Mesoscale Model for Ostwald Ripening of Catalyst Nanoparticles

Data structure

The network model consists of three elements: vertex, satellite and edge (see Fig. 2). A vertex is located at the center of a catalyst particle. It contains information on the current particle radius and the local concentration of Pt-ion governed by the electrochemical model of Pt-dissolution and plating. The spherical surface of each particle is partitioned by a Voronoi tessellation, and each spherical-polygon mesh element is called satellite. All the satellites belonging to the same spherical shell are associated to one vertex. The satellites are the medium in which dissolved Pt (adjacent to a a particle's surface) diffuse around a particle—each satellite communicates with its neighbors similar to surface diffusion. In addition, satellites can be thought of as a local particle atmosphere that communicates material to a particle via a Butler-Volmer current. The separation between vertex and satellites allows to treat separately transport and electrochemical reactions, and it provides a representation for the boundary-layer, i.e., a surface layer characterized by local excess concentration of Pt-ions in solution. Each satellite contains information about its relative surface area, the relative distance to and border-length with neighbouring satellites. Additionally, a satellite has its own time-dependent concentration.

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A packing model is used to generate a representative volume of the CCL microstructure (as illustrated in the left panel of Fig. 1) and determine the position of the Pt-particles over the carbon support. The microstructure in Fig. 1 consists of spherical Vulcan-type carbon particles with mean size of 30 nm, and coated by an ionomer layer of thickness 3−5 nm. The Pt-particles decorate the carbon support, and the Pt/C ratio was fixed at 0.2 weight ratio.

A 3D Delaunay triangulation having nodes located at the particles' center is used to construct the edges in the network model and represent the connectivity among neighbouring particles. The triangulation connects the particles' center with straight lines and does not take into account the curvature of the carbon support. This approximation can be removed in future implementations of the model. To account for ionomer poisoning of the catalyst, a scaling factor between the geometric surface area and the ECSA could be defined for each Pt-particle depending on its interaction with the ionomer.

Edges connect satellites belonging to different particles. All the edges and satellites together form a fully connected diffusion network that allows for the redistribution of mass of dissolved platinum. The complete list of information contained in the data-structure is listed in Table A·I of the Appendix.

Diffusion and electrochemical model

Upon voltage cycling, electrochemical dissolution tend to release Pt-ions, allowing them to diffuse from one particle to another and even leave the cathode catalyst layer. The dissolved ions tend to deposit on larger particles, resulting in growth of large particles at the expense of smaller ones. Platinum mass loss and redistribution leads to loss in catalytic surface area available for the oxygen reduction reaction. The transport of platinum ions is mediated by the water generated from the electrochemical reaction at the cathode and from vapor condensation.

The model accounts for three mechanisms of Pt-ion transport with the definition of three fluxes: the flux J^ptct within the boundary layer, the flux J^sat over the particle surface, and J^edge along the edges connecting particle-pairs. The steady-state fluxes are in nanomoles per unit of time and their definitions are listed in Table I.

Table I. Equations for electrochemical dissolution/plating of platinum and transport of dissolved platinum through the network.

The flux Eqs. in Table I have the following general form

$$\begin{eqnarray*}\begin{array}{rcl}{\mathrm{flux}}_{{AB}} & = & \mathrm{diffusivity}* \mathrm{concentration}\ \mathrm{gradient}* \mathrm{cross}-\mathrm{section}\\ & = & \mathrm{diffusivity}({{nm}}^{2}/s)\left(\displaystyle \frac{{\mathrm{nanomoles}}_{B}({nmol})}{{\mathrm{volume}}_{B}({{nm}}^{3})}\right.\\ & & \left.-\displaystyle \frac{{\mathrm{nanomoles}}_{A}({nmol})}{{\mathrm{volume}}_{A}({{nm}}^{3})}\right)\displaystyle \frac{{\mathrm{section}}_{{AB}}({{nm}}^{2})}{{\mathrm{distance}}_{{AB}}({nm})}\end{array}\end{eqnarray*}$$

For instance the gradient along the edge is computed from the difference in concentration at the end satellites divided by the edge length. The gradient is multiplied by the cross-section of the edge, in turn expressed as a fraction of the satellite's surface area through S_link.

The expression of the flux through the boundary layer J^ptct derives from the product between the gradient across the boundary layer (n_{i ν} − n_i )/δ and the satellite surface ${S}_{\nu }\pi {r}_{i}^{2}$, divided by the volume of the spherical shell $4\pi {r}_{i}^{2}\delta$.

