Reactions directives

Several types of elementary reaction steps can be specified, depending on the first keyword of each listing. An overview is provided in the Table below.

Directive	Number of Columns	Description
&AR, &ARQ, &AR-L, &ARQ-L	6-7 without Q, 8-9 with Q	Arrhenius-type elementary reaction step. -L uses the ZFH-M lateral interaction model. Q tag enables pre-exponential factor calculation based on temperature and quotient of partition functions of TS and IS/FS.
AR-E, ARQ-E	8-9	Arrhenius-type elementary reaction step for electrochemistry.
ARL, ARQL	12-13	Arrhenius-type step. L uses ZFH-S lateral interaction model. Q enables pre-exponential factor calculation based on partition function quotients.
MIG	13-14	Lateral interaction for migration steps.
HK, HK-L, HK3, HK3-L	8-9	Hertz-Knudsen adsorption/desorption kinetics. Default for linear adsorbates; use 3 for non-linear, -L for ZFH-M model.
HK, HKN-L	15-16	Hertz-Knudsen kinetics with Shomate equation for entropy. -L uses ZFH-M model.
HKNL	21-22	Hertz-Knudsen kinetics with Shomate equation and ZFH-S lateral interaction model.
ADS	21-22	Hertz-Knudsen kinetics with Shomate equation. Rates use quotients of vibrational partition functions.

Reaction objects

Below, we explain upon the different types of reaction objects, their arguments and how reaction rate constants are derived from these arguments.

AR tag

Arrhenius-type of reactions are typically for surface-reactions where minimal changes in the entropy between the reactants are expected. Their forward and reverse reaction rate constants are given by

\[ k_{f} = \nu_{f} \exp \left( -\frac{\Delta E_{\text{a,f}}}{RT} \right) \]

and

\[ k_{b} = \nu_{b} \exp \left( -\frac{\Delta E_{\text{b,f}}}{RT} \right) \]

where \(\nu_{f}\) and \(\nu_{b}\) are the forward and backward pre-exponential factors in units of reciprocal second and \(E_{\text{a,f}}\) and \(E_{\text{a,b}}\) are the activation energies in units of J/mol.

Below, an example is provided for a unimolecular reaction

#                      vf      vb      Eaf      Eab
AR; {A*}      => {B*}; 1e13;   1e13;   100e3;   100e3

and here for the CO dissociation reaction.

#                               vf      vb      Eaf     Eab
AR; {CO*} + {*} => {C*} + {O*}; 1e13;   1e13;   80e3;   120e3

Important

When using AR kinetics, the pre-exponential factor is considered to be temperature-independent.

HK tag

The HK tag indicates a Hertz-Knudsen type of reaction. This reaction should be chosen for adsorption/desorption elementary reaction steps. The forward and backward rate constants are given by

\[ k_{f} = \frac{pA_{\text{site}}}{\sqrt{2 \pi m k_{b} T}} S \]

and

\[ k_{b} = \frac{k_{b}T^{3}}{h^{3}} \frac{A_{\text{site}}(2 \pi m k_{b})}{\sigma \theta_{\text{rot}}} \exp \left( -\frac{\Delta E_{\text{des}}}{k_{b}T} \right) \]

where the parameters correspond to

• \(A\): Size of the active site
• \(m\): Mass of the adsorbent in amu
• \(S\): Sticking factor (typically set to 1)
• \(\sigma\): Symmetry factor of a molecule
• \(\theta_{\text{rot}}\): Rotational temperature (\(=\frac{h^{2}}{8 \pi^{2} I k_{b}}\))
• \(\Delta E_{\text{des}}\): Desorption energy

Below, an example is given

#                        m^2    amu   K  sigma sticking  J/mol
HK; {A} + {*} => {A*}; 1e-19;   28;   88;   2;   1;      100e3

Important

• When using HK type of reactions, always write the reaction as an adsorption, i.e., with the gas-phase components on the left hand side of the reaction and the adsorbed state on the right hand side. For example {CO} + {*} --> {CO*} is correct.
• The default HK setting is for linear adsorbates. For non-linear adsorbates, one needs to use HK3.

HK3 tag

When the adsorbate is a non-linear molecule, one should use HK3 instead of HK which changes the equation for reaction rate constant in the reverse direction to

\[ k_{b} = \frac{k_{b}T^{7/2}}{h^{3}} \frac{A_{\text{site}}(2 \pi^{3/2} m k_{b})}{\sigma \sqrt{\theta_{\text{rot}}}} \exp \left( -\frac{\Delta E_{\text{des}}}{k_{b}T} \right) \]

Importantly, here \(\theta_{\text{rot}}\) is the product of the three rotational temperatures (per principle rotational axis).

