Density Functional Theory Calculations of H/D Isotope Effects on Polymer Electrolyte Membrane Fuel Cell Operations

3.1 Model Hydrogen Species in the Electrode Catalyst Phase and their RPFRs

The optimised structures of Pt₁₀–H⁺ and Pt₁₀–H are shown in Figure 2a–d and their computed physical parameters are summarised in Table 1.

Figure 2:

Optimised structures of Pt₁₀–H and Pt₁₀–H⁺. (a) Pt₁₀–H with H absorbed on the central Pt atom, (b) Pt₁₀-H with H absorbed on a peripheral Pt atom, (c) Pt₁₀-H⁺ with H⁺ absorbed on a center Pt atom, and (d) Pt₁₀–H⁺ with H⁺ absorbed on a peripheral Pt atom. No significance is attached to the relative sizes of the spheres.

Two types of optimised structures were obtained. In one structure, an H⁺ ion (or an H atom) is bonded to the central Pt atom of the Pt₁₀ cluster surface, as shown in Figure 2a or c. In the other structure, the H⁺ (H) is bonded to a peripheral Pt atom of the Pt surface, as shown in Figure 2b or d. Even in the cases where the H⁺ (H) was positioned above the midpoint of two adjacent Pt atoms in the Gaussian input, the final converged structure was either of the two structures. We may thus conclude that the optimised structures shown in Figure 2 are the most stable ones for the Pt₁₀–H⁺ and Pt₁₀–H systems and that the H⁺ (H) directly interacts with one of the Pt atoms and does not form bonds of similar strength with multiple Pt atoms at a time.

The values of the Mulliken charge on the H⁺ ion (and a H atom) are listed in the seventh column of Table 1. The value of the Mulliken charge on the H⁺ ion is larger than that on the corresponding H atom, simply because the total charge of the Pt₁₀–H⁺ is set to 1.0 whereas it is zero for Pt₁₀–H. It is also seen that the values of Mulliken charge for the H⁺ ion (the H atom) bonded to the central Pt atom of the cluster are larger than that bonded to the peripheral Pt atom of the cluster. In addition, the SM has little influence on the Mulliken charge of H⁺ (H).

The experimental Pt-H distance of a PtH molecule is reported to be 1.529 Ǻ [7]. The computed Pt–H distance of the PtH molecule at the UB3LYP/LanL2dz level of theory is 1.543 Ǻ, which agrees with the experimental one within the 1 % error. The present Pt–H⁺ and Pt-H bond distances for the Pt₁₀–H⁺ and Pt₁₀–H systems range from 1.559 to 1.568 Ǻ and are slightly larger than that of the PtH molecule. This difference may be attributable to the difference in the number of Pt atoms involved in the two systems. It is also seen in Table 1 that The Pt–H⁺ and Pt–H bond distances are independent of the value of the SM and the kind of H species, H⁺ or H.

The experimental harmonic wave number of the PtH molecule is reported to be 2295 cm⁻¹ [7]. On the other hand, the normal mode analysis of the same molecule at the UB3LYP/LanL2dz level of theory yields 2288 cm⁻¹, which is not very different from 2295 cm⁻¹. In normal mode analyses of the Pt₁₀–H⁺ and Pt₁₀–H species, 27 wave numbers are obtained for all of the considered species, and among them, three wave numbers are real and the other 24 are zero as expected. The H/D RPFRs at 25 °C and 39 °C were calculated by using six real wave numbers calculated for the pairs of the H and D species and (6), and the results are summarised in the ninth and tenth columns of Table 1. The values of RPFRs at 25 °C for the Pt₁₀–H⁺ and Pt₁₀-H species range from 3.8764 to 4.3056 and from 3.7158 to 4.8707, respectively. For both Pt₁₀–H⁺ and Pt₁₀–H, the RPFR value for the H species bonded to the central Pt atom is larger than that bonded to a peripheral Pt atom. The SM has little effect on the RPFR value except for the cases of entries 1 and 2, for which the SM value of 4 gives a larger RPFR value than the SM value of 2.

The value of RPFR in general becomes larger for a more stiffly bonded atom [2], and the stiffness of the bond may primarily be reflected in the Pt–H bond distance in the present case. That is, the H species with a shorter H–Pt bond distance is expected to have a larger RPFR value. This general view seems to be directly applied to the Pt₁₀–H species. For the given SM value, the H atom bonded to the central Pt has a larger RPFR value that that bonded to the peripheral Pt atom, because the former has a shorter Pt–H bond distance than the latter. The general view, however, is not applied to the Pt₁₀–H⁺. A shorter bond distance does not necessarily resulted in a larger RPFR value for a given SM value. For the Pt₁₀–H⁺, the number of second closest Pt atoms may be important. That number is six for the H⁺ bonded to the central Pt atom, while it is only two for the H⁺ bonded to the peripheral Pt atom, which may account for the larger RPFR value of the former H⁺ than that of the latter H⁺.

