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Thermochimica Acta

journal homepage: www.elsevier.com/locate/tca

Kinetics deconvolution study of multi-component pyrotechnics

Anirudha Ambekar, Jack J. Yoh⁎

Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, 151-742, Republic of Korea

A R T I C L E I N F O

Keywords:PyrotechnicsChemical kineticsDeconvolutionNon-linear regressionVariable activation energy

A B S T R A C T

This study reports an experimental investigation into the chemical kinetics of several commercial pyrotechniccompositions. A differential scanning calorimeter was utilized to elucidate the thermo-kinetic characteristics offive multicomponent pyrotechnic compositions. The multicomponent nature of practical pyrotechnics results ina complex interaction between the various components as well as condensed phase reactions. The thermo-kineticstudy was carried out to approximate the heating experienced by the pyrotechnics before the combustion zone.The chemical kinetic parameters such as overall activation energy (Ea) and pre-exponential factor (A) corre-sponding to these reactions were estimated using Friedman and Starink method. The large variation in Ea at-tributed to multiple parallel reactions was addressed through the application of deconvolution techniques uti-lizing the Frazer-Suzuki function as well as nonlinear mechanistic modelling. The observations from the DSCdata and the comparison between the deconvolution techniques provided an insight into the phenomenology ofthe combustion process of energetic pyrotechnics.

1. Introduction

Pyrotechnics are granular heterogeneous porous composite en-ergetic materials utilized in a wide range of applications. These multi-component energetic materials undergo a complex combustion processinvolving multiple parallel and competing reactions. The reaction ki-netics is typically characterized by an accurate estimate of the ‘kinetictriplet’, which consists of the chemical activation energy (Ea), the pre-exponential factor (A), and the reaction model (f(α)). An accurate es-timation of the kinetic triplet is essential to obtain reasonable predic-tions of burning rate, reaction progress, ignition temperature, storagelife, and heat generated during the process. Reaction kinetics of en-ergetic materials is often studied by thermal analysis techniques, whichsimulate the temperature conditions of actual combustion wave.Subsequently, a kinetic analysis may be performed on the experimentaldata to obtain the kinetic triplet.

The well-known isoconversional methods are most commonly uti-lized due to their lack of the a priori assumptions. The study by Starink[1] reveals the so-called Type B-1.92 method [1] and Friedman method[2] to be highly accurate. Both techniques evaluate the activation en-ergy (Ea) as a function of reaction progress (α). However, the predic-tions of the isoconversion methods may not be accurate for cases be-yond strictly single step reactions, and additional mechanisticmodelling is often necessary. The average Ea value may be used as thenominal activation energy independent of α if the relative variation in

the isoconversional Ea-α relationship is less than 10% [3], signifying asingle step reaction [4]. However, in case of complex processes, a largevariation in the Ea-α relationship [5,6] as well as dependence of the Eavalues on the heating rates [7] has been related to competing oroverlapping reactions.

In order to address the meaningfulness of the variable activationenergy calculated by an isoconversional technique, a number of single-step reactions may be envisaged with individual constant activationenergy values [8]. Thus, in cases involving a complex reaction me-chanism, the kinetic analysis must identify the constituent reactions inthe overall process in a physically meaningful way. Typically, the ex-perimentally measured overall reaction rate is deconvoluted by re-presenting it as a sum of a set of kinetic equations.

The overlapping curves of a complex process may be deconvolutedusing standard statistical functions such as Fraser-Suzuki (FS) function[9–11] shown in Eq. (1), followed by a kinetic analysis of the separatedpeaks.

= ⎧⎨⎩

− × ⎡⎣

+ − ⎤⎦

⎫⎬⎭

( )y aa

a x aaexp ln 2 ln 1 20

32 3 1

2

2

(1)

where a0, a1, a2, a3 are the parameters corresponding to the amplitude,position, half-width and asymmetry of the curve, respectively. While, xand y indicate arbitrary independent and dependant variable respec-tively.

Alternatively, nonlinear regression (NLR) techniques are sometimes

https://doi.org/10.1016/j.tca.2018.07.007Received 8 May 2018; Received in revised form 9 July 2018; Accepted 11 July 2018

⁎ Corresponding author.E-mail address: jjyoh@snu.ac.kr (J.J. Yoh).

