J. Phys. Chem. C 2008, 112, 14209–14215
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Ab Initio Calculations of the Condensation of Phosphonic Acid and Methylphosphonic Acid: Chemical Properties of Potential Electrolyte Materials for Fuel Cell Applications Berit Heggen,* Sudip Roy,§ and Florian Mu¨ller-Plathe‡ Eduard-Zintl Institut fu¨r Anorganische und Physikalische Chemie, Technische UniVersita¨t Darmstadt, Petersenstrasse 20, 64287 Darmstadt, Germany ReceiVed: April 24, 2008; ReVised Manuscript ReceiVed: June 22, 2008
Organic and polymeric phosphonic acid derivatives are promising candidates as proton-conducting electrolytes in high-temperature (>100 °C) polymer electrolyte membrane fuel cells (PEM-FC). The proton conduction is adversely affected by the tendency of the amphoteric phosphonic acid groups to condensate (Steininger, H.; Schuster, M.; Kreuer, K. D.; Kaltbeitzel, A.; Bingo¨l, B.; Meyer, W. H.; Schauff, S.; Brunklaus, G.; Maier, S.; Spiess, H. W. Phys. Chem. Chem. Phys. 2007, 9, 1764) to dimers, trimers, or higher polycondensates. Ab initio calculations have been performed to obtain the geometry of phosphonic acid (phosphorus acid), methylphosphonic acid, and their condensation products. The reaction energies for the formation of the dimer, the trimer, and the cyclic trimer have been calculated, including various corrections due to zero-point vibrations, entropy, and solvation. The results show that the formation of dimers and trimers is energetically possible both for unsubstituted phosphonic acid and for organic phosphonic acids, whereas the formation of the cyclic trimer is unfavorable because of the ring strain. Similarly, a direct mechanism for the dimerization of two molecules of phosphonic acid or methylphosphonic acid can be ruled out because of the high transition-state energies. In the condensed liquid, the condensation step proceeds most likely with the help of other protondonating or proton-accepting phosphonic acid moieties. I. Introduction Polymer electrolyte membrane (PEM) fuel cells are promising mobile energy sources.1 A key element in these fuel cells is a proton-conducting polymer membrane whose proton conductivity must be high to reduce the inner electrical resistance of the cell. At the same time, transport of other compounds (gases, water, solvent) should be minimized. The standard membrane material is Nafion, a proton conducting perfluorosulfonic acid.2,3 The proton transport mechanism of this material is based on the presence of a water phase, and the operating temperature is limited to a maximum of 100 °C. Therefore, numerous attempts have been made to replace water by proton-carrying amphoteric heterocycles like imidazole, which have a higher boiling point.4 It has also been suggested to test new materials that combine the effects of a protogenic group and proton solvatization.5 Polymers like polystyrene were functionalized with amphoters such as imidazole or phosphoric acid groups. Phosphoric acid has already been known for fuel cell applications, since one type of fuel cell uses neat phosphoric acid as proton conductor.6 The operating temperature of these fuel cells is around 200 °C.7 The immobilization of phosphoric acid on a polymer, however, is not stable toward hydrolysis, as the linkage is realized via a C-O bond. To avoid such labile chemistry, the use of organic phosphonic acids was suggested. Phosphonic acids are bound via a much more stable C-P bond and show comparable proton conducting properties.1 The mechanism of proton transfer in phosphonic acids has been investigated theoretically as well as experimentally. Theoretical calculations have compared sulfonic acids, imida* To whom correspondence should be addressed. E-mail: bheggen@ fschemie.tu-darmstadt.de. § E-mail:
[email protected]. ‡ E-mail:
[email protected].