In this version of the model, Pt-ions can diffuse within the network, and the total mass is conserved. The equilibrium condition for vertices

$$\begin{eqnarray}&&{J}_{i}^{\mathrm{ptcl}}+{J}_{i}^{\mathrm{diss}}=0,\quad \forall i\in \mathrm{ptcls}\end{eqnarray} \tag{ 6 }$$

is a balance between the dissolution flux and the flux through the boundary layer. The mass-conservation condition at the satellites

$$\begin{eqnarray}&&\begin{array}{l}\sum _{i,j\,\in \ \mathrm{all}\ \mathrm{edges}}{J}_{i\alpha \to j\alpha }^{\mathrm{edge}}+\sum _{i\,\in \ \mathrm{ptcls}}\left(\sum _{\alpha \,\in \ \mathrm{sat}\ \mathrm{of}\ i}{J}_{i\alpha }^{\mathrm{sat}}-{J}_{i}^{\mathrm{ptcl}}\right)=0\end{array}\end{eqnarray} \tag{ 7 }$$

includes the contribution of the neighbouring satellites, the flux from the particle surface and the diffusion along the edge. A sink mechanism can be included to simulate Pt-mass loss due to reaction with hydrogen-gas cross-over.

The numerical implementation of the model was coded in the Wolfram Language and simulated using Mathematica. ¹⁸

The main transport parameters are: the diffusivity D_TL of Pt-ion along the triple line (represented by the edge), the surface diffusivity D_S , and the diffusivity through the boundary layer D_BL . The vehicle for Pt-ion diffusion and mass redistribution is not well understood. The value for the edge diffusion D_TL is an empirical representation of the Pt-ion diffusivity within the CCL microstructure. It is not know if the preferred diffusion pathway for Pt-ions is through liquid water or ionomer, or both. Therefore, D_TL may depend on all of the following properties

• diffusivity the Pt-ion in bulk water,
• liquid-water saturation of cathode catalyst layer (which in turn depends on the relative humidity and on the wetting property of the substrate),
• diffusivity the Pt-ion in Nafion as a function of its water saturation
• ionomer dispersion

The bulk diffusivity of Pt-ions in water may be in the order of 10⁻⁵ cm² s⁻¹ according to the values estimated for other transition metals like Ni. ¹⁹ Atomic surface diffusion on Pt nanoparticles has been quantified by high-resolution transmission electron microscopy (TEM) in vacuum to be in the order of 10⁻¹⁷ − 10⁻¹⁶ cm² s⁻¹ . ²⁰ It is yet to be determined whether Pt diffuse faster in Nafion and in what form. In the numerical analyses presented here, we assumed the following values for the diffusivity constants D_TL = 1000 nm² s⁻¹, D_S = 1 nm² s⁻¹, D_TL = 10 nm² s⁻¹, but these values will need to be adjusted as we learn more about the physics of Pt-mass redistribution.

The electrochemical adsorption/desorption of platinum is modeled by a Butler-Volmer equation (see Eq. 4 in Table I). In Eq. 4, the parameter k^diss represents the surface reaction kinetics, and U_Pt (r_p ) is the particle-size dependent dissolution potential. The rate of change of the particle radius follows from the dissolution flux according to Eq. 5-b.

The dissolution flux grows exponentially with decreasing particle size. To prevent numerical instabilities the model includes a cut-off radius (here set to 0.5 nm). Once the particle size approaches the cut-off radius, the remaining mass of platinum is assumed to dissolve within the next time step. This simple approach prevents the dissolution flux to grow unbounded as the radius r_p → 0.

The dissolution potential U_Pt diverges from its reference values ${U}_{{Pt}}^{o}=1.18\,{\rm{V}}$ ²¹ as the particle size decreases (see Fig. 3a). The Gibbs-Thomson expression relates the particle radius to the chemical potential of its Pt crystallite. The bigger the particle, the smaller its chemical potential. The Gibbs Thomson relation reads

$$\begin{eqnarray}&&{\rm{\Delta }}\mu =2{\rm{\Omega }}{\gamma }_{{Pt}}/{r}_{p}.\end{eqnarray} \tag{ 8 }$$

This expression assumes spherical Pt particles which is a simplification to the faceted structure of Pt nanoparticles (NPs). In fact, Pt has a face-centered-cubic structure (FCC) and the thermodynamic equilibrium shape of its NPs is governed by the surface energies of its high symmetry planes (γ₍₁₁₁₎ < γ₍₁₀₀₎ < γ₍₁₁₀₎). ²² Hence, the equilibrium shape of Pt-NPs is a truncated octahedron and the NP surface is predominantly (111) faceted (see Fig. 4). While empirical evidence suggests that the Gibbs-Thomson effect applies to faceted NPs, ¹⁵ it is unclear what the range of validity of Eq. 8 is and what value of γ_Pt is to be used especially at the limit of small NP sizes. The increased chemical potential could be computed with the anisotropic analog of surface tension multiplied by by curvature—the weighted-mean-curvature, ²³ but it becomes increasingly challenging when we transition to nanoclusters.