Thermochimica Acta 667 (2018) 27–34

0040-6031/ © 2018 Elsevier B.V. All rights reserved.

T

employed which assume a certain reaction scheme, and the overallreaction rate is represented by the sum of each component reaction step[12,13] as shown in Eq. (2).

∑= ⎛⎝

− ⎞⎠=

dαdt

c AE

RTf αexp ( )

i

n

i ia i

i i1

,

(2)

where, ∑ == c 1in

i1 , ∑ == c α αin

i i1 , and the corresponding kinetic para-meters are optimized iteratively to minimize the difference between theexperimental and simulated curves.

Numerous investigations on simple binary pyrotechnic composi-tions [14–20] have been reported in the literature, which typicallyadopt the simplified Kissinger method [21]. Although, the thermo-ki-netic phenomena occurring in composite energetic materials can beexpected to be more complex compared to the idealized binary mix-tures, relatively fewer investigations [22,23] pertaining to practicalmulticomponent pyrotechnic compositions or their chemical kineticscan be found.

The current study aims to investigate a set of commercial multi-component pyrotechnic compositions in order to elucidate theirthermo-chemical behavior and extract the thermo-kinetic parameters.The study employs Differential Scanning Calorimetry (DSC) in combi-nation with currently established isoconversional techniques in order toevaluate the Ea-α variation. The broad variation in the Ea-α relationshipfor each pyrotechnic was addressed by attempting to obtain a mean-ingful split of the overall reactions into several single step reactions.The study compares the method of deconvolution using the FS functionagainst the so-called Bayesian approach of mechanistic deconvolution.The deconvolution technique fitted a number of FS functions to re-present the effective overall reaction rate obtained by averaging thethermograms at several heating rates. As the onset and endset of theprocess corresponds to a low reactivity, a weightage corresponding tothe normalized reaction rate at each point was utilized to allow fittingof the most significant parts of the peak with a higher degree of accu-racy. In case of the mechanistic deconvolution, known data relevant tothe kinetics under consideration are collected to form the a priori re-action mechanism and adjustable factors associated with the kineticparameters were numerically optimized.

2. Experimental study

The current study follows the ICTAC Kinetics Committee re-commendations for collecting experimental data [24]. The optimalsample mass was pre-determined to be between 1.2–1.5mg in order toprevent self-heating of the sample while providing sufficient sensitivityto resolve various peaks. The particle size of the samples was closelycontrolled using a 200 mesh (75 μm) particle sieve. According theICTAC Kinetics Committee recommendations for thermal analysis anddata processing [25], several distinct steps were followed in this study.The first step is concerned with obtaining good quality experimentalkinetic data at multiple heating rates. The second step involves using anisoconversional method to evaluate the Ea-α relationship. The third stepis evaluating the kinetic triplet of the single step or multi-step reactionthrough either model fitting or mechanistic techniques. While thefourth and final step is stated to be validation of the simulated curvesagainst the experimental data.

DSC experiments were carried out with a Mettler Toledo DSC 3instrument under constant heating rates of 1, 2, 5, 10, 20, 40, and50 °C/min with a purge flow of nitrogen maintained at 80mL/min. Thesamples were contained within the standard 40 μL aluminum panssealed with an aluminum lid with a perforation of approximately0.5 mm dimeter. The sample mass was spread across the bottom of thepan in a thin layer to ensure rapid and uniform distribution of heat andminimize the temperature gradients inside the sample. Duplicate ex-periments were carried out for the extreme heating rates in order toensure the repeatability of the thermograms and accuracy of the

calculated parameters. Furthermore, in case of a thermogram showingpeculiar variations, the particular experiment was repeated twice. Theresulting minor variation in the appearance of peaks in the thermo-grams was attributed to the slight inhomogeneity in the sample andsmall variations in the particle size distribution.

The mass based composition of the pyrotechnics currently beingstudied has been reported previously [26,27] and reproduced inTable 1. These compositions, typically utilized in display pyrotechnics,have been designated according to the color of the flame they generate.

3. Results and discussion

Typical DSC analysis elucidates the heat flow associated with thechemical reaction occurring within a given sample. The data from theDSC thermograms were used for understanding the chemical kinetics ofthe pyrotechnics.

