate v of energetic materials. The transformations in thesolid phase occur at much lower rates and are likely tohave no noticeable effect on the character of combus-tion [13]. A necessary condition for studying the kinet-
b, K/s 0.01 0.1 0.5 1.0 5.0
m0, mg >50 15.6 4.6 2.6 0.7
2r, mm >2 0.73 0.21 0.12 0.033
0
450
W*, W, K–1
T, K
0.03
500
0.02
0.01
550
12
3
4
5
m0, mg
b, K/s10–2
10–1
100
101
102
1
2
3
5
4
Fig. 2. Functions W and W* versus programmed tempera-ture at m0 = 10 mg (2r = 0.46 mm), b = 0.0167 K/s, and Ψ =750 K/W: (1) W*(T) calculated using Eqs. (3), (4), and (6);(2) baseline; (3) W*(T) corrected for the baseline drift;(4) best approximation of curve 3 by the function W(T) =
exp exp (corresponds to
E = 192.3 kJ/mol and Z = 1.72 × 1016 s–1); and (5) W(T) cal-culated at E = E0 and Z = Z0.
Zb--- E
RT-------–⎝ ⎠
⎛ ⎞ Zb---– E
Rζ------– dζexp
T0
T
∫⎝ ⎠⎜ ⎟⎛ ⎞
Fig. 3. Approximate admissible region of heating rates andsample weights in studying the kinetics of thermal decom-position of HMX by DSC. Straight lines 1 and 2 are deter-mined by the device sensitivity. The position of straight line3 was found from the condition that samples must not melt.Straight lines 4 (Ψ = 75 K/W) and 5 (Ψ = 750 K/W) corre-spond to the relative deviation ε ≈ 5% of the calculated Eand lnZ values from given E0 and lnZ0 values.
10–3
DOKLADY PHYSICAL CHEMISTRY Vol. 416 Part 2 2007
APPLICATION OF THERMAL ANALYSIS METHODS TO STUDYING THE KINETICS 269
ics of decomposition of energetic materials in the meltis to reach quite high heating rates. The lower estimateof b on the surface of energetic materials in steady com-bustion can be written as
where Tw is the temperature on the combustion surfaceand a is the thermal diffusivity. For HMX, Tw ≈ 700 Kand a ≈ 1.0 × 10–7 m2/s. Taking v ≈ 4 × 10–4 m/s at apressure of p1 = 0.1 MPa and v ≈ 10–2 m/s at p2 =6.0 MPa [14], we obtain bmin = 6.4 × 102 and 4 × 105 K/s,respectively. The actual rates of heating of the surfaceof energetic materials can exceed 106 K/s [13]. Kineticstudies at such b using available TA devices are impos-sible. Even at b = 1.0 K/s, the error in determining E fora nonmelting energetic material with characteristicsidentical to those of HMX using an “ideal” DSC devicewith Ψ = 75 K/W exceeds 6–10%.