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We employ simple molecular mechanics energy minimization to evaluate the potential energies of different size crystalline Pt-NP at their equilibrium shape using the Wulff construction. Calculations were conducted using the large-scale atomic/molecular massively parallel simulation (LAMMPS) tool. ²⁴ We use a commonly applied Embedded Atom Method (EAM) force field (Interatomic Potentials Repository—National Institute of Standards and Technology NIST) ²⁵ and a recently developed reactive one; namely, Charge Optimized Many Body (COMB) potential ^26,27 for comparison. Surface energies of the high symmetry facets were evaluated using the Fiorentini and Methfessel method ²⁸ where U_slab = 2γ_slab + NU_bulk , in which N is the total number of atoms in the slab and γ_slab is the surface energy. By plotting the energy of a multi-layer slab vs N it is possible to derive 2γ_slab as the intersect and bulk energy U_bulk as the slope of the slab potential energy U_slab vs N curve.

A summary of the γ_slab for different planes and force fields is provided in Table II. Both force fields estimate similar U_bulk and lattice constant. In addition, the ranking of γ_slab values for different facets was as expected for high symmetry planes, with γ₍₁₁₁₎ having the lowest magnitude. However, different γ_slab ratios would results in a slightly different equilibrium NP shapes.

Table II. Evaluated bulk and surface properties of Pt.

Property	EAM	COMB
Lattice constant (A)	3.91	3.9
Bulk energy (eV atom−1)	5.77	5.774
γ(111) (J m−2)	2.08	1.71
γ(110) (J m−2)	2.39	2.27
γ(100) (J m−2)	2.17	2.2
γPt (J m−2)	2.796	2.776

We define the chemical potential of the Pt atoms extracted from atomistic simulations as follows

$$\begin{eqnarray}&&{\rm{\Delta }}{\mu }_{{atomistic}}=\displaystyle \frac{{U}_{{NP}}}{N}-{U}_{{bulk}}\end{eqnarray} \tag{ 9 }$$

where U_NP is the potential energy of the NP. Here, Δμ_atomistic should be inversely proportional to the particle radius, as suggested by the Gibbs-Thomson relation, but the definition of a spherical radius is ambiguous for a faceted NP. One approach to address that is to employ the quantity R_g (radius of gyration) to evaluate an equivalent spherical radius. ^29,30 The R_g can be evaluated as:

$$\begin{eqnarray}&&{R}_{g}=\sqrt{\displaystyle \frac{1}{N}\sum _{i}{\left|{R}_{i}-{R}_{{cm}}\right|}^{2}}\end{eqnarray} \tag{ 10 }$$

In Eq. 10, R_i denotes the coordinate of theith atom of the nanoparticle and R_cm is the coordinate of the NP center of mass. We propose employing a modified equivalent spherical radius of the NP that is calculated according to

$$\begin{eqnarray}&&{r}_{{NP}}={R}_{g}\sqrt{5/3}-{r}_{{Pt}-{atom}}\end{eqnarray} \tag{ 11 }$$

with r_Pt−atom being the radius of the Pt atom. The shift using r_Pt−atom resulted in a good match with the continuum theory as discussed below.

In Fig. 4, the plot of Δμ_atomistic vs r_NP shows a smoothly decreasing function that can be fit by a power law. The COMB potential results in power of −0.99 while EAM data decays with a power of −0.95, both in very good agreement with the Gibbs-Thompson exponent of −1. The results in Fig. 4 show that the Gibbs-Thompson relation can be extended to faceted nanoparticles by defining an equivalent spherical particle which is derived from the radius of gyration (please refer to Appendix for direct energy evaluation of NP of varying sizes). We believe that the r_Pt−atom shift is originated from the nonlinear relation between the number of atoms along different directions and their interaction energies especially in small atomic clusters. ³¹ Hence, a small shift on the order of atomic radius, Gibbs-Distance, ^31,32 would be needed to achieve the proper transition from the atomistic to the continuum model. In addition, a single intrinsic value of free energy γ_Pt can be employed to describe the state of NPs in continuum models. The extracted value of γ_Pt is considered as a weighted mean surface energy.