zoles, and phosphonic acids as proton conductors, the polymer being mimicked by a methyl group.8 These ab initio calculations showed that the proton transfer between two molecules of methylphosphonic acid has a lower energy barrier compared, for example, to methylsulfonic acid or 2-methylimidazole. The stronger amphoteric character of phosphonic acid could be shown. Joswig et al. studied proton transfer between two molecules of neat phosphonic acid.9 They show that it is similar to proton transfer in water. The proton conduction is facilitated by building and breaking of hydrogen bonds in a network. The dynamics of this network formation has also been studied by molecular dynamics calculations on heptylphosphonic acid,6 which confirmed a picture of formation, breaking and fast reformation of phosphonic acid clusters. Experimentally, proton transfer has been studied with 2H solid state NMR in combination with molecular dynamics simulations.10 The system treated was poly(vinylphosphonic acid) PVPA. The assumption of proton transfer by fast reorganization of hydrogen bonding networks was experimentally supported. Measurements of proton conduction in “nominally dry” PVPA indicate that a high concentration of phosphonic acid groups is necessary to allow sufficient aggregation and to enable building of extended hydrogen bonding networks.1 These measurements have not been carried out under truly dry conditions: condensation of phosphonic acid groups occurs to a certain extent, thereby providing moisture.
2R-PO(OH)2 f R-PO(OH)-O-PO(OH)-R + H2O The effect is a reduction in proton conductivity. Therefore, this reaction is allowing water in the material that is not desired for the ultimate production of dry phosphonic based membrane for PEM fuel cells. Lee et al.21 has shown in their paper on poly(vinyl phosphonic acid) that at high temperature range
10.1021/jp803589w CCC: $40.75 2008 American Chemical Society Published on Web 08/16/2008
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Figure 1. Optimized structures (HF/6-31G*) of the monomer (PA), dimer (DPA), trimer (TPA), cyclic trimer (CPA), and a tetramer (4mer-PA) of phosphonic acid.
(150-300 °C) the polymer loses weight because of the condensation reaction between highly concentrated phosphonic acid. Therefore, the intention of this work is to study possible condensation pathways and estimation of the reaction barrier of phosphonic acid (or phosphorous acid, H-PO(OH)2) and methylphosphonic acid (CH3-PO(OH)2). The methyl derivative of phosphonic acid was used to study the influence of organic side chains on the phosphonic acids as encountered in a membrane polymer. As no geometrical data of the condensation products were available, molecular geometries have been calculated at various levels of theory. Reaction energies in the gas phase as well as with the use of solvation models have been obtained via thermochemistry. Additionally, the transition states have been considered. II. Computational Methods Ab initio self-consistent field (SCF) calculations were performed using Gaussian 03.11 Geometry optimizations have been carried out at the Hartree-Fock level with the 6-31G* basis set for all structures. An average number of 20 different starting structures has been optimized to find the global minimum. These initial structures are created by changing different dihedrals, angles, and bond lengths that may be near the global minimum. Optimization of the geometries was followed by calculation of single point energies with varying methods and basis sets: Hartree-Fock with 6-31G**, 6-311G**, 6-311+G**, B3LYP with 6-31G* and MP2 with 6-31G** basis set. In addition, all structures were also optimized at the B3LYP level with the 6-31G* basis set. For all structures, a frequency analysis was carried out with similar level of theory as for optimization. The frequency analysis also included the calculation of the zeropoint correction to the energies and thermochemical information at 298.15 K and 1 atm. For calculations of solvated molecules the polarized continuum model (PCM) was used with the parametrization for water.12,13 In general, the standard united atom model UA0 was used, which treats only the heavy atoms individually, whereas hydrogen atoms are merged. We chose water as the reference solvent rather than bulk phosphonic or methylphosphonic acid in order to establish how large solvation effects can possibly become and, in this sense, obtain an upper boundary. Transition states searches at the HF/6-31G* and
B3LYP/6-31G* levels for the dimerization of phosphonic acid and methylphosphonic acid were started from several structures. Transition state optimizations were carried out by finding the saddle point using the Berny optimization program, which is already included in Gaussian 03.11 Here, the same PCM solvation model has been used but with UFF in which hydrogen atoms are also treated individually because hydrogen atoms can transfer. III. Results A. Geometries. The structures treated were the monomer (PA), dimer (DPA), trimer (TPA), cyclic trimer (CPA), and tetramer (4mer-PA) of phosphonic acid (phosphorous acid) and methylphosphonic acid (MPA, MDPA, MTPA, MCPA, 4merMPA). The structures optimized using HF/6-31G* are shown in Figures 1 and 2. In Table 1 bond lengths and angles are given for the monomers of phosphonic and methylphosphonic acid optimized with HF/6-31G* as well as with B3LYP/6-31G*. All structures have C1-symmetry. Additional reference data from the literature are given. Geomerical data for the dimeric, trimeric, and cyclic phosphonic and methylphosphonic acid are given in the Supporting Information 1-3. Several points are worth noting. Bond lengths and angles of the B3LYP/6-31G* optimization agree well (see Supporting Information 1-3) with literature values from correlated calculations (MP2 and CBS-Q, which is a correlated extrapolation method).1,14,15 Hartree-Fock bond lengths tend to be shorter, in particular the P-O single bonds and the O-H bonds. This holds for both PA and MPA. On the basis of this comparison, we decided to go on with HF and B3LYP calculations for most of this work rather than to use correlated methods. Compared to room-temperature X-ray data,16 the calculated length of P-O single bonds usually seems to be overestimated substantially by all calculations, Hartree-Fock being in closer agreement with experiment than either correlated calculations or density-functional theory. This overestimation seems to be a trend for a variety of phosphorus-oxygen compounds17 and sometimes also for P-C single bonds. The origin could be a systematic difficulty of quantum chemical calculations (in which case it would disappear again only at a much higher level of theory) or a problem of the rather old X-ray structures.