3.1. DSC thermogram analysis

Fig. 1 shows the characteristics DSC thermograms obtained at var-ious heating rates for the pyrotechnics under study. Despite utilizingpotassium perchlorate as the primary oxidizer and magnesium as theprimary fuel, a definite variation in the appearance of the thermogramsof each composition was observed. However, certain features of purecomponent thermochemistry are evident in the DSC thermogramsshown in Fig. 1. The phase transition of pure potassium perchlorate wasmanifested as an endothermic peak between 301–307 °C, which iscomparable to the values reported in the literature [14,28].Fig. 1(a)–(d) show the variation of the phase change temperature withheating rates for each pyrotechnic composition.

The oxidizer-fuel reactions in pyrotechnics are typically triggered bymelting or decomposition of either the fuel or the oxidizer [15,16]. Inthe current study, the onset of the primary exothermic peak was typi-cally observed to occur between 360–390 °C, significantly lower thanthe melting point of pure KClO4 [14], as shown in Fig. 1(a)–(d). Thisreduction in the temperatures has been attributed [17–19] to the solid-state reactions, increased inter-diffusion of reactive components beyondthe Tammann temperature, and inadvertent formation of local hotspotsin the sample, as well as crystal defects such as distortions, dislocations,and cracks. Additionally, the primary exothermic event exhibits anappearance of an exothermic peak superimposed with the endothermicpeaks corresponding to the decomposition of various additives. This hasbeen further elaborated through the detailed view in Fig. 1(a). Thisbehavior indicates the parallel nature of decomposition process ofcertain additives.

The extent of reaction (α), which varies from 0 to 1, for each heating

Table 1Composition of pyrotechnics used in current study.

Component Red Green Blue Yellow

KClO4 45 % 16 % 48 % 40 %Mg 20 % 18 % 2.5 % 18 %SrCO3 20 % – – 12 %(C2H3Cl)n 8 % 8 % 6 % 7 %C10H11Cl17 7 % 8 % 8 % 5 %CuO – – 23 % –Ba(NO3)2 – 50 % – –S – – 12.5 % –Na3AlF6 – – – 18 %Pyrotechnic

Flames

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rate curve at time t may be equated to the ratio of cumulative energyreleased until time t and the total energy released as shown in Eq. (3).

∫ ∫= − −α t S t B t dt S t B t dt( ) ( ) ( ) [ ( ) ( )]t

t

t

tend

0 0 (3)

where, S t( ) indicates the DSC signal, B t( ) indicates the baseline, whilet0 and tend represent the time of start and end point of reaction re-spectively. Furthermore, the reaction rate (dα dt) for each heating ratecan be obtained using numerical differentiation of the data. Fig. 2shows the calculated reaction rates for five different heating rates. Incontrast with Fig. 1 which shows the thermal response of the samplesover a wide temperature range, Fig. 2 focuses on the temperature zoneover which the primary reactions of interests occur. Furthermore, Fig. 2provides a comparison between the chemical reactions at various

heating rates on a common base-line.

3.2. Isoconversional reaction kinetics

Chemical kinetic parameters for each pyrotechnic composition wereevaluated from the DSC thermograms using the Friedman method andthe nonlinear method by Starink using a custom MATLAB code. Theparameters were verified against the AKTS thermo-kinetics software[29] which allows calculation using differential isoconversional methodby Friedman [2]. The analysis was restricted to the alpha range of 0.02to 0.98 in order to exclude the extreme values of conversion with highermeasurement uncertainty. Fig. 3 shows the Ea-α distribution for eachpyrotechnic composition evaluated using the Friedman and Starinkmethod as well as the variation of the logarithmic term ln(A× f(α))calculated by the Friedman method. The Ea profiles show a significant

Fig. 1. DSC thermograms at various heating rates for (a) red (b) green (c) blue and (d) yellow pyrotechnic compositions. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article).

Fig. 2. Reaction rates at various heating rates for (a) red (b) green (c) blue and (d) yellow pyrotechnic compositions. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this article).

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variation with α while the results obtained by the isoconversionalmethods were comparable. This is in contrast with the earlier resultsobtained for RDX based energetic material with fewer components[30].

