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60c7418fbdbb892f24a3837f | 21 | Here the ∆M s = 0 label indicates that the density matrix γ ∆Ms=0 is computed for the spinconserving excitations. Importantly, this applies not only to the EOM-EE ansatz, but also to the EOM-SF, EOM-IP, and EOM-EA variants for the transitions between the states with the same spin projections. Non-zero spin blocks are denoted as γ ∆Ms=0 αα and γ ∆Ms=0 |
60c7418fbdbb892f24a3837f | 22 | , where the spin label indicates the α or β spin-orbitals in Eq. . It is easy to see that γ ∆Ms=0 triplet matrix is constructed through a † pα a qαa † pβ a qβ excitations between the bra and ket states. Although it is a triplet excitation operator, for the consistency with other triplet excitation operators, the γ ∆Ms=0 triplet matrix should be further divided by √ 2, as per definition of T 1,0 in Eq. . If the spin projections of bra and ket differ by ±1, the corresponding excitation operators are spin-flip triplet operators, and no decomposition is needed. However, for spintensor form they should be multiplied by the appropriate phase factors from Eqs. ( ) and . We note that for these spin-flipping excitations right-to-left and left-to-right density matrices gain opposite phase factors. Once the density matrix between the states with the same spin projection is computed, one can easily calculate u from Eq. ( ) as |
60c7418fbdbb892f24a3837f | 23 | and the full SO matrix is a product of Clebsh-Gordan's coefficients to these reduced matrix elements. The prefactors in the expressions above arise because the spin part is taken in spin-tensor form whereas the orbital momentum is taken in L + /L -, L z form, which is not a spin-tensor form. The overall workflow of the algorithm is shown in Fig. . |
60c7418fbdbb892f24a3837f | 24 | FIG. The workflow of the present algorithm (shown by green arrows) and a previously reported scheme implemented within the CI framework (red arrows). In the latter, the reduced matrix elements are computed from three matrix elements. In our scheme, we start from one transition density, extract the triplet part, make the spinless density, and form three reduced matrix elements. |
60c7418fbdbb892f24a3837f | 25 | SOMF approximation considerably reduces the cost of the calculation of the twoelectron part of SOC while introducing negligible errors. Briefly, SOMF entails consider-ing only the contributions from the separable part of the two-particle density matrix. In CC/EOM-CC, the separable part of the two-particle density matrix, Γ, has the following form : |
60c7418fbdbb892f24a3837f | 26 | where the operators P + (pr, qs) and P -(p, q) generate symmetrized/antisymmetrized expressions, respectively. Here ρ is the density of the reference determinant, which is diagonal in case of canonical Hartree-Fock molecular orbitals. When contracted with a two-electron operator A = 1 2 pqrs A pqrs p † q † sr, the separable part of two-particle density matrix yields: |
60c7418fbdbb892f24a3837f | 27 | As often is the case with mean-field treatments, construction of mean-field effective operators for open-shell references might lead to artifacts. Below we discuss intrinsic issues of SOMF in the case of open-shell references. Note that these problems arise only when the reference, not the target state, have open-shell character. For example, EOM-IP/EA calculations of doublet states using closed-shell references are not affected by this issue, however, EOM-SF calculations using high-spin references are. The analysis is based on considering the second-quantized form of the mean-field spin-orbit operator in a spin-restricted spinorbital basis (meaning that the spatial parts of spin-orbitals φ pα and φ pβ are the same, e.g. |
60c7418fbdbb892f24a3837f | 28 | In the case of closed-shell references, the SOMF approximation of the two-electron part of SOC can be treated in the same way as the one-electron part. To illustrate the problem of open-shell references, let us first discuss the contributions from the h SOM F,2e z terms. In the case of a closed-shell reference, α and β parts of the HF density (or, in general, the density of the reference determinant) are the same, which makes h SOM F,2e z,pαqα equal to -h SOM F,2e z,pβqβ . |
60c7418fbdbb892f24a3837f | 29 | for an open-shell reference determinant in which the number of α and β electrons is different. In this case, α and β parts of the HF density are not the same; therefore, the mean-field Fock-like matrices from Eq. ( ) also do not have a proper symmetry of the different spin blocks. The h SOM F,2e z terms acquire additional unphysical contributions from the singlet part of the transition density matrix, which leads to artifacts illustrated numerically in Section III D. |
60c7418fbdbb892f24a3837f | 30 | One possible solution, which we adopt in this work, is to antisymmetrize the h SOM F,2e z integrals with respect to the αα and ββ parts and take a trace triplet transition density matrix for SOCs. This operation eliminates the unphysical contribution from the singlet transition densities. By doing so, one can take the h SOM F,2e z integrals out of parenthesis, making the form of the SOMF expression, Eq. ( ), the same as for the one-electron part, Eq. ( ). |
60c7418fbdbb892f24a3837f | 31 | The contribution of the two remaining terms to SOC in case of the open-shell references require the formation of h SOM F L ± not from the full x and y matrices, as in Eqs. ( ) and ( ), but from their parts that will be contracted with transition density matrices of a † pβ a qα and a † pα a qβ types, respectively (see appendix B in Ref. ). This leads to the partitioning of the terms as in Eqs. ( ) and . Different α and β parts of the HF density deteriorate the symmetry of the resulting matrix elements as well. The impact of this violation and a possible fix are discussed in Section III D. |
60c7418fbdbb892f24a3837f | 32 | A common strategy for computing interstate properties within EOM-CC is to take a geometric average of individual matrix elements . Numerical examples illustrate that, as expected, the phases of A → B and B → A matrix elements are not exactly conjugated. This leads to an ambiguity in geometric averaging: one can either average absolute values of the matrix elements and assign phases in some way, or average the Cartesian components of the matrix. We observe that in the absence of point group symmetry geometric averaging of the absolute values violates the rotational invariance of SOCC. In contrast, the arithmetic average satisfies the requirements above while producing the same value regardless of whether the spherical or Cartesian components were averaged. |
60c7418fbdbb892f24a3837f | 33 | In set (2), the target states have either one electron or one hole in two degenerate orbitals (π or E). To achieve a balanced treatment of the corresponding configurations, we used EOM-IP-CCSD to remove an electron from degenerate orbitals of a closed-shell reference to describe diatomic cations (Figure ) and EOM-EA-CCSD to add an electron to an empty degenerate orbitals of a cationic reference to describe calcium derivatives (Figure ). |
60c7418fbdbb892f24a3837f | 34 | We used Dunning's cc-pVDZ and cc-pVTZ basis sets . Core electrons were frozen in all calculations. Unrestricted HF references were used in the calculations using open-shell references. For diradicals, we used the same geometries as in Ref. . The calculations for set 2 were carried out at their ground-state doublet geometries. The structures of AsN + and GeO + were optimized with EOM-IP-CCSD/cc-pVDZ from the neutral reference; CaF and CaOCH 3 were optimized with EOM-EA-CCSD/cc-pVDZ from the cationic reference. |
60c7418fbdbb892f24a3837f | 35 | The structure of a model system representing SMM was derived from the original one by replacing the mesitylene groups in the chelating ligand, tpa Mes , by hydrogenes. We refer to the resulting ligand as tris(pyrrolylmethyl)amine (tpa). The ground state of the neutral (tpa)Fe has 5 unpaired electrons, as in Fe 3+ ion. The geometry of the neutral (tpa)Fe complex was optimized with ωB97X-D/cc-pVDZ for the hextet state. We used this geometry for the calculations of the anion as well. |
60c7418fbdbb892f24a3837f | 36 | The results in Table show anticipated trends: the magnitude of SOCC increases for heavy elements. Moreover, for these systems the contribution of the two-electron part is smaller than for typical organic molecules. The magnitude of SOCCs between the degenerate pairs of states ( 2 Π or 2 E) is slightly larger than between π-and σ manifolds, which probably can be rationalized by considering the shapes of the respective MOs and El-Sayed's rule . |
60c7418fbdbb892f24a3837f | 37 | Table shows atomic SOCCs. For atoms Eq. ( ) no longer gives a rotationally invariant constant. Therefore, here we summed the matrix elements not only by spin projections, but also by the projections of orbital angular momentum. This also leads to large atomic SOCCs. These values can be compared with experimental data within Landé interval rule , as explained below. To compare these values with experiment, one should divide the SOCC by L(L + 1) S(S + 1). The resulting quantity can be compared with Russell-Sauders LS coupling constant (ζ) extracted from splittings through Landé interval rule . For example, SOMF gives an estimate for 5 D g levels of Mn + of ζ = 68.29 cm -1 , while the values extracted from experiment are within 58.66-66.99 cm -1 interval . The averaged experimental value of ζ for Fe 2+ is 100 cm -1 , while the SOMF estimate from Table The SMM studied in Ref. has d 6 electronic configuration, corresponding the overall negative charge. As explained below, this electronic configuration gives to electronic degeneracy and Jahn-Teller distortions. EOM-CC methods are capable of tackling Jahn-Teller and pseudo-Jahn-Teller effects very well , however, this particular system is different and its description is affected by the non-Hermitian nature of EOM-CC theory . |
60c7418fbdbb892f24a3837f | 38 | The high-spin hextet neutral (tpa)Fe state has electronic configuration d 5 and, therefore, can be well described by a single Slater determinant, similarly to Fe 3+ and Mn 2+ . This state is not degenerate and it has a geometry with C 3 point group of symmetry, as shown in Fig. . |
60c7418fbdbb892f24a3837f | 39 | Although this group is Abelian, it has one-and two-dimensional real irreducible representations (irreps), which can give rise to Jahn-Teller effect for certain electronic configurations of the two-dimensional irrep . For example, the quintet anion (tpa)Fe -has doubly degenerate states at a C 3 structure. Unlike C 3v group, C 3 group does not have planes of symmetry, which would split a two-dimensional irrep into a symmetric and antisymmetric irrep with respect to the plane. Therefore, the two Jahn-Teller states would not fall into two different irreps, as it happens in a majority of symmetry-imposed degenerate states . The only Abelian subgroup of C 3 is C 1 , therefore the two degenerate states belong to the same irrep, giving rise to general conical intersection problem. As documented numerically and theoretically , the description of true conical intersections is problematic in EOM-CC due to the non-Hermitian nature of the theory. In our case this leads to issues with finding left vectors at the symmetric geometry. To overcome this problem, we run calculations at slightly asymmetric geometry, coming from DFT optimization (listed in SI). Although this geometry is asymmetric only at the magnitude of a symmetry threshold, it leads to a small artificial energy splitting (∼ 0.01 eV in the EOM-EA calculation from a neutral reference) of the states that should be degenerate. This artifact is small enough to be neglected in typical photochemical applications, but it affects the magnitude of spin-orbit splitting. To mitigate this issue, we average the energies of the states that should be degenerate by symmetry. it also splits the degenerate irreps (the symmetry of the system is no longer described by point group symmetry; it is now described by double group symmetry), and the lowest of them looks very similar to the structure shown in Fig. . We followed the state-interaction procedure in computation of the ZFS splittings: the arithmetically averaged SOC blocks were plugged in the matrix Hamiltonian and the unperturbed state energies were on the diagonal. Then the entire matrix was diagonalized to yield energy-split multiplet states. |
60c7418fbdbb892f24a3837f | 40 | To study convergence with respect to the number of states included in the calculation, we computed the splittings between 2, 3, and 5 multiplets (10, 15, 25 electronic states with different spin-projections, respectively). This sequence was chosen to include the degenerate irreps of the point group. The values in in Table are very close to the experimentally derived result for (tpa Mes )Fe -of 158 cm -1 . |