The value of the H/D RPFR at 25 °C for the Pt–H⁺ in which surfaces of the Pt catalyst were approximated by a single Pt atom was calculated to be 3.6381 in our previous study [1]. This value is slightly raised to 3.6505 if we set the mass of the Pt atom at 10¹⁶ as in the case of the present study. This enhancement in the RPFR value with increasing Pt mass is reasonable, since only the H⁺ ion moves in the Pt-H stretching and other vibrational modes when the Pt atoms are given an artificially large mass. The present model of the H⁺ on the Pt catalyst surface, Pt₁₀-H⁺, yielded the RPFR value of 3.8764 ∼ 4.3056 at 25 °C, larger than the RPFR value for the Pt–H⁺. This difference in the RPFR value between the present and previous models may be ascribable to the fact that, while only the Pt–H stretching mode contributes to the H/D RPFR in the previous model, the Pt-Pt-H bending modes also contribute to the RPFR in addition to the stretching mode.

To check the basis set dependence of the value of the H/D RPFR, the structures of the Pt₁₀–H⁺ species were optimised and their RPFR values at 25 °C were calculated using the frequencies obtained at the optimised structures at the UB3LYP/cep-4g level of theory. The value computed was 4.2394, 4.2786, 3.9045 and 3.8937 for entries 5, 6, 7, and 8 of Table 1. The mean difference in the RPFR value between UB3LYP/LanL2dz and UB3LYP/cep-4g was 1.6 %.

3.2 Model Hydrogen Species in the Electrolyte Membrane Phase and Their RPFRs

The optimised structure of the model of the electrolyte membrane phase, C₅F₁₁-SO₃⁻···H₃O⁺ (H₂O)₇, is depicted in Figure 3, in which the H₃O⁺ ion is circled with the dotted line, and some structural parameters are summarised in Table 2. In the optimised structure, one of the three Hs of H₃O⁺ interacts with one of the O atoms of the −SO₃⁻ group. That H atom is indicated by an arrow (→) in Figure 3. The H₃O⁺ part of H₃O⁺(H₂O)₇ can be distinguished from the H₂O parts since, while H₂O has only two O–H covalent bonds with the bond distance of ca 1 Ǻ, H₃O⁺ has three O–H covalent bonds. However, it is impossible to specify which H among the three Hs of H₃O⁺ is actually H⁺.

Figure 3:

Optimised structure of C₅F₁₁-SO₃⁻···H₃O⁺(H₂O)₇ as the model hydrogen species in electrolyte membrane phase. The H₃O⁺ ion is shown in the dotted circle. The H with an arrow directly interacts with an O of the −SO₃⁻ group. No significance is attached to the relative sizes of the spheres.

In the present calculations, the O–H bond distance of H₃O⁺ range from 0.995 to 1.043 Ǻ, and that of H₂O range from 0.971 to 1.017 Ǻ. The O···H intermolecular bond distance, i.e., the hydrogen bond (HB) distance among the H₃O⁺ ion and the seven H₂O molecules are from 1.313 to 1.635 Ǻ. The O···H bond distance between the H of H₃O⁺ and the O of −SO₃⁻ is 1.687 Ǻ.

The values of the Mulliken charge for the H atoms of H₂O molecules surrounding the H₃O⁺ ion and the −SO₃⁻ group range from 0.3708 to 0.4852, and a tendency is observed that the H atom interacting with the O atom of a neighbouring H₂O molecule has a larger Mulliken charge value than those with no such interaction. Against our surmise, a tendency that the three H atoms of H₃O⁺ have larger Mulliken charge values than the Hs of the H₂O molecules is not observed. The Mulliken charge value of the H atom of H₃O⁺ interacting with the O atom of the −SO₃⁻ group is 0.4476 and smaller than that of the other two Hs of H₃O⁺, 0.4885 and 0.4899.

For six H atoms from seventeen H atoms of C₅F₁₁-SO₃⁻···H₃O⁺ (H₂O)₇, the H/D RPFR values at 25 °C and 39 °C were calculated and results are listed in the 5^th and 6^th columns of Table 2. The selected six H atoms are categorised into three types; 1) those directly interacting with the O atom of the neighbouring H₂O molecule, 2) those directly interacting with an O atom of the −SO₃⁻ group, and 3) those without interaction, as shown in the 3^rd column of Table 2. Among the RPFR values computed at 25 °C, the largest value of 15.5758 is obtained for the H atom (the one marked with an arrow in Fig. 3) of H₃O⁺ interacting with an O atom of the −SO₃⁻ group. The result indicates that this H atom strongly interacts with the O atom, though the value of the Mulliken charge on this H atom is not very specific compared with those of other H atoms.

