pH-Dependent Distribution of Functional Groups on Titanium-Based MXenes

MXenes are a new rapidly developing class of two-dimensional materials with suitable properties for a broad range of applications. It has been shown that during synthesis of these materials the surfaces are usually functionalized by O, OH, and F and further suggested that controlling the surface allows controlling the material properties. However, a proper understanding of the surface structure is still missing, with a significant discrepancy between computational and experimental studies. Experiments consistently show formation of surfaces with mixed terminations, whereas computational studies point toward pure terminated surfaces. Here, we explain the formation of mixed functionalization on the surface of titanium-based two-dimensional carbides, Ti2C and Ti3C2, using a multiscale modeling scheme. Our scheme is based on calculating Gibbs free energy of formation by a combination of electronic structure calculations with cluster expansion and Monte Carlo simulations. Our calculations show formation of mixtures of O, OH, and F on the surface with the composition depending on pH, temperature, and the work function. On the other hand, our results also suggest a limited stable range of compositions, which challenges the paradigm of a high tunability of MXene properties.

We adopt the notation and conceptual framework that were proposed by Todorova and Neugebauer, 1 where the absolute chemical potential is denoted µ(i) and µ(i) is referenced to standard conditions. Consequently, and at reference conditions µ i = 0. The chemical potential can be obtained directly from the Gibbs free energies, e.g. µ • (O) = 1 2 G • (O 2 ). Next, we need to determine the chemical potentials for H, O, and F mimicking the experimental conditions. O can only come from water. H and F come from the solvated HF. HF is gaseous, but in aqueous solution ionizes as HF + H 2 O ← − → H 3 O + + F − and H 3 O + ← − → H 2 O + H + . We should have H + , F − , H 2 O, whereas the concentration of OH − should be fairly small when pH is low.
Oxygen is easier as no ions are involved and µ(O) = µ(H2O) − 2µ(H). µ(H2O) here is solvated water. For starters, we can just use the experimental Gibbs free energy of formation where ∆ f G(H 2 O) = −237.14 kJ/mol = −2.458 eV. 2 Next, we need to determine the chemical potential of H. The first option would be in the form of H 2 molecule. On the other hand, H could be in the form of H+ ions, in which case we also need to consider the electron chemical potential. The formation energy of H+ ion is written as where µ e is given with respect to vacuum, i.e., µ e = µ e − µ vac e . The energy of solvated ions is obtained from the ion heat of formation in gas phase (breaking of H 2 and ionization (to vacuum(?))) plus hydration energy: Ion heats of formation are taken from NIST-JANAF thermochemical tables and the hydration energies from Refs. 3,4 It is convenient to connect this to pH, which is directly related to the H+ concentration in the solution: where c 0 = 55.55 mol/l is the concentration of H 2 O molecules in water. From this and by using Eq. 7 we obtain We could also connect it to SHE, but that's not really necessary here as we have a closed system and not electrochemical system where external circuit could deplete or replenish charges. From 12 we obtain the final form the H chemical potential as Lastly, for F chemical potential, basic equations are similar to H and were already written out in Eqs. 7 and 9. We assume that majority of H+ ions arise from the solvated HF and thus concentration of H+ should be equal to F-, c H+ = c F − , and from Eq. 11 we again obtain and finally

XC-functional benchmarks
Several exchange-correlation functionals were tested in order to choose the most reasonable one for further calculations via comparison to experimental enthalpy of formation. We considered PBE, PBE-D3(BJ), PBEsol, PBEsol-D3(BJ), HSE06, SCAN, and SCAN+rVV10. In addition we examined the effect of Ti semicore p-and s-electrons.

Free energy contributions
The enthalpies of formation were determined following the model described in the main text, i.e., by evaluating the enthalpies of all the molecules and solids that take part in the formation. This requires evaluating or extracting several correction terms on top of the DFT energies. These various terms are collected in Table S1 For the solids, Ti and TiO 2 , the enthalpy change and entropy contain only the phonon contributions and are obtained from first-principles calculations, similar to those for MXenes described in the main paper. Both are calculated using PBEsol functional and Ti PAW with 3s and higher electrons in the valence (Ti sv setup). We used 4x4x4 supercell, 12x12x12 k-point mesh for Ti and 3x3x3 supercell, 8x8x8 k-point mesh for TiO 2 To integrate the thermal properties, we used 400x400x400 k-mesh over the first Brillouin zone.

TiO 2
The formation energy and enthalpy for TiO 2 are written as described in the main paper: Using the data in Table S1, the zero-point energies sum up to 0.0809 eV, the enthalpy change to -0.0449 eV, and in total 0.0360 eV. The formation energy of titanium dioxide was calculated using the bunch of different exchange correlation functionals. Moreover, several calculations were performed including pv and sv semi-core states for titanium in order to find the best option. The results are listed in Table S2 and show a wide variation of enthalpy of formation from -9.0569 eV up to -10.5663 eV. However, the best match with experimental value is obtained using PBEsol exchange correlation functional including sv semi-core states (-9.7025 ev compared to -9.7300 eV).

Water
Enthalpy of water formation could be obtained as: Again, from Table S1, we get 0.2414 eV for difference of zero-point energies, and -0.0354 eV for the difference in enthalpies, and in total 0.2061 eV. Note, that we consider water in the gas phase. Experimental enthalpy of formation from NIST-JANAF is -2.5063 eV in the gas phase (and -2.9623 eV in the liquid phase). As one can see from the table S3, PBEsol and PBEsol-D3(BJ) give the closest match to experimental values, differing by about 0.05 eV.

Cluster expansion
The degree of improvement of the cluster expansion as more clusters are included in the expansion are illustrated in Fig. S1. Effective cluster interactions from Ti 3 C 2 case are plotted as a function of the cluster diameter in Fig.  S2. The surface structures and Gibbs free energies of formation for Ti 3 C 2 are shown in Fig. S3 and S4, similar to Figs 2, 3, and 6 in the main paper for Ti 3 C 2 .
The SQoS structures have been uploaded to the NOMAD repository and can be found following the link in Ref.

MXene properties
The mixing energy, work function, and lattice constant of Ti 2 C as a function of the surface composition are shown in Fig. S5, similar to what was presented for Ti 3 C 2 in Fig. 4 of the main paper.   The density of states for all Ti 2 C SQoS and for the pure terminations are shown in Fig. S7, similar to what was presented for Ti 3 C 2 in Fig. 9 of the main paper.