60c7418fbdbb892f24a3837f | 41 | Because the α part of Hartree-Fock density ρ is not equal to the β part for open-shells, the symmetry between h SOM F,2e ± is broken. Table shows numerical consequences of such violation. Fortunately, regardless of the size of the system or of the number of unpaired electrons in the reference, the extent of such violation does not exceed 0.05 cm -1 for the reduced matrix elements. To eliminate its influence on SOCC and splittings, we used arithmetic averaging for S||H L -||S and S||H L + ||S , similar to Eq. . |
60c7418fbdbb892f24a3837f | 42 | Table shows the impact of singlet component of transition density matrix. The singlet part is small for transitions between states with different multiplicities. It is not zero because of small spin contamination of the electronic states. The magnitude of singlet part can be significant for the transitions between states of the same multiplicities. Usage of the triplet component of transition density matrix for calculation of matrix elements avoids this issue. |
60c7418fbdbb892f24a3837f | 43 | To conclude, the arithmetic averaging between S||H L -||S and S||H L + ||S * restores the correct symmetry of these elements. Usage of the triplet transition density does not lead to contamination of S||H Lz ||S matrix element with the unphysical singlet contribution. We presented a theory for calculating matrix elements of the BP SOC operator based on the application of Wigner-Eckart's theorem to reduced one-particle density matrices. The key equations are given in the second-quantized form. The algorithm is ansatz-agnostic and can be used with any electronic stricture method for which state and transition density matrices are available. The current implementation is based on the EOM-CCSD suite of methods. The approach allows one to compute the SOC matrix for the entire multiplet from just one transition density matrix, which solves the problem of accessibility of states of different spin projections. It also addresses, by constriction, the phase problem due to Born-Oppenheimer's separation of the nuclear and electronic degrees of freedom. We also highlighted an important aspect of calculation of SOCCs within a non-Hermitian theory and proposed using arithmetic averaging of the EOM-CC matrix elements, which leads rotationally invariant SOCCs. |
60c7418fbdbb892f24a3837f | 44 | The current implementation treats the two-electron part of the BP Hamiltonian via the SOMF approximation. We discuss special aspects of the application of SOMF to openshell references and propose practical solutions. In particular, L + and L -reduced matrix elements are slightly different. This issue can be solved by averaging. Contribution of the (unphysical) singlet part of the transition density can be important. This issue does not occur if only the triplet component is used. |
66105b8821291e5d1d2b6d2e | 0 | Pioneered by Prof. Jens Nørskov and co-workers in their early works, descriptor-based analyses have now become an indispensable tool used by researchers within the heteroge-1 neous catalysis community. This philosophy is based on the hypothesis that appropriately identified descriptors, which are often binding energies of key intermediates, can help in rationalizing experimentally observed trends across catalyst compositions, thereby accelerating the design of novel functional materials for the target reaction. The continuing impact of Prof. Nørskov's contributions in this field is evidenced by several recent reviews on these topics. A defining characteristic of this Nørskovian-philosophy is the ability to describe the complexity of multistep reactions, often involving several surface-bound intermediates, using just one or two adsorption-based descriptors. Historically, this simplification was necessary to alleviate the computational costs of calculating hundreds of reaction barriers for every elementary step, and then repeating this process for several different catalyst compositions. Indeed, the concept of linear scaling relationships, which is based on the well-known bond order conservation principle, now forms the core vocabulary used within this field. Although the calculation of reaction barriers remains cumbersome, recent work by us and others has demonstrated how machine learning potentials (MLPs) can be used to overcome this bottleneck. For instance, using Cu-exchanged zeolites as a prototypical example, we have explicitly calculated the transition state geometries and reaction barriers of methane activation for thousands of [CuOCu] 2+ sites across 52 zeolites. While most sites show linear trends between the C-H activation barrier and the H binding energy, our analysis identifies several important factors (e.g., confinement and accessibility) that cause deviation from the expected universal scaling behaviour. Similarly, building upon the foundation provided by scaling relationships, other groups have proposed novel strategies to further improve the accuracy of these approximations. Analogous to how the development of MLPs has enabled the calculation of reaction barriers (which are usually intractable with density functional theory (DFT)), in this study, we focus on the related but less explored phenomena of surface diffusion. Indeed, surface phenomena such as adsorption, diffusion, and desorption of molecular and atomic adsorbates are of wide interest in the field of catalysis. Several studies have shown that slow diffusion processes can have a significant impact on the overall rate of catalytic reactions. While the typical approach within the field relies on using nudged elastic band calculations and harmonic transition state theory (hTST) to estimate diffusion barriers, here, we propose an alternative strategy that uses sufficiently long, MLP-based molecular dynamics simulations to measure adsorbate diffusivities without making any prior assumptions. Specifically, we calculate temperature-dependent diffusivities of key adsorbates using Ag(111) as an example of an industrially relevant catalyst. We use a learning curriculum that combines ab-initio molecular dynamics (AIMD) simulations of gas-phase species along with AIMD simulation of the bound adsorbates to obtain a transferable DeepMD-Kit-based model. Importantly, we also show that the resulting model is easily transferable to related species that were not included in the original training dataset. Taken together, this study highlights the wide versatility and growing relevance of MLP-based approaches in investigating phenomena that are typically not possible using traditional DFT simulations. |
66105b8821291e5d1d2b6d2e | 1 | Our MLP development protocol involves several different types of DFT calculations. We use single-point energy evaluations (DFT/SPE), geometry optimizations (DFT/OPT) and abinitio molecular dynamics (AIMD) simulations. All DFT calculations are performed using the Vienna ab-initio simulation package (VASP). We use the RPBE functional 41 with Grimme's D3 dispersion corrections using a Becke-Johnson damping scheme. A plane wave energy cutoff of 500 eV is used. Ionic relaxation steps are terminated when the forces on all atoms are less than 0.05 eV/ Å. Our AIMD simulations are performed within an NVT ensemble using the Nose-Hoover thermostat. |
66105b8821291e5d1d2b6d2e | 2 | A 2-stage iterative approach is used for MLP development. Stage A uses a series of short AIMD simulations (298 K, 2,000 steps, 0.5 fs timestep) of 678 gas-phase adsorbates obtained as their SMILES notation from Reuter et al. (27,000 configurations, Γ-point only) along with AIMD simulations of these adsorbates bound to a constrained 1-layer Ag(111) slab (27,000 configurations). All slab calculations use 4x4x1 k-points with > 10 Å vacuum spacing between periodic images in the z-direction. These AIMD configurations are sampled every 50 steps to obtain an initial DFT training dataset. RDKit 47 and the Atomic Simulation Environment (ASE) are used to streamline the workflows. We emphasize that the 1-layer slab model is physically unrealistic. However, the DFT data from these simulations provides a useful preliminary MLP model (denoted as DP 0 prelim ) to accelerate subsequent active learning iterations in Stage B. |
66105b8821291e5d1d2b6d2e | 3 | Additionally, we acknowledge that any of the several publicly available MLP models, which have been pre-trained on larger DFT datasets, could have been used here instead of DeepMD-kit. While these architectures may show better performance in predicting energies and/or forces, previous work by Jaakkola et al. emphasizes the importance of using predicted physiochemical properties (e.g., diffusivity) as a metric of model performance. |
66105b8821291e5d1d2b6d2e | 4 | We use an active learning approach for model refinement. For example, MD configurations obtained from the i th DP model (i.e., DP i /MD, 400 K, 2 ns, 0.5 fs timestep, using LAMMPS) are down-selected based on their estimated model uncertainties. Similar to previous work by us and others, we use the ϵ t metric to quantify model uncertainty. Here, ϵ t for a given configuration R R R t is defined by equation 1. |
66105b8821291e5d1d2b6d2e | 5 | where, F w,j denotes the force on the atom with index j predicted by the model w. This metric measures the maximal standard deviation in force predictions obtained through an ensemble of models. Here, we use an ensemble of 4 models differing in the seed value used to initialize the weights of the neural nets. This uncertainty quantification approach underpins our active learning protocol. |
66105b8821291e5d1d2b6d2e | 6 | Molecular dynamics simulations are used to calculate the self-diffusivities of several Agbound adsorbates. LAMMPS 57 is used as the MD simulation engine as it interfaces smoothly with DeepMD-kit. Specifically, we perform 2 ns NVT MD simulations (0.2 ns equilibration, 0.5 fs timestep) at three different temperatures (300 K, 350 K and 400 K). The resulting unwrapped coordinates are used to calculate a windowed mean square displacement (MSD) as a function of all possible lag-times using equation 2. Here, ⃗ r i refers to the positions of the atoms at time t, m is the lag-time (m < m max , the length of the MD trajectory), and N is the maximum number of possible lag times. This averaging strategy mitigates the MSD fluctuations that arise due to the small number of diffusing species in our system. All MSD-based analyses use the open-source MDAnalysis toolkit. M SD(m |
66105b8821291e5d1d2b6d2e | 7 | Here, n p is the number of equivalent diffusion paths, l is the jump length, and ν 0 is the attempt/jump frequency of the adsorbate that is related to their vibrational free energies. Results and Discussion The computational efficiency of the above MLP development protocol depends on the sampling strategy used to select configurations for DFT/SPE calculations. In this work, we employ a wider sampling range compared to previous studies. Specifically, we use 0.2 eV/ Å and 0.7 eV/ Å as the lower and upper ϵ t bounds for sampling, respectively. Within these bounds, configurations from the MD trajectory are randomly sampled to obtain a maximum of 100 possible configurations for each adsorbate. If fewer than 100 configurations lie within the chosen ϵ t bounds, then all available configurations are used for DFT/SPE. To minimize sampling of configurations that are already well described by the model, a tighter sampling procedure (0.3 -0.7 eV/ Å) was used for the final two iterations. |
66105b8821291e5d1d2b6d2e | 8 | The number of configurations sampled for each adsorbate during the training curriculum is shown in Fig. . The first few iterations of our active learning protocol result in uniform sampling across different adsorbates. However, as expected, we observe that the number of configurations sampled per iteration decreases during model training. Ethyl requires the largest number of iterations and sampled configurations to converge. Taken together, the final DP 10 model is trained on ∼27,000 gas-phase, ∼27,000 1-layer and ∼6,000 4-layer DFT data points. Although not used here, we note that analogous workflows can now be implemented through the DPGen interface. 56 |
66105b8821291e5d1d2b6d2e | 9 | Figs. show the performance of the final DP 10 model for the test dataset. The MLPpredicted energies and forces show good agreement with DFT; we observe mean absolute errors (MAEs) of 0.3 eV (for energies) and 0.04 eV/ Å (forces). To further test the predictive power of the model, we ran 2 ns MD simulations for all 8 adsorbates at 3 different temperatures using LAMMPS (NVT ensemble, 0.5 fs timestep, 0.2 ns equilibration, 5 repeats). |
