# Modeling and Simulation for Fuel Cell Polymer Electrolyte Membrane

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Calculation Method

_{2}–CF

_{2}–CF

_{2}–CF

_{2}–, –O–CF

_{2}–CF[CF

_{3}]–O– and –CF

_{2}–CF

_{2}–SO

_{3}H were defined as beads A, B and C, respectively, and H

_{2}O were represented by W beads, with each W bead containing four water molecules [13]. The volume of the beads was around 0.12 nm

^{3}.

#### 2.1. DPD and MC

_{ij}

^{C}, the dissipative force F

_{ij}

^{D}and the random force F

_{ij}

^{D}, each of which is described in Equations (1–3).

_{c}, the relative position and velocity between particle i and j is given by r

_{ij}= r

_{i}– r

_{j}and v

_{ij}= v

_{i}– v

_{j}, ω

^{D}and ω

^{R}are weight functions, and the randomly fluctuating variable θ

_{ij}introduces randomness into the system with Gaussian statistics. For particles bonded to each other within the same polymer chain, Equation (4) is added as the bonding force F

_{ij}

^{S}.

^{3}, this results in the interaction distance in of r

_{c}= 0.71 nm.

_{ij}in Equation (1). They were derived by a

_{ij}= a

_{ii}+ 3.27 χ

_{ij}. By setting the diagonal terms at a value of a

_{ii}= 104, the water compressibility is reproduced for a system solely composed of W beads. Table 1 also shows the χ parameters in parenthesis which were calculated from atomistic simulations of mixing energies [13]. The simulation cells were cubic with box edge lengths of more than 35 times the interaction distance r

_{c}. In all DPD calculations, equilibrium phase-separated structures that served as input for the MC calculations were obtained at more than 20,000 time steps, using a time step of 0.05.

_{c}in DPD so that the phase-separated structure of DPD could be rebuilt. The type of each node was defined as that type of bead which is most close to that node. As a result, every node belongs to the polymer phase or the water phase. When investigating proton conductivity, the pore network is defined as a summation of nodes with a nearest bead being W or C. For examining errors by conversion, it was confirmed that the difference between the pore bead fraction within the DPD and pore node fraction of the MC grid was within 0.5%.

**Table 1.**Dissipative particle dynamics (DPD) repulsions used for Nafionand corresponding χ parameters in parenthesis.

Interaction parameter | A | B | C | W |
---|---|---|---|---|

A | 104 (0) | – | – | – |

B | 104.1 (0.02) | 104 (0) | – | – |

C | 114.2 (3.11) | 108.5 (1.37) | 104 (0) | – |

W | 122.9 (5.79) | 120 (4.90) | 94.9 (−2.79) | 104 (0) |

**R**

_{i}(t) is the position of tracer particle i on the lattice at time t. According to Equation (5), the slopes obtained by plotting the mean square displacement (MSD) against MC step (MSC), are proportional to the diffusion constant. For pure water, each jump trial is accepted resulting in slopes equal to 1. For diffusion through the hydrated membranes the slopes will be <1, since not each jump trial is accepted. Therefore, for the model membranes, the calculated diffusion constants are expressed relative to the pure water case and are given by D'

_{rel.}= d(MSD/d(MCS). A comparison with experimental values for diffusion in Nafion was achieved by multiplication of D'

_{rel.}with the pure water diffusion constant of 2.3 × 10

^{−5}cm

^{5}/s.

#### 2.2. CGMD

_{ij}is averaged over all orientations.

_{LJ}and distributions P, energy calculations more than 10,000 times for potential energy and molecular dynamics simulations longer than 10 ns for calculating the distributions were performed for sampling, where COMPASS [23] force field was applied to all atoms. From obtained U and P, the potentials derived from Equations (9–11) were fitted to the function forms of Equations (6–8) to determine the parameters. Thus, obtained fitted parameters are listed in Table 2. These values are normalized by the energy of 0.533 kcal/mol and the length of 0.544 nm and the angle is measured as external angle (the angle of linearly connected atoms is 0 degree).

