Rate Constant Parameterizations

This page describes the different parameterizations available in Cantera for calculating the forward rate constant \(k_f\) for a reaction.

Arrhenius Rate Expressions

An Arrhenius rate is described by the modified Arrhenius function:

\[ k_f = A T^b e^{-E_a / RT} \]

where \(A\) is the pre-exponential factor, \(T\) is the temperature, \(b\) is the temperature exponent, \(E_a\) is the activation energy, and \(R\) is the universal gas constant.

YAML Usage

An Arrhenius rate can be specified for a reaction in the YAML format by providing an Arrhenius rate expression for the reaction’s rate-constant field.

Falloff Reactions

A falloff reaction is one that has a rate that is first-order in the total concentration of third-body colliders \([\t{M}]\) at low pressure, like a three-body reaction, but becomes zero-order in \([\t{M}]\) as \([\t{M}]\) 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 [Lindemann, 1922]:

\[ k_f(T, [\t{M}]) = \frac{k_0 [\t{M}]}{1 + \frac{k_0 [\t{M}]}{k_\infty}} \]

In the low-pressure limit, this approaches \(k_0 [\t{M}]\), and in the high-pressure limit it approaches \(k_\infty\).

Defining the non-dimensional reduced pressure:

\[ P_r = \frac{k_0 [\t{M}]}{k_\infty} \]

The rate constant may be written as

\[ k_f(T, P_r) = k_\infty \left(\frac{P_r}{1 + P_r}\right) \]

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 \(F(T, P_r)\):

\[ k_f(T, P_r) = k_\infty \left(\frac{P_r}{1 + P_r}\right) F(T, P_r) \]

This expression is used to compute the rate coefficient for falloff reactions. The function \(F(T, P_r)\) is the falloff function.

YAML Usage

A falloff reaction may be defined in the YAML format using the falloff reaction type.

The Troe Falloff Function

A widely-used falloff function is the one proposed by Gilbert et al. [1983]:

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

YAML Usage

A Troe falloff function may be specified in the YAML format using the Troe field in the reaction entry. The first three parameters, \((A, T_3, T_1)\), are required. The fourth parameter, \(T_2\), is optional; if omitted, the last term of the falloff function is not used.

Tsang’s Approximation to \(F_\t{cent}\)

Wing Tsang presented approximations for the value of \(F_\t{cent}\) for Troe falloff in databases of reactions, for example, Tsang and Herron [1991]. Tsang’s approximations are linear in temperature:

\[ F_\t{cent} = A + BT \]

where \(A\) and \(B\) are constants. The remaining equations for \(C\), \(N\), \(f_1\), and \(F\) from the Troe falloff function are not affected.

YAML Usage

A Tsang falloff function may be specified in the YAML format using the Tsang field in the reaction entry.

Added in version 2.6.

The SRI Falloff Function

This falloff function is based on the one originally due to Stewart et al. [1989], which required three parameters \(a\), \(b\), and \(c\). Kee et al. [1989] 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:

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

In keeping with the nomenclature of Kee et al. [1989], we will refer to this as the SRI falloff function.

YAML Usage

An SRI falloff function may be specified in the YAML format using the SRI field in the entry.

Chemically-Activated Reactions

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

\[ \t{Si + SiH_4 (+M) \leftrightarrow Si_2H_2 + H_2 (+M)} \]

which competes with:

\[ \t{Si + SiH_4 (+M) \leftrightarrow Si_2H_4 (+M)} \]

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 proportional to the low-pressure rate constant:

\[ k_f(T, P_r) = k_0 \left(\frac{1}{1 + P_r}\right) F(T, P_r) \]

and the optional blending function \(F\) may be described by any of the parameterizations allowed for falloff reactions.

YAML Usage

Chemically-activated reactions can be defined in the YAML format using the chemically-activated reaction type.

Pressure-Dependent Arrhenius Rate Expressions (P-Log)

This parameterization represents pressure-dependent reaction rates by logarithmically interpolating between Arrhenius rate expressions at various pressures. Given two rate expressions at two specific pressures:

\[ \begin{align}\begin{aligned} P_1: k_1(T) = A_1 T^{b_1} e^{-E_1 / RT}\\P_2: k_2(T) = A_2 T^{b_2} e^{-E_2 / RT} \end{aligned}\end{align} \]

The rate at an intermediate pressure \(P_1 < P < P_2\) is computed as

\[ \log k(T,P) = \log k_1(T) + \bigl(\log k_2(T) - \log k_1(T)\bigr) \frac{\log P - \log P_1}{\log P_2 - \log P_1} \]

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.

Caution

Negative A-factors can be used for any of the rate expressions at a given pressure. However, the sum of all of the rates at a given pressure must be positive, due to the logarithmic interpolation of the rate for intermediate pressures. When a P-log type reaction is initialized, Cantera does a validation check for a range of temperatures that the sum of the reaction rates at each pressure is positive. Unfortunately, if these checks fail, the only options are to remove the reaction or contact the author of the reaction/mechanism in question, because the reaction is mathematically unsound.

YAML Usage

P-log reactions can be defined in the YAML format using the pressure-dependent-Arrhenius reaction type.