66105b8821291e5d1d2b6d2e | 10 | Our results, shown as ϵ t violin plots in Fig. for the 400 K run, show that the predicted model uncertainties are well below the 0.3 eV/ Å threshold. Analogous analysis for the other temperatures is shown in the SI (Fig. ). Taken together, this analysis confirms the ability of the model to reliably access simulation timescales (2.2 ns) that are typically inaccessible with DFT. |
66105b8821291e5d1d2b6d2e | 11 | To further emphasize the value of the above MD simulations, Fig. compares the adsorbate surface density histograms for all 8 diffusing species obtained from 2 ns MD simulations performed at 400 K. Here, the adsorbate surface density is calculated by analyzing the MLP/MD trajectories and averaging the site occupancy using a 0.57 Å × 0.57 Å grid. We observe that atomic species such as C, H and O remain localized to the 3-fold sites, with lower occupancy of the bridge sites, and an almost complete exclusion of the on-top positions. In contrast, other species (e.g., methyl, hydroxyl and ethyl) diffuse more freely across the entire surface. These trends, which are correlated with the denticity of the adsorbates, suggest that the diffusion of (say) methyl on Ag(111) could follow several different pathways. On the other hand, the diffusion of O and other atomic adsorbates is largely limited to an activated hoping step across the bridge sites. As hTST/NEB-based diffusivity calculations have historically focused on specific diffusion pathways, these results could have important implications on the accuracy of hTST/NEB estimates, especially for weakly-bound species such as methyl. |
66105b8821291e5d1d2b6d2e | 12 | Table shows the diffusion coefficients calculated using equation 3 at three different temperatures for the 8 adsorbates considered here. An example of the MSD fitting procedure is shown in Fig. . Consistent with increased adsorbate mobility at higher temperatures, we obtain larger diffusion coefficients at 400 K compared to 300 and 350 K. Comparing across different adsorbates, we observe that methyl shows the highest diffusivity (2.38 x10 -7 m 2 /s at 400 K) that is about two orders of magnitude higher than strongly bound adsorbates such as C, O and CH. The diffusivity of other species (i.e. H, OH, CH 2 , and ethyl) lies in between these two extremes. Using equation 4, the temperature-dependent diffusion coefficients are used to obtain apparent activation energies for the overall diffusion process. Consistent with the adsorbate surface density histograms discussed previously, we observe near-perfect Arrhenius behaviour for the atomic adsorbates (Fig. .) As shown in Fig. , slightly larger deviations are observed for molecular adsorbates. |
66105b8821291e5d1d2b6d2e | 13 | Consistent with previous DFT-based studies, we note that the calculated diffusion barrier for CH 3 (i.e., E app d = 0.18 eV) is higher than that for CH 2 (0.08), CH (0.12) and C (0.08). However, CH 3 is the most mobile species, where the higher diffusivity is due to a much larger pre-exponential factor for methyl (i.e., e 8.7 )compared to other adsorbates. |
66105b8821291e5d1d2b6d2e | 14 | Although not explicitly included in the Arrhenius form, we hypothesize that these trends can be rationalized based on the higher attempt to diffuse frequency of CH 3 as shown in equation 5. Specifically, equations 4 and 5, show that the rate of surface diffusion is correlated with the diffusion jump length and attempt frequency. It is also interesting to note that C shows low diffusivity despite a small diffusion barrier. We expect that this behaviour is likely due to the jump frequency being quite low for carbon as observed by the low intercept, implying carbon rarely jumps between sites leading to a lower average E app d . Since this E app d intrinsically includes a measure of the diffusion attempt frequency and is not restricted to a particular pathway as would be the case with an NEB, these values are likely the "true" diffusion barriers for an adsorbate over the Ag (111) surface. Similar arguments could explain the higher apparent E app d for CH than C. The apparent diffusion energy barrier for CH is higher than that of C. We hypothesize that this is likely because not all valence electrons are engaged in binding to Ag and this was previously observed by Mavrikakis et al. Taken together, the above discussion further highlights the advantages of using MD simulations (rather than a hTST framework) for investigating surface diffusion. |
66105b8821291e5d1d2b6d2e | 15 | As a point of comparison with traditional NEB-based studies, we used the DP 10 model to run an NEB calculation. Specifically, using O as a prototypical example, we compare the apparent diffusion energy barrier obtained from the MD simulation to the one that is obtained from a NEB calculation. We studied the diffusion of O on Ag (111) starting from the fcc site, the most stable site 65 diffusing to the hcp site. We observe a small energy mismatch between the DFT-predicted barrier and ML-predicted barrier, this is possibly due to higher total energy MAE error arising from the variety of systems and model training being more focused on reducing the force loss. The accuracy is also likely limited by the parameters and frameworks chosen for the MLP, we can sufficiently reduce the error with more optimal hyperparameters to reach near DFT-level accuracy. |
66105b8821291e5d1d2b6d2e | 16 | These discrepancies however do not impact the overall conclusions of this study as the xyz coordinates obtained from MLP-based NEB are in excellent agreement with DFT calculations. For instance, single point energy calculations on MLP/NEB-derived geometries are almost identical with full DFT/NEB optimization across the entire reaction coordinate (Fig. ). We observe that the barrier predicted from MD is ∼0.11 eV lower than the barrier evaluated for the fcc to hcp path explored using NEB. As previously mentioned, this energy difference likely stems from the fact that our calculated E app d takes into account the likelihood of diffusion whereas an NEB does not. A more detailed analysis using the ensemble of DP models is provided in the SI (Fig. ). |
66105b8821291e5d1d2b6d2e | 17 | The above diffusion studies used the final DP 10 model. However, the active learning curriculum used for model training allows us to compare the trends in model performance and reliability across different training iterations. As shown in Fig. and the violin plots in S4-11, we observe a general trend where the rate of model convergence is correlated with the chemical complexity of the adsorbate. For instance, Fig. compares the model uncertainty (as measured by ϵ t ) over multiple training iterations for O and OH. While the model learns to describe the dynamics of atomic adsorbates such as O (Fig. ) in a few iterations, more complex behaviour is observed for OH (Fig. ). Specifically, the MD simulation using the iteration 2 model (i.e., DP 2 /MD) explores a wider configuration space resulting in a higher ϵ t than the previous iteration, which is then well-described in the subsequent DP 3 model. This ability to self-correct is also observed for ethyl (Fig. ) and is a major advantage of a curriculum-based training approach. Specifically, ethyl requires the longest number of iterations to converge, which is also reflected in the higher number of sampled configurations across all training iterations (Fig. ). Similar plots and trends for the other adsorbates can be found in the SI (Fig. ). While it is possible that graph-based MLP architectures may require fewer training data, these comparisons are beyond the scope of this work. |
66105b8821291e5d1d2b6d2e | 18 | This work demonstrates an active learning approach to develop a transferable machine learning-based potential (MLP) that can be used to study surface diffusion phenomena at ab initio accuracy. Specifically, we focus on modelling the surface dynamics of 8 commonly studied adsorbates on a prototypical Ag(111) surface. The resulting diffusivities, obtained by fitting mean square deviations from molecular dynamics simulations, highlight potential shortcomings of traditional NEB-based estimates. The active learning approach allows the model to identify configurations that require further refinement; this strategy is particularly advantageous for describing the dynamics of complex adsorbates. Furthermore, the model demonstrates good transferability and can be fine-tuned to study unseen adsorbates by including small amounts of additional DFT data. Thus, this work illustrates a valuable MLP-based workflow for studying surface diffusion that can be easily generalized to other |
62c302e27b3b3079bc0cb448 | 0 | The climate change crisis requires large-scale human behaviour changes and the rapid deployment of a range of greenhouse gas emissions mitigation technologies. Among these, carbon dioxide capture and storage is an important technology that can reduce emissions from point sources. In this approach, CO2 is captured from gas emissions at industrial sources that may include power stations, hydrogen production plants, cement factories and steel factories. This is achieved using a capture material that selectively absorbs CO2 from the mixture of emitted gases (Figure ). The CO2 is then collected from the capture material via the application of heat (and/or a vacuum) and is subsequently stored in the ground where it can be permanently sequestered. A closely related technology is direct air capture, where a capture material is instead used to capture and remove carbon dioxide directly from the atmosphere. If the carbon dioxide is then collected and stored securely, this technology can offer "negative emissions", which are increasingly thought to be important in climate change mitigation pathways. A good carbon capture material for must generally have (i) large CO2 capture capacities, (ii) highly selective CO2 uptake, (iii) fast uptake kinetics, (iv) low energy consumption (joules per kg CO2 captured), and (v) long term stability to repeated cycling. The most established technology for point source carbon capture employs aqueous amine solutions to capture CO2 and is used in a number of large-scale demonstration projects. However, this technology has some limitations including relatively large energy consumption for regeneration, corrosion of the steel containers in which the amines are housed, and amine degradation (e.g. by oxidation). An active area of research is, therefore, to design new capture materials with improved performance. A large range of new carbon capture materials are under consideration, not limited to advanced amine solvents, ionic liquids and solid sorbents. In this article we focus on solid sorbents, which are generally porous materials that bind CO2 at their internal surfaces (inside their pores). A wide range of porous materials including metal-organic frameworks (MOFs), porous silicas, activated carbons and porous polymers have been explored for CO2 capture. For example, a MOF was recently reported to have excellent performance for CO2 capture via the physical adsorption of CO2 in the pores of the material. Another promising strategy is to functionalise porous materials with reactive functional groups such as amines or hydroxides. For example, amine-functionalised MOFs have shown promising performance and have enabled new adsorption mechanisms with unusual adsorption thermodynamics (see below). The design of improved carbon dioxide capture materials requires a detailed understanding of how carbon dioxide binds and moves within capture materials. Characterisation of carbon capture chemistry is often challenging and requires the use of a wide range of experimental techniques as well as simulation methods. Among these methods, nuclear magnetic resonance (NMR) spectroscopy has emerged as a powerful probe of binding chemistry and molecular dynamics. NMR experiments can be performed on liquids, solids and gases, as well as samples containing a variety of phases. In the case of solid materials, there is no requirement for crystallinity and amorphous materials can readily be studied. In this article, we review the practical considerations for carrying out NMR studies of carbon dioxide capture. We then show three case studies of how NMR methods have been developed to understand CO2 adsorption and diffusion mechanisms in MOFs. |