It is observed that the H atom with a HB has a larger RPFR value that the one without a HB. That is, hydrogen bonding enhances the RPFR value of a H atom of a H₂O in an aqueous system, which is consistent with the results of our previous study, in which H/D RPFRs are estimated for water clusters [8, 9].

The smallest value of RPFR at 25 °C, 12.7422, is observed for an H atom of H₃O⁺ (entry 1 of Tab. 2) with a HB whose distance is 1.444 Ǻ, which is within the range of computed hydrogen bond distance of 1.313–1.635 Ǻ. This value is smaller than those of H atoms with no HB, which is against the general view above that hydrogen bonding enhances the RPFR value of a H atom of a H₂O molecule. The reason may be ascribable to the lengthened O-H covalent bond of 1.043 Ǻ caused by the formation of the HB.

As we reported in the previous article [1], the value of the H/D RPFR for the isolated H₂O molecule or a H₂O molecule in vacuo, is 12.2181 at 25 °C, which is smaller than 13.4098–14.6566 obtained for H₂O molecules with no direct interaction in the structure of C₅F₁₁-SO₃⁻···H₃O⁺ (H₂O)₇. This indicates that the RPFR of an H atom with no direct interaction is slightly enhanced by the presence of H₂O molecules and the −SO₃⁻ group around the H₂O molecules the H atom belongs to, possibly through the O-H covalent bond. As for the H₃O⁺ ion, the value of the RPFR for the H₃O⁺ ion in vacuo is 14.4640 [1], which is in between the values obtained for the interacting Hs of H₃O⁺ in C₅F₁₁-SO₃⁻···H₃O⁺ (H₂O)₇ in the present study.

3.3 H/D Isotope Exchange Equilibria at 39 °C between the Electrode Catalyst Phase and the Electrolyte Membrane Phase

During FC operations, H (and D) atoms trapped on surfaces of the Pt catalyst are expected to be oxidised to H⁺ (D⁺) ions and transferred to the electrolyte membrane phase. This means that the chemical form of hydrogen on the catalyst surface in equilibrium with hydrogen in the electrolyte membrane phase is the H⁺ (D⁺) ion. Thus, we regard the average of the four RPFR values at 39 °C evaluated for the Pt₁₀-H⁺ species (entries 5, 6, 7, and 8 in Tab. 1) as the RPFR value for the H species in the electrode catalyst phase. The value is 3.7503 at 39 °C.

All six hydrogen species shown in Table 2 can be candidates for H species expected to exist in the electrolyte membrane phase during FC operations. Therefore, we consider 12.1378, the simple arithmetic average of six RPFR values at 39 °C shown in Table 2 as the RPFR value for the H species in the electrolyte membrane phase.

The equilibrium constant of Reaction (2), K₂, is given as

(9)K2 = felec/fcat, (9)

where the values of f_cat and f_elec are 3.7503 and 12.1378, respectively, as estimated. So, K₂ at 39 °C is calculated as 3.2365. The value of K₂ is larger than unity, which means that the heavier isotope D is preferentially fractionated into the electrolyte membrane phase. Moreover, it is close to the value of the experimentally obtained S, which ranged from 3.46 to 3.99 at 39 °C [1].

In our previous study [1], the model H species for the electrode catalyst phase and the electrolyte membrane phase were the isolated Pt-H⁺ ion, and the isolated H₂O molecule and the isolated H₃O⁺ ion, respectively, the simplest ones among conceivable models. RPFR values at 39 °C for Pt-H⁺, H₂O and H₃O⁺ were calculated as 3.3814, 10.7055 and 12.5265 respectively. Using these values, K₃ of (7) and K₄ of (8) at 39 °C are 3.1660 and 3.7045, respectively. The present K₂ value of 3.2365 is in between those two values. This means that the value of equilibrium constant of Reaction (2) little depends on the kind of adopted models. In other words, the effect of interactions like hydrogen bonding on the H/D RPFR is very localised; the interaction only affects the RPFR of the H species that is directly involved in that interaction and has little influence on the RPFRs of neighbouring H species. This result is consistent with that obtained for the H/D RPFRs of water clusters [8].

Two ways to check the validity of the aforementioned considerations are conceivable. One is to compare the calculated wave numbers with experimentally obtained vibration data. To obtain vibrational data experimentally, however, seems difficult even with the state-of-the-art equipment. Another is to carry out calculations using systems that are bigger and closer to real systems than the systems employed in this study. Those systems should include more than one hydrogen atoms and/or hydrogen ions adsorbed on the Pt catalyst surface. We have already started the study based on the second approach.