First Principles Analysis of Ethylene Oligomerization on Single-site Ga 3+ Catalysts Supported on Amorphous Silica

Amorphous, single site, silica-supported main group metal catalysts have recently been found to promote olefin oligomerization with high activity at moderate temperatures and pressures (~250°C and 1 atm). Herein, we explore the molecular-level relationship between active site structures and the associated oligomerization mechanisms by developing amorphous, silica-supported Ga3+ models from periodic, first-principles calculations. Representative Ga3+ sites, including three- and four-coordinated geometries, are tested for multiple ethylene oligomerization pathways. We show that the three-coordinated Ga3+ site promotes oligomerization through a facile initiation process that generates a Ga-alkyl intermediate, followed by a Ga-alkyl-centered Cossee-Arlman mechanism. The strained geometry of a three-coordinated site enables a favorable free energy landscape with a kinetically accessible ethylene insertion transition state (1.7 eV) and a previously unreported β-hydride transfer step (1.0 eV) to terminate further C-C bond formation. This result, in turn, suggests that Ga3+ does not favor polymerization chemistry, while microkinetic modeling confirms that ethylene insertion is the rate-determining step. The study demonstrates promising flexibility of main group ions for hydrocarbon transformations and, more generally, highlights the importance of the local geometry of metal ions on amorphous oxides in determining catalytic properties.


(S.1) Ga 3+ site creation and formation energies
Each Ga 3+ site was created by replacing an Si atom with Ga. A proton was added to the adjacent oxygen atom to maintain charge balance. Five Si atoms were tested and replaced, with four possible oxygen atoms per site, yielding a total of 20 distinct Ga site configurations ( Figure S1).

(S.3) Dehydration of constrained Ga sites
For a 4CN Ga site, a Si-OH moiety is close to the Ga site due to the constrained environment. The activation of a C-H bond of ethylene on these sites then generates a second Si-OH moiety. The proximity of the two Si-OH groups can lead to a dehydration step, which results in an oxygen bridged Si pair and a water molecule. Figure S2 shows the pathways after the dehydration step, and Figure S3 shows the schematics of the dehydration step of a 4CN site. With the absence of the Si-OH moiety, the proton transfer step, which reforms the original 4CN Ga site, is not possible. Two other pathways, including β-hydride elimination and β-hydride transfer, are also examined. Figure S5 shows the free energy landscape of the dehydrated 4CN Ga site, demonstrating that the β-hydride transfer pathway is more energetically favorable. Figure S2. Two site activation pathways after a dehydration step of a constrained 4CN Ga site Figure S3. Site activation pathways after a dehydration step on a constrained 4CN Ga site. The two adjacent Si-OH groups are represented by H1 and H2. Figure S4. Free energy diagram of the 4CN site dehydration, followed by the β-hydride elimination and β-hydride transfer pathways (T = 523 K). The adsorption energies are referenced to empty site and appropriate amounts of gaseous ethylene molecules at 1 atm. Table S3. Energy information of ethylene oligomerization (T = 523 K), the adsorption energies are referenced to Ga-ethyl species and appropriate amounts of gaseous ethylene molecules at 1 atm.

(S.7) Construction of microkinetic model
Adsorption of gas molecules are described using collision theory. The rate constant of this process is represented by: where A0 is the area of the surface site, and Λ is the probability of adsorbate sticking, which is assumed to be one.
The forward rate constant of a surface reaction is calculated using the following equation: It can be noticed that transition state theory is employed to incorporate the free energy of the transition state. The potential energy, ZPE, and entropy effects have been included in the free energy.
The equilibrium constants for each elementary step are defined by: For surface reactions, the reverse rate constant can be obtained by: In a differential reactor with a low conversion, the equation that describes the mass balance of each gaseous species is: where term b is related to site normalization. To obtain a reaction rate comparable to the experiments, the rates are normalized by the number of Ga sites in the catalyst.
The differential equation that describes the mass of each surface species (using A* as an example) is: The differential equations, together with the site balance equation, are solved in MATLAB until steady state is reached.

(S.8) Calculation of entropy using mode decompositions
Harmonic vibrational modes form the basis for calculating most adsorbate entropies. However, for vibrational modes with low wavenumbers (< 150 cm -1 ), particle-in-a-box (PIB) and free rotor schemes are used for calculating their entropic contributions. For a low frequency that corresponds to translation of a molecule, the PIB model is used with a length scale corresponding to the size of the cavity where the Ga site is located; the theoretical treatment produces a lower entropy limit of the vibrational mode. On the other hand, for a low frequency vibration that resembles rotation of a molecule, the free rotor model is used and likely represents a small overestimate of the corresponding entropy. Finally, the entropies of the transition states are approximated as being equivalent to the entropies of the corresponding reactants or products, depending on whether the geometry of the transition state more closely resembles that of reactants or products.  (2) Step 2 is irreversible and rate-determining, and (3) the coverage of empty sites is negligible.
The overall reaction rate per Ga site is: where overall is the rate per unit surface area, N denotes the total number of reactive Ga sites, and denotes the fractional coverage of species A (ethylene physisorbed on Ga-ethyl). If we apply the assumptions on the other steps, we have: where P is the gas-phase partial pressure, and * is the coverage of empty sites. We can write the total balance of sites as: 1 = * + + Therefore: = 2 1 (1 + 1 + 3 D ) (S.10) Arrhenius plot at 523 K, 1 atm, and pure ethylene feed From the Arrhenius plot, where the temperature was varied, an apparent activation barrier of 1.52 eV is obtained. This corresponds well to the value obtained from the free energy analysis. Figure S10. Arrhenius plot at 523 K, 1 atm, and pure ethylene feed on a 3CN Ga site 1000×T -1 , K -1