62c302e27b3b3079bc0cb448 | 1 | The first practical consideration is which nucleus to study. Many studies directly probe the bound CO2 molecules by C NMR spectroscopy. As a spin 1/2 nucleus 13 C can readily be studied with a wide range of NMR techniques, though the low natural abundance of 13 C (1.11 %) may require the use of signal enhancement techniques, especially since T1 relaxation times are often long for C. The use of C-enriched CO2 is generally very helpful for increasing the signal to noise ratios and enabling more advanced experiments. Beyond 13 C NMR, O NMR may be used to directly study the CO2 molecules. The very low natural abundance of this isotope (0.037 %) generally means that enrichment is likely to be essential for most experiments. Since 17 O is a spin 5/2 quadrupolar nucleus, solid-state experiments require the use of high magnetic fields and multi-dimensional experiments to resolve closely related chemical environments (see later). Beyond directly studying the captured CO2, insight into the capture behaviour can be gained by studying additional NMR-active nuclei present in the capture material. For example, N NMR experiments have been used in amine-based materials, where reaction generally occurs at the nucleophilic nitrogen centres. A critical further consideration for NMR spectroscopy studies of carbon dioxide capture is how to prepare and study the sample in the presence of CO2. NMR experiments of carbon dioxide capture may be categorized according to whether they are performed with sample magic angle spinning (MAS), or with static sample conditions (Figure ). Experiments may then be further categorised into ex situ and in situ experiments, where ex situ refers to experiments on samples that are dosed with CO2 and sealed in the laboratory prior to NMR studies, while in situ refers to dosing the sample with gas inside the magnet while simultaneously studying it with NMR methods (Figure ). Each of these approaches has advantages and disadvantages, and the optimum method will depend on the exact materials studies and the desired information. Importantly, safety must always be the top priority. CO2 is a colourless and odourless gas at ambient conditions. Exposure to CO2 may cause headaches, dizziness, confusion and loss of consciousness. At high levels of exposure, death by asphyxiation could occur (note that CO2 is heavier than air, and can accumulate in the laboratory in some scenarios). Figure shows a common ex situ approach for studying solid-adsorbents for CO2 capture, which we term "ex situ MAS". In this approach, a MAS rotor is packed with the solid adsorbent and is transferred to a gas-dosing manifold, with the rotor left open. The sample is then evacuated before dosing CO2 gas. Following a period of equilibration, the sample is sealed, for example by using a mechanical plunger, and is then removed for MAS magnetic resonance experiments. An advantage of this approach is that standard MAS rotors and probes are used, which can greatly improve spectral resolution for solid samples. If the gas-dosing manifold is designed to have a low volume, isotopically enriched gases can also be readily used (e.g. 13 CO2) to improve signal intensities. A clear disadvantage of this approach is that it is time-consuming to study multiple gas-dosing conditions (e.g. different gas-dosing pressures, different gas-dosing temperatures). A further disadvantage is that the dosed gas may escape from the MAS rotor over time (depending on how strong the adsorption is, and how leak-tight the rotor is). In cases where bespoke rotor dosing equipment is not available, simple alternatives such as a gas-filled glovebag or chamber may also be employed, provided appropriate safety measures are taken to avoid CO2 exposure. The ex situ MAS approach has been applied to a range of solid-adsorbent materials to study carbon capture modes. A number of studies have explored carbon capture chemistry in amine-functionalised silicas, 20-26 metal-organic frameworks, 8,27-32 amine-functionalised metal-organic frameworks, zeolites, polymers, and porous carbons. The disadvantages of the ex situ MAS approach can be partly resolved by the "in situ MAS" approach (Figure ). Here MAS NMR experiments are performed on the solid adsorbent sample with the simultaneous delivery of CO2 gas to the sample via a hole in one of the rotor caps. This approach has the advantages that the sample is studied directly under operating conditions, and that the gas dosing conditions (sample temperature and partial pressure of CO2) can be varied continuously. |
62c302e27b3b3079bc0cb448 | 2 | Kinetic studies should also be accessible with this approach. One potential disadvantage is the potentially high cost of flowing isotopically enriched gases unless a gas recovery system can be employed. The flow of gases also requires more careful planning with regard to safety issues surrounding gas exposure. When setting up in situ experiments, a further safety consideration is that one must avoid bringing any magnetic parts close to the magnet (e.g. for the gas delivery system). |
62c302e27b3b3079bc0cb448 | 3 | Initial experiments with the in situ MAS approach on amine-functionalised silicas highlight the great promise of this approach. Two further approaches may be considered that employ NMR measurements on static samples. These can be applied to both liquid absorbents and solid adsorbents. In the "ex situ static" approach, CO2 is introduced to the sample in an NMR tube, either by bubbling CO2 through a liquid or by dosing CO2 into a solid with a gas manifold (Figure ). After the sample has equilibrated, the tube is sealed and taken for NMR experiments. A key advantage of this approach is that only conventional solution-state NMR spectroscopy equipment is required and that both liquid absorbents and solid adsorbents can be studied. A clear disadvantage is that MAS experiments are not possible. However, we do note that MAS is not required for liquid absorbents (where molecular motion averages out anisotropic interactions), nor for many solid adsorbents, where the considerable motion of CO2 (especially for physisorbed CO2, with only non-covalent CO2-adsorbent interactions) can lead to relatively narrow peaks for adsorbed CO2. This method has been used to study a wide range of materials including ionic liquids and metal-organic frameworks. Finally, the "in situ static" approach extends the above approach to directly dose CO2 gas into the static sample during the acquisition of NMR data (Figure ). This again has the advantage of enabling the capture material to be studied during operation, while also enabling a range of gas pressures and sample temperatures to be studied efficiently. One excellent example of these measurements explored the CO2 absorption pathways in a wide range of state-of-the-art aqueous amine CO2 absorbents. 50 CO2 gas was bubbled through the aqueous amine solutions inside the NMR magnet, and NMR measurements were recorded as a function of time. These experiments revealed the time-dependent uptake mechanisms, which varied significantly depending on the exact choice of amine. For example, monoethanolamine initially captures CO2 via the formation of ammonium carbamate, but at longer reaction times ammonium bicarbonate products began to dominate at the expense of the ammonium carbamates. A further excellent example used the "in situ static" approach to study mixed gas adsorption in a solid MOF adsorbent. The role of framework flexibility on the selectivity of gas adsorption was explored. |
62c302e27b3b3079bc0cb448 | 4 | MOFs are a class of microporous materials typically constructed from metal nodes and organic linkers to give three-dimensional framework structures. Their use as CO2 sorbents is highly favourable owing to their high surface areas and the possibility to precisely control the pore chemistry. One emerging class of MOFs that have shown promise for CO2 capture are amine-appended frameworks based on the MOF-74 structure, such as M2(dobpdc) where M = Mg, Mn, Fe, Co, Ni, Cu, Zn, Cd, and dobpdc = 4,4′-dioxidobiphenyl-3,3′-dicarboxylate (Figure ). In this structure, the metal ion has a vacant coordination site along the microporous channels. The appendage of an organic amine compound to these vacant metal sites (Figure ) gives rise to large adsorption capacities for selective and reversible CO2 uptake. As the nature of both the metal cation and the amine can be varied, the adsorption thermodynamics can be tuned for a desired application. 12, Amine-appended MOF-74 analogues may be advantageous over other sorbent materials, as they display step-shaped adsorption isotherms, suggesting that CO2 uptake/release is sudden upon exposure at a threshold pressure. As such, these materials have lower regeneration energies than traditional amine scrubbing technologies. Basic characterisation methods such as powder X-ray diffraction and solution state 1 H NMR of digested frameworks can be used in order to ascertain the long-range ordering of the material and to assess the amine loading, respectively. In these materials a 1:1 metal:amine ratio can be achieved, indicating that all initially vacant metal sites are occupied by amines (Figure ). X-ray crystallography studies on large single crystals of amine-appended-Zn2(dobpdc) materials have shown that at least two types of CO2 adsorption products can be formed in these materials. The first product is ammonium carbamate chains, where CO2 inserts itself into the Metal-N bond to form a negatively charged carbamate (Figure ) and a neighbouring diamine forms a positive ammonium. Adjacent carbamates and ammoniums form ion pairing and hydrogen bonding interactions to produce chains, explain the cooperative adsorption in these materials, and accounting for the step-shaped adsorption isotherms. The ammonium carbamate chain product has been observed for many diamine-Zn2(dobpdc) analogues, and is thought to be the most prevalent product. In (dmpn)2-Zn2(dobpdc) (dmpn = 2,2-dimethyl-1,3-diaminopropane), however, the formation of carbamic acid pairs was reported, whereby the CO2 molecule reacts with a "dangling" primary amine, and two carbamic acids interact via hydrogen bonding to form a pair (Figure ). Unfortunately, structure solution by X-ray crystallography generally requires large single crystals which are difficult to synthesise for many analogues of M2(dobpdc). The case studies presented here use multinuclear NMR spectroscopy to explore CO2 adsorption on the molecular level without the need for large single crystals and can, therefore, be applied to many different amine analogues in these MOFs. Indeed, we will show how solid-state NMR studies on (dmpn)2-Mg2(dobpdc) materials revealed that CO2 can adsorb by a third mechanism where both ammonium carbonate chains and carbamic acids are formed on adjacent metal sites, stabilised by hydrogen bonding (Figure ). Case Study 1: Elucidating Chemisorption in MOFs with |
62c302e27b3b3079bc0cb448 | 5 | As discussed previously, C NMR experiments can be implemented easily and provide a wealth of information about the number and type of carbon environments present in sorbent materials. More detailed information about the binding mechanisms can be gained by exploring the local connectivity of the carbon atoms through correlation experiments. However, this approach is typically limited by the low natural abundance of C and as such, the use of isotopically enriched 13 CO2 gas and/or more complex sensitivity-enhanced experiments. In solids, the cross polarisation (CP) experiment can be used to transfer magnetisation via dipolar couplings from a high abundance, high g nucleus, such as 1 H, to a low abundance, low g nucleus, such as C, thus increasing the sensitivity and decreasing spectral acquisition time. The build-up of magnetisation on the low g nuclei is governed by the distance between the coupled spins, as well as molecular dynamics. By using short transfer pulses, only spins in close proximity will be selectively enhanced in the NMR spectrum and therefore the CP experiment can be used as a spectral editing technique. Longer-range couplings can be explored by increasing the transfer time, however, this build-up can be impeded by the T1r relaxation of the high g nuclei and as such, the absolute intensities of signals in these experiments should be treated with care. Furthermore, cross polarisation can be used to transfer magnetisation in two-dimensional HETCOR experiments, which allows for the observation of correlations between coupled spins. For these experiments, an ex situ CO2 loading approach is favoured to gain high-resolution spectra of the final adsorption products under MAS conditions. Here we present a case study showing how one-dimensional C MAS and two-dimensional 1 H- C HETCOR experiments can be used to investigate binding mechanisms in MOF materials. |