Phosphonic and Methylphosphonic Acids
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Figure 2. Optimized structures (HF/6-31G*) of the monomer (MPA), dimer (MDPA), trimer (MTPA), cyclic trimer (MCPA), and tetramer (4merMPA) of methylphosphonic acid.
TABLE 1: Geometries of Phosphonic Acid (Phosphorous Acid) and Methylphosphonic Acid: Bond Length (Å), Angles (deg), and Energy (hartree)
energy P-O1 P-O2 PdO3 O1-H O2-H P-H/P-C C-H P-O1-H P-O2-H O1-P-O2 O1-PdO3 O2-PdO3 H-P-O1/C-P-O1 H-P-O2/C-P-O2 H-PdO3/C-PdO3 H-C-P a
PA HF/ 6-31G*
PA B3LYP/ 6-31G*
-567.1174 1.586 1.590 1.453 0.952 0.951 1.373
-568.8771 1.616 1.625 1.481 0.973 0.972 1.398
113.204 112.273 105.209 115.139 113.657 100.504 102.334 118.124
111.905 109.517 103.976 117.641 113.654 97.688 103.239 118.246
PA expa 1.54 1.54 1.47
102 113 116
PA HF/ 4-31G*b -566.4536 1.57 1.57 1.45 0.95 0.95 1.37 125.74 125.74 105.28 114.76 114.76 100.84 100.84 118.31
PA MP2/ 6-311++G**c 1.61 1.63 1.49 0.97 0.97 1.39
97.90 103.81 117.33
MPA HF/ 6-31G*
MPA B3LYP/ 6-31G*
MPA HF/6-31G*, CBS-Qd
-606.1724 1.596 1.596 1.458 0.951 0.951 1.790 1.083 112.054 112.028 105.253 113.042 113.003 102.630 102.648 118.758 109.525
-607.2112 1.630 1.630 1.486 0.972 0.972 1.804 1.093 109.490 109.487 105.458 113.573 113.579 101.491 101.476 119.471 109.326
-607.4865 1.62 1.62 1.48 0.97 1.79 1.09 112.2 105.9 113.7
100.7 120.1 109.6
From ref 16. b From ref 14. c From ref 9. d From ref 15.
