## Modeling Chemical Reactions in Cantera¶

Here, we describe how Cantera calculates chemical reaction rates for various reaction types.

## Reactions with a Pressure-Independent Rate¶

The reaction() entry is used to represent homogeneous reactions with pressure-independent rate coefficients and mass action kinetics. Examples of reaction entries that implement some reactions in the GRI-Mech 3.0 natural gas combustion mechanism [3] are shown below:

units(length = 'cm', quantity = 'mol', act_energy = 'cal/mol')
...
reaction( "O + H2 <=> H + OH", [3.87000E+04, 2.7, 6260])
reaction( "O + HO2 <=> OH + O2", [2.00000E+13, 0.0, 0])
reaction( "O + H2O2 <=> OH + HO2", [9.63000E+06, 2.0, 4000])
reaction( "O + HCCO <=> H + 2 CO", [1.00000E+14, 0.0, 0])
reaction( "H + O2 + AR <=> HO2 + AR", kf=Arrhenius(A=7.00000E+17, b=-0.8, E=0))
reaction( equation = "HO2 + C3H7 <=> O2 + C3H8", kf=Arrhenius(2.55000E+10, 0.255, -943))
reaction( equation = "HO2 + C3H7 => OH + C2H5 + CH2O", kf=[2.41000E+13, 0.0, 0])


## Three-Body Reactions¶

A three-body reaction is a gas-phase reaction of the form:

\begin{equation*} \mathrm{A + B + M \rightleftharpoons AB + M} \end{equation*}

Here $$\mathrm{M}$$ is an unspecified collision partner that carries away excess energy to stabilize the $$\mathrm{AB}$$ molecule (forward direction) or supplies energy to break the $$\mathrm{AB}$$ bond (reverse direction).

Different species may be more or less effective in acting as the collision partner. A species that is much lighter than $$\mathrm{A}$$ and $$\mathrm{B}$$ may not be able to transfer much of its kinetic energy, and so would be inefficient as a collision partner. On the other hand, a species with a transition from its ground state that is nearly resonant with one in the $$\mathrm{AB^*}$$ activated complex may be much more effective at exchanging energy than would otherwise be expected.

These effects can be accounted for by defining a collision efficiency $$\epsilon$$ for each species, defined such that the forward reaction rate is

\begin{equation*} k_f(T)[A][B][M] \end{equation*}

where

\begin{equation*} [M] = \sum_{\mathrm{k}} \epsilon_{\mathrm{k}} C_{\mathrm{k}} \end{equation*}

where $$C_{\mathrm{k}}$$ is the concentration of species $$\mathrm{k}$$. Since any constant collision efficiency can be absorbed into the rate coefficient $$k_f(T)$$, the default collision efficiency is 1.0.

A three-body reaction may be defined using the three_body_reaction() entry. The equation string for a three-body reaction must contain an 'M' or 'm' on both the reactant and product sides of the equation. The collision efficiencies are specified as a string, with the species name followed by a colon and the efficiency.

Some examples from GRI-Mech 3.0 are shown below:

three_body_reaction("2 O + M <=> O2 + M", [1.20000E+17, -1, 0],
"AR:0.83 C2H6:3 CH4:2 CO:1.75 CO2:3.6 H2:2.4 H2O:15.4 ")

three_body_reaction("O + H + M <=> OH + M", [5.00000E+17, -1, 0],
efficiencies="AR:0.7 C2H6:3 CH4:2 CO:1.5 CO2:2 H2:2 H2O:6 ")

three_body_reaction(
equation = "H + OH + M <=> H2O + M",
rate_coeff=[2.20000E+22, -2, 0],
efficiencies="AR:0.38 C2H6:3 CH4:2 H2:0.73 H2O:3.65 "
)


As always, the field names are optional if the field values are entered in the declaration order.

## Falloff Reactions¶

A falloff reaction is one that has a rate that is first-order in $$[M]$$ at low pressure, like a three-body reaction, but becomes zero-order in $$[M]$$ as $$[M]$$ increases. Dissociation/association reactions of polyatomic molecules often exhibit this behavior.

The simplest expression for the rate coefficient for a falloff reaction is the Lindemann form [2]:

\begin{equation*} k_f(T, [{\mathrm{M}}]) = \frac{k_0[{ \mathrm{M}}]}{1 + \frac{k_0{ [\mathrm{M}]}}{k_\infty}} \end{equation*}

In the low-pressure limit, this approaches $$k0{[\mathrm{M}]}$$, and in the high-pressure limit it approaches $$k_\infty$$.