62c302e27b3b3079bc0cb448 | 6 | The 13 C NMR spectrum of (e-2)2-Mg2(dobpdc) (Figure , e-2 = N-ethylethylenediamine) shows two peaks at 162.1 and 124.7 ppm, corresponding to chemisorbed and physisorbed CO2, respectively. The chemisorbed CO2 signal is enhanced using a CP experiment and this signal is assigned to the formation of an ammonium carbamate chain upon CO2 adsorption. In contrast, the (dmpn)2-Zn2(dobpdc) shows two chemisorbed CO2 peaks with an intense signal at 161.2 ppm, assigned to carbamic acid pairs and a weaker signal at d = 164 ppm, assigned to a small quantity of ammonium carbamates (Figure ). It is clear that the number and type of environments in (e-2)2-Mg2(dobpdc) and (dmpn)2-Zn2(dobpdc) are different, a range of C chemical shift values are reported in the literature for ammonium carbamate chains and carbamic acids. The small differentiation in chemical shifts between the similar C environments in these mechanisms, therefore, means that the confidence in the assignments of these binding modes by C MAS NMR spectroscopy alone is low. |
62c302e27b3b3079bc0cb448 | 7 | To improve confidence in the peak assignments two-dimensional 1 H- C HETCOR experiments at short contact times were performed, allowing for the investigation of 1 H nuclei located near the 13 CO2 adsorption product. The 1 H-13 C HETCOR experiment for (e-2)2-Mg2(dobpdc) (Figure ) shows a strong correlation of the chemisorbed CO2 to a 1 H environment at d = 4.3 ppm. This 1 H signal can be assigned to HNCOO -, confirming the reaction of the CO2 to the e-2 primary amine species. A weaker correlation can also be seen with a 1 H signal at d = 13.2 ppm, arising from hydrogen bonding between the adsorption product and the secondary ammonium on an adjacent metal site. As such, these correlations provide greater evidence for ammonium carbamate chain formation. Conversely, in the |
62c302e27b3b3079bc0cb448 | 8 | Whilst 1 H and C NMR spectroscopy can be useful probes of binding modes in amine-appended MOFs, the similar local environments surrounding the C atom makes unambiguous assignment of adsorption products difficult. There is a need to develop new techniques to unequivocally characterise these materials. From the adsorption mechanisms shown in Figure , it can be seen that each binding mode has two unique oxygen sites and the type of oxygen site present varies drastically depending on the binding mechanism. As such, O NMR spectroscopy should be a highly diagnostic probe to study these materials. As discussed previously, the low natural abundance and quadrupolar nature of spectra. Furthermore, the quadrupolar interaction introduces a second-order broadening which cannot be removed by MAS. This coupling can be quantified by its magnitude (CQ), defined as CQ= eQVzz/h, and the asymmetry (hQ), defined as hQ = Vxx-Vyy/Vzz, where e is the electronic charge, Q is the nuclear quadrupolar moment, h is Planck's constant and Vxx, Vyy and Vzz are the principle components of the electric field gradient tensor. The 17 O diso, CQ and hQ can be determined by fitting the 17 O MAS spectrum. For spectra containing many overlapping O signals, fitting the MAS spectrum can be challenging. This can be overcome by using two-dimensional MQMAS 53-55 or STMAS experiments. The use of O NMR spectroscopy, therefore, provides an additional set of parameters compared to C NMR spectroscopy, which can be used to classify CO2 adsorption products. One drawback of this method is the relatively high cost of O-enriched CO2. This, therefore, precludes the use of in situ experiments where a flow system is used. Using an ex situ MAS method, acquisition of a high-resolution O MAS NMR spectrum for amine-functionalised MOF materials can be achieved in approximately 30 minutes by dosing the sample at ~1 bar. This translates to a cost of £50 per sample, although this value could be reduced by optimising the design (i.e., reducing the volume of gas in the system) of the experimental set up shown in Figure . |
62c302e27b3b3079bc0cb448 | 9 | To ascertain the value of O NMR as a probe, the 17 O NMR parameters were calculated using DFT calculations for amine appended Mg2(dobpdc) containing a wide range of organic diamines. The use of DFT calculations provides an excellent opportunity to explore and compare both known and novel adsorption products. Of the three adsorption mechanisms explored, ammonium carbamate chains, carbamic acid pairs and the mixed adsorption mechanism, four unique O environments were identified: M-OCO and M-OCO -, resulting from ammonium carbamate chains and COOH and COOH, resulting from the formation of carbamic acids. Each environment can be identified by their O NMR parameters, as shown in Table . The key distinguishing feature for carbamic acids is the lower chemical shift, large CQ and small hQ values for the OH. Furthermore, in general, the metal bound oxygen can be distinguished from the carbamate O by a lower chemical shift. These calculations suggested that O NMR spectroscopy is a powerful probe into these adsorption mechanisms. Experiments were then performed to explore this further. ammonium carbamate chains and therefore the O NMR spectroscopy confirms this binding mechanism for (ee-2)2-Mg2(dobpdc). The second amine-appended Mg2(dobpdc) material of interest is dmpn-Mg2(dobpdc). As discussed previously, dmpn-Mg2(dobpdc) is thought to capture CO2 via a mixed adsorption mechanism, however, the assignment of this binding mechanism is not conclusive by 1 H and 13 C NMR spectroscopy owing to the overlap of signals for similar C environments. Excitingly, the O MAS spectrum (Figure ) for dmpn-Mg2(dobpdc) showed a broad overlapping of multiple signals corresponding to chemisorption of CO2 and two sharp resonances corresponding to physisorbed CO2. |
62c302e27b3b3079bc0cb448 | 10 | To aid in the deconvolution of this spectrum, an O MQMAS NMR spectrum was acquired (Figure ). This spectrum shows 3 signals and the extracted NMR parameters were consistent with the calculated diso, CQ and hQ values for COO and M-COO, suggesting the formation of carbamate chains and a C=O in the mixed adsorption structure (Table ). However, when fitting these signals to the (Table ). Overall, the work presented in case study 2 shows that O NMR spectroscopy is a powerful probe to explore CO2 binding mechanisms in metal-organic frameworks. |
62c302e27b3b3079bc0cb448 | 11 | CO2 diffusion is an important physical phenomenon that influences the CO2 capture performance of porous solid adsorbents. Following on from earlier studies that measured CO2 diffusion in MOFs with pulsed-field gradient NMR (PFG NMR) methods, we explored CO2 diffusion in the material Zn2(dobdc) (dobdc 4-= 2,5-dioxidobenzene-1,4-dicarboxylate). Similar to the M2(dobpdc) materials discussed above, this material features one-dimensional hexagonal channels in an ordered lattice (Figure ). The Zn(II) ions have a vacant coordination site, which is the primary binding site for CO2. The one-dimensional pores of this material, and the lack of connectivity between different pores in the crystal structure, should result in a significant diffusion anisotropy, i.e. CO2 self-diffusion along the pores (D//) is anticipated to be much faster than diffusion between pores D ⊥ . |
62c302e27b3b3079bc0cb448 | 12 | First, large rod-like crystals of Zn2(dobdc) with lengths of 100s of microns were synthesised via a solvothermal method (Figure ). In these crystals the hexagonal pores run along the length of the rods. The availability of such large crystals ultimately simplifies the PFG NMR analysis, since exchange of CO2 between different crystals and with free CO2 gas becomes negligible on the experimental timescales (typically tens of milliseconds for PFG NMR experiments). With large crystals in hand, the "ex situ static" approach was used whereby a large number of crystals were loaded into a valved NMR tube, which was dosed with 13 CO2, and then taken for static C NMR measurements (Figure ). |
62c302e27b3b3079bc0cb448 | 13 | A static C NMR spectrum revealed a signal with a chemical shift consistent with physisorbed CO2 as expected, and with a lineshape consistent with an anisotropic chemical shift observed for the adsorbed CO2 (Figure ). While the lineshape is much narrower than that reported for solid CO2, the observed chemical shift anisotropy suggests that CO2 molecules have a preferred average orientation relative to the framework pores, consistent with other studies of similar frameworks. For the conditions studied in Figure , signal intensity at the left hand edge of the spectrum with a chemical shift d// could be assigned to CO2 adsorbed in crystals oriented parallel to the applied magnetic field, while signal intensity at the right hand edge could be assigned to CO2 adsorbed in crystals oriented perpendicular to the applied magnetic field d ⊥ . As was realised in earlier PFG NMR studies, the observation of a chemical shift anisotropy lineshape provides an excellent opportunity for the measurement of anisotropic diffusion. Indeed, as the gradient strength was increased in a series of PFG NMR experiments, clear lineshape changes are observed (Figure ), which arise from the anisotropic diffusion of CO2. In these experiments, the pulsed field gradient was applied along the laboratory z-axis, i.e. co-linear with the main applied magnetic field, B0. Therefore, the rapid decay of signal intensity at the left hand edge of the spectrum corresponds to the relatively fast self-diffusion along the length of the rod-like crystals, i.e. along the hexagonal pores of the metal-organic framework. On the other hand, the much slower decay of signal intensity at the right hand edge of the spectrum reports on the much slower diffusion of CO2 perpendicular to the MOF pores. Inspection of the spectra in Figure therefore confirms the anticipate result that D// >> D ⊥ . The above arguments were also substantiated by additional experiments in which the magnetic field gradients were instead applied along the laboratory x-axis. |
62c302e27b3b3079bc0cb448 | 14 | Inspired by earlier work, 58 spectral simulations were then performed to enable a more quantitative analysis (Figure ). The only free parameters in the simulations are D// and D ⊥ . The simulations shown in Figure suggested that CO2 diffusion along the framework pores takes a value of D// = D// = 1.5(4) × 10 -9 m 2 s -1 . This value is comparable to liquid water diffusion at ambient conditions, and is 4 orders of magnitude slower than the diffusion of free CO2 gas. For diffusion of CO2 between the MOF pores, we were only able to conclude an upper limit of D ⊥ < 10 -13 m 2 s -1 , and it was not possible to conclude whether or not D ⊥ had non-zero value. These findings show that the CO2 diffusion between the MOF channels is at least 10,000 times slower than diffusion along the MOF channels, i.e. the diffusion is very strongly anisotropic. This finding has implications for the practical applications of these materials, and suggests that rod-like crystals are in-fact not desired for practical applications due to their long diffusion pathways, and the possibility of poor blockage leading to inaccessible regions of the material. |
62c302e27b3b3079bc0cb448 | 15 | Finally, we note that these measurements have also been applied to the larger-pore Zn2(dobpdc) framework. Qualitatively similar spectra were obtained, though the increase in pore size from 15 to 22 Å led to an increase in D// by a factor of 4 to 6 across a range of CO2 dosing pressures. These measurements therefore highlighted a straightforward means to tune transport properties through modification of the framework structure. Measurements and analysis for the Zn2(dobpdc) sample interestingly also supported non-zero diffusion between the MOF pores, which was tentatively attributed to defects in the MOF structure enabling additional transport pathways. |
62c302e27b3b3079bc0cb448 | 16 | The case studies above highlight the power of NMR spectroscopy in revealing the chemistry of carbon dioxide capture. A particular success of the approach has been in studying the adsorption mechanisms that operate and the speciation of CO2 in the captured state. In some cases NMR studies have revealed new adsorption mechanisms in CO2 capture materials that were not readily accessible with other techniques. Moreover, diffusion measurements have been used to probe CO2 transport mechanisms in capture materials. |
62c302e27b3b3079bc0cb448 | 17 | A key limitation of much of the work to date is that it has largely been carried out under pure CO2 conditions. For real-world CO2 separations a mixture of gases is always present, not limited to CO2, N2, O2, H2O, CH4, NOX and SOX. A number of initial efforts have been carried out to study mixed gas adsorption, including CO2/H2O mixtures, as well as CO2/CH4 mixtures. The role of H2O should be studied in particular, since various studies have highlighted that H2O can have a significant impact on adsorption thermodynamics. We further note that few studies have investigated CO2 capture materials in the presence of O2, despite O2 often playing an important role in the oxidative degradation of many carbon capture materials. Studies of adsorbent degradation in general have also been lacking. |
