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On the Controlling Mechanism of the Upper Turnover States in the NTC
Regime
Weiqi Ji a, Peng Zhao b,c,*,†, Tanjin He d, Xin He a,d, Aamir Farooq e, Chung K. Law a,c,*,‡
a

Center for Combustion Energy, Tsinghua University, Beijing 100084, China
Department of Mechanical Engineering, Oakland University, Rochester, MI, 48309, USA
c
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ
08544, USA
d
State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing
100084, China
e
Clean Combustion Research Center, Division of Physical Sciences and Engineering, King
Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
b

*Corresponding authors
†
Email: pengzhao@oakland.edu
‡
Email: cklaw@princeton.edu

1

Abstract
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Using n-butane, n-heptane and iso-octane as representative fuels exhibiting NTC
(negative temperature coefficient) behavior, comprehensive computational studies with
detailed mechanisms and theoretical analysis were performed to investigate the upper
stationary point, denoted as turnover states, on the NTC curve near the higher temperature
regime, where the ignition delay τ exhibits a local maximum. It is found that the global
behavior of the turnover states exhibits distinctive thermodynamic and kinetic characteristics
under different pressures, in that the ignition delay at the turnover states shows an Arrhenius
dependence on the temperature T and an approximate inverse quadratic power law
dependence on the pressure P. These global behaviors imply that the temperature and
pressure of the turnover states are not independent and can be correlated by an Arrhenius
dependence, as ln P ∝ 1/T. Further theoretical analyses demonstrate that such turnover states
result from the competition between the low-temperature chain branching reactions and the
decomposition of the intermediate species, and therefore correspond to a critical value, α, of
the ratio of OH production from low-temperature chemistry. In addition, the ignition delay at
the turnover state can be well correlated by the analytical expression derived by Peters et al.,
with the further demonstration that the pressure dependence of the turnover ignition delay
mainly result from the H2O2 decomposition reaction. Comparison of the present results with
the literature experimental data of n-heptane ignition delay time shows very good agreement.
Key Words: NTC; Turnover; Auto-ignition; Chain branching ratio; Low temperature
chemistry

2

1. Introduction
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The phenomenon of negative temperature coefficient (NTC) is an essential feature of the
oxidation kinetics of large hydrocarbons, and is closely related to such practically important
phenomena and processes as low-temperature ignition [1], cool flames [2, 3], flame
stabilization [4], fuel processing [5], engine knock [6] and the development of HCCI
(homogenous charge compression ignition) engines [7]. It refers to the effect that, for a given
pressure and in an initial temperature regime much lower than the adiabatic flame
temperature of the reacting mixture, the ignition delay could increase with increasing initial
temperature. The NTC behavior in auto-ignition has been extensively observed and studied in
homogenous systems such as the shock tube [8], jet-stirred reactor [9], flow reactor [10] and
rapid compression machine (RCM) [11].
It is widely accepted that the NTC behavior is due to the peroxy-chemistry pathways in
the low-to-intermediate temperature regimes, as discussed in [12-14]. As an example, Fig. 1
shows a schematic reaction path for the oxidation of large alkanes in such regimes.
Specifically, fuel consumption occurs via H-atom abstraction from the fuel molecule to form
the alkyl radical, R. Since β scission is slow at low temperatures, O2 addition to the R radical
is dominant, forming the RO2 radical. Then isomerization of RO2 forms the
hydroperoxy-alkyl radical (QOOH). Further O2 addition to QOOH forms a transition-ring
complex (O2QOOH) that isomerizes and decomposes to form ketohydroperoxide (KET) and
the hydroxyl radical (OH) (red arrows in Fig. 1). The KET then decomposes to carbonyl,
aldehyde and generates another OH radical, leading to the key chain branching of
low-temperature chemistry. However, with increasing temperature, all the oxygen addition
reactions shift towards the decomposition side, and as such the extent of low-T chain
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branching is reduced. Meanwhile, at higher, intermediate temperatures (800K < T < 900K, 20
atm for n-heptane), dissociation of the low-temperature intermediates (e.g., RO2 and QOOH)
and the R radical through β scission gradually becomes important and forms different
products, including alkenes, HO2 and other alkyl radicals (blue arrows in Fig. 1), which
further compete with the low-temperature chain branching pathway. The backward shift of
the low-temperature reaction channel, together with the favored β scission at intermediate
temperatures, results in globally reduced chemical reactivity at moderate temperatures in the
NTC regime.
At intermediate-to-high initial temperatures (900K < T < 1200K, 20 atm for n-heptane),
the major chain branching that drives the system runaway is hydrogen peroxide (H2O2)
decomposition [15] forming two OH radicals, with its rate constant increasing with
temperature. Consequently, the ignition delay starts to decrease again with increasing
temperature beyond a threshold value, termed the “turnover temperature”, corresponding to
the local maximum of the ignition delay time.
In this work, the characteristics of the turnover states including the thermodynamic and
chemical kinetic behaviors are studied. The motivations are three-fold: First, since the
turnover state also corresponds to the intermediate-to-high temperature boundary of the NTC
regime, identification of the mechanism controlling it will lead to new insights on the NTC
behavior, and can bring about significant convenience in the description of auto-ignition.
Second, sensitivity of the ignition delay to temperature variation is one of the key factors
affecting the flame propagation mode in HCCI engines [16] and relevant RCM experiments
[17, 18], deeper understanding of the underlying mechanism of the temperature-insensitive
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nature of the ignition delay at the turnover state could be helpful in interpreting the various
combustion modes under HCCI conditions. Third, correlations of the ignition delay time as a
function of initial pressure and temperature usually show large deviations near the turnover
temperature [8, 19]. Thus, developing global scaling for the turnover temperature can help to
reduce the uncertainties and provide an accurate reference for the selection of experimental
data at the relevant experimental conditions.
The present study is computational. The fuels investigated are n-butane, n-heptane and
iso-octane, which represent typical short and long straight-chain alkanes as well as branched
alkanes. The mechanisms used are the NUIG n-butane mechanism [20], the LLNL n-heptane
mechanism version 3.1 [12], and the LLNL iso-octane mechanism version 3 [21],
respectively. The global behaviors of the ignition delay at turnover temperatures under
different pressures are investigated for all of the three fuels. It is noted that although the
pressure dependence of the rate constants of the oxygen addition reactions, alkylperoxy
radical (RO2) dissociation and isomerization reactions, are not considered, it has been
recently demonstrated in [22, 23] that the high-pressure limits for these reactions are
sufficient to describe the low-temperature chemistry at the conditions of this study. The
mechanism for n-heptane updated by Karwat et al. [24], based on the rate rules from Villano
et al. [22, 23], is also referred to show that the behavior demonstrated in this study is
mechanism independent and with general implications on the low-temperature kinetics.
In the following sections, we shall first identify the global system behavior at the
turnover states under different pressures. We shall then analyze the coupling between
pressure and the turnover temperature, as well as the power-law dependence on the pressure
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for the ignition delay at the turnover states. Finally, the characteristics of the turnover states
are validated against available ignition delay data in the literature, demonstrating good
agreement.

2. Global behavior of the turnover states
Figure 2 shows the ignition delay as a function of temperature for a homogeneous,
stoichiometric mixture of n-heptane and air under 2-60 atm pressure, using detailed
mechanism [12]. It is seen that for each pressure, the total ignition delay, τ, first decreases
and then increases with increasing initial temperature, T, in the NTC regime, and finally
decreases again when the temperature is above the turnover temperature, Tm , which, marked
by symbols, is seen to shift to higher values as pressure is increased.
We next note that the ignition delay, τm , at the turnover state exhibits unique dependences
on the corresponding temperature, Tm, and the system pressure, P. Specifically, its linear
variation with Tm in the plot of Fig. 2 suggests an Arrhenius correlation, namely
ln τ m  1 / Tm , while Fig. 3 shows it also varies nearly quadratically with the system

pressure, as τ m  P 2 . These results then suggest that Tm and P can be related by
ln P  1/ Tm , as shown in Fig. 4, yielding a thermodynamic correlation between the

turnover temperature and the system pressure. It is further noted that the data in Figs. 3 and 4
include those for φ = 0.5, 1 and 2, demonstrating that these observations are also insensitive
to the equivalence ratio. We shall demonstrate in the following the underlying kinetic
mechanism affecting such a behavior.

