Small Angle X‐Ray Scattering Study for Investigating the 3D Packing Structure of Pt Catalysts on Gd‐Doped CeO2 Supports for Fuel Cells

A 3D structural model for fuel cell catalysis systems is constructed, which consist primarily of Pt and CeO2 nanoparticles, to fit observed small angle X‐ray scattering (SAXS) patterns by using the reverse Monte–Carlo (RMC) method. The observed SAXS patterns are well reproduced by those of the simulations. Pore size distributions using the obtained structure models are compared with the Barrett–Joyner–Halenda (BJH) analysis for the nitrogen gas adsorption data, and the lower quartiles and medians of the pore diameters are reasonably consistent. In addition, analysis of the SAXS patterns indicates that the number of nanometer‐size Pt particles is much smaller than that of the introduced amount. This suggests that most Pt particles are not uniformly distributed in the catalysis system.


Introduction
3][4][5][6][7][8][9][10][11][12][13][14][15] The most critical component of a PEFC is its catalysis layers on both sides of the polymer electrolyte.The layers must have good electric conductivity, as well as a high gas diffusion coefficient.The former characteristic is important for supplying fuel H 2 and O 2 gases and discharging resultant H 2 O molecules.Shi, et al. have proposed superior oxygen reduction catalysis on the bases of Pt nanorods on Gd-doped ceria (GDC) supports. [16]Transmission and scanning electron microscopy (TEM and SEM, respectively) are usually used to investigate their microscopic structures.They can provide precise DOI: 10.1002/adts.202300713information about the local structure of materials, such as how the nanometer-size Pt catalysts combine with GDC particles, how the GDC particles connect to each other, and so on. [7]However, if we try to study the complex, aggregated 3D structure of entire layers of a catalyst, which should be critical to determine electric conductivity and gas diffusion coefficients, the materials can be easily destroyed by sample preparation for those measurements.In addition, it is very difficult to construct a real 3D structure, because TEM and SEM can only provide 2D images of very thin specimens or the surface.
Small angle X-ray scattering (SAXS) is a powerful technique for investigating nanometer-scale structure without destroying specimens because X-rays have large penetration length and transmit through materials as is.The X-ray scattering intensity is calculated from the Fourier transform of the electron density of materials, and the calculation is straightforward when the electron density is known.According to the reverse Monte Carlo (RMC) procedure, [17,18] positions of the constituent particles are shifted randomly and SAXS patterns are calculated to fit the experimental pattern.This procedure is repeated many times (typically 10 7 -10 8 iterations) until there is no further improvement in the match between calculated and experimental patterns.The constructed 3D model enables us to simulate precise structural features of complex materials.This technique is used to study structures that are easily destroyed by sample preparation, such as block polymer micelles, [19] aggregate formations of nanoparticles in solvents and polymer nanoparticles, [20] and to investigate magnetic nanochain formations under a magnetic field. [21]However, these approaches were introduced to avoid problems such as the effect of particle size distribution, computational cost in all interparticle pair calculations, and the low-q oscillations caused by the finite box size effect.For example, according to Olds's study [22] of efficient algorithms for calculating small-angle scattering from large model structures, the key to speeding up the calculation was to reduce the cost in all interparticle pair calculations by using a number density function with the same form factor.In our previous study, [23] we arrived at a simple solution that uses individual form factors for the short-distance correlations, the number density function for long-distance correlations, and the average density in distances longer than half the cell size.In this study, we have constructed a 3D structural model that consists of primary Pt catalysts on GDC support nanoparticles, to evaluate the coordination number of Pt particles, and pore size distribution.
First, we explain how to construct the 3D model of the Pt/GDC catalysis system for the SAXS simulation.Next, a brief review of our RMC calculation for matching experimental SAXS patterns is presented.Then, we present analysis methods for the constructed 3D structure and their results.Comparison of the obtained results with that of the gas adsorption method and TEM are presented.Finally, the potential of the present nondestructive method for investigating complex 3D structures, which are easily destroyed by any of the sample preparations, is discussed.

Principle of the Calculation
We propose real space modeling using the RMC procedure. [23]he model contains a considerable number of primary particles in a cell with size L × L × L, and the procedure searches for appropriate positions of all the particles in the calculation cell to fit the observed SAXS pattern.SAXS intensity is calculated by the following modified Debye-equation: where, q = 4  sin is the scattering vector,  is the X-ray wavelength, and 2 is the scattering angle.F i (q) is the form factor of the ith primary particle and r ij is the distance between the ith and jth primary particles.N is the total number of primary particles and L is the cell size of the model.We have assumed when interparticle distance r is longer than R, the individual form factor F i (q)F j (q) can be replaced by its average , and the local number density . (r) can be assumed to be the average density  0 at large distances, where r ≥ L 2 .In other words, we must select a cell size L that is large enough so that (r > L 2 ) becomes a constant value  0 = N∕L 3 .We also take into account the resolution of a real X-ray instrument used for the measurements, Δq = Δ   , which is determined by the incident beam divergence and resolution of the detecting system, where Δ is the convoluted angular resolution of the measurement system.For simplicity, Equation (1) can be expressed separately in two parts, inside/outside of the cell, by using the distance L 2 as follows: I eu (q) = I in (q) + I out (q) (2) r 2 H (q, r) dr (4)   where, H(q, r) ≡ sin(q r) q r e − Δq 2 r 2 2 . To analyze catalysis systems, we have to extend the previous formulation to make it capable of treating multi-constituent systems by introducing M different types of primary particles as follows: where, N  and N  are the total numbers of type  and  primary particles, respectively, in the considered calculation volume.
⟨F  (q)F  (q)⟩ = 1 , and   (r) is a local number density of type  particle relative to type  particle.When r > L 2 , the number density   is assumed to be the constant value √ N  N  ∕L 3 .To minimize calculation time, we have introduced a threshold distance R  at which the large number of primary particles are included in the volume Δv = 4R 2  Δr and the individual form factors F i (q)F j (q) can be replaced by their average ⟨F  (q)F  (q)⟩.
In the present case, the constituent primary particles are Pt catalysts and GDC supports, which are indicated as P and C, respectively.The typical diