Lateral Interactions

Lateral interactions are implemented via a lateral interaction potential which is systematically added to the activation energy to (de-)stabilize intermediates based on the current coverage. This is done during the time-integration and ensuring that the overall thermodynamics remains conserved.

In the work of Zijlstra et al. ¹, two lateral interaction models have been constructed:

The ZFH lateral interaction matrix model
The simplified lateral interaction model

Both these models will be described in the text below.

Hint

When no explicit coverage-dependent information is known for your system, it is recommended to use the simplified lateral interaction model.

ZFH-M: lateral interaction matrix model

Description

In the ZFH lateral interaction matrix model ¹ lateral interactions are described via a linear and an exponential dependency in the coverage, the former effect attributed to the sharing of multiple metal atoms by the adsorbates and the latter assigned to the through-space repulsion of the adsorbate species. For like adsorbates, the lateral interaction potential is given by the following equation

\[ E_{\textrm{ads}}^{\textrm{int}} = \sum_{i} A_{i} \cdot \theta_{i} + \epsilon \left[ \exp \left(\sum_{j} B_{j} \cdot \theta_{j} Z_{ij} \right) - 1 \right] \]

wherein $A_{i}$ is the linear lateral interaction parameter for species $i$, $B_{j}$ the exponential repulsion factor, $Z_{ij}$ is the cross-term for intercomponent interactions and $\epsilon$ a dimensional correction factor that holds the same units as $A_{i}$. Typically, units of J/mol are used. Note that $Z_{ij}=1$ in the case that $i=j$, which signifies the repulsion between like species. For repulsion between unlike species, i.e. when $i \neq j$, $Z_{ij}$ can have values other than unity.

Implementation

The lateral interactions are defined by means of two tags in the input file: &lateral and lateral-het. Under the former tag, a list of species and their values of $A$ and $B$ are provided. For the latter tag, the values for $Z_{ij}$ are given. For all species and interactions not listed under these two tags, the values of $A$, $B$ are set to 0.0 whereas all undefined values for $Z_{ij}$ are assumed to be 1.0.

&lateral
{CO*}; 21.7477; 4.5838
{H*};   2.631 ; 2.2335
&lateral-het
{CO*}; {H*}; 0.42

To implement a lateral interaction potential to a elementary reaction step, the step has to be supplemented by the -L affix. For example, to utilize a lateral interaction potential for CO and H2 adsorption, the following two lines would be used.

HK-L; {CO}      +   {*} =>  {CO*}       ;1e-20  ;28   ;  2.73;  1   ;   1;  151e3;  1
HK-L; {H2}      +   2{*}    =>  2{H*}   ;1e-20  ; 2   ;    88;  2   ;   1;   85e3;  1

ZFH-S: Simplified lateral interaction model

Description

In the simplified model, rather than building a lateral interaction potential on the basis of a large set of cross-interactions as described by the matrix $\mathbf{Z}$, the lateral interaction is described on a per-atom basis. For hydrocarbon catalysis, this would entail defining the lateral interaction on the basis of the number of C and H species in each adsorbate rather than defining the interaction potential between each type of CH_x adsorbate.

In this model, the through-space lateral repulsion is based on the total coverage rather than a sum of individual contributions of partial coverages as used above. For a given element $x`, the lateral interaction potential is given by the equation below

\[ E_{x}^{\textrm{lat}} = E_{x}^{\theta=\theta_{\textrm{total}}} \cdot \frac{\left(101^{\theta_{\textrm{lat}}} - 1 \right)}{100} \]

It can be readily seen that when the total coverage $\theta_{0}$ is zero, the lateral interaction potential vanishes.

It is assumed that all species contribute equally to the lateral interaction potential with the exception of H due to its significantly smaller size. Moreover, we also allow for a mapping of the lateral interaction between a specific coverage range by introduction of a set of lower and upper bound parameters $\theta_{\textrm{LB}}$ and $\theta_{\textrm{UB}}$, yielding the following overall equation for $\theta_{\textrm{lat}}$:

\[ \theta_{\textrm{lat}} = \frac{\left( \theta_{\textrm{total}} - \theta_{*} - \frac{1}{2} \theta_{\textrm{H}} \right) - \theta_{\textrm{LB}}}{\theta_{\textrm{UB}} - \theta_{\textrm{LB}}} \]

For example, setting $\theta_{\textrm{LB}}$ = 0.25 ML and $\theta_{\textrm{UB}}$ = 0.75 ML scales the lateral interaction coverage such that the lateral penalty is always zero below a coverage of 0.25 ML. The penalty then increases exponentially, reaching $E_{x}^{\theta=1}$ at e.g. 0.75 ML CO, or at 0.5 ML CO and 0.5 ML H.

Implementation

Hertz-Knudsen adsorption with Shomate-equation-based entropy corrections

To explain upon the input parameters, consider CO adsorption:

HKNL; {CO} + {*t} => {CO*t}; 120e3 ; 0 ; 0.25 ; 0.75 ;  1.0  ; {*t},0.5{H*t}; 5.384e-20;  28; 1; 25.56759 ; 6.09613 ; 4.054656; -2.671301;  0.131021; -118.0089  ; 227.3665  ; -110.5271;    79.4709129; 140e3;1

The line as shown above contains 21 semicolon-delimited entries which are discussed below. For the Shomate eqaution coefficient, please also see the gas phase thermochemistry data page of the NIST database.

Elementary reaction step
Lateral interaction potential (kJ/mol). The maximum lateral interaction penalty obtained at full coverage.
$\Delta E_{x}^{\theta=\theta_{\textrm{total}}}$ Lateral interaction direction parameters (0 = only affect backward reaction; 1 = only affect forward reaction). For HK reactions, a typical and sensible value is 0 as we assume that the forward reaction (adsorption) does not carry an adsorption penalty as this reaction event is non-activated.
$\theta_{\textrm{LB}}$ Lateral interaction lower bound. This is the minimum repulsive surface coverage at which a lateral interaction penalty starts to occur.
$\theta_{\textrm{UB}}$ Lateral interaction upper bound. This is the saturation point of the lateral interaction. Any repulsive surface coverage beyond this concentration does not further increase the lateral interaction penalty.
$\theta_{\textrm{total}}$ Maximum repulsive surface coverage at full coverage. This value is typically taken to be 1.
Species which determine the repulsive lateral interaction coverage. In the example here, two species are given {*t} and 0.5{H*t}. The repulsive surface coverage is calculated as 1 - {*t} - 0.5{H*t}. This expression basically means that all surface species are equally repulsive, with the exception of hydrogen, which is only half as repulsive as all other species.
$A$: effective adsorption site in m².
$m$ Mass of the species in amu.
$S$ Sticking factor. This value is typically taken to be 1.
Shomate coefficient A
Shomate coefficient B
Shomate coefficient C
Shomate coefficient D
Shomate coefficient E
Shomate coefficient F
Shomate coefficient G
Shomate coefficient H
$Q_{\textrm{ads}} = \prod_{i} q_{i}$ Total vibrational partition function for the adsorbed state, calculated as the product of the individual vibrational partition functions of the normal modes of the adsorbed species.
$\Delta E_{\textrm{des}}$ Desorption energy at zero coverage in kJ/mol.
DRC flag. Indicates whether a DRC analysis needs to be performed for this elementary reaction step. 1 = Yes; 0 or absent = No.

Surface reactions with Shomate-equation-based entropy corrections

Note

This part is still needs to be written.

To assign the simplified lateral interaction potential to an elementary reaction step, one uses the affix L. For example,

HKL; {CO}      +   {*} =>  {CO*}       ;1e-20  ;28   ;  2.73;  1   ;   1;  151e3;  1
HKL; {H2}      +   2{*}    =>  2{H*}   ;1e-20  ; 2   ;    88;  2   ;   1;   85e3;  1

ARQL; {A*} + {B*} => {C*} + *; lat_fac ; lat_dir; lat_lb; lat_ub; lat_start; lat_cmps ; vf ; vb; Eaf; Eab

Warning

For the matrix-model, -L is used while for the simplified model, L. Mind the dash!

Microkinetic modeling of the Fischer-Tropsch synthesis on cobalt, Bart Zijlstra, PhD thesis, chapter 3 ↩↩