It is noted that despite the difference in surface energies in EAM and COMB force fields and hence the slight difference in the NP Wulff shape, the magnitude of γ_Pt only differs by 0.7% (see Table II). Such small difference might stem from the homogenizing effect of transforming a faceted NP into a sphere which minimizes the intricate differences in the particle shape. The value of γ_Pt is high compared to the γ_slab of the high symmetry planes. This could be attributed to the rough surface of the reconstructed equivalent sphere for an FCC atomic structure with predominantly under-coordinated atoms at the surface. ³³

Rigorous evaluation of the faceted NP surface energy can, in principal, be attained using the weighted mean curvature; ²³ however, our approach provides a simple and efficient estimate of the NP surface energy.

For a given voltage V, the equilibrium concentration of Pt in solution varies locally as a function of the particles size. The equilibrium concentration around each particle of radius r_p can be computed by the Nernst equation as follows

$$\begin{eqnarray}&&{c}^{\mathrm{eq}}(V,{r}_{p})=\mathrm{Exp}\left[2\displaystyle \frac{V-{U}_{\mathrm{Pt}}({r}_{p})}{{RT}/F}\right]\end{eqnarray} \tag{ 12 }$$

The dependency of the Nernst concentration on the particle size is illustrated in Fig. 3b. For a given voltage in the range of 0.9 − 1.2 V, the value of c^eq(V, r_p ) spans over several orders of magnitude as a function of the particle size. (In Fig. 3b, we assumed a typical PEM fuel cell operating temperature of 80 °C.)

When the voltage is held above the dissolution potential, each particle dissolves until the local concentration of Pt in solution reaches the equilibrium value. A perfectly uniform particle size distribution represents a metastable state: it leads to uniform Pt-ion concentration and no mass diffusion within the system. However, any small perturbation in particle size, and hence local concentration, can initiate coarsening. The amplitude of the perturbation grows over time at a rate that is limited by mass transport and surface kinetics. Local fluxes tend to accumulate Pt-ions around the largest particles, thereby causing supersaturation of the solution around the particle surface and electroplating.

In the limit of zero Pt-volume fraction, the LSW equation for the kinetics of coarsening in the diffusion-limited regime is

$$\begin{eqnarray}&&{K}_{{LSW}}^{D}=\displaystyle \frac{8D{\gamma }_{{Pt}}{{\rm{\Omega }}}^{2}{c}^{{eq}}\left(V,\infty \right)}{9{RT}}\end{eqnarray} \tag{ 13 }$$

with K_LSW ^D being the prefactor in the power law for the average radius $\langle r{\rangle }^{3}=\langle {r}_{0}{\rangle }^{3}+{K}_{{LSW}}^{D}t$, and ${c}^{{eq}}\left(V,\infty \right)$ the equilibrium concentration of platinum in solution for a perfectly flat solid/liquid interface. In the microstructural case, the Pt-diffusivity D includes a tortuosity factor due to the network of transport pathways, and it represents an empirical average of the three diffusion mechanisms embedded in the network model.

Following the strong dependence of the equilibrium concentration on the the applied voltage, the LSW kinetic coefficient will also be sensitivity to the voltage. In the next section, the coarsening kinetics extracted from the simulations is compared to the scaling predicted by the LSW theory. Equation 4 and Eq. 5 are similar to the Darling and Meyers model, ³⁴ but a factor of 2 was applied to Eq. 6 of Ref. 34 for consistency with the Gibbs-Thompson Eq. 8. In this implementation we did not consider the effect of the oxide coverage. A variety of approaches have followed the initial interpretation of Darling and Meyers as purely passivating layer. ³⁵ Recent electrochemical models ^35,36 employed a non-ideal solid solution theory to describe the observed coupling between the equilibrium dissolved Pt concentration and potential. Activity coefficients of Pt and PtO_x were correlated to oxide coverage. In general, electrochemical models for Pt-dissolution with different levels of details can be hosted by the vertices of this data-structure.

The effect of carbon corrosion can be included by modeling its affect on the position and interactions of Pt-nanoparticles. In terms of modeling, carbon corrosion may alter the inter-particle distance, cause mass-loss due to particle detachment, and may require significant re-meshing of the network during the simulation. Migration of Pt-nanoparticles over the substrate can be described as a a thermal fluctuation of the particle location, leading to a random walk over the curved carbon surface.

By decreasing the distance between particles, both carbon corrosion and particles migration may increase the probability of agglomeration and coalescence. A simple approach to modeling particle collision includes fixing a minimum distance between particles. Once such distance is reached, the entire mass of the smaller particle is transferred to the larger one (in one time step or gradually as a result of mass transport along the edge bridging the particles).