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3. Chemical Kinetic Analysis on the Turnover States
3.1 Low-Temperature Chain Branching Ratio at the Turnover State
Figure 5 shows the evolution of the intermediate species at the turnover states, using
stoichiometric n-butane/air mixtures under 20 atm for demonstration because less isomers are
involved in its oxidation process. It is seen that many radicals of interest, largely remain
quasi-steady for an extended period between the starting chain initiation and the final ignition
runaway. Take OH as an example, the OH consumption rate is about 10-5 ~ 10-4 mole/cm3s as
shown in Fig. 6, while the net OH production rate at the same condition is below 10-9
mole/cm3s, four orders of magnitude lower. As such, the OH mole fraction is kept at a low
level (less than 10-8) before ignition as shown in Fig. 5 and could be assumed to be in
quasi-steady in the plateau period. Therefore, the total production and consumption rates of
OH should approximately balance. Figure 6 then plots the rates of the important reactions
consuming OH and the fuel molecule, showing that OH is mainly consumed by H-abstraction
from the fuel molecule (nearly 90% of OH consumption at the instant of 75% τm in Fig. 6a),
while fuel consumption is also mainly through the H-abstraction by OH (nearly 90% of fuel
consumption at the instant of 75% τm by the four reactions in Fig. 6b). Consequently, the OH
production rate can be estimated by the consumption rate of the fuel molecule. We shall
demonstrate in the following how these observations identify the major kinetic characteristics
at the turnover states.
The controlling reactions for the turnover temperature were identified by performing
sensitivity analysis, with the sensitivity coefficient calculated by perturbing the A-factor of
each elemental reaction rate by 30%, following the definition:
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S

d ln(T ) ln(T1.3k )  ln(Tk )

d ln(k ) ln(1.3k )  ln( k )

(1)

A step size of 0.1 K was used to search the turnover temperature, and the perturbation of
30% was selected to allow sufficient change of the turnover temperature for accurate
evaluation. We shall only present the sensitivity for n-butane as the mechanisms for
n-heptane and iso-octane are based on similar reaction classes and rate rules.
Figure 7 shows the sensitivity for the stoichiometric n-butane/air mixture at 2 and 10 atm.
The most sensitive reactions are shown to be the chain termination reactions from the β
scission of RO2 and QOOH, as well as the isomerization from RO2 to QOOH and the
subsequent chain branching and propagation reactions. However, it is seen that the turnover
temperature is not sensitive to the HO2 chain termination reaction and the H2O2 chain
branching reaction. Consequently, the turnover state is mainly controlled by the competition
between the β scission of low-temperature intermediates as chain termination reactions
producing

the

stable

intermediates

and

HO2,

and

the

low-temperature

chain

branching/propagation reactions producing OH. It should be noted that the turnover
temperature is insensitive to the perturbation of the rate of oxygen addition to QOOH reaction
as long as its equilibrium constant is not perturbed. Such a behavior is expected as the oxygen
addition to QOOH is in the partial equilibrium state, and the low-T chain branching is only
affected by the equilibrium constant. Therefore, the result by perturbing the oxygen addition
to QOOH using reversible reactions is not presented in Fig. 7, which otherwise might cause
misleading and underestimate its role at the turnover states.
We note in passing the worthwhile studies on such competitions in the first-stage ignition
controlled by the low-temperature chemistry. Specifically, Peters et al. [25] defined a ratio
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between the R + O2 ⇌ RO2 with subsequent isomerization RO2 ⇌ QOOH and the β scission
of R to determine the crossover temperature, at which the concentration of KET reaches its
peak value and the first-stage ignition occurs. Merchant et al. [26] used a similar concept to
study the first-stage ignition of propane, and included the HO2 chemistry, namely RO2 →
Alkene + HO2, which was not included in [25, 27]. However, these ratios were derived from
a reduced mechanism, and only specific isomers of R, RO2 and QOOH were taken into
account. Thus, these definitions are applicable only for limited conditions and for certain
mechanisms, in the context of two-stage ignition. However, for the concept of the turnover
state of current interest, sensitivity analysis has shown that the turnover temperature is also
sensitive to the decomposition of QOOH to produce OH. Consequently, we shall define a
ratio of the OH production rate, instead of the KET production rate, to the total production
rate of R, as a criterion to evaluate the net contribution from the low-temperature chain
branching:

α

OH production rate from low-T chain-branching
total production rate of R
OH production rate from low-T chain-branching
total rate of H-abstraction from RH

The ratio α can be readily determined from the detailed mechanism including all isomers of R,
RO2, and QOOH. In particular, we have followed the reaction flux from RH to OH, as shown
in Fig. 8, and have considered the effect from β scission at each step, defined by the
following ratios:

ηi 

rate of RH+OH  i-R+H 2O
rate of RH  i -R

total rates of RH  R total rates of RH+OH  R+H 2O

βi 

net rate of i -R  i -RO 2
total production rate of i-R
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γi , j 

net rate of i -RO 2  i, j -QOOH
total production rate of i -RO2

θi , j 

net rate of i, j -QOOH  OH
total production rate of i, j -QOOH

The subscript or prefix i indicates the index of the isomer of R, and those of i, j indicate the
index of the isomer of QOOH. Thus, α is determined by

α   ηi βi  γi , j θi , j
i

(2)

j

In the Appendix, we present a systematic determination of α by first deriving it using a
simplified skeletal mechanism with major reaction steps involving only one isomer, shown in
Fig. 1 and Table 1. We shall then extend the analytical expressions to detailed kinetics
involving all of the isomers.
We shall now show that the turnover state can be defined well by a constant value of α
under different pressures. Following the definition of Eq. (2), the computed α as a function of
temperature under various pressures for the three fuels are shown in Figs. 9 a, b and c,
respectively. As expected, α decreases with increasing temperature, and eventually to zero at
high enough temperatures. This shows the progressively decreasing significance of the
low-temperature chain branching reactions with increasing temperature. Similar to the shift of
NTC, the α curves shift to higher temperature with increasing pressure.
Based on α, we define a second ratio, αRO 2 , which corresponds to the ideal case where all R
radicals participate in the oxygen addition reaction without β scission, i.e., βi = 1:

αRO2   ηi  γi , j θi , j
i

j

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(3)

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which is also plotted in Fig. 9. Consequently, this definition isolates the effect of the
competition between β scission and oxygen addition to R. It is seen that αRO2 is equal to α at
low temperatures, but becomes larger than α when the temperature reaches a critical value at
intermediate to high temperatures. These results then indicate that β scission of the R radical is
negligible in the low-to-intermediate temperature regime before the turnover state is reached.
In Fig. 9, the values of α corresponding to the turnover temperatures are marked by
symbols in the evolution with increasing temperature. It is then of interest to note that the
turnover temperatures for different pressures largely correspond to a narrow range of α,
which, for a given fuel, slightly depends on the pressure range. Figure 10 further shows that
the turnover states calculated using a constant α agree well with those obtained with detailed
mechanism, thereby supporting the suggestion that the turnover states can be correlated with
a critical state of the low-temperature chain branching ratio, with the associated reactions
involved in the determination of α. The critical value of α is found to be less than one, and is
different for each fuel studied here, potentially depending on the fuel structure and the range of
the turnover temperatures. The pressure effect on α is mainly through the equilibrium shift of
oxygen addition reactions, as implied from Eq. (A11) in the Appendix.
It is worth noting that the α ratio developed in the current work is based on the widely used
rate rules in low temperature chemistry, and it can be readily extended to future mechanisms
with more accurate rate coefficients. Recently, extensive work has been done on the
uncertainty quantification in kinetic analysis and combustion modeling [28-30]. Following the
idea of uncertainty propagation, the value of α in this study is rate dependent and will be
affected by the uncertainty in the rate coefficients, however, such uncertainty will not affect the
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scaling characteristics and the controlling mechanism identified for the turnover states. On the
other hand, there are some other factors that could affect the quantitative determination of the α
value, such as the transient effects of species concentration and other possible reaction
channels, however, those effects are secondary with negligible influence compared with the
leading order quasi-steady assumptions and dominant reaction channels. It is found that the
correlation of the turnover state with a constant α value is valid and general for different fuels
and mechanisms, as demonstrated by both detailed calculations and experimental validations
in Sec. 4.
3.2 Pressure Scaling of Ignition Delay at Turnover States
With the thermodynamic condition identified at the turnover state, we next investigate
the reactions controlling the pressure-scaling factor for the turnover ignition delay, shown in
Fig. 3. The sensitivity on the turnover ignition delay, defined as S  d (ln τ ) / d (ln k ) and
shown in Fig. 11 for n-butane and n-heptane (results for iso-octane are similar), demonstrates
that the most sensitive reactions are those involving the HO2 and H2O2 chemistry, while the
secondary important reactions are those involved in the low-T chemistry that determine α.
Peters et al. [25] have derived an analytical solution for the ignition delay time τ in the
intermediate temperature range based on the quasi-steady-state approximation for HO2, as
1/3