Table 2 shows the range of activation energy and relative deviationfor each method. The large relative deviation values obtained from theresults of isoconversional analysis clearly establish that the experi-mental Ea-α relationship represents the effective activation energy ofmultiple simultaneous reactions. Furthermore, despite the large varia-tion with α for bothe methods, the goodness of fit indicated by the R2

values for the Starink method was observed to be higher, indicating ahigher degree of accuracy. Furthermore, it has been clearly established[1] that the Starink method provides more reliable results overFriedman method in terms of handling the potential uncertainty in thebaselines or the experimental thermal analysis data. Thus, the resultsobtained from Starink method were utilized for the subsequent nu-merical analysis.

3.3. Deconvolution of reaction rate using FS function

The reaction rate curves shown in Fig. 2 may be deconvoluted byexpressing these curves as a summation of several FS function shown inEq. (1). The technique equates the independent variable x with tem-perature and the dependant variable y with the reaction rate. Anaverage reaction rate (RR) was evaluated from the interpolated data ofeach heating rate curve. This averaged reaction rate calculated from theDSC data was fitted with a number of FS functions depending on thenumber of apparent exothermic and endothermic peaks. The number ofpeaks as well as the FS function parameters a0–a3 were numericallyoptimized to match the summation of FS function with the averagedexperimental reaction rate. Fig. 4 shows the comparison between ex-perimental and simulated reaction rates as well as deconvoluted peaksfor various pyrotechnics.

The parameters associated with the deconvoluted peaks have beenlisted in Table 3. The negative value of amplitude (a0) indicates anendothermic process that may be attributed to melting or decomposi-tion of various additives, while the peak showing highest positive valueindicates the primary exothermic reaction.

The accuracy of the simulated peaks was estimated through theresidual between the experimental and simulated values. The initialportion of the reaction rate curve was observed to be fitted with a loweraccuracy while the central prominent peak was found to be fitted withhigh accuracy. The residuals for each pyrotechnic have been plottedagainst temperature in Fig. 5.

3.4. Mechanistic deconvolution of variable activation energy

A mechanistic deconvolution process was also proposed in thecurrent study. A set of reactions was postulated for the current analysisbased on the known components of the pyrotechnic composition. Thesereactions include combustion reaction between Mg and KClO4 (R1a),combustion reaction between Mg and Ba(NO3)2 (R1b), combustion re-action between Mg and CuO (R1c), decomposition reaction of pureKClO4 (R2), decomposition reaction of pure SrCO3 (R3), and thepolymer decomposition reactions (R4). The kinetic triplets representingthese reactions have been reported in the literature and listed inTable 4. The f(α) for R1a and R1c was assumed to be the same as re-ported by Umbrajkar et al. [31]. Although, both R3 and R4 are known

Fig. 3. Ea-α relationship for (a) red (b) green (c) blue and (d) yellow composition. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article).

Table 2Variation in activation energy according to Friedman and Starink method.

Friedman method

Ea RangekJ/mol

Deviation R2

Red 158–380 83% 0.95± 0.03Green 163–270 55% 0.99± 0.01Blue 115–234 76% 0.95± 0.04Yellow 159–225 36% 0.97± 0.02

Starink method

Ea RangekJ/mol

Deviation R2

Red 131–346 100% 0.98± 0.02Green 152–244 51% 0.99± 0.01Blue 125–181 42% 0.98± 0.02Yellow 167–204 21% 0.98± 0.01

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to consist of two sub-reactions, in the current study, the activationenergy and pre-exponential factor for these sub-reactions were re-presented by a single pair of parameters while the reaction model wasrepresented by a summation of the two f(α) functions.

Based on the composition of each pyrotechnic and the proposed setof reactions, the overall process followed by red and yellow composi-tions was postulated to consist of the reactions R1a, R2, R3, and R4. Thegreen composition was assumed to follow the reactions R1a, R2, R1b,and R4 while the blue composition was proposed to undergo the re-actions R1a, R1c, R2, and R4. A typical NLR technique would define aminimization function based on the constituent reaction kinetics mod-ified by adjustable parameters. Subsequent minimization of this func-tion would present with certain values of fitting variables. However,such techniques face the problem of the kinetic compensation, whichsignifies that a given set of experimental data may be fitted accuratelyby multiple pairs of kinetic parameters due to which the fitted values ofactivation energy may be rendered meaningless and non-unique.

