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:
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]:
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:
The rate constant may be written as
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 \(F(T, P_r)\):
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]:
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:
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:
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:
which competes with:
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:
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 [Gou et al., 2011]. Given two rate expressions at two specific pressures:
The rate at an intermediate pressure \(P_1 < P < P_2\) is computed as
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:
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
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.
Linear Burke Rate Expressions#
Linear Burke rate expressions employ the reduced-pressure linear mixture rule (LMR-R). This mixture rule is used to evaluate the rate constants of complex-forming reactions, and is a weighted sum of the bath gas rate constants (when pure) evaluated at the reduced pressure (\(R\)) and temperature (\(T\)) of the mixture.
where the reduced pressure, \(R_{\text{LMR}}\), in its most general form
and the fractional contribution of each component to the reduced pressure, \(\tilde{X}_{i}\)
can be cast in terms of the absolute value of the least negative chemically significant eigenvalue of the master equation for the \(i^{th}\) collider (when pure) in the low-pressure limit, \(\Lambda_{0,i}(T)[M]\), and high-pressure limit, \(\Lambda_{\infty}(T)\), and \([M]\) is the total concentration.
Evaluating all rate constants at the reduced pressure (\(R\))—instead of the pressure (\(P\))—of the mixture takes advantage of the fact that rate constants (and their chemically significant eigenvectors) for different colliders are usually far more similar at the same \(R\) than the same \(P\). In practice, since rate constants are usually expressed with respect to pressure \(P\) (which has units of Pa, Torr, bar, atm, etc.) rather than reduced pressure \(R\) (which is dimensionless), one needs to find the effective pressure for the \(i^{th}\) collider, \(P_{i}^{\text{ eff}}\) (with units of \(P\)), such that the reduced pressure of pure collider \(i\) is equal to the reduced pressure of the mixture. This can be shown to be
such that an alternate version of the generalized LMR-R equation can be written as
It is worth noting two convenient implications of this change in basis. First, when LMR-R is implemented with the above equation, it is not necessary to specify \(\Lambda_{\infty}(T)\), which cancels out in evaluating \(P_{i}^{\text{ eff}}\). Second, only ratios of \(\Lambda_{0,i}(T)\) appear in the calculations of \(\tilde{X}_{i,\text{LMR}}(T,P,\boldsymbol{X})\) and \(P_{i,\text{LMR}}^{\text{ eff}}(T,P,\boldsymbol{X})\), such that third-body efficiencies \(\epsilon_{0,i}(T)=\Lambda_{0,i}(T)/\Lambda_{0,\text{M}}(T)\) (where the user must assign \(\epsilon_{0,\text{M}}(T)=1\), as this is true by definition), can be implemented by the user in lieu of \(\Lambda_{0,i}(T)\).
While full implementation of LMR-R via the above equation would require \(k_i(T,P)\) be specified in addition to \(\Lambda_{0,i}(T)\) or \(\epsilon_{0,i}(T)\) for each important collider, often data for \(k_i(T,P)\) (that is, the complete \(T,P\)-dependence) for each collider is not available even when \(\Lambda_{0,i}(T)\) or \(\epsilon_{0,i}(T)\) have available data or can be estimated using typical values (as is typically done in kinetic models for reactions in modified Lindemann expressions). Therefore, for colliders with unique \(\Lambda_{0,i}(T)\) (or \(\epsilon_{0,i}(T)\)) but without \(k_i(T,P)\), the same reduced-pressure dependence as M (that is, \(k_{i}(T,R)=k_{M}(T,R)\)) is assumed:
where the sum over \(n\) is only for the colliders for which unique \(k_n(T,P)\) are available. Each \(k_n(T,P)\) can be specified in the user’s choice of Troe, Plog, or Chebyshev formats. For the Troe format, the effective third-body concentration is calculated by dividing \(P_{i}^{\text{ eff}}\) by the temperature and ideal gas constant. For the other formats, \(P_{i}^{\text{ eff}}\) is implemented directly as the ‘pressure’ of interest.
While not required if unique \(k_i(T,P)\) data are available, this approximation, like LMR-R, takes advantage of the fact that rate constants for colliders with even very different third-body efficiencies often are much more similar at the same reduced pressure (\(R\)) than at the same pressure (\(P\)) and, in fact, are exactly the same if they differ in only their collision frequency (but have the same energy- and angular-momentum-transfer kernel). This equation forms the basis of the computational implementation of LMR-R in Cantera via the LinearBurkeRate reaction class, as it enables the most accurate representation of \(k_{\text{LMR-R}}(T,P,\boldsymbol{X})\) possible given limitations in the completeness of the dataset at any given moment. Further description of the LMR-R theory and computational method is available in Singal et al. [2024].
YAML Usage
Linear Burke rate expressions can be defined in the YAML format using the
linear-Burke reaction type.
Added in version 3.1.
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 \(\Delta H\):
where
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
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:
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:
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:
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:
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.
Electron Collision Plasma Reactions#
The electron collision plasma reaction rate uses the electron collision data and the electron energy distribution to calculate the reaction rate. Hagelaar and Pitchford Hagelaar and Pitchford [2005] define the reaction rate coefficient (Eqn. 63) as,
where \(\gamma = \sqrt{2/m_e}\) (Eqn.4 in Hagelaar Hagelaar [2015]), \(m_e\) [kg] is the electron mass, \(\epsilon\) [J] is the electron energy, \(\sigma(\epsilon)\) [m²] is the reaction collision cross section, \(F_0(\epsilon)\) [\(\t{J^{-3/2}}\)] is the normalized electron energy distribution function, and \(k\) has units of [m³/s].
To recast this in terms of moles rather than molecules, and letting \(\epsilon_V\) be the electron energy expressed in volts, the forward rate coefficient can be rewritten as
where \(e\) is the elementary charge [C] and \(N_A\) is the Avogadro constant [\(\t{kmol^{-1}}\)].
Added in version 3.1.
YAML Usage
Electron collision reactions can be defined in the YAML format by specifying
electron-collision-plasma as the reaction type
and providing lists with the cross-sections and corresponding energy-levels.