We performed numerical experiments with an ensemble of 80 particles dispersed in a volume of 100 nm in each spatial dimension. A packing model was used to generate a cluster of primary carbon particles of 30 nm average size, and coated by a ionomer layer of thickness 3−5 nm. From the microstructure shown in the left image of Fig. 1, a subset of Pt-particles was extracted to form the network structure illustrated in center image of Fig. 1. The average inter-particle distance, from Pt-particle center to Pt-particle center, is 35 nm. The linear distance is a simplification to the curved path taken by the Pt-atoms due to carbon particle curvature. Unless noted otherwise, the Pt-particle radius distribution is uniform in the range of 0.8−2.0 nm.

First, voltage-hold tests were performed to analyze the kinetics of particle coarsening as a function of the voltage. In the simulation, the evolution in each particle radius and the local concentration of platinum in solution were tracked. Figure 5a illustrates the relative amount of dissolved platinum (as nanomoles of Pt in solution normalized by the total nanomoles of platinum available in the system). Each curve in Fig. 5a shows a similar trend. The concentration of Pt in solution increases until the mean-field concentration for each applied voltage is reached asymptotically. The mean field concentration is computed as the average of the equilibrium concentration around each particle i as ${\sum }_{i}{c}^{{eq}}\left(V,{r}_{i}\right)/N$, with N being the total number of particles at time zero. Coarsening begins as the peak mean concentration is reached. The completed dissolution of a particle causes sudden increase of Pt concentration in solution, this is represented by the spikes in the descending part of the curves in Fig. 5a. This is clearly illustrates in Fig. 5b, with the overlay of the plot of the number of dissolved particles vs time. The gray vertical lines show a perfect correspondence between an increment in the number of dissolved particles (orange line) and the concentration peaks (black line). The exponential decays following the spikes are controlled by mass transport and indicate the approaching of a new mean equilibrium concentration. On average the mean field concentration decreases during coarsening, as the average particle radius increases.

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To study the effect of voltage cycling, initially two voltage cycling tests (with voltage ranging from 0.95 V to 1.05 V) were performed and the results compared with the voltage-hold experiment at 1 V. In the inset to Fig. 6a, the triangular wave (TW) and square wave (SW) tests are illustrated. In both cases, the voltage oscillates around the mean value of 1 V, with a frequency of 20 s for the full cycle. Figure 6a shows the amount of dissolved platinum with the peak concentration shifted earlier in time for the TW and SW cycling tests. Figure 6b shows the details of the descending part of the dissolution curve for the triangular wave case. The orange curve in Fig. 6b is a staircase function that shows the number of dissolved particles over time. Each step is marked by a vertical gray line to show the correspondence with some of the dissolution peaks. The remaining peaks are caused by the voltage cycling and are characterized by uniform frequency.

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The trend in the surface area loss is investigated with respect to several factors. In Fig. 7a, the four voltage hold experiments clearly indicate that even a 20 mV shift in voltage causes a significant change in the coarsening rate. The kinetics and time scale of coarsening are both voltage-dependent, as indicated by the slope and intercept of the four curves in the logarithm-scale plot. A similar trend can be observed also in Fig. 8a, where the relative number of particles is plotted as a function of time. The system simulated here contains 80 particles at beginning of the simulation and in the most aggressive conditions, only 10−15 particles are left after 3 hours.

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The effect of voltage cycling on the loss of active surface area and on the particle dissolution is illustrated in Figs. 7b and 8b respectively. It is interesting to note that in both cases the slope of the curves for the TW and SW results follows approximately the same time scaling (same slope) as the constant voltage experiment at 1 V. The green (TW) and yellow curve (SW), are shifted with respect to the red curve, representing the baseline case of 1 V hold. The coarsening kinetics are accelerated by the voltage cycling, in particular in the case of square wave cycling, leading to more rapid aging. This example indicates how the proposed model can be used to predict aging under realistic driving cycle conditions, given that any functional form can be used to prescribe the voltage as function of time. In a future study we will explore the effects of the amplitude and frequency of the voltage wave on the coarsening kinetics and ECSA loss.

Two additional tests were performed to show the predictive capability of the model with respect to accelerated stress tests (ASTs). Triangular-wave (TW) and square-wave (SW) tests between 0.6 and 1 V with 8 s half-cycle are typical ASTs used to to probe the durability of the cathode catalyst layer. The experimental data is extracted from Fig. 5 of Harzer et al. ³⁷ for the samples with 0.4 mg_Pt /cm². Figure 9 shows a comparison between the measured relative ECSA loss (marked by points) and the predicted values (marked by curves).

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We did not perform an extensive parametrization of the model, we simply used the same parameters as in the previous examples and only tuned the dissolution kinetics in Eq. (4) (increasing it by 5-fold) to improve the fit. We also employed the same particles spacing used in the previous analyses (in the experiments a Vulcan-carbon support was used). To approximately reproduce the PSD shown in the inset of Fig. 5 in Harzer et al., ³⁷ we assumed a normal particle size distribution with a mean radius of 1.5 nm standard deviation of 0.25 nm. Figure 9 shows that the model is able to capture the relative trend in surface area loss for both the TW (shown in black) and SW tests (in orange) in the first 10000 and 3000 cycles respectively. The simulation was interrupted when 90% of the initial particles were dissolved.