For the condensates, no reference data are available. Condensation, in general, leads to shorter single bonds between phosphorous and hydroxyl oxygens, whereas the bonds from P to the bridging oxygens are elongated. The PdO double bonds, as well as the single bonds between P and C, are largely unaffected by condensation. This is found for the dimers and continues with further condensation toward the trimers and tetramers. Cyclization of the trimers causes a shortening of the PdO double bond but has no other discernible effect. The dimers and the trimers (Figures 1 and 2) are stabilized by intramolecular hydrogen bonding. The typical distance between oxygen and hydrogen forming hydrogen bond is 2.3 Å for the dimers and 1.9 Å for the trimers. The closer approach of the hydrogen bonds in the trimers is due to their higher intrinsic flexibility. Replacing the hydrogen on phosphorus with a methyl group has the expected effect of substitution of a smaller group by a bigger group. In general, the constitution is preserved. Because of the larger spatial volume requirement of the methyl group, the P-O-bonds are stretched and the bond angles (see Supporting Information) shifted in a rather foreseeable way. The geometries were optimized at the Hartree-Fock 6-31G* level. At the optimized geometries the energies were recalculated
using several methods and basis sets for comparison (Table 2). Larger basis sets decrease the energies slightly for all structures. Further decreases were found for correlated calculations (Møller-Plesset perturbation theory, MP2/6-31G**). The densityfunctional energies (B3LYP/6-31G*) are also lower than the corresponding HF values. Reoptimizing the geometry at the B3LYP/6-31G* level does not lead to dramatic energy improvements. On the basis of these results, further energy calculations were carried out using HF/6-31G*, as it allows the performance of calculations with sufficiently good accuracy in reasonable time. Finally, the effects of solvation were studied. Single-point energy calculations were performed at the HF/6-31G* level. An aqueous environment was simulated by using the polarizable continuum model (PCM).12,13 Solvation is energetically favored, the solvation energy (gas phase f solution) being between -75 and -103 kJ/mol for the phosphonic acids and between -56 and -77 kJ/mol for the methylphosphonic acids. Reoptimization of the geometries under PCM solvent conditions is expected to yield only a minor improvement of the solvation energy. Already this simple model shows that the gain in energy by solvation is larger for the monomer and the cyclic trimer than for the dimer and the open-chain trimer as can be seen from Table 2. The
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TABLE 2: Total Energies (hartree) for Several Basis Sets and Methods
PA DPA TPA CPA 4mer-PA MPA MDPA MTPA MCPA 4mer-MPA H 2O a
HF/6-31G* optimized
HF/6-31G** single-pointa
HF/6-311G** single-pointa
HF/6-311+G** single-pointa
MP2/6-31G** single-pointa
B3LYP/6-31G* single-pointa
B3LYP/6-31G* optimized
-567.1174 -1058.2237 -1549.3350 -1473.2737 -2040.4146 -606.1724 -1136.3366 -1666.5029 -1590.4457 -2196.6433 -76.0107
-567.1338 -1058.2423 -1549.3553 -1473.2802
-567.2140 -1058.3792 -1549.5545 -1473.4567 -2040.6929 -606.2780 -1136.5105 -1666.7502 -1590.6558 -2196.9578 -76.0469
-567.2208 -1058.3894 -1549.5683 -1473.4693 -2040.7117 -606.2845 -1136.5203 -1666.7638 -1590.6688 -2196.9770 -76.0532
-567.7938 -1059.3732 -1550.9558 -1474.6847
-568.7938 -1061.3386 -1553.8083 -1477.3489 -2046.2447 -608.2075 -1140.0102 -1671.8143 -1595.3585 -2203.5905 -76.4081
-568.8771 -1061.3462 -1553.8191 -1477.3575 -2046.6582 -608.2112 -1140.0179 -1671.8252 -1595.3672 -2203.6040 -76.4090
-606.1918 -1136.3614 -1666.5327 -1590.4317 -2196.6788 -76.0236
-606.9955 -1137.7805 -1668.8566 -1592.2938 -76.2194
HF/6-31G* single-point,a solvatedb -567.1491 -1058.2526 -1549.3661 -1473.3130 -2040.4667 -606.2020 -1136.3581 -1666.5257 -1590.4736 -76.0237
b
Single-point energy calculation at the optimized HF/6-31G* geometry. Single-point calculation in solution using the PCM-water model.