Let us evaluate the possibilities of studying thekinetics of thermal decomposition of energetic materi-als by the DSC method in the isothermal mode. Thetemperature is limited from below for the signal-to-noise ratio N to be high enough and from above lest thedegree of sample decomposition before reaching themode T = const be too high. Let us assume that we cancreate the following sample heating mode: (1) linearheating (b = const) until a given programmed tempera-ture Tp is reached at the moment of time τb (when thedegree of decomposition is ηb and the heat flux poweris qb) and (2) treatment of the sample at Tp = const fromthe moment of time τb for a time that ensures a suffi-ciently informative change in η. The first stage isdescribed by Eqs. (3) and (4) under the initial condi-tions q(0) = 0 and η(0) = 0. The second stage is charac-terized by the set of equations
(10)
(11)
under the initial conditions q(0) = qb and η(0) = ηb,where time is counted from the moment of reaching theisothermal mode. Calculation at a rate of reaching theisothermal mode of b = 0.1 K/s for HMX samples atE = E0, Z = Z0, and f(η) = 1 – η gives the following val-ues for qb and ηb at various T = Tp:
Figure 4 illustrates the dependence η(τ) at variousTp (time is counted from τb). Figure 4 also presents theη versus time curves describing the “pure” kinetics
Tp, K 473 493 513 533 548
qb, W 1.26 × 10–4 2.1 × 10–5 –4.4 × 10–4–2.1 × 10–3–4.3 × 10–3
ηb 0.0013 0.0077 0.0388 0.1686 0.4196
bmin Tw T0–( )v2
a------,≈
ΨCsdqdτ------ q– m0QZ
ER T p qΨ–[ ]----------------------------–⎝ ⎠
⎛ ⎞ f η( ),exp–=
dηdτ------ Z
ER T p qΨ–[ ]----------------------------–⎝ ⎠
⎛ ⎞ f η( )exp=
(when the mode T = const is reached instantaneously)at T = 493, 503, 513, and 533 K; the calculation wasperformed using the formula
(12)
The minimal temperature at which the measurementaccuracy is satisfactory is found from the condition q ≥q*Nmin. At the above q* and Nmin, a suitable q value forQm0 = 2 J is reached at Tp ≥ 493 K (when q ≥ 0.1 mWfor η ≤ 0.24). At the other Tp, the admissible signal lev-els correspond to times τ limited from the right by bro-ken line B in Fig. 4. Assuming |qΨ| Tp and introduc-
ing the notation V ≡ and V* ≡ – , we transform
Eq. (10) to the form
(13)
Equation (13) implies that the description of the rateof change in the degree of decomposition of energeticmaterials by the experimental function V* is the moreaccurate the smaller the ΨCs and Em0QΨ values. Theerrors in determining E in the isothermal mode dependvery strongly on the errors in determining the currentdegree of decomposition δη. For example, in calculat-ing E from two curves η(τ), ε may exceed 5% even atδη = 0.025.
η 1 ZE
RT-------–⎝ ⎠
⎛ ⎞ τexp–⎩ ⎭⎨ ⎬⎧ ⎫
.exp–=
dηdτ------ q
m0Q-----------
V* V V*Em0QΨ
RT p2
--------------------⎩ ⎭⎨ ⎬⎧ ⎫
exp ΨCsdV*dτ
----------.–≈
0.2
0 2000
η
τ, s
1.0
4000
0.8
0.6
0.4
6000 8000 10000
1
23
4
56
B5'
4'
2'3'
Fig. 4. Degree of decomposition of HMX samples of weightm0 = 1 mg at Ψ = Ψ0 versus thermostating time as calcu-lated using (1–6) Eqs. (10) and (11) and (2'–5') Eq. (12) atT = (1) 473, (2, 2') 493, (3, 3') 503, (4, 4') 513, (5, 5') 533,and (6) 548 K. The time for DSC curves 1–6 is countedfrom the moment of reaching the isothermal mode at a rateof b = 0.1 K/s. The admissible time region q is to the left ofline B.
270
DOKLADY PHYSICAL CHEMISTRY Vol. 416 Part 2 2007
KOPTELOV et al.
Analogues of W* in studying the kinetics of decom-position of energetic materials in the mode b = const byDTA and TG methods are the functions =
(Ts − Tr)/(Qm0bΨ) and = (Ts – Tp)/(Qm0bΨ),respectively; the latter formula follows from analyzinga TG model [9] that assumes Newtonian heat transfer inthe sample–heater system. Unlike the DSC model, theheat resistances Ψ in these formulas appear explicitly,generally take much higher values than in DSC devices,and are nonlinear functions of temperature. These basicfeatures of DTA and TG pose additional problems inanalyzing experimental thermograms and are sourcesof serious errors in determining kinetic constants.
Thus, satisfactory accuracy in determining thekinetic constants for the reaction of thermal decompo-sition of energetic materials in the condensed phase byTA methods is reached within very narrow ranges ofheating rates and temperatures. Significant limitationsare also imposed on the size and weight of samples andthe quality and reproducibility of heat-transfer condi-tions in the sample–measuring cell system. Violation ofthese limitations is one of the most obvious causes ofthe wide scatters of published estimates of kinetic con-stants.
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W1*
W2*
