Proton Transport Barrier

Alignment

Angle between N N axes

Angle between plane

nitrogen axis in plane axis Rotation Axes:

Molecule Axes:

nitrogen axis methyl axis

Figure 3.4: Schematic figure: definition of the imidazole molecule axis.

The approximated potential energy barrier for the proton transport between two pro-ton conducting species is obtained from re-strained energy minimisations in a system of either two methyl imidazole molecules, two deprotonated methyl sulphonic acid molecules or two water molecules. Each sys-tem contains an additional proton, which leads to a total of (-1) electronic charge in the case of the sulphonic acid system and a total of (+1) electronic charge for the other systems.

Computational Details

Besides the choice of proton conducting groups, a further parameter characterising the system is the distance between the groups, which is restrained by position re-straints on either one nitrogen atom or one oxygen atom of each molecule, respectively.

To avoid rotations of methyl imidazole and methyl sulphonic acid, the carbon atom of the methyl groups is restrained in addition. For each system with fixed distance of the pro-ton conducting groups, the energy characteristics along the path of the propro-ton between the groups is scanned stepwise by restrained energy minimisations, keeping not only the distance between the groups fixed, but also the position of the proton. The difference between energy

minimum and energy maximum, i.e. the proton transport barrier, is given as a function of the distance between the species for each system.

In the case of methyl imidazole molecules, the influence of the angle between the molecule axes has been analysed. The axes are defined in Fig. 3.4, where a schematic view of two adjacent molecules from three different perspectives is shown with the blue circles marking the nitrogen positions. Two orthogonal molecule axes are defined: the light blue axis is called nitrogen axis, as it is defined by the nitrogen position of each imidazole molecule, and the light red axis is called methyl axis, as it is defined by the bond between the methyl group and the imidazole ring. The following arrangements between the two imidazole molecules are considered: both molecular axes parallel (0/0); 90^◦ angle or 180^◦ angle between the methyl axes with parallel nitrogen axes, noted (90/0) and (180/0); parallel methyl axes with 45^◦ angle or 90^◦ angle between the nitrogen axes, noted (0/45) and (0/90).In order to keep the orientation fixed during the minimisation, all nitrogen positions were fixed.

Results

The Fig. 3.5 shows the potential energy barrier for the proton transport. The barrier for imi-dazole obtained by DFTB is underestimated by about 2 kcal/mol compared to DFT(B3LYP functional and basis function set: cc-pVZQ for water and 6-31g for sulphonic acid and imi-dazole). Proton transport in sulphonic acid systems and in water compares even better with the ab initio values. The energy barrier for all species increases with the distance between the groups, reaching 10 kcal/mol at about 0.30 nm for the imidazole system, at a shorter distance than 0.29 nm for water and at about 0.28 nm for sulphonic acid. For each proton conducting species, a distance exists where the barrier vanishes: for imidazole at a nitrogen distance of about 0.27 nm, for sulphonic acid at an oxygen distance of about 0.25 nm, for water at an oxygen distance of about 0.26 nm. In water systems a positive charged complex of two water molecules bound via an additional interstitial hydrogen atom is called Zundel ion. Following this, the biatomic protonated configurations of imidazole and sulphonic acid are also called Zundel like configuration.

Zundel like imidazole complexes are already reported in the literature^[75][72]. The inter-molecular distance between nitrogen atoms is about 0.28 nm. The complex is chacterised by an elongation of the distance between the interstitial/intermediate hydrogen and the nitrogen atoms. The formation energy gain of a Zundel like imidazole complex is about 17 kcal/molⁱ, while for water it is only 8 kcal/mol. For sulphonic acid two protonation states are consid-ered, a dimer with a proton hole and a dimer with an excess proton. The gain is in both cases about 24 kcal/mol. The Zundel like aggregation of two protonated imidazole molecules involves the alignment of all nitrogen atoms and their bonded hydrogen atoms, as well as mirror symmetry at the plane orthogonal to the nitrogen axes through the interstitial hydro-gen. The hydrogen is delocalised between two imidazole molecules. The angle between the imidazole molecules formed by the nitrogen atoms influences significantly the energy barrier for proton transport. The alignment of the axes favours proton transport, as reported in Fig. 3.5.

In Fig. 3.6, the minimal and maximal potential energy along the proton transport path is plotted. The abscissa is a measure for the fixed nitrogen distance and equals the distance between the position of the proton at the minimum and maximum of the energy barrier.

Different arrangements of imidazole groups are compared. The alignment of imidazole means

iA Zundel like imidazole complex is compared with an isolated neutral imidazole molecule and an isolated protonated imidazole molecule.

3.1. PROTON CONDUCTING SPECIES - VALIDATION 47

energy barrier [kcal/mol]

distance (N−N/O−O) [nm]

water imidazole sulphonic acid line − DFTB symbol − DFT

0 2 4 6 8 10 12 14 16

0.24 0.25 0.26 0.27 0.28 0.29

Figure 3.5: Potential energy barrier for proton transport between two proton conducting species:

water - red; methyl imidazole - blue; methyl sulphonic acid - green; straight lines derived from DFTB calculation, symbols from DFT calculation with B3LYP functional. The lines serve as guidance to the eye.

max./min. energy [kcal/mol]

N−N axis rotation

(180/0)

(0/90) (0/45) (0/0) (90/0) in plane axis

rotation

distance minimum to maximum [nm]

−30

−20

−10 0 10 20 30

0.0 0.5 1.0

Figure 3.6: The maximum and minimum potential energy for proton transport between two methyl imidazole molecules, depending on the alignment of the two rings. The angle between the nitrogen axes of both molecules and between the methyl axes, see above: red both molecular axis aligned; green -angle between methyl axis; blue - -angle between N-N axis. The lines serve as guidance to the eye.

that both molecular axes are parallel, i.e. the nitrogen axis and the methyl axis as introduced above. The rotation of the ring around the axes defined by the nitrogen atoms leads to an increase in total energy and transport barrier, as the formation of a Zundel like symmetric protonated complex is suppressed. Energy minimisation was performed under constraints of the nitrogen position. The aligned configuration and the configuration with an angle of 90^◦ between the methyl axes show the lowest total energy. The slight increase of total energy for small distances in the aligned configuration is caused by the restraints, as the system is not allowed to relax freely and is hindered from forming the Zundel like configuration.