Chebyshev Reaction Rate Expressions

Chebyshev rate expressions represent a phenomenological rate coefficient \(k(T,P)\) in terms of a bivariate Chebyshev polynomial. The rate constant can be written as:

\[ \log k(T,P) = \sum_{t=1}^{N_T} \sum_{p=1}^{N_P} \alpha_{tp} \phi_t(\tilde{T}) \phi_p(\tilde{P}) \]

where \(N_T\) is the order of the polynomial in the temperature dimension, \(N_P\) is the order of the polynomial in the pressure dimension, \(\alpha_{tp}\) are the constants defining the rate, \(\phi_n(x)\) is the Chebyshev polynomial of the first kind of degree \(n\) evaluated at \(x\), and

\[ \begin{align}\begin{aligned} \tilde{T} \equiv \frac{2T^{-1} - T_\t{min}^{-1} - T_\t{max}^{-1}} {T_\t{max}^{-1} - T_\t{min}^{-1}}\\\tilde{P} \equiv \frac{2 \log P - \log P_\t{min} - \log P_\t{max}} {\log P_\t{max} - \log P_\t{min}} \end{aligned}\end{align} \]

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

A Chebyshev rate expression is specified in terms of the coefficient matrix \(\alpha\) and the temperature and pressure ranges.

Caution

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.

YAML Usage

Chebyshev reactions can be defined in the YAML format using the Chebyshev reaction type.

Blowers-Masel Reactions

In some circumstances like thermodynamic sensitivity analysis, or modeling heterogeneous reactions from one catalyst surface to another, the enthalpy change of a reaction (\(\Delta H\)) can be modified. Due to the change in \(\Delta H\), the activation energy of the reaction must be adjusted accordingly to provide accurate simulation results. To adjust the activation energy due to changes in the reaction enthalpy, the Blowers-Masel rate expression is available. This approximation was proposed by Blowers and Masel [2000] to automatically scale activation energy as the reaction enthalpy is changed. The intrinsic activation energy \(E_a^0\) is defined as the activation energy when \(\Delta H = 0\). The activation energy can then be written as a function of \(\Delta H\):

\[\begin{split} E_a = \begin{cases} 0 & \text{if } \Delta H \leq -4 E_a^0 \\ \Delta H & \text{if } \Delta H \geq 4 E_a^0 \\ \frac{\left( w + \frac{\Delta H }{2} \right) (V_P - 2 w + \Delta H) ^2} {V_P^2 - 4 w^2 + \Delta H^2} & \text{otherwise} \end{cases} \end{split}\]

where

\[ V_P = 2 w \frac{w + E_a^0}{w - E_a^0}, \]

and \(w\) is the average of the bond dissociation energy of the bond breaking and that being formed. Note that the expression is insensitive to \(w\) as long as \(w \ge 2 E_a^0\), so we can use an arbitrarily high value of \(w = 1000\text{ kJ/mol}\).

After \(E_a\) is evaluated, the reaction rate can be calculated using the modified Arrhenius expression

\[ k_f = A T^b e^{-E_a / RT}. \]

Added in version 2.6.

YAML Usage

Blowers Masel reactions can be defined in the YAML format using the Blowers-Masel reaction type.

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 \(\theta_k\) of one or more surface species. The forward rate constant for a reaction of this type is:

\[ k_f = A T^b \exp \left( - \frac{E_a}{RT} \right) \prod_k 10^{a_k \theta_k} \theta_k^{m_k} \exp \left( \frac{- E_k \theta_k}{RT} \right) \]

where \(A\), \(b\), and \(E_a\) are the modified Arrhenius parameters and \(a_k\), \(m_k\), and \(E_k\) are the coverage dependencies from species \(k\).

YAML Usage

In the YAML format, surface reactions are identified by the presence of surface species and support several additional options.

The surface reaction type defaults to interface-Arrhenius, where the rate expression uses the Arrhenius parameterization (see YAML documentation).

Added in version 2.6: As an alternative, Cantera also supports the interface-Blowers-Masel surface reaction type, which uses the Blowers-Masel parameterization (see YAML documentation).

Sticking Reactions

Sticking reactions represent a special case of surface reactions, where collisions between gas-phase molecules and surfaces result in the gas-phase molecule sticking to the surface. This process can be described as a reaction which is parameterized by a sticking coefficient:

\[ \gamma = a T^b e^{-c/RT} \]

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:

\[ k_f = \frac{\gamma}{\Gamma_\t{tot}^m} \sqrt{\frac{RT}{2 \pi W}} \]

where \(\Gamma_\t{tot}\) 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.

YAML Usage

Sticking reactions can be defined in the YAML format by specifying the rate constant in the reaction’s sticking-coefficient field.

The sticking reaction type defaults to sticking-Arrhenius, where the rate expression uses the Arrhenius parameterization (see YAML documentation).

Added in version 2.6: As an alternative, Cantera also supports the sticking-Blowers-Masel surface reaction type, which uses the Blowers-Masel parameterization (see YAML documentation).

Two-Temperature-Plasma Reactions

The two-temperature-plasma reaction is commonly used for non-equilibrium plasmas. The reaction rate of a two-temperature-plasma reaction depends on both gas and electron temperature [Kossyi et al., 1992], and can be expressed as:

\[ k_f = A T_e^b \exp \left( - \frac{E_{a,g}}{RT} \right) \exp \left(\frac{E_{a,e}(T_e - T)}{R T T_e}\right), \]

where \(A\) is the pre-exponential factor, \(T\) is the temperature, \(T_e\) is the electron temperature, \(b\) is the electron temperature exponent, \(E_{a,g}\) is the activation energy for gas, \(E_{a,e}\) is the activation energy for electron and \(R\) is the gas constant.

Added in version 2.6.

YAML Usage

Two-temperature plasma reactions can be defined in the YAML format by specifying two-temperature-plasma as the reaction type and providing the two activation energies as part of the rate-constant.