62c302e27b3b3079bc0cb448 | 18 | A further limitation is that most of the NMR work to date has been carried out under ex-situ rather than in situ conditions. Time consuming ex-situ experiments often lead to a limited number of experimental conditions being sampled, and may result in temperature and pressure dependent phenomena being missed. Further method development will help to make in situ techniques more accessible and practical. Importantly, in situ measurements are well-suited to gas mixtures, and should also enable kinetic studies, which have been rare to date. We also note that low field NMR methods are being developed for the study of adsorption phenomena. The lower cost of these approaches are particularly attractive, and may be well-suited for industrial applications. An important remaining challenge is to differentiate closely related adsorption products. As discussed above, the 13 C isotropic chemical shifts generally provide poor differentiation of ammonium carbamate, carbamic acid and bicarbonate species. Approaches that utilise additional NMR parameters, such as 13 C chemical shift anisotropies, 25 as well as O NMR 14,70 may help to resolve closely related adsorption products in the future. The availability of increasingly high magnetic field strengths may make O measurements particularly attractive. Finally, signal enhancement methods based on dynamic nuclear polarisation (DNP) may enable complex NMR measurements to be carried out without the need for isotopic enrichment. The use of exogenous radicals as polarising agents may perturb the CO2 capture modes in some cases, though advances in the use of endogeneous radicals for DNP may be well suited for studies of CO2 capture. DNP methods may also allow for the enhancement of signals from species adsorbed at adsorbent particle surfaces, which may impact surface transport barriers. |
61310da7fc08e3e0548929f2 | 0 | The rates of chemical reactions are usually affected by both temperature and pressure. The effect of temperature on the rate of a chemical reaction is measured as the activation energy. For thermally activated reactions (no tunneling 3 involved), the larger the activation energy 𝐸 ! , the greater the increase in the reaction rate with temperature. Activation energies are measured experimentally by treating the temperature dependence of the rate constant using the Arrhenius equation. The activation energy is the slope of the plot of 𝑙𝑛𝑘 vs 1/𝑇, 𝐸 ! = 𝑅 * "#$% "('/)) + + , where 𝑘 is the rate constant, 𝑇 the temperature, 𝑅 the ideal gas constant, and 𝑝 the pressure. |
61310da7fc08e3e0548929f2 | 1 | , where ∆𝐺 ‡ is the Gibbs energy of activation, according to the thermodynamic formulation of the transition state theory. While activation energies are almost always positive, activation volumes can commonly be both positive and negative. Reactions with negative and positive ∆𝑉 ‡ are accelerated and decelerated by pressure, respectively. Usually, the signs of activation volumes can be intuitively predicted. For example, in the case of a bond formation reaction between two molecules (Figure ), a negative ∆𝑉 ‡ is expected since the unimolecular transition state is more compact and hence has a smaller volume than the two separated reactant molecules. The opposite is usually true for bond cleavage reactions. |
61310da7fc08e3e0548929f2 | 2 | Measurements of ∆𝑉 ‡ are a valuable tool to distinguish different reaction mechanisms. A notable example is the case of several Diels-Alder reactions whose activation volumes have given evidences in favor of a concerted one-step mechanism over the competing two-step mechanism. Another exemplary case of competing mechanisms is revealed by the illuminating study of the effect of pressure on the thermal dimerization of 1,3-cyclohexadiene by Klärner et al in 1986. At 1 atm, thermal dimerization of 1,3-cyclohexadiene yielded five reaction products-two [4+2] cycloadducts of the endo and exo configurations, two [2+2] cycloadducts of the syn and anti configurations, and one [6+4]-ene adduct (Figure ). It was also observed that, under pressure up to 7 kbar, the reactions producing all five products reactants transition state product |
61310da7fc08e3e0548929f2 | 3 | are accelerated, indicating negative activation volumes for all these reactions. Considering the bond-formation nature of these reactions, the observed acceleration under pressure is not surprising. Apparently, different mechanisms are in operation in this dimerization, for both the symmetry-allowed [4+2] and [6+4] and forbidden [2+2] dimers, according to the Woodward-Hoffmann rules, were observed. The symmetry-allowed [4+2] cycloadducts and the [6+4]-ene adduct could, in principle, be formed through either a concerted or stepwise mechanism, whereas the symmetry-forbidden [2+2] cycloadducts are usually expected to be formed via a stepwise mechanism under thermal conditions (Figure ). The main difference between the concerted and stepwise mechanisms, in terms of activation volume, is that two bonds are formed at once in a concerted mechanism and only one bond is initially formed in a stepwise mechanism, so that the concerted transition state (TS) structure is usually more compact than the stepwise TS structure. Therefore, the concerted reaction usually has a more negative activation volume than a stepwise reaction. |
61310da7fc08e3e0548929f2 | 4 | product ratio ΔV ‡ (cm The measured activation energies 𝐸 ! for the reactions giving the five products in Figure are within 4 kcal/mol. An even smaller difference in 𝐸 ! is seen between the endo and exo [4+2]-cycloadditions, though the former likely follows a concerted mechanism while the latter a stepwise mechanism, based on the measured difference in ∆𝑉 ‡ . Interestingly, it seems that the more negative the ∆𝑉 ‡ , the lower the 𝐸 ! of the reaction, though the two quantities do not necessarily have a correlation. |
61310da7fc08e3e0548929f2 | 5 | In 2008, Ess et al. calculated the TS structures and reaction pathways for the dimerization of 1,3-cyclohexadiene in the gas-phase. Calculations at B3LYP, CASPT2 and CBS-QB3 levels show that the computed reaction barriers for the concerted [4+2]cycloadditions, concerted [6+4]-ene reactions, and the stepwise additions (Figure ) are within 5 kcal/mol. This small difference is consistent with the difference in experimental 𝐸 ! , confirming the competitive nature of the concerted and stepwise mechanisms. |
61310da7fc08e3e0548929f2 | 6 | The Ess et al. work focused on the activation energies and didn't deal with the activation volumes of the reaction. Theoretical calculations of ∆𝑉 ‡ , especially from firstprinciples, have been a challenge. A simple and intuitive way to compute ∆𝑉 ‡ is to calculate the difference in van der Waals (vdW) volumes of the molecule at the reactant and transition states. The vdW volume is the volume of interlocking vdW spheres (often with scaled vdW radii) centered on the constituting atoms of the molecule. However, likely due to the negligence of intermolecular interactions and solvent effect, ∆𝑉 ‡ computed by this method are always too small in magnitude. An empirical packing coefficient was introduced to account for such negligence and to correct the underestimated ∆𝑉 ‡ , computed by this method. A new method for more rigorous ∆𝑉 ‡ calculations is the recently-developed extreme pressure polarizable continuum model (XP-PCM). As an extension of the popular polarizable continuum model (PCM) that tackles molecular solution energies at the standard condition of pressure, the XP-PCM allows for quantum chemical calculations of the energy profiles of chemical reactions under pressure. The effect of the pressure is introduced in XP-PCM via a repulsive interaction between the reactive molecular system and the surrounding solvent medium. Within XP-PCM, ∆𝑉 ‡ can be computed, according to the transition state theory, as the derivative of activation free energy with respect to pressure. The XP-PCM method has been applied to the calculations of the energy profiles of a subset of pericyclic reactions. Interesting phenomena such as a shift of the transition state and a switch of the rate determining step have been discovered by the calculations. Furthermore, the computed ∆𝑉 ‡ are in reasonable agreement with experimental value. Recent work by Fukuda and Nakatani applied the XP-PCM method to a retrocycloaddition. The necessary details of the physical basis and computational protocol of the XP-PCM method are given in the Computational Methodology section. In additional to reaction profiles, the XP-PCM method has also been applied to the studies of the effect of pressure on a variety of molecular properties, such as equilibrium geometries, vibrational frequencies, electronic excitation energies. We note here relevant approaches derived from the mechanochemistry field for high-pressure calculations on molecules and reactions. Notably, the recent GOSTSHYP method from the Stauch group 44 is capable of calculating activation volumes of reactions. Another approach for simulating high-pressure organic reactions employs a simulation box with periodic boundary condition, filled with solvated reactant molecules. |
61310da7fc08e3e0548929f2 | 7 | The solvent molecules are explicitly included in the simulation box, in contrast to the implicit solvation approach in the XP-PCM method. Molecular dynamics (MD) simulations or Monte Carlo simulations (an early example by Klärner et al. ) are performed to obtain reaction profiles at high pressures and activation volumes. Due to the large size of the system using a simulation box, the MD simulations were usually done with force fields, as illustrated in the works from the Weinberg group, or by a hybrid quantum mechanics/molecular mechanics (QM/MM) approach, as shown in the work by Plotnikov and Martinez 49 and a recent work by Loco et al. We now report a thorough consideration of the potential energy surface (PES) of the thermal dimerization 1,3-cyclohexadiene, including [4+2]-ene pathways that were not considered previously. In addition, we report, for the first time, XP-PCM calculations on the activation volumes for the various dimerization reactions of 1,3-cyclohexadiene, offering a new approach for computationally studying the competing mechanisms of this reaction. |
61310da7fc08e3e0548929f2 | 8 | Gas-phase calculations. Gas-phase geometries were optimized at the ωB97XD 51 /def2-TZVP 52 level of theory. Frequency analyses were performed at the same level to verify the optimized structure to be either a minimum or a transition state, and to obtain the zero-point vibrational, thermal, and entropic corrections, necessary in calculating enthalpies and free energies. For open-shell singlets that appear in the stepwise mechanism, broken-symmetry DFT with spin-projection by the Yamaguchi-Houk procedure was used. On the ωB97XD/def2-TZVP optimized structures, single-point calculations were performed using the strongly contracted N-electron valence state perturbation theory (SC-NEVPT2) and coupled-cluster CCSD(T) methods with the same def2-TZVP basis set. In the SC-NEVPT2 calculations, a (4,4) active space was used for 1,3-cyclohexadiene, which comprises the valence π-type orbitals and the valence π electrons in the molecules. An (8,8) active space was used for the transition states, which consists of the same set of π-type orbitals and π electrons of two molecules of 1,3cyclohexadiene; however, some of the π-type orbitals become more of σ-type orbitals in the bond formation regions in the transition states. The ωB97XD and CCSD(T) calculations were computed using the Gaussian 16 program package. The SC-NEVPT2 calculations were done with the ORCA 4.2.1 program package. |