Defining the non-dimensional reduced pressure:

\begin{equation*} P_r = \frac{k_0 [\mathrm{M}]}{k_\infty} \end{equation*}

The rate constant may be written as

\begin{equation*} k_f(T, P_r) = k_\infty \left(\frac{P_r}{1 + P_r}\right) \end{equation*}

More accurate models for unimolecular processes lead to other, more complex, forms for the dependence on reduced pressure. These can be accounted for by multiplying the Lindemann expression by a function $$F(T, P_r)$$:

\begin{equation*} k_f(T, P_r) = k_\infty \left(\frac{P_r}{1 + P_r}\right) F(T, P_r) \end{equation*}

This expression is used to compute the rate coefficient for falloff reactions. The function $$F(T, P_r)$$ is the falloff function, and is specified by assigning an embedded entry to the falloff field.

### The Troe Falloff Function¶

A widely-used falloff function is the one proposed by Gilbert et al. [1]:

\begin{equation*} \log_{10} F(T, P_r) = \frac{\log_{10} F_{cent}(T)}{1 + f_1^2} \end{equation*}
\begin{equation*} F_{cent}(T) = (1-A) \exp(-T/T_3) + A \exp (-T/T_1) + \exp(-T_2/T) \end{equation*}
\begin{equation*} f_1 = (\log_{10} P_r + C) / (N - 0.14 (\log_{10} P_r + C)) \end{equation*}
\begin{equation*} C = -0.4 - 0.67\; \log_{10} F_{cent} \end{equation*}
\begin{equation*} N = 0.75 - 1.27\; \log_{10} F_{cent} \end{equation*}

The Troe() directive requires specifying the first three parameters $$(A, T_3, T_1)$$. The fourth parameter, $$T_2$$, is optional, defaulting to 0.0.

### The SRI Falloff Function¶

This falloff function is based on the one originally due to Stewart et al. [4], which required three parameters $$a$$, $$b$$, and $$c$$. Kee et al. [5] generalized this function slightly by adding two more parameters $$d$$ and $$e$$. (The original form corresponds to $$d = 1$$ and $$e = 0$$.) Cantera supports the extended 5-parameter form, given by:

\begin{equation*} F(T, P_r) = d \bigl[a \exp(-b/T) + \exp(-T/c)\bigr]^{1/(1+\log_{10}^2 P_r )} T^e \end{equation*}

In keeping with the nomenclature of Kee et al. [5], we will refer to this as the “SRI” falloff function. It is implemented by the SRI() directive.

## Chemically-Activated Reactions¶

For these reactions, the rate falls off as the pressure increases, due to collisional stabilization of a reaction intermediate. Example:

\begin{equation*} \mathrm{Si + SiH_4 (+M) \leftrightarrow Si_2H_2 + H_2 (+M)} \end{equation*}

which competes with:

\begin{equation*} \mathrm{Si + SiH_4 (+M) \leftrightarrow Si_2H_4 (+M)} \end{equation*}

Like falloff reactions, chemically-activated reactions are described by blending between a “low pressure” and a “high pressure” rate expression. The difference is that the forward rate constant is written as being proportional to the low pressure rate constant:

\begin{equation*} k_f(T, P_r) = k_0 \left(\frac{1}{1 + P_r}\right) F(T, P_r) \end{equation*}

and the optional blending function $$F$$ may described by any of the parameterizations allowed for falloff reactions. Chemically-activated reactions can be defined using the chemically_activated_reaction() directive.

An example of a reaction specified with this parameterization:

chemically_activated_reaction('CH3 + OH (+ M) <=> CH2O + H2 (+ M)',
kLow=[2.823201e+02, 1.46878, (-3270.56495, 'cal/mol')],
kHigh=[5.880000e-14, 6.721, (-3022.227, 'cal/mol')],
falloff=Troe(A=1.671, T3=434.782, T1=2934.21, T2=3919.0))


In this example, the units of $$k_0$$ (kLow) are m3/kmol/s and the units of $$k_\infty$$ (kHigh) are 1/s.

## Pressure-Dependent Arrhenius Rate Expressions (P-Log)¶

The pdep_arrhenius() class represents pressure-dependent reaction rates by logarithmically interpolating between Arrhenius rate expressions at various pressures. Given two rate expressions at two specific pressures:

\begin{equation*} P_1: k_1(T) = A_1 T^{b_1} e^{E_1 / RT} \end{equation*}
\begin{equation*} P_2: k_2(T) = A_2 T^{b_2} e^{E_2 / RT} \end{equation*}

The rate at an intermediate pressure $$P_1 < P < P_2$$ is computed as

\begin{equation*} \log k(T,P) = \log k_1(T) + \bigl(\log k_2(T) - \log k_1(T)\bigr) \frac{\log P - \log P_1}{\log P_2 - \log P_1} \end{equation*}

Multiple rate expressions may be given at the same pressure, in which case the rate used in the interpolation formula is the sum of all the rates given at that pressure. For pressures outside the given range, the rate expression at the nearest pressure is used.