In order to address this issue, the activation energy values in thecurrent study were assumed to remain unchanged and the deviationfrom ideal kinetics in the complex pyrotechnic mixture was represented

through a change in the pre-exponential factor values. Furthermore, thereaction model represented by the f(α) function was proposed to use afraction of α related to the total value through an adjustable parameter.The effective activation energy resulting from the superposition of theconstituent reactions in this manner with modified kinetic parameterswas proposed following the relationship in Eq. (4).

=∑ ′

∑ ′

−

−

( )( )

EE A f α

A f α

exp ( )

exp ( )α

a i iERT i i

iERT i i

,a i

a i

,

,(4)

where, the subscript i represents various reactions in the reactionscheme. The modified reaction progress are defined as shown in Eq. (5).

∑= × =α α z zwhere, 1i i i (5)

where, ′A i and zi indicate the adjustable parameters.Appropriate adjustable parameters were evaluated through nu-

merically minimizing the error function given by = −error E Ea α. Thistechnique was implemented using a custom MATLAB code and utilizesthe known data regarding the kinetics and reactive behavior of thecomponents instead of representing the constituent reactions througharbitrary parameters. The optimization was carried out using the pat-tern search technique with the ideal values of the activation energy asthe initial guesses and the limits of search for the parameter zi identifiedfrom the known mass based composition of the pyrotechnics. The re-sults obtained through this exercise have been depicted in Fig. 6, whichshows a good match between the experimental and numerical fits of Ea-α relationships.

The values of the adjustable parameters for various pyrotechnicsand postulated constituent reactions have been shown in Table 5. Themodel adjustable parameters were obtained through constrained opti-mization with the constraints determined from the chemical kinetics ofthe constituent reactions as well as the composition of the pyrotechnic.The values reported in Table 5 unsurprisingly differ from the valuesreproduced in Table 4 due to the deviation of the current compositionsfrom the stoichiometric composition typically reported in the literatureas well as due to the multicomponent nature of the overall process.

The current values of the parameters may be further improved inaccuracy through a global search and additional constraints based onthe additional modelling. Despite the known limitations, the para-meters evaluated in the current study provide certain insight into the

Fig. 4. Plots showing experimental and simulated reaction rates as well as deconvoluted peaks for (a) red (b) green (c) blue and (d) yellow composition. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

Table 3FS function parameters for deconvoluted peaks.

a0 a1 a2 a3

RedPeak 1 0.019282 774.15 23.34 0.02Peak 2 0.016248 753.15 31.86 −0.56

GreenPeak 1 0.006853 761.33 33.12 −0.1Peak 2 0.000649 711.19 8.33 −0.04Peak 3 0.000434 731.16 4.5 0.34Peak 4 −0.00059 690.81 0.8 0.02Peak 5 0.009258 716.08 57.08 0.75

BluePeak 1 0.030698 691.49 12.05 0Peak 2 0.027488 678.94 22.49 0.42

YellowPeak 1 0.010082 746.29 24.46 0.68Peak 2 −0.00218 699.23 1.62 −0.94Peak 3 0.010539 726.23 40.21 −0.31

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combustion process of the pyrotechnics. Based on the numerical results,combustion process may be said to primarily driven by the decom-position reaction of KClO4 (R2) as indicated by the typically large va-lues of yi corresponding to R2 for each pyrotechnic. This dominance ofthe KClO4 decomposition reaction on overall combustion process wasalso captured in the reduced order model for the burning rate predic-tion of these pyrotechnic compositions [35].

In certain cases, the optimization process was observed to result invery small value of the adjustable parameters. Although this value maybe quantitatively inaccurate, qualitatively it indicates a low significanceof the reaction to the overall process. Similar observation regarding thequalitative nature of the reaction mechanism can be made with highconfidence. The combustion reaction of Mg with KClO4 was found to bethe second most dominant reaction in case of red pyrotechnic while thereaction between Mg and CuO was more prominent for the blue pyr-otechnic. In case of green and yellow pyrotechnic, the decomposition ofpolymers was found to significant.