SCHEME 1: Possible Condensation Reactions of Phosphonic Acid
latter two can stabilize themselves by intramolecular hydrogen bonds even as isolated molecules and thus lower their energy in the gas phase. In the solvent model, the internal hydrogen bonds lead to some intramolecular charge compensation, and the polarization of the surrounding continuum model is less effective. B. Energetics of Condensation. Scheme 1 illustrates possible condensation pathways of phosphonic acid and, by analogy, for methylphosphonic acid. Thermodynamic properties like enthalpy and Gibbs energy as well as the zero-point correction have been estimated for all molecules at 298.15 K and 1 atm in the gas phase and in water (calculated with PCM) via harmonic frequency analysis and molar volume calculation at the HF/631G* level. The energies obtained are reported in the Supporting Information 4. Reaction energies and Gibbs energies, as well as various corrections to them, have also been estimated (Table 3). A few
trends are visible: (i) The qualitative findings are not very sensitive to the basis set. The gas-phase reaction energies calculated at the HF/6-21G* level (column 2) are already reasonable approximations. The single-point energy calculations at these geometries confirm that neither the inclusion of changing to a triple-valence basis and polarization functions on the hydrogens (column 3) nor the inclusion of diffuse functions (column 4) changes them significantly. But exceptions are visible for the condensation of the monomer and dimer to the openchain trimer (reaction III) and condensation of open chain trimer and monomer to tetramer as well as dimer to tetramer (reactions IV and V). This is easily explained: The flexible trimer and tetramer is stabilized in the gas phase by internal hydrogen bonds. The description of hydrogen bonds benefits from a larger basis and polarization functions disproportionately, which causes the reaction to appear more exothermic. In the condensed phase of a fuel cell membrane this effect will, however, be compen-
18.9 1.9 60.5 56.9 39.9 6.5 -3.8 53.5 46.773425 36.5
15.0 1.3 66.2 32.4 18.6 1.8 -4.8 60.4 22.2 15.6
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a
-23.2 -25.8 126.5
OPT: geometry optimization at this theory level. SP: single-point energy calculation at HF/6-31G* geometry. PCM: polarizable continuum model.
56.4 43.5 -55.2 -5.7 -18.6 65.5 40.5 -47.5 16.1 13.5 -49.8 -4.4 -6.9 13.2 10.5 -46.8 0.15 -2.6 0.9 1.0 -8.5 1018.4 1018.5 0.8 0.4 -8.4 1096.7 1096.2 0.7 -0.6 -13.8 -7.8 -9.1 -0.6 -2.3 -13.3 -7.5 -9.2 -2.7 -12.6 138.3 -973.9 -983.8 -11.7 -13.3 128.9 -1067.2 -1068.9 -13.2 -21.6 135.9
-2.8 -29.6 120.2 -956.8 -983.6 -11.8 -31.9 109.8 -1051.0 -1071.1 4.8 -21.7 133.3 -945.3 -971.8 -3.4 -22.6 124.7 -1036.4 -1055.6 1.2 -12.0 132.6 -949.3 -962.5 -6.9 -12.3 122.1 -1042.6 -1047.9
12, ∆G 11, ∆G
HF/6-31G* 2+7+8+9 HF/6-31G* PCM HF/6-31G* HF/6-31G* HF/6-31G* B3LYP/6-31G* OPT MP2/6-31G** SP HF/6-311+G** SP HF/6-311G** SP HF/6-31G* OPT
theory level approximationa reaction I: PA + PA II: PA + DPA III: TPA cycl IV: TPA+PA V: DPA + DPA I: MPA + MPA II: MPA + MDPA III: MTPA cycl IV: MTPA + MPA V: MDPA + MDPA
10, solvation correction 9, -T∆S 8, p∆V 2, ∆E
3, ∆E
4, ∆E
5, ∆E
6, ∆E
7, zero-point energy correction
TABLE 3: Reaction Energies and Corrections (298.15 K, 1 bar) for the Condensation Reactions of Scheme 1 (All Values in kJ/mol)
B3LYP/6-31G* 6+7+8+9
Phosphonic and Methylphosphonic Acids
sated by the possibility to form hydrogen bonds with surrounding molecules. (ii) The inclusion of electron correlation (column 5) does not change the qualitative picture for posphonic acid and dimerization of methylphosphonic acid. (iii) There is no significant difference between the gas-phase reaction energies calculated with Hartree-Fock (column 2) and hybrid densityfunctional (B3LYP) theory (column 6). This is another confirmation that electron correlation is not too important. (iv) The zero-point corrections (column 7) do not have a decisive effect. Similarly, the contribution of the molecularvolume change to the reaction enthalpy p∆V (column 8, only exception for reactions IV and V) is comparatively small so that the reaction enthalpy (not shown) would be similar to the reaction energy. (v) The entropic contributions are reported (column 9), although they assume the general approximations of thermochemistry: harmonic vibrations, rigid-rotor approximation, vacuum environment. In particular, the latter is a severe approximation for the condensed-phase environment of a fuelcell membrane. For the dimerizations and trimerizations (reactions I and II), the entropic contributions are small and positive. It is large and negative for the cyclization reaction (III), probably from the translational entropy (one particle