HKN[L,-L] tags

When using the tags HKN, HKN-L or HKNL, gas phase entropies are calculated using the Shomate equation. Using \(\Delta S_{\textrm{gas}}\), the forward reaction rate constant is given by

\[ k_{f} = \frac{pA}{\sqrt{2 \pi m k_{b} T}} S \]

and the backward reaction rate constant is calculated from the Gibbs free energy of adsorption as given by

\[ K = \exp \left( - \frac{\Delta G_{\textrm{ads}}}{RT} \right) = \frac{Q_{\textrm{ads}}}{Q_{\textrm{gas}}} = \frac{k_{f}}{k_{b}} \]

which yields

\[ \begin{align} k_{b} &= \frac{k_{f}}{K} \\ &= \frac{pA}{\sqrt{2 \pi m k_{b} T}} S \cdot \exp \left( \frac{\Delta H_{\textrm{ads}}}{RT} - \frac{\Delta S_{\textrm{ads}}}{R} \right) \\ &= \frac{pA}{\sqrt{2 \pi m k_{b} T}} S \cdot \exp \left( \frac{\Delta H_{\textrm{ads}}}{RT} - \frac{R \ln Q_{\textrm{ads}} - \Delta S_{\textrm{gas}}}{R} \right) \\ &= \frac{pA}{\sqrt{2 \pi m k_{b} T}} S \cdot \exp \left( \frac{\Delta H_{\textrm{ads}}}{RT} \right) \exp \left( \frac{\Delta S_{\textrm{gas}}}{R} \right)\frac{1}{Q_{\textrm{ads}}} \end{align} \]

where \(Q_{\textrm{vib}}\) corresponds to the product of the vibrational partition functions in the adsorbed configuration and \(\Delta H_{\textrm{des}}\) corresponds to the desorption enthalpy defined as

\[ \Delta H_{\textrm{ads}} = \Delta E_{\textrm{elec}} + \Delta E_{\textrm{zpe}} + \frac{N}{2}RT \]

where N is the number of rotational and vibrational degrees of freedom of the adsorbate in the gas phase.

ADS tag

When using the tag ADS, the procedure is comparable to HKN[-L,L] as shown above, yet the parameter corresponding to the sticking coefficient S is used as a weighing coefficient in the manner as explained below.

Using \(\Delta S_{\textrm{gas}}\), the reaction rate constant for adsorption is given by

\[ k_{f} = \frac{k_{B}T}{h} \cdot S_{0}^{(1-S)} \cdot S_{1}^{S} \]

wherein

\[ S_{0} = \exp \left( -\frac{\Delta S_{\textrm{gas}}}{R} - \ln Q_{\textrm{vib}} \right) \]

and

\[ S_{1} = \frac{pA}{\sqrt{2\pi m k_{B}T}} \left(\frac{k_{B}T}{h}\right)^{-1} \]

The reaction rate constant for desorption is given by

\[ k_{b} = \frac{k_{B}T}{h} \cdot S_{1}^{S} \exp \left( -\frac{\Delta H_{\textrm{des}} + \Delta S_{\textrm{gas}}}{RT} \right) \]

where

\[ S_{1} = \exp \left( -\frac{\Delta S_{\textrm{gas}}}{R} - \ln Q_{\textrm{vib}} \right) \left(\frac{k_{B}T}{h}\right)^{-1} \left(\exp \left( -\frac{\Delta S_{\textrm{gas}}}{R} - \ln Q_{\textrm{vib}} \right)\right)^{-1} \]

Special tags

In the listing as presented above, a few important tags can be encountered whose meaning is explained in more detail below.

Q-tag

In the earlier versions of MKMCXX, the pre-exponential factors were set by the input file and were considered to be independent of temperature. Using the Q tag, this behavior can be changed.

Without a Q tag, the reaction rate constant is calculated using

\[ k_{f} = \nu_{f} \exp \left( \frac{-\Delta E_{f}}{RT} \right) \]

whereas with a Q-tag, it is calculated using

\[ k_{f} = \frac{k_{B}T}{h} \frac{Q_{TS}}{Q_{IS}} \exp \left( \frac{-\Delta E_{f}}{RT} \right) \]

L-tag

When the L tag is used, the ZFH-S lateral interaction model is used. This lateral interaction model modifies the reaction barriers on the basis of the elements present for each compound. Each reaction that uses this model requires 4 additional arguments which are provided directly after the description of the elementary reaction step.

1. Correction energy
2. Direction
3. Lower bound
4. Upper bound

N-tag

When the N tag is present, the Shomate equation is used to calculate the gas phase entropy. The user is required to provide the Shomate coefficients A-G.

Tip

The Shomate coefficients can be easily found via the NIST database. For example, the Shomate coefficients for water are found here