Further analyses were carried out on the effect of particle radius distribution (PRD). In addition to the baseline case with particle radius uniformly distributed in the range of 0.8−2.0 nm, two additional tests were performed at a fixed voltage of 1 V. First, a normal distribution with mean radius of 1.75 nm and standard deviation of 0.25 nm; second, a bimodal distribution centered around 1.45 nm and 2 nm. The particle locations and the initial total surface area were the same in each simulation. Figure 7c shows that the surface area loss for the normal PRD (yellow curve) converges to the one for the uniform PRD (red curve) after the first 1000 s. A similar trend is observed for the time evolution of the number of Pt nanoparticles (see Fig. 8c). We expect that the time to reach this steady state condition, where the coarsening is not sensitive to the initial PRD, increases with decreasing voltage and it also depends on the particular PRD. The bimodal distribution, consisting of a set of smaller particles and a set of larger ones, shows a different trend. The smaller particles tend to dissolve more rapidly, leading to a more extensive surface area loss before 1000 s (see green curve in Fig. 7c). Following this initial accelerated coarsening, there is a stalling period, where the surface evolves slightly and the number of particles stays constant.

The simulations were ended when the relative number of particles approaches 0.1 (i.e., 10% of the initial distribution), and at this point the three simulations have converged to approximately the same number of particles and relative surface area, independently of the initial PRD. The time scale for ripening to converge to the same trend depends on the specific operating conditions and model parameters.

The analysis of the average mean radius as function of time provides a comparison with the limiting regimes predicted by the LSW theory. A power law of the form ${\left(1+{kt}\right)}^{p}$ is used to fit the relative mean radius 〈r〉/〈r₀〉 vs time t data, with 〈r₀〉 being the mean radius at time 0. The data extracted from the coarsening simulations and the power-law fit are plotted in Fig. 10a. In the case of coarsening limited by inter-particle diffusion, the LSW theory predicts p = 1/3. Conversely, when surface kinetics is the rate limiting phenomenon the exponent is expected to be p = 1/2. With the assumption of dissolution kinetics k^diss = 10⁻¹¹ mol/cm² · s, and triple-line diffusion D_TL = 10³ nm²/s, the model's data produces best-fit values of the exponent in the range 0.21 ≤ p ≤ 0.25. The value of p scales lineraly with the voltage. This may be interpreted as the relative contribution of dissolution kinetics and mass transport to be dependent on the applied voltage. However, this hypothesis needs to be investigated further with a larger ensemble of particles. Also the the sensitivity on the parameters $\left({k}^{{diss}},{D}_{{TL}}\right)$ should be considered in future studies.

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Here we analyze the scaling of the coefficient k_i in the power-law fitting ${\left(1+{k}_{i}t\right)}^{p}$, with i = 0,...,3 for the four voltage-hold experiments. In Fig. 10b, the black line represents the value of ${K}_{{LSW}}^{D}(V)$ computed according to Eq. 13, normalized with respect of ${K}_{{LSW}}^{D}(0.95V)$. This provides the scaling with respect to the applied voltage, without taking into account the coefficients in Eq. 13, as those depend on the regime of coarsening, and on the parameters $\left({k}^{{diss}},{D}_{{TL}}\right)$. Similarly, the four large points in Fig. 10b represent the values k_i /k₀, with i = 0...,3 extracted from the power law fits shown in Fig. 10a. The scaling with respect to the voltage predicted by our model is one half of the one predicted by the LSW theory.

The assumptions of LSW may apply if the edge diffusivity and inter-satellite diffusivity are large compared to the Butler-Volmer current. In this case, each particle is more likely to be exchanging mass with the mean field produced by all of the particles. However, because of inter-particle diffusion (i.e., restricted to the region between carbon particles) is reduced, subsequent analysis may show that the power-law exponents are affected by the diffusion's reduced dimensionality.

On the other hand, if edge diffusivity is slow, each particle is interacting primarily with its proximate neighborhood: LSW would not apply. If the inter-satellite diffusivity and Bultler-Volmer kinetics are slow, the proximity effect would be exacerbated. A particle would not be interacting with the average characteristics of its average neighborhood–it would be interacting with the extreme values of the radii of its neighborhood

The coarsening kinetics for the voltage cycling tests is also marked in Fig. 10b to highlight the increase in coarsening rate (a factor of 3.49 and 5.6 respectively for TW and SW) with respect to the constant voltage test. However, the power law exponent for the square wave test is 0.17 and for the triangular wave is 0.18, both lower than the exponents 0.24 obtained for the voltage-hold tests (see Fig. 10a).