LJ | ε | σ | bond | k_{r} | r_{0} | |
---|---|---|---|---|---|---|

A–A | 1.00 | 1.00 | A–A | 3850 | 0.994 | |

A–B | 1.33 | 0.962 | A–B | 2270 | 0.741 | |

A–C | 2.38 | 0.898 | B–C | 5080 | 0.806 | |

A–W | 0.831 | 0.987 | – | – | – | |

B–B | 1.87 | 0.919 | – | – | – | |

B–C | 2.86 | 0.874 | angle | k_{φ} | φ_{0} | |

B–W | 0.789 | 1.00 | A–A–A | 17.9 | 10.8 | |

C–C | 5.29 | 0.850 | A–A–B | 16.8 | 77.1 | |

C–W | 8.67 | 0.854 | A–B–C | 34.4 | 84.4 | |

W–W | 5.89 | 0.880 | – | – | – |

## 3. Results and Discussion

#### 3.1. Proton Conductivity

**Figure 3.**Water diffusion in hydrated Nafion as function of water volume fraction: (○) experimental data [8]; (●) DPD and MC calculation.

_{16}C

_{4}}

_{1}, which had the longest hydrophilic and hydrophobic block chain lengths, well-connected water clusters with a large diameter were seen. For{A

_{8}C

_{2}}

_{2}, for which the block lengths were two times smaller, water clusters were connected, but their diameter was small. On the other hand, for {A

_{4}C

_{1}}

_{4}, extremely small clusters that seem to be badly connected were seen and some of them were even isolated. The calculated water diffusion coefficient increases with an increase of hydrophilic block length. In order to explore the effect of water cluster shapes on water diffusion, the cluster radius and the separation between clusters were estimated from the pair correlation function of the W beads. The inter-cluster distance is defined as the position where the second peak in the pair correlation function occurs, while the pore radius is defined as that distance for which the pair correlation function of the W beads starts to drop below the value 1.

**Figure 5.**DPD structures obtained for the sequences {A

_{16}C

_{4}}

_{1}, {A

_{8}C

_{2}}

_{2}and {A

_{4}C

_{1}}

_{4}. A bead is in gray, C bead in white and W bead in black.Reproduced with permission from [3]. Copyright 2010 Royal Society of Chemistry.

**Figure 6.**(

**a**) Pore radius and (

**b**) inter-cluster distance derived from pair correlation function plotted against D'

_{rel.}. Reproduced with permission from [3]. Copyright 2010Royal Society of Chemistry.

**Figure 7.**Pore morphologies obtained for the sequences {A

_{7}[A

_{1}C]}

_{4}and {A

_{1}[A

_{7}C]}

_{28}. DPD bead representations of two repeat units are included. Reproduced with permission from [3]. Copyright 2010Royal Society of Chemistry.

**Figure 8.**W bead fraction contained within the largest water cluster vs. length which defines connection between W beads. Reproduced with permission from [3]. Copyright 2010Royal Society of Chemistry.

**Figure 9.**D'

_{rel.}plotted against y/x (the ratio of side chain length and branching point distance). Reproduced with permission from [3]. Copyright 2010Royal Society of Chemistry.

#### 3.2. Gas Permeability

_{Nafion}is the volume fraction of Nafion.

_{2}and O

_{2}in dry Nafion, respectively. These equations were derived from the experimental data [20]. The H

_{2}and O

_{2}solubility in water were deduced from the temperature dependencies of the molar fraction solubility [24,25] and water mass density [25]. Thus obtained temperature dependent solubilities are given by Equations (14a) and (14b).

^{−5}cm

^{2}/s in water and 0.014 × 10

^{−5}cm

^{2}/s in Nafion.

_{Nafion}/D

_{water}. The gas solubility within the Nafion phase is always larger than within the water phase. Therefore, for a gas particle that tries to leave a water node and enter a nearest selected Nafion node, the trial is always accepted, while a jump trial in the reverse direction is reduced and becomes S

_{water}/S

_{Nafion}. These jumping rules ensured that the diffusion of the gas particles within both phases was well described, and that during the simulations, the gas particle concentration within each phase matched exactly the experimental pure phase component solubility. The gas diffusion constants within the membrane were obtained by multiplication of the slopes derived from the MSD curves with the diffusion constant of oxygen and hydrogen in the pure water phase given by Equations (17) and (18), respectively. The permeability K was calculated by multiplying of the thus-obtained diffusion constant with the solubility values given by Equation (12).