3
1  α  k11



τ ξ
2 2 
2
c
(
k
)
k
 Fuel  12,i 13 
i



(4)

where ξ is a proportional constant, α is defined by Eq. (2), cFuel is the fuel concentration,
and k11, k12,i, k13 correspond to the rate constants of 2HO2 → H2O2 + O2 (R11), RH + HO2 →
i-R + H2O2 (R12), and H2O2 (+M) → 2OH (+M) (R13), respectively. Figure 12 shows the
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turnover ignition delay calculated using both detail mechanism and the correlation of Eq. (4).
It is seen that very good agreement is achieved between the two, and the pressure-scaling
factor for all of three fuels are well reproduced.
Equation (4) employs the rates of three types of elemental reactions, namely R11, R12,
and R13 to correlate the HO2 and H2O2 chemistry on the ignition delay. The ratio α is also
employed in Eq. (4) to account for the contribution from low-temperature chemistry on the
ignition delay. Since ξ and α are both constants, the change of τm is therefore dominated by
the rate change of R11, R12, and R13. Especially, the k13 corresponding to the decomposition
of H2O2 with inherent pressure dependence is described by a Troe formula [28] and varies
nearly quadratically with changes of the turnover temperature and pressure in the fall-off
regime, shown in Fig. 13. Therefore, given the fact that τm and k13 vary nearly quadratically in
terms of pressure with power index being about -2 and 2 respectively, it could be shown from
Eq. (4) that the rate change of H2O2 decomposition reaction R13 contributes to about
two-thirds of the pressure dependence for τm, while the rate changes of R12 by the cFuel ,
i.e. cFuel ~P, and the effect of temperature variation on k11, k12,i contribute to the rest
one-third.

4. Experimental validation
We now assess the validity of the above two global behaviors and correlations against the
ignition delay data available in literature, with the ignition delay for n-heptane/air mixture
with φ = 1, covering the turnover temperature under 4 – 55 atm, reported in shock tube [8, 31,
32] and in RCM [11].
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Gauthier et al. [8] reported a pressure scaling factor of -1.64 for the ignition delay under
the pressure range of 13-55 atm by using a quadratic fitting to determine the conditions at the
turnover state, while the high temperature ignition data show a scaling factor of -0.55 [33].
However, simulation using the LLNL n-heptane mechanism [12] shows that the ignition
delay at the turnover state acquired from quadratic fitting covering the data points in regimes
of both high temperature and near the turnover states is lower than the actual value. Thus, a
cubic fitting covering only the points near the turnover temperature is adopted here, as shown
in Fig. 14, because computation shows that the cubic relation also cannot hold far from the
turnover state. Higher order fitting is also precluded due to the limited number of data points
of each dataset.
The conditions corresponding to the turnover states derived from the fitting curve are
plotted in Figs. 15 and 16. Strong Arrhenius dependence is found between pressure and
temperature. The predictions using the LLNL n-heptane mechanism [12] and the updated
n-heptane mechanism from Karwat et al. [24] are also plotted in Fig. 15 for comparison. Both
mechanisms reproduce the Arrhenius behavior between the turnover temperature and pressure,
and the “slopes” predicted from both mechanisms are close to those from the experimental
data. The prediction using α = 0.64 from the LLNL n-heptane mechanism [12] is plotted in
Fig. 14, and agrees with the experimental data very well. Strong power law dependence is
also found between ignition delay and pressure with τ m  P