61310da7fc08e3e0548929f2 | 9 | High-pressure calculations. High pressure calculations were performed using the XP-PCM method at the ωB97XD/def2-ΤZVP level of theory in Gaussian 16 and an in-house Julia script. The XP-PCM is a quantum chemical method aimed to introduce the effects of the pressure on the calculation of the electronic energy 𝐺 /0 of a molecular system in a dense medium via a Pauli exchange-repulsion interaction between the molecular system and the external medium. Such a Pauli-exchange repulsion is motivated by the fact that at high pressure, the reduction of the volume of a dense medium forces the intermolecular distances below the van der Waals contacts, in a domain where the intermolecular interactions are dominated by the Pauli exchange-repulsion. The increase of the pressure is modeled by simply shrinking the volume 𝑉 1 of the cavity hosting the molecular system so as to increase the overlap between the electron densities of the system and of the external medium (Figure ). This molecular cavity is built up starting from the envelope of vdW spheres centered on the nuclei of the reactive system and with scaled vdW radii. In studying chemical reactions at high pressure, the effective potential energy for the motion of the nuclei of the reactive system, 𝐺 232 (𝑝), corresponds to the electronic energy 𝐺 /0 (𝑝), supplemented by a contribution of the so-called cavitation Gibbs energy that corresponds to the work necessary to create the void cavity hosting the molecular solute at the given condition of pressure 𝑝: |
61310da7fc08e3e0548929f2 | 10 | This work also includes a configurational entropic contribution from the external medium. The effective potential energy profile for a reaction at a given pressure 𝑝 is then determined by computing 𝐺 232 (𝑝) for a set of selected structures along a suitable reaction coordinate. According the transition state theory, the activation volume ∆𝑉 ‡ is determined from the slope of the corresponding activation energy ∆𝐺 232 ‡ (𝑝) as a function of the pressure, In this work, the effective potential energy profile for a reaction at a given pressure 𝑝 is then determined by computing 𝐺 232 (𝑝). The corresponding activation volume ∆𝑉 ‡ is computed by the same computational protocol used previously for the study of the effect of pressure on a selection of pericyclic reactions, where reasonable agreement of the computed activation volumes with experiment values was obtained. We recommend that interested readers consult ref 17 and the SI of the current paper for a detailed tutorial about the protocol. All the XP-PCM calculations are performed for selected structures (reactants, transition state, product) along the gas phase intrinsic reaction coordinates. |
61310da7fc08e3e0548929f2 | 11 | The [4+2]-ene pathways were not considered in previous calculations by Ess et al. For the concerted [4+2]-cycloadditions and [6+4]-ene reactions, each of these pathways can generate two stereo isomers as a result of different relative orientations of the two cyclohexadiene rings in the TS structures. These isomers are termed endo and exo for the [4+2] cycloaddition adducts and threo (also could be called rac or dl) and erythro (also could be called meso) for the [6+4] ene-adducts. |
61310da7fc08e3e0548929f2 | 12 | For concerted [4+2]-ene reactions, two pathways are possible. In each of them, there exist configurational isomers for the TS, for example, TS-5 and TS'-5. However, both of these two TS structures lead to the same dimer (i.e., 5). The configurational isomerism in the TS disappears in the dimer because the bottom ring in the adduct becomes 1,4cyclohexadiene that does not contain any stereocenter. |
61310da7fc08e3e0548929f2 | 13 | For stepwise pathways, we considered those beginning with a C-C bond formation between two 1,3-hexadiene molecules at the terminus of the diene moieties and generating a diallyl intermediate, rac-8 or meso-8. Either of the two intermediates contains two stereocenters, but the meso-8 isomer is achiral due to the presence of an inversion center in the structure. Other stepwise pathways involving bond formation at internal sites of the diene moiety are likely to be unfavorable due to the generation of isolated (i.e., not in conjugation with a double bond) radical sites; these pathways are not considered. In the second step, a rotation about the first formed C-C bond followed by a radical recombination or a H-transfer leads to a cycloaddition or ene product. and erythro-4 can be formed from it, though erythro-4 was not observed experimentally. |
61310da7fc08e3e0548929f2 | 14 | Figure shows the potential energy surface of various dimerization pathways of 1,3-cyclohexadiene at 1 atm, calculated at the ωB97XD/def2-TZVP level of theory. Table reports the computed reaction barriers at ωB97XD, multireference perturbation NEVPT2, and coupled-cluster CCSD(T) levels. The barrier of the exo pathway is 1 kcal/mol higher, likely due to the absence of secondary orbital interactions in TS-exo-2. ωB97XD/def2-TZVP calculations also show that the activation enthalpy of the threo [6+4]-ene reaction going through TS-threo-4 is 3 kcal/mol higher than that of the endo [4+2] cycloaddition. However, CCSD(T)/def2-TZVP single- |
61310da7fc08e3e0548929f2 | 15 | ΔH 298K in kcal/mol point calculations give opposite results that the threo [6+4]-ene pathway is the most favored pathway, with TS-threo-4 being 1 kcal/mol lower than TS-endo-2. The CCSD(T) computed activation enthalpies agree excellently with the experimental activation energies (Table ). Calculations show that the entropy of TS-threo-4 is 5 J/mol•K smaller than TS-endo-2; this entropic difference corresponds to a difference of 1. formed C-C bond (~180 degrees, see Figure ) is necessary in order to reach a geometry for the second bond to form to give syn-3. This might be the reason that syn-3 has a lower product ratio than exo-2 in experiment. |
61310da7fc08e3e0548929f2 | 16 | Many structures in the stepwise pathways are open-shell singlet states with significant multi-reference character. Single-reference DFT calculations using the brokensymmetry approach, which is what we used to compute the energies of, for example, TS_meso and TS_rac (Table ), introduce spin contamination from the triplet to the openshell singlet wave functions, as indicated by the <S 2 > values from broken-symmetry DFT calculations, for instance <S 2 > = 0.62 for TS-meso. The spin-contamination can be removed by the Yamaguchi and Houk spin-projection procedure. However, either the spin-contaminated or the spin-projected energies shows that the stepwise TS-meso is lower in energy than the concerted TS-endo-2, in contradiction to the experimental activation energies 𝐸 ! in Table . In order to compare the computed barriers of the stepwise pathways with the concerted pathways at a consistent level of theory, we resorted to NEVPT2 calculations. These NEVPT2 calculations are based on CASSCF (8,8) wavefunctions and, in addition to the fully-accounted dynamic electron correlation within the active space, the dynamic electron correlation outside of the active space is recovered through the 2 nd order perturbation theory. As shown in Table , TSmeso was computed to be 1 kcal/mol higher in enthalpy than TS-endo-2 at the NEVPT2(8,8)/def2-TZVP//(U)ωB97XD/def2-TZVP level, consistent with the experimental 𝐸 ! . However, the 3-5 kcal/mol difference between TS-meso and TS-rac at ωB97XD and NEVPT2 levels is larger than the experimental 𝐸 ! , though this large difference is consistent with previous calculations. The [4+2] cycloadducts endo-2 and exo-2 may be formed by either a concerted or stepwise mechanism as shown in Figure . The NEVPT2 calculations in Table suggest that these two [4+2]-cycloadducts are formed by different mechanisms. For endo-2, the concerted mechanism is computed to be enthalpically more favorable whereas for exo-2, the stepwise mechanism is more favorable. |
61310da7fc08e3e0548929f2 | 17 | • The above calculations are in good agreement with the previous calculations by Ess et al. In addition, we explored the plausible [4+2]-ene mechanisms that were not considered before. The large computed barriers (> 31 kcal/mol at ωB97XD/def2-TZVP level of theory) are consistent with the fact that the corresponding [4+2]-ene products 5 and 6 were not observed experimentally. and the exo cycloaddition is given in the SI. The profiles at 0 GPa are from the intrinsic reaction coordinate calculations; the highpressure profiles are computed using the XP-PCM method based on the 0 GPa structures. |
61310da7fc08e3e0548929f2 | 18 | A first look shows that the four types of reaction have very different profiles, for example, in the location of the TS structure along the reaction coordinate and in the shape of the ∆𝐺 232 and volume profiles. They do share a common feature-the ∆𝐺 232 profile decreases as the pressure increases, which is consistent with the expected negative activation volumes of these dimerizations. |
61310da7fc08e3e0548929f2 | 19 | TS at about 2/3 of the entire reaction course. The reaction profile is similar to other [4+2] cycloadditions previously studied using the same XP-PCM method. In the current endo-cycloaddition, the first 1/4 of the PES is rather flat, where two isolated 1,3cyclohexadiene molecules reorient themselves in a proper geometry for the cycloaddition and begin to approach one another; the latter motion would lead to a decrease of the separation between the molecules. At high pressure, a minimum of a vdW complex develops prior to the TS, as noted previously by us and by Loco et. al. The minimum is very shallow at low pressures (may be difficult to see in the figure, but it is there) but becomes apparent at 5.7 GPa. This minimum also shifts towards the TS as the pressure increases. The emergence of a pre-TS minimum and the shift of it towards the TS could be explained by the pressure-enhanced vdW complex formation. The higher the pressure, the shorter the vdW separation between the molecules, and the smaller the volume of such complex. The favorable pV term of the enthalpy leads to the appearance of a minimum for the complex. This phenomenon of a pre-TS minimum seems to be common in bimolecular reactions; the other three reactions in Figure The vdW cavity of TS-threo-4 is 142 cm 3 /mol, compared with 145 cm 3 /mol for TS-endo-2. |
61310da7fc08e3e0548929f2 | 20 | Note that 1 cm 3 /mol difference in volume corresponds to 0.24 kcal/mol difference in enthalpy at 1 GPa. The greater overlap of the rings and the highly-ordered staggered conformation in TS-threo-4 is correlated with the small pre-exponential factor (i.e., small entropy of the TS) in the Arrhenius equation for the threo [6+4]-ene reaction (as discussed in the 1 atm PES section), compared with other reactions. |
61310da7fc08e3e0548929f2 | 21 | Stepwise addition 2.00 Å late TS. The cavity volume of the system decreases from the beginning of the reaction to about the point at which the H-transfer completes (i.e., a bit pass the TS). Then the cavity increases, as the ene-adduct opens up to adapts to its optimal conformation at 0 GPa. At high pressures, a pre-TS vdW complex minimum develops, similar to the case of the [4+2] cycloaddition. But this minimum in the [6+4]-ene reaction is much deeper compared with that in the [4+2] cycloaddition. A most likely reason is that the separated reactant molecules occupy a larger volume (164 cm 3 /mol) than in the [4+2] cycloaddition (158 cm 3 /mol), so that a greater volume reduction is possible in forming the vdW complex. |
61310da7fc08e3e0548929f2 | 22 | The flat TS region of the PES at 0 GPa becomes tilted with the post-TS region lowering in enthalpy at high pressures, due to the monotonical volume decrease in this region. A clear shift of the TS towards the reactant is seen at high-pressures. The shift of TS at high pressure is consistent with the Hammond postulate 74 that the TS shifts towards the reactant when the reaction becomes more exothermic. However, note that the TS shift is apparent only in the [6+4]-ene reaction, not so in the other three reactions in Figure , though all these reactions become more exothermic (or less endothermic) at high pressures. One distinct feature of the [6+4]-ene reaction is that the cavity volume decreases in the TS region, while in the other three reactions, the cavity volume stays almost constant in the TS region. Therefore, the effect of pressure is larger on the PES in the TS region in the [6+4]-ene reaction, leading to an apparent shift of the TS towards the reactant. The shift of TS along reaction coordinate under pressure has been noted previousely. Concerted [4+2]-ene reaction. In contrast to the flat TS region in the [6+4]-ene reaction, the [4+2]-ene reaction has a sharp maximum in the TS region (Figure ). We inspected the TS structures in the two reactions to understand the reason. In the [6+4]-ene TS-threo-4, the C-C bond is almost fully formed at a distance of 1.68 Å while the H is barely transferred with distances of 1.20 and 1.90 Å for the breaking and forming C-H bonds (Figure ). On the contrary, in the [4+2]-ene TS-6, the H-transfer is substantial with 1.50 and 1.28 Å for the two C-H bonds but the C-C bond to be formed is at a large distance of Stepwise addition. Moving on to the stepwise mechanism, Figure shows the reaction profiles of the meso addition. Compared with the above-discussed concerted reactions, this meso addition has a rather short course, especially in the post-TS region. The C-C distance of the formed bond changes from 2.00 Å in TS-meso to 1.57 Å in the adduct meso-8. The cavity volume of the system decreases in the meso-addition, but with a much less volume reduction compared with the above concerted reactions where two bonds are formed. Consequently, pressure provides much less reduction of the reaction barrier for the meso addition. The meso-adduct meso-8 is calculated to be higher in enthalpy than the reactants at pressures lower than 6.2 GPa. |