becomes two), and negative but very small for tetramerization. The Gibbs energies (Hartree-Fock energies plus Hartree-Fock corrections, column 11, and density-functional energies plus Hartree-Fock corrections, column 12) belong also to the gas phase and are only reported for completeness. (vi) The phosphonic acid species and the corresponding methylphosphonic acid species show similar reaction energies and corrections at all levels of theory. Notable exceptions are the dimerization (reaction I) energies, which come out more exothermic for the methyl derivatives. (Similarly, the cyclization of the methyl trimer is less endothermic than for the unsubstituted trimer. Because of the high energy, however, this possible reaction is probably irrelevant anyway.) What can be learned from these calculations regarding the possible condensation reactions in a phosphonic acid containing fuel cell membrane? First, condensations are definitely energetically possible. The dimerization (reaction I) is energetically neutral to slightly exothermic, depending on details of the theory and the species. As the methylphosphonic acid dimerizes more easily, we may conclude that also larger organic or polymeric phosphonic acids (checked until tetramerization) will form dimers easily if they are not hindered sterically. Second, the formation of the open chain trimer (reaction II) in a condensed medium probably has a reaction energy similar to the dimerization (reaction I). In the gas phase studied here, the extra exothermicity is probably exaggerated by the absence of solvating molecules: The trimer and tetramer can form favorable internal hydrogen bonds, whereas the monomer and dimer cannot form compensating hydrogen bonds to other molecules, which they would in the condensed phase. (The internal hydrogen bonds of the dimer, suffer from more strain, as is evident from their longer distances.) Third, by induction, higher condensates are probably also feasible in the condensed phase. Finally, the formation of the cyclic trimer (reaction III) is energetically unfavorable compared to the formation of open-chain polycondensates. The cyclic trimer simply has too much ring strain, and it is almost impossible that in phosphonated polymer membrane cyclization will take place. Dimerization (reaction I) and trimerization (reaction II) become more exothermic by a polarizable environment (column 10) probably because water molecules are released. The cyclization (reaction III) becomes more endothermic by solva-
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Figure 3. (a) Transition state of the dimerization reaction of phosphonic acid (schematic). (b) Calculated transition state for phosphonic acid dimerization. (c) Same as (b) but for methylphosphonic acid.
tion because of the formation of a very symmetric molecule with low polarity. The estimated solvation corrections are upper bounds, for the chosen environment corresponds to water, which probably has a higher dielectric constant than any phosphonic acid compound studied here (these values do not seem to be available from experiment). Therefore, the solvation energies are only trend values. They confirm, however, that the tendency to form dimers and open trimers will most likely be higher, rather than lower, in a condensed phase compared to the gas phase. The cyclic trimer, which is improbable already in the gas phase, will be even more unlikely in a condensed environment. Condensation reactions were observed indirectly in experiment, too. For poly(vinylphosphonic acid) it was found that measurements could be conducted only under “nominally dry” conditions, as there is always a certain amount of condensation water in the sample. A ratio of very approximately 0.15 water/ phosphonic acid has been mentioned.1 Assuming that the water comes predominantly from dimerization (no higher polycondensates), this would translate into a Gibbs energy of about 10 kJ/mol for the dimerization. This value confirms the order of magnitude of our calculated ∆G. Given the uncertainties in both calculation and experiment, as well as the different conditions (gas phase vs condensed phase, small-molecule phosphonic acid vs phosphonic acid containing polymer, different temperatures), this can be considered a success. C. Transition State of the Dimerization. We also searched for a reaction path of condensation reaction I via a transition state. The transition states are illustrated in Figure 3 schematically (part a) and from calculations (parts b and c). The first stage of the reaction is the formation of a hydrogen bond between a hydroxyl group (OH) and a double bonded oxygen (labeled 1 in Figure 3a). The second stage is the approach of the oxygen of the second hydroxyl group to the phosphorus of the other molecule and