The evolution of the particle radius distribution over time is shown in the Fig. 11 for the cases with an initial normal PRD (images a, b) and bimodal PRD (images c, d). The time flow is represented by the color of the curve, going form blue (t = 0) to red. According to the LSW theory, a steady state normalized distribution function is approached asymptotically. For diffusion-limited coarsening, the steady-state normalized curve has the following equation (with ρ = r/〈r〉, and h₀ = 0.014419) ³⁸

$$\begin{eqnarray}&&h(\rho )=\displaystyle \frac{1}{{h}_{0}}\displaystyle \frac{{\rho }^{2}}{{\left(\tfrac{3}{2}-\rho \right)}^{11/3}{\left(\rho +3\right)}^{7/3}}\exp \displaystyle \left(\frac{3}{2\rho -3}\right)\end{eqnarray} \tag{ 14 }$$

and it is plotted in black in images b and d of Fig. 11

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The cases illustrated in Fig. 11 exemplify a trend that we observe also in the other examples that we analyzed, namely the normalized PRD tends to converge to a steady state curve that is similar to the LSW one but with the peak shifted to the left of ρ = 1. Also, a few particles larger than the cut-off value of 1.5〈r〉 are observed. Further investigation is needed to understand if a unique steady-state PRD can be defined for this system as a function of the relative contribution of the dissolution and transport kinetics, and what is the time-scale for convergence.

Under the assumption that the total volume is conserved, an approximate relationship can be derived between the number of particles and the average radius as they evolve upon coarsening. If the particle radius distribution at time t has converged to the LSW steady-state function (Eq. 14), the following relationship holds

$$\begin{eqnarray}&&E\left[{r}^{3}(t)\right]={\left(E\left[r(t)\right]\right)}^{3}\end{eqnarray} \tag{ 15 }$$

between the expectation value for the radius ant its third power. In other words 〈r³(t)〉 = 〈r(t)〉³. It follows that the total volume of Pt-nanoparticles can be expressed as a function of the mean radius and the number of particles

$$\begin{eqnarray}&&{V}_{{NP}}(t)=\sum _{i=1}^{N(t)}\displaystyle \frac{4}{3}\pi {r}_{i}^{3}(t)=N(t)\displaystyle \frac{4}{3}\pi \langle r(t){\rangle }^{3}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{V}_{{NP}}(t=0)=\sum _{i=1}^{{N}_{0}}\displaystyle \frac{4}{3}\pi {r}_{0i}^{3}={N}_{0}\displaystyle \frac{4}{3}\pi \langle {r}_{0}{\rangle }^{3}\end{eqnarray} \tag{ 17 }$$

Here N(t), N₀ are the number of particles at a generic time t and at time t = 0 respectively; r_i (t) is the radius of particle i = 1,...,N(t) at time t and r_0i is the radius of particle i = 1,...,N₀ at t = 0.

If we account for the mass of dissolved platinum at time t, and if not mechanism of Pt-mass loss is considered the total volume of Pt-nanoparticles is conserved. This leads to a scaling law between the evolution of the relative number of catalyst nano-particle and the average radius. Under the assumption that the relative average radius follows a power law scaling with exponent p, the number of particles scaling is

$$\begin{eqnarray}&&\displaystyle \frac{N(t)}{{N}_{0}}\sim {\left(\displaystyle \frac{\langle {r}_{0}\rangle }{\langle r(t)\rangle }\right)}^{3}\sim {\left(1+{kt}\right)}^{-3p}\end{eqnarray} \tag{ 18 }$$

In the case of a normal PRD, Eq. 15 is replaced by the following

$$\begin{eqnarray}&&\displaystyle \frac{E\left[{r}^{3}(t)\right]}{{\left(E\left[r(t)\right]\right)}^{3}}=1+3{\left(\displaystyle \frac{\sigma }{\mu }\right)}^{2}\end{eqnarray} \tag{ 19 }$$

with μ being the mean and σ the standard deviation of the PRD. For small values of σ, Eq. 16 can still be considered a good approximation. Numerical analyses show that Eq. 18 is approximately satisfied with the exponent in the range between − 3p and − 2.5p.