1.79

, and the correlation using

Eq. (4) agrees well with the measurements in Fig. 16.
It is nevertheless also noted that the fitted turnover temperature under 12 atm from Shen
et al. [31] and under 20 atm from Gauthier et al. [8] show large deviations from the Arrhenius
14

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correlation with pressure, due to the lack of data near the turnover state. The deviation is
smaller for the power law between the ignition delay and pressure, as the ignition delay is not
sensitive to temperature near the turnover state.
The validation demonstrates the potential to use these two behaviors and correlations to
check the consistence among different ignition delay data. The correlations for the two
behaviors can also be used for interpolating measured sample data to other conditions.

5. Conclusions
The current work investigated the thermodynamic and chemical kinetic behavior of the
upper turnover states of the NTC regime, namely, the Arrhenius behavior between the
turnover temperature and pressure, and the power law between the ignition delay and
pressure at the turnover states. The underlying mechanism for these two behaviors was
further derived from the detailed mechanism. It is found that the turnover states under
different pressures correspond to a near-constant ratio α between the OH production from the
low-temperature chemistry and the total fuel consumption, and that the pressure-scaling
between the ignition delay at the turnover states can be correlated well using the analytical
solution from Peters et al. [25]. The analytical expression for the ignition delay only employs
the rates of three types of reactions on HO2 and H2O2 chemistry and the ratio α.
The two behaviors and correlations were further validated against the ignition delay data
for n-heptane/air mixture with φ = 1, covering the turnover states under 4 – 55 atm;
demonstrating quantitative agreement.
The correlations for the two global behaviors at turnover states provide insights into the
mechanism controlling the transition from low-temperature to intermediate-to-high
15

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temperature regimes. The behaviors and their controlling mechanism can also assist in the
design and validation of measured ignition delay data near turnover states, provide reference
points for developing correlation of ignition delay from experimental data, and help to
identify the regime where the ignition delay is not sensitive to temperature. Furthermore,
these correlations can provide targets for optimizing the reaction rate rules for the NTC
chemistry.

Acknowledgements
This work was primarily supported by the National Natural Science Foundation of
China 91441202. The work at Princeton University was supported by the Combustion Energy
Frontier Research Center, an Energy Frontier Research Center funded by the US Department
of Energy, Office of Basic Energy Sciences under Award Number DESC0001198. Peng Zhao
was also supported by the startup funding at Oakland University. The guidance on the NUIG
n-butane mechanism from Dr. Kuiwen Zhang from NUIG is very much appreciated.

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18

Appendix: Determination of the Low Temperature Chain Branching Ratio α
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Species time histories plotted in Fig. 5 show that it is justified to set R, RO2, QOOH,
O2QOOH and KET into quasi-steady state, for which the production rates balance the
consumption rates, yielding:

w1  w2 w3

w ,3



3

w
 w 3  w 4 w5

w5



w5  w5  w6  w7  w8  w8
w8  w8  w9 , w9  w10

(A1, A2)
(A3)
(A4, A5)

From the definition of θ and applying Eq. (A3-A5), we have

θ

w7  w9  w10
w7  2w9

w5
w5  w6  w7  w9

(A6)

Introducing Eq. (A4) and the rate expressions into Eq. (A6), and further define
K8  k8 / (k8  k9 ) , we obtain

θ

k7  2k9 K8 [O 2 ]
k5  k6  k7  k9 K8 [O 2 ]

(A7)

Recall the definition of γ, and assume w5  w5 , we obtain

γ

w5  w5
w5
k5


w3  w3 w4  w5 k4  k5

(A8)

which is only temperature dependent.
Recall the definition of β:

β

w3  w3
w3  w3

w1
w2  w3  w3

Substitute Eq. (A2) into Eq. (A9) with rate expressions, and further define
K3  k3 / (k3  k4  k5 ) , we have β as

19

(A9)

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β

(k4  k5 ) K 3[O 2 ]
k2  (k4  k5 ) K 3 [O 2 ]

(A10)