ultimately the formation of the new P-O bond (labeled 2). In the third stage, the hydrogen of the attacking oxygen forms a hydrogen bond (labeled 3) to the nearest hydroxyl group, so a water molecule is formed, still bound to the phosphorus atom. In the fourth stage, not considered here, this water molecule leaves the phosphorus. The reaction is assumed to start from the hydrogen-bonded complex, i.e., stage 1 completed. For the simulation of stage 2, the distance between P and O forming the new bond was decreased from 2 to 1.6 Å in steps of 0.1 Å. This distance was constrained, while all other geometrical parameters were optimized. In Figures 4 and 5, this stage is represented by reaction steps 1-5. Stage 3, the formation of hydrogen bond number 3, happens automatically and synchronously. It does not need to be forced. Stage 4, the leaving of a water molecule, is modeled by increasing the constrained distance between P and the oxygen of the leaving water molecule from 1.55 to 1.85 Å in steps of 0.1 Å, while optimizing the rest of the structure. These are reaction steps 6-9 in Figures 4 and 5. The hydrogen
Figure 4. Profile of the potential energy of the dimerization reaction of phosphonic acid on the gas phase (triangles for B3LYP; circles for HF) and in an aqueous environment modeled by PCM (squares for HF). The reaction steps are described in the text.
Figure 5. Profile of the potential energy of the dimerization reaction of methylphosphonic acid on the gas phase (triangles for B3LYP; circles for HF) and in an aqueous environment modeled by PCM (squares for HF). The reaction steps are described in the text.
that was supposed to move from one oxygen to the other was always placed between both for the input but not as a constraint. Figure 4 shows the potential energy curves (HF/6-31G* and B3LYP/6-31G*) for phosphonic acid along the reaction path described above in the gas phase and in an aqueous environment modeled by the PCM. In the gas phase, the activation barrier is about 192 kJ/mol (HF) and 163 kJ/mol (B3LYP) and, in water, about 192 kJ/mol. The picture is similar for methylphosphonic
Phosphonic and Methylphosphonic Acids acid (Figure 5), where the activation energy is about 172 kJ/ mol (HF) and 166 kJ/mol (B3LYP) in the gas phase and 172 kJ/mol in water. High and also similar energy barriers are observed in a condensed-phase environment. If other amphoteric phosphonic acid groups are present in the neighborhood, they can act as proton donors or acceptors and lower the activation barrier in this way. Ways to study the mechanistic of the dimerization in the liquid phase could be quantum mechanical/ molecular mechanical (QM/MM) methods, a combination of quantum mechanical simulations for the reaction, and molecular dynamics simulations for the surrounding18-20 or Car-Parrinello calculations.10 Both are beyond the scope of this contribution. IV. Conclusions Using phosphonic (with out any substitution) acid and methylphosphonic acid as model compounds, we have studied by ab initio and density-functional quantum chemical calculations the condensation behavior of organic phosphonic acid groups, as they are present in modern polymer electrolyte membrane fuel cells. The gas-phase calculations show that dimerization is energetically neutral or slightly favorable. Also, the formation of trimers and larger polycondensates should be favorable, as long as the structures are open. The cyclic trimer, in contrast, is energetically not feasible because of internal ring strain; also entropic and other corrections cannot compensate this, so this structural moiety can be excluded. Comparing corrections (zero-point vibrations, molecular-volume work) calculated in the standard way shows them to be too small in magnitude to alter these findings qualitatively. The entropic contributions, obtained in the standard thermochemical gasphase approximations, have a small disfavoring effect on the condensation. Studying the reaction energetics also with continuum models for a condensed-phase environment shows that the trends are likely to be stronger in the condensed phase: dimerization and trimerization favored, cyclic trimers virtually impossible. Comparison of the estimated Gibbs energy for the dimerization with (rather indirect) experimental evidence shows experiment and simulation to be of the same order of magnitude. A first guess at the reaction mechanism shows that for the dimerization of an isolated pair of phosphonic acid or methylphosphonic acid molecules, the transition state energy is far too high to make such a mechanism feasible in a liquid or polymeric environment. Here, the dimerization is likely to proceed under the catalyzing influence of third phosphonic acid groups.
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