There are several ways to analyze the effect of the local environment on the competitive coarsening process. For instance, the correlation function f_sd illustrated in Fig. 12a is a distance weighted average of the relative size of the neighbouring particles. If all the particles sharing an edge with a given particle are larger in size than the particle itself than f_sd takes a positive value. Conversely, if a particle is surrounded by smaller particles, the value of f_sd for that particle will be negative. In Fig. 12a, the vertical axis shows the value of the spatial correlation function at the beginning of coarsening and the horizontal axis represents the initial radius. The clear segregation of red points (indicating particles that are decreasing in size) and black points (for growing particles) in the positive and negative regions of the plot suggests that the function f_sd is a good indicator for the system behaviour. Figure 12a is representative of all the constant-voltage simulations with uniform particle size distribution. The function f_sd appears also to have some correlation to the particle size, but this is dependent on the model parameters (i.e., Pt-diffusivity, kinetics of dissolution and surface energy). Figure 12b shows, for each of the 80 particles, the evolution of its radius over time with applied voltage of 1 V. The rainbow coloring scheme of the curves illustrates the relationship between the initial particle size at the coarsening behaviour. While it appears that the smallest particles tend to dissolve first (blue curves) and the largest particles to grow, the trend for the intermediate cases is less clear and it depends on the local environment.

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The results extracted from the simulations are not in accord with predictions from mean-field coarsening-theories. Fitting the mean radius to $\dot{\langle r\rangle }={\left(1+{kt}\right)}^{p}$ produces values p from 0.21 → 0.25 that scale linearly as the voltage varies from 0.95 → 1.02. The mean-field LSW theories would produce exponents of $\tfrac{1}{2}$ for interface-limited kinetics and $\tfrac{1}{3}$ for diffusion-limited kinetics. Furthermore, the mean-field exponent would not depend on voltage.

It is possible that the limited number particles in the simulation is not sufficient to obtain good statistics. This requires further study. It is also possible that the selected diffusivities do not fall in either of the two limiting cases of LSW. However, if this is the case, then p (which is $\approx \tfrac{1}{5}$) should be between $\tfrac{1}{3}$ and $\tfrac{1}{2}$.

It is also possible that the reduced dimensionality of diffusion in our microstructural model does not conform to the LSW model. If this is the case, then we conclude that LSW should not be applied to coarsening in composite microstructures in which diffusivity is small in one of the phases–a microstructural model such as ours is necessary. We expect that trends like power-laws could be obtained by repeated Monte Carlo experiments with our method.

The presented model is attractive because it divides the simulation method into two parts. The first part is the data structure (particles, satellites, edges) that depend on connectivity and morphological features of the microstructures. The second part is the physics of dissolution and diffusion that is embedded into the data structure.

Numerical tests performed on a representative particle ensemble indicate a strong sensitivity of the coarsening rate on the applied voltage and on the surface tension.

Processes that alter the surface tension or creates a diffusion barrier on the particle surface would reduce coarsening, however they may also affect catalytic efficiency. For instance, high relative humidity is generally favorable for the performance of PEMFC, however it tends for increase the water saturation of of cathode catalyst layer, allowing for higher Pt-ion mobility and Pt-mass redistribution. Surface treatments that increase cathode hydrophobicity could compensate for water condensation and would be beneficial for durability.

A combination of experiments and modeling is required characterize the structural changes in platinum-based catalyst particles, and distinguish between several catalyst deactivation mechanisms and transport pathways. Specialized liquid cell TEM techniques can provide direct imaging of motion and volume changes in individual catalyst nanoparticles during electrochemical cycling under realistic operating conditions. ³⁹ Measurements of nanoparticle dissolution in operando with high spatial and temporal resolution would provide the data necessary for accurate parametrization of this model.

Appendix. Data-structure for the Network Model of Ostwald ripening

Appendix. Limitations of direct surface energy calculation for nanoparticles

A plot of a directly calculated NP surface energy γ for different NP sizes demonstrates the coupling between r_NP and the corresponding γ. Here

$$\begin{eqnarray}&&\gamma =\displaystyle \frac{{U}_{{NP}}-{{NU}}_{{bulk}}}{4\pi {r}_{{NP}}^{2}}\end{eqnarray} \tag{ A·1 }$$

Figure 13 shows that γ is not an intrinsic property of the NP, but it varies with particle size, independent of the force field. The value of γ for a 0.7 nm particle is approximately 3.6 J/m², but drops to a stable value at large particle sizes (>2.5 nm) of 2.15 J/m² (COMB) or 2.52 J/m² (EAM). The variation of γ is the highest in the relevant size range of NP in PEMFC (1 − 3 nm). Similar behavior is observed if γ is calculated in units of [eV/atom] by replacing the denominator in Eq. 20 by the number of surface atoms in the NP (data not shown). The coupling between γ and particle size aligns with previous reports on crystalline NPs. ^33,40 Such ambiguity in evaluating γ bring difficulty is supporting continuum models of NPs evolution by atomistic simulations.

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