4. Conclusions

The study reports a calorimetric study of five multicomponent en-ergetic pyrotechnic compositions. The pyrotechnic compositions withcolored flame used for display pyrotechnics were selected. TheDifferential Scanning Calorimetry data was utilized to gain an insightinto the combustion mechanism of the pyrotechnics. The non-iso-thermal DSC experimental technique was utilized to evaluate the

apparent chemical kinetics parameters for these propellants. This studyprovides an insight into how the pure component thermal behavior isreflected in the behavior of the mixture. The DSC thermograms for eachpyrotechnic were deduced to be an approximate superimposition of thethermo-chemical response of each pure component. This unsurprisingoccurrence of multiple simultaneous reactions in complex pyrotechnicswas conspicuously quantified by the large variation of activation energyreported in the results of isoconversional methods. The variable acti-vation energy reported by the isoconversional methods was found toindicate a complex reaction mechanism. This observation of largevariation in activation energy may be utilized as a litmus test for ade-quateness of the preliminary isoconversional analysis. In cases wherethe presence of multiple simultaneous reactions is detected, elaboration

Fig. 5. Plots showing residual in deconvoluted peaks for (a) red (b) green (c) blue and (d) yellow composition.

Table 4A priory kinetic data for the proposed reactions.

Sr. No. Reactions Ea (kJ/mol) lnA (1/s) Reaction model

R1a [14] + → +4Mg KClO 4MgO KCl4 153.6 21.5 f(α) = (1–α)3.9R1b [32] + → + +3Mg Ba(NO ) 3MgO BaO 2NO3 2 110.82 7.71 f(α) = [-ln(1–α)]2/3

R1c [15] + → +Mg CuO MgO Cu 120 8.64 f(α) = (1–α)3.9R2 [33] → +KClO KCl 2O4 2 309.22 16.03 f(α) = 3(1–α)[-ln(1–α)]2/3R3 [34] → +SrCO SrO CO3 2 241 12.1 f(α) = 3(1–α)2/3

f(α) = (3/2)(1–α)1/3R4 [9] → −

+−

m(C H Cl) (C H Cl) (C H )

HCln n n m2 3 2 3 2 2 160.5 19.28 f(α) = (1–α)[-ln(1–α)]1/2

f(α) = (3(1–α) 2/3)/(2[1–(1–α)1/3])

The thermal decomposition of polymers such as PVC and chlorinated rubber is known to be extremely complex and the reaction shown here represents thedehydrochlorination step of the complex mechanism.

Fig. 6. Experimental and numerical Ea-α relationship for pyrotechnics.

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of the constituent reactions and their kinetic parameters is importantfor understanding the combustion process as well as practical applica-tions. Such a situation may be typical for various multicomponentpyrotechnics as well as other energetic materials.

The current study reports the application of two separate methodsof deconvolution for the complex reaction mechanisms. The deconvo-lution technique using Frazer-Suzuki equations was found to yield afeasible set of parameters describing the Ea-α relationship through aarbitrary reaction mechanism. This method provided an accurate de-scription of the process without any assumptions regarding the reactionmechanism. However, the physical meaning of such parameters is ty-pically unknown. In certain cases, the knowledge of the reaction me-chanism in terms of the concentration of reaction constituents or extentof certain reaction may be desirable. Hence, a mechanistic non-linearregression technique was implemented in order to elucidate the fun-damental steps constituting the overall reactions. The prominent reac-tion from the set of proposed reactions was discerned and the KClO4

decomposition reaction was found to be the driving reaction in eachcase. The chemical kinetic parameter values evaluated through such atechnique would be instrumental in theoretical estimation of theburning rate of the pyrotechnics.

Acknowledgments

This work was supported by Brain Korea 21 Plus Program and theAgency for Defence Development through IAAT at Seoul NationalUniversity. Additional funding came from the Advanced ResearchCenter (NRF-2013R1A5A1073861) contracted through the NextGeneration Space Propulsion Research Center at Seoul NationalUniversity.

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Table 5Numerically optimized adjustable parameters.

Red Blue

′Aln ( )i (1/s) zi ′Aln ( )i (1/s) zi

R1a 29.93 0.45 R1a 21.50 0.49R2 52.32 0.20 R2 50.32 0.12R3 3.02 0.20 R1c 21.85 0.26R4 2.89 0.15 R4 20.14 0.13

Green Yellow

′Aln ( )i (1/s) zi ′Aln ( )i (1/s) zi

R1a 8.76 0.16 R1a 10.00 0.38R2 42.62 0.16 R2 50.00 0.33R1b 1.00 0.50 R3 38.24 0.15R4 17.66 0.16 R4 26.00 0.15

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