As there is only one isomer of R is involved, we have η = 1. By substituting Eq. (A7), Eq.
(A8) and Eq. (A10) into Eq. (2), we obtain

α

(k4  k5 ) K3[O 2 ]
k5
k7  2k9 K8 [O2 ]
k2  (k4  k5 ) K3[O 2 ] k4  k5 k5  k6  k7  k9 K8 [O 2 ]

(A11)

Clearly, it depends on both temperature and pressure, with the pressure dependence mainly
through the oxygen concentration and therefore oxygen addition reactions.
To determine α using detailed mechanism, the reaction path and rate corresponding to each
isomer should be taken into account for different low-temperature intermediates. In the
following, we shall present the adaptive definitions of βi, γi,j and θi,j considering isomeric
effects.
Merchant et al. [26] noticed that the decomposition of O2QOOH to KET and subsequent
reactions may lead to three OH in some cases, e.g., the case where CH2CHO is in the products
of KET decomposition. Besides, detailed mechanism shows that R7 and subsequent reactions
may lead to more than one OH in some cases, e.g., the reaction of C7H14OOH1-3 →
C4H7OOH1-4 + NC3H7 will lead to two OH radicals in the LLNL n-heptane mechanism [12].
For generality, the number of OH produced by R7 and R9 with their subsequent reactions are
denoted as n7, n9, respectively, and θi,j is determined by rewriting Eq. (A7) as

 n7 k7i , j  n9 k9i , j K8i , j [O 2 ] 
θi , j   i , j i , j i , j i , j i , j

 k5  k6  k7  k9 K8 [O2 ] i , j QOOH

(A12)

where the superscript i,j indicates that the rate constant corresponds to the specific isomer of
i,j-QOOH.
As i-RO2 can isomerize to several isomers of QOOH, γi,j can be determined by including
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all of the isomerization reactions from i-RO2 and rewriting Eq. (A8) as



k5i , j


γi , j   i
i ,m 
 k 4   k5 
m

i  RO2  i , j QOOH

(A13)

i, j

where k5 corresponds to the rate of isomerization from i-RO2 to i,j-QOOH.
The βi can be determined by specifying the rates in Eq. (A10) for each isomer of i-R as.

 (k4i   k5i ,m ) K3i [O 2 ] 


m
βi   i

i
i ,m
i
 k2  (k4   k5 ) K 3[O 2 ] 
m

i  R

(A14)

w1i
k1i
ηi 

 w1m  k1m

(A15)

Similarly, ηi is determined by

m

m

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List of Tables
Table 1. Reactions used for determining α.
Label
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10

Reaction
RH + OH → R + H2O
R → R1 + Alkene
R + O2 ⇌ RO2
RO2 → Alkene + HO2
RO2 ⇌ QOOH
QOOH → Alkene + HO2 b
QOOH → Epoxide + OH c
QOOH + O2 ⇌ O2QOOH
O2QOOH → KET + OH
KET → P + OH

Rate coefficient
k1
k2
k3, k-3 a
k4
k5, k-5
k6
k7
k8, k-8
k9
k10

Rate
w1
w2
w3, w-3 a
w4
w5, w-5
w6
w7
w8, w-8
w9
w10

a

k-3 is corresponding to the reverse rate of k3

b

Decomposition reaction of QOOH that does not lead to OH production is also classified in R6

c

Decomposition reaction of QOOH that leads to one OH radical is also classified in R7