**Figure 10.**Calculated permeability value in hydrated Nafion compared with experimental data [26]: (●) hydrogen; (▲) oxygen.

_{water}and K

_{N}

_{afion}are the permeability of the gas species in the pure water and Nafion components, respectively.

**Figure 11.**Permeation through heterogeneous media. Relation between permeability and internal heterogeneous structure.

**Figure 13.**Tragectory of gas particlesduring diffusion process through hydrated Nafion at 40 degrees Celsius and 0.7 volume fraction of Nafion.

#### 3.3. Mechanical Strength

^{6}time steps to obtain equilibrium structures of hydrated membranes. The time step was set at 0.006 so that relative errors of the temperature were less than 0.3%. When evaluating elastic modulus from a stress-strain curve by extending the hydrated membranes, the cells were deformed by 5% with a speed of 0.0176 m/s along one direction while keeping the cell volume constant under the deformation. To save the calculation cost the deformation rate used here is almost twenty times higher as the deformation rate in the experiment [29]. The general coarse graining molecular dynamics simulator OCTA/COGNAC [30] was used for the calculation.

λ = N_{H20}/N_{SO3H} | Number of Nafion chains | Number of W particles |
---|---|---|

0 | 41 | 0 |

3 | 128 | 2397 |

4 | 37 | 919 |

4 | 124 | 3103 |

5 | 121 | 3780 |

8 | 33 | 1663 |

14 | 98 | 8518 |

20 | 26 | 3201 |

^{3}[31] while our calculated value was 1.94 g/cm

^{3}. Under the water absorption conditions at λ = 3, λ = 4, λ = 5 and λ = 14, the mass densities shown in Figure 14 were obtained, this indicates that the calculated bulk density agreed well with the experimental values. On the other hand, absorbed water forms cluster network structures inside the membranes and the separation distance between the clusters have already been estimated from small-angle X-ray scattering experiments [7,8]. We estimated the cluster separations from the radial distribution function of the coarse-graining water particles, and the results are compared with the experimental results in Figure 15. Overall, the inner structures were reproduced well quantitatively for a wide range of hydration levels, which reveals that the calculation method used in this study is adequate. At high water volume fraction the obtained cluster spacing is slightly lower than the experimental values. The reason for this might be that the atomistic force field used for the water molecules to obtain the coarse-grained potentials is the 3-site model, which might somewhat underestimate the interaction between water, resulting in more dense water and therefore smaller cluster spacings at high hydration levels.

**Figure 14.**Density of hydrated Nafion as function of water content: (○) experimental data [31]; (●) coarse-grained calculation.

**Figure 15.**Cluster spacing of hydrated Nafion as function of water volume fraction: (○) experimental data [8]; (●) coarse-grained calculation.

**Figure 16.**Elastic modulus of hydrated Nafion as function of water activity: (○) experimental data [29]; (●) coarse-grained calculation.

## 4. Conclusions

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**MDPI and ACS Style**

Morohoshi, K.; Hayashi, T. Modeling and Simulation for Fuel Cell Polymer Electrolyte Membrane. *Polymers* **2013**, *5*, 56-76.
https://doi.org/10.3390/polym5010056

**AMA Style**

Morohoshi K, Hayashi T. Modeling and Simulation for Fuel Cell Polymer Electrolyte Membrane. *Polymers*. 2013; 5(1):56-76.
https://doi.org/10.3390/polym5010056

**Chicago/Turabian Style**

Morohoshi, Kei, and Takahiro Hayashi. 2013. "Modeling and Simulation for Fuel Cell Polymer Electrolyte Membrane" *Polymers* 5, no. 1: 56-76.
https://doi.org/10.3390/polym5010056