An example of a reaction specified in this format:

pdep_arrhenius('R1 + R2 <=> P1 + P2',
[(0.001315789, 'atm'), 2.440000e+10, 1.04, 3980.0],
[(0.039473684, 'atm'), 3.890000e+10, 0.989, 4114.0],
[(1.0, 'atm'), 3.460000e+12, 0.442, 5463.0],
[(10.0, 'atm'), 1.720000e+14, -0.01, 7134.0],
[(100.0, 'atm'), -7.410000e+30, -5.54, 12108.0],
[(100.0, 'atm'), 1.900000e+15, -0.29, 8306.0])


The first argument is the reaction equation. Each subsequent argument is a sequence of four elements specifying a pressure and the Arrhenius parameters at that pressure.

## Chebyshev Reaction Rate Expressions¶

Class chebyshev_reaction() represents a phenomenological rate coefficient $$k(T,P)$$ in terms of a bivariate Chebyshev polynomial. The rate constant can be written as:

\begin{equation*} \log k(T,P) = \sum_{t=1}^{N_T} \sum_{p=1}^{N_P} \alpha_{tp} \phi_t(\tilde{T}) \phi_p(\tilde{P}) \end{equation*}

where $$\alpha_{tp}$$ are the constants defining the rate, $$\phi_n(x)$$ is the Chebyshev polynomial of the first kind of degree $$n$$ evaluated at $$x$$, and

\begin{equation*} \tilde{T} \equiv \frac{2T^{-1} - T_\mathrm{min}^{-1} - T_\mathrm{max}^{-1}} {T_\mathrm{max}^{-1} - T_\mathrm{min}^{-1}} \end{equation*}
\begin{equation*} \tilde{P} \equiv \frac{2 \log P - \log P_\mathrm{min} - \log P_\mathrm{max}} {\log P_\mathrm{max} - \log P_\mathrm{min}} \end{equation*}

are reduced temperatures and reduced pressures which map the ranges $$(T_\mathrm{min}, T_\mathrm{max})$$ and $$(P_\mathrm{min}, P_\mathrm{max})$$ to $$(-1, 1)$$.

A Chebyshev rate expression is specified in terms of the coefficient matrix $$\alpha$$ and the temperature and pressure ranges. An example of a Chebyshev rate expression where $$N_T = 6$$ and $$N_P = 4$$ is:

chebyshev_reaction('R1 + R2 <=> P1 + P2',
Tmin=290.0, Tmax=3000.0,
Pmin=(0.001, 'atm'), Pmax=(100.0, 'atm'),
coeffs=[[-1.44280e+01,  2.59970e-01, -2.24320e-02, -2.78700e-03],
[ 2.20630e+01,  4.88090e-01, -3.96430e-02, -5.48110e-03],
[-2.32940e-01,  4.01900e-01, -2.60730e-02, -5.04860e-03],
[-2.93660e-01,  2.85680e-01, -9.33730e-03, -4.01020e-03],
[-2.26210e-01,  1.69190e-01,  4.85810e-03, -2.38030e-03],
[-1.43220e-01,  7.71110e-02,  1.27080e-02, -6.41540e-04]])


Note that the Chebyshev polynomials are not defined outside the interval $$(-1,1)$$, and therefore extrapolation of rates outside the range of temperatures and pressure for which they are defined is strongly discouraged.

## Surface Reactions¶

Heterogeneous reactions on surfaces are represented by an extended Arrhenius- like rate expression, which combines the modified Arrhenius rate expression with further corrections dependent on the fractional surface coverages $$\theta_{\mathrm{k}}$$ of one or more surface species. The forward rate constant for a reaction of this type is:

\begin{equation*} k_f = A T^b \exp \left( - \frac{E_a}{RT} \right) \prod_{\mathrm{k}} 10^{a_{\mathrm{k}} \theta_{\mathrm{k}}} \theta_{\mathrm{k}}^{m_{\mathrm{k}}} \exp \left( \frac{- E_{\mathrm{k}} \theta_{\mathrm{k}}}{RT} \right) \end{equation*}

where $$A$$, $$b$$, and $$E_a$$ are the modified Arrhenius parameters and $$a_{\mathrm{k}}$$, $$m_{\mathrm{k}}$$, and $$E_{\mathrm{k}}$$ are the coverage dependencies from species $$\mathrm{k}$$. A reaction of this form with a single coverage dependency (on the species H(S)) can be written using class surface_reaction() with the coverage keyword argument supplied to the class Arrhenius():

surface_reaction("2 H(S) => H2 + 2 PT(S)",
Arrhenius(A, b, E_a,
coverage=['H(S)', a_1, m_1, E_1]))