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Figure Captions
Fig. 1. Schematic mechanism for the low temperature oxidation of a large alkane, RH.
Fig. 2. Total ignition delay as a function of the initial temperature T for stoichiometric
n-heptane/air mixture under various pressure. The turnover ignition delays are marked by the
triangle symbols, showing an Arrhenius dependence on the turnover temperature for different
pressures.
Fig. 3. Computed ignition delays at turnover temperatures for stoichiometric n-butane,
n-heptane and iso-octane in air. The turnover ignition delays show a power law dependence
on the pressure, to the order of -2, approximately.
Fig. 4. Dependence of pressure and temperature at the turnover states for stoichiometric
n-butane, n-heptane and iso-octane in air.
Fig. 5. Computed species and temperature profiles for stoichiometric n-butane/air mixture at
the turnover temperature under 20 atm.
Fig. 6. Magnitudes of the rates of important reactions producing or consuming OH (a) and
fuel molecule (b) for stoichiometric n-butane/air mixture at the turnover temperature under 20
atm.
Fig. 7. The brute force sensitivity on the turnover temperature for stoichiometric n-butane/air
mixture.
Fig. 8. Reaction flux from RH to OH and the definition of ηi, β i, γi,j, θi,j. Only one of the
isomers at every step is presented as an example.
Fig. 9. Computed α and αRO2 as a function of temperature under various pressures.
Fig. 10. Turnover temperature calculated using detailed mechanism and obtained at a constant
α.
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Fig. 11. The brute force sensitivity on the ignition delay at turnover temperature for n-butane
and n-heptane.
Fig. 12. The ignition delays from simulation using detailed mechanism and from the
correlation using Eq. (4).
Fig. 13. The rates of H2O2 (+M) → 2OH (+M) as a function of pressure at turnover
temperature, and those rates at various temperature and pressure.
Fig. 14. Ignition delay data for stoichiometric n-heptane/air mixture under various pressure.
All the data are scaled to the listed pressures using τ  P 1 to account for deviations in
experimental pressure [31]. Each dataset is fitted using a cubic polynomial and covering the
points near turnover temperature.
Fig. 15. The pressure and turnover temperature of each dataset.
Fig. 16. The ignition delay from measurements and those from correlation using Eq. (4).

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Fig. 1. Schematic mechanism for the low temperature oxidation of a large alkane, RH.

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Fig. 2. Total ignition delay as a function of the initial temperature T for stoichiometric
n-heptane/air mixture under various pressure. The turnover ignition delays are marked by the
triangle symbols, showing an Arrhenius dependence on the turnover temperature for different
pressures.

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Fig. 3. Computed ignition delays at turnover temperatures for stoichiometric n-butane,
n-heptane and iso-octane in air. The turnover ignition delays show a power law dependence
on the pressure, to the order of -2, approximately.

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Fig. 4. Dependence of pressure and temperature at the turnover states for stoichiometric
n-butane, n-heptane and iso-octane in air.

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Fig. 5. Computed species and temperature profiles for stoichiometric n-butane/air
mixture at the turnover temperature under 20 atm.

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(a)

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(b)
Fig. 6. Magnitudes of the rates of important reactions consuming OH (a) and fuel molecule (b)
for stoichiometric n-butane/air mixture at the turnover temperature under 20 atm. The
presented reactions contribute nearly 90% to the total consumption rate at the instant of 75%
τm.

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Fig. 7. The brute force sensitivity on the turnover temperature for stoichiometric n-butane/air
mixture.

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Fig. 8. Reaction flux from RH to OH and the definition of ηi, β i, γi,j, θi,j. Only one of the
isomers at every step is presented as an example.

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(a)

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(b)

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(c)
Fig. 9. Computed α and αRO2 as a function of temperature under various pressures for
different fuels.

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Fig. 10. Turnover temperature calculated using detailed mechanism (symbols) and obtained at
a constant α (lines).

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(a)

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(b)
Fig. 11. The brute force sensitivity on the ignition delay at turnover temperature for n-butane
and n-heptane.

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Fig. 12. The ignition delays from simulation using detailed mechanism (symbols) and from
the correlation using Eq. (4).

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Fig. 13. The rates of H2O2 (+M) → 2OH (+M) as a function of pressure at turnover
temperature for stoichiometric n-butane, n-heptane and iso-octane in air (symbols), and those
rates at various temperature and pressure (dash lines). The rate constants are adopted from
each detailed mechanism.

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Fig. 14. Ignition delay data for stoichiometric n-heptane/air mixture under various pressure.
All the data are scaled to the listed pressures using τ  P 1 to account for deviations in
experimental pressure [31]. Each dataset is fitted using a cubic polynomial and covering the
points near turnover temperature.

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Fig. 15. The dependence of pressure on turnover temperatures for stoichiometric
n-heptane/air mixture obtained from different methods.

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Fig. 16. The ignition delay from measurements and those from correlation using Eq. (4).

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