61310da7fc08e3e0548929f2 | 23 | In addition to activation energy, activation volume also provides information about the transition state, arguably, in a more tangible manner. The volume/size of a TS structure is directly related to its geometry. While competing mechanisms of a reaction can sometimes be difficult to distinguish based on activation energies, they may be distinguishable by activation volumes when the competing TS structures have very different geometries. This is especially useful in studying the stepwise vs concerted mechanisms of cycloadditions, given that the stepwise mechanism often has a less negative activation volume. In order to calculate the activation volume of a reaction, the barriers computed by XP-PCM are plotted against the pressure (Figure ). The ∆𝐺 232 ‡ (𝑝) data shows a nice linear relationship. The slope of the fitted line gives the activation volume, according to eq. ( ) . |
61310da7fc08e3e0548929f2 | 24 | Note that the free energy ∆𝐺 232 ‡ (𝑝) does not contain zero-point vibrational correction and thermal correction, assuming their negligible contributions to ∆𝑉 ‡ . The components of ∆𝐺 ‡ are the electronic energy and cavitation energy computed by XP-PCM (eq. 1). Note that for the ∆𝑉 ‡ calculations, ∆𝐺 232 ‡ at different pressures are computed using as the reference isolated reactant molecules, not the vdW complexes (pre-TS minima) in Figure . This is to ensure a common reference (isolated reactants) in the ∆𝑉 ‡ comparison for different mechanisms. |
61310da7fc08e3e0548929f2 | 25 | This work is part of our continuing efforts to apply a new quantum chemical method, the extreme pressure polarizable continuum model (XP-PCM), to the study of reaction energy profiles under pressure, and to calculate activation volumes. The method has been tutorially reviewed and applied by the present authors and Hoffmann 17 in 2017 to a set of pericyclic reactions where interesting phenomena on reaction energy profiles under pressure were revealed by calculations, such as the shift of the transition state, changes in the rate-determining step, etc. In addition, a reasonable agreement of the computed activation volumes with the experimental values was discovered. The evolution of the van der Waals volume of the reactive system upon proceeding from the reactants to products, as a chemical relevant diagnostic to analyze the effect of the pressure on the reaction energy profile, was emphasized. |
61310da7fc08e3e0548929f2 | 26 | In this work, the XP-PCM computational method is applied to the study of the pressure effects on the competing mechanisms of the thermal dimerization of the 1,3-cyclohexadiene, providing, for the first time, accurate calculations of activation volumes, compared with the experimental ones. The activation volumes are of primary relevance for the determination of the competing mechanisms in this reaction. |
61310da7fc08e3e0548929f2 | 27 | As in the case of dimerization of the 1,3-cyclohexadiene the experimental activation volumes (and not the activation energies) have been the primary basis for the heuristic suggestion of the mechanisms in operation for this reaction, we may consider the agreement between the computed and experimental activation volumes a more direct validation of the suggested mechanisms. |
63f4de869da0bc6b334ea018 | 0 | Asphaltenes are made of polycyclic aromatic hydrocarbon rings with peripheral alkane chains and are among the most surface-active and polarizable components of crude oil. They are characterized as a solubility class that is insoluble in n-heptane but soluble in aromatics such as toluene. Molecular structure analysis using atomic force and scanning tunneling microscopy methods show poly-disperse asphaltenes with a variety of molecular structures . Following the widely accepted Yen-Mullins model for asphaltenes, the average molecular weight of a molecule is around 750 g/mol and the asphaltenes can exist as a monomer, nano aggregate or clusters based on their concentration in oil . |
63f4de869da0bc6b334ea018 | 1 | Asphaltene adsorption at an oil-water interface is known to increase the stability of crude oilwater emulsions, impacting the oil-water separation process and consequently affecting oil production . The interfacial interaction of asphaltene-rich crude oil and reservoir rock is also known to influence reservoir wettability and oil recovery process. Other research on kerogen, which is an organic matter similar to asphaltene, showed that kerogen can make the reservoir oil-wet or water-wet depending on its composition (Jagadisan and Heidari 2018a; Jagadisan and Heidari 2018b; Jagadisan and Heidari 2019; Jagadisan and Heidari 2020; Jagadisan and Heidari 2022). In addition, asphaltene precipitation and deposition during different stages of the oil production process can result in reduced reservoir recovery and increasing costs associated with downtime in production facilities. Therefore, understanding the interfacial properties of asphaltenes is important for efficient management of petroleum production. |
63f4de869da0bc6b334ea018 | 2 | There are several studies focused on understanding the physicochemical effects of asphaltenes at interfaces. Even with a wealth of research on this topic, considerable debate on how asphaltenes adsorb at surfaces still exists. The source of asphaltenes in these studies are different leading to variability in composition and structure in the asphaltenes used. A recent study demonstrated the molecular structure of asphaltenes, which is dependent on their origin, impacts the interfacial behavior of asphaltenes . Additionally, extracted asphaltenes are resolubilized in model solvents in many studies, the results of which are compared with studies performed original crude oil with high viscosities. |
63f4de869da0bc6b334ea018 | 3 | Here I discuss recent advances on surface science of asphaltenes focusing on phenomena related to oil-water interfaces to better understand the mechanism of asphaltene adsorption and emulsion stability. This review summarizes the results obtained from different evaluation techniques adopted by researchers, including indirect and direct experimental approaches, and simulation approaches. I attempt to explain the potential reasons for discrepancies in results, which could serve as a guide to adopting appropriate strategies and control variables for pursuing further investigations on this topic. |
63f4de869da0bc6b334ea018 | 4 | The dynamic interfacial tension of a water-model oil interface decreases with adsorption of asphaltenes . Initially the interfacial tension decreases rapidly and varies linearly with the square root of time, indicating diffusion-controlled adsorption . Asphaltene adsorption onto stainless-steel surface measured by a quartz crystal microbalance (QCM) technique showed that at initial times the adsorbed mass of asphaltenes on the stainless-steel surface varies linearly with the square root of time, implying a diffusion-controlled process . This is consistent with dynamic interfacial tension of asphaltenes at the oil-water interface ). |
63f4de869da0bc6b334ea018 | 5 | The shift from diffusion-controlled adsorption to an adsorption barrier regime, brought on by the steric hindrance effect between already adsorbed molecules particles, occurs for low solubility asphaltene solutions (asphaltene solutions in mixtures of toluene and aliphatic base oil). Surface coverage asymptotically approaches the 2-D packing limit of polydisperse disks, and the equilibrium interfacial tension is independent of asphaltene concentration in low solubility aliphatic oil. . In contrast, the interfacial tension for high solubility asphaltene solutions (asphaltene in toluene) depends on the concentration of asphaltenes in the solution, and the equilibrium surface coverage can be calculated using the Ward-Tordai equation . Low surface pressures and higher critical nano-aggregate concentration (CNAC) are obtained in high solubility asphaltenes for the same quantity of asphaltene, which is another impact of solubility . According to , equilibrium interfacial tension is achieved more quickly in aromatic solvents and more slowly in aliphatic solvents. Less asphaltene is needed to stabilize the emulsion in the aliphatic solvent . However, at lower concentrations, regardless of asphaltene solubility, emulsion stability is observed at a surface concentration of ~4 mg/m2 . At higher bulk asphaltene concentrations, surface coverage increases with increase in asphaltene concentration . Table shows a list of surface coverages obtained at different asphaltene concentrations. ). Rane and coworkers, however, find that transition to 'non-diffusion'-controlled kinetics is not in agreement with the conformational relaxation based on observations of transition time and the deviation from the diffusion-controlled regime . A later study reported that the interfacial tension and the dilational rheology data always follow the main features of the diffusion control regime, and the deviations could be explained by the poly-disperse nature of asphaltenes and distribution in adsorption coefficients within the asphaltenes class . A ternary mixture model for asphaltenes could explain most of the experimental results for dynamic interfacial tension and dilatational rheology at different times . |
63f4de869da0bc6b334ea018 | 6 | Particle-laden interfaces between two immiscible liquids can be studied by deformations in the dilatational mode . Surface pressure rises as the interfacial layer is compressed due to an increase in surface concentration. Using pendant droplet apparatus as a Langmuir trough, droplet aged in low solubility asphaltene solution were rapidly expanded to the point of droplet detachment, and interfacial tension measured as a function of interfacial area. |
63f4de869da0bc6b334ea018 | 7 | Figure shows the relative coverage against interfacial tension for two base oils at different asphaltene concentrations and different aging times. The results show that for asphaltene adsorption results could be explained by a unique equation of state. Figure shows that the relationship between interfacial tension and surface coverage could be interpreted by a Langmuir equation of state (EOS) expressed as, |
63f4de869da0bc6b334ea018 | 8 | The fitting parameter for Langmuir EOS Γ ∞ : surface excess coverage, was found to be 3.2 molecules/nm 2 corresponding to a cross-section area of 0.3125 nm 2 , which is in agreement with the reported range by Chaverot and coworkers . These results are consistent with the Yen-Mullins model (Mullins 2010, ) and corresponds to the average size of polyaromatic hydrocarbon core in asphaltenes (6-8 rings) and to their conformation at the interface with aromatic cores lying flat at the interface and alkyl chains tending perpendicular into the oil phase. Asphaltene adsorption studies on solid surfaces also indicate similar results, that absorption occurs face-on on hydroxyl substrates in order to maximize the polar-polar interactions between the hydroxyl groups and aromatic center of asphaltenes . Copyright (2013) American Chemical Society.) |
63f4de869da0bc6b334ea018 | 9 | The applicability of the Langmuir EOS is also established for petroleum derived asphaltenes extracted from Kuwaiti UG8, coal-derived, and synthetic asphaltenes (Figure ). Analysis of surface excess coverage for petroleum-derived, coal -derived and synthetic asphaltenes yielded approximately 6 ring core, 4 ring core and a 13-ring core, respectively. The estimates of mean number of rings in aromatic core from interfacial measurements are consistent with the Asphaltene adsorption at oil-water interface is found to be primarily of asphaltene monomers and suggest the lack of interfacial activity of nano-aggregate as: (a) below the CNAC, the slope of interfacial tension decay evolves similarly to asphaltene monomer concentration from nuclear magnetic resonance (NMR). (b) The values of the slopes are consistent with those for adsorption of monomers with higher diffusivities compared to that of nanoaggregates ). These differences in views and observations of the structure of the adsorbed layers still remain to be reconciled. |
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