For a reaction with multiple coverage dependencies, the following syntax is used:

surface_reaction("2 H(S) => H2 + 2 PT(S)",
Arrhenius(A, b, E_a,
coverage=[['H(S)', a_1, m_1, E_1],
['PT(S)', a_2, m_2, E_2]]))


### Sticking Coefficients¶

Collisions between gas-phase molecules and surfaces which result in the gas- phase molecule sticking to the surface can be described as a reaction which is parameterized by a sticking coefficient:

\begin{equation*} \gamma = a T^b e^{-c/RT} \end{equation*}

where $$a$$, $$b$$, and $$c$$ are constants specific to the reaction. The values of these constants must be specified so that the sticking coefficient $$\gamma$$ is between 0 and 1 for all temperatures.

The sticking coefficient is related to the forward rate constant by the formula:

\begin{equation*} k_f = \frac{\gamma}{\Gamma_\mathrm{tot}^m} \sqrt{\frac{RT}{2 \pi W}} \end{equation*}

where $$\Gamma_\mathrm{tot}$$ is the total molar site density, $$m$$ is the sum of all the surface reactant stoichiometric coefficients, and $$W$$ is the molecular weight of the gas phase species.

A reaction of this form can be written as:

surface_reaction("H2O + PT(S) => H2O(S)", stick(a, b, c))


### Reaction Orders¶

Explicit reaction orders different from the stoichiometric coefficients are sometimes used for non-elementary reactions. For example, consider the global reaction:

\begin{equation*} \mathrm{C_8H_{18} + 12.5 O_2 \rightarrow 8 CO_2 + 9 H_2O} \end{equation*}

the forward rate constant might be given as [6]:

\begin{equation*} k_f = 4.6 \times 10^{11} [\mathrm{C_8H_{18}}]^{0.25} [\mathrm{O_2}]^{1.5} \exp\left(\frac{30.0\,\mathrm{kcal/mol}}{RT}\right) \end{equation*}

This reaction could be defined as:

reaction("C8H18 + 12.5 O2 => 8 CO2 + 9 H2O", [4.6e11, 0.0, 30.0],
order="C8H18:0.25 O2:1.5")


Special care is required in this case since the units of the pre-exponential factor depend on the sum of the reaction orders, which may not be an integer.

Note that you can change reaction orders only for irreversible reactions.

Normally, reaction orders are required to be positive. However, in some cases negative reaction orders are found to be better fits for experimental data. In these cases, the default behavior may be overridden by adding negative_orders to the reaction options like the following:

reaction("C8H18 + 12.5 O2 => 8 CO2 + 9 H2O", [4.6e11, 0.0, 30.0],
order="C8H18:-0.25 O2:1.75", options=['negative_orders'])


Some global reactions could have reactions orders for non-reactant species. One should add nonreactant_orders to the reaction options to use this feature:

reaction("C8H18 + 12.5 O2 => 8 CO2 + 9 H2O", [4.6e11, 0.0, 30.0],
order="C8H18:-0.25 CO:0.15",
options=['negative_orders', 'nonreactant_orders'])


References

 [1] R. G. Gilbert, K. Luther, and J. Troe. Ber. Bunsenges. Phys. Chem., 87:169, 1983.
 [2] Lindemann. Trans. Faraday Soc., 17:598, 1922.
 [3] Gregory P. Smith, David M. Golden, Michael Frenklach, Nigel W. Moriarty, Boris Eiteneer, Mikhail Goldenberg, C. Thomas Bowman, Ronald K. Hanson, Soonho Song, William C. Gardiner, Jr., Vitali V. Lissianski, , and Zhiwei Qin. GRI-Mech version 3.0, 1997. see http://www.me.berkeley.edu/gri_mech.
 [4] P. H. Stewart, C. W. Larson, and D. Golden. Combustion and Flame, 75:25, 1989.
 [5] (1, 2) R. J. Kee, F. M. Rupley, and J. A. Miller. Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics. Technical Report SAND89-8009, Sandia National Laboratories, 1989.
 [6] C. K. Westbrook and F. L. Dryer. Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames. Combustion Science and Technology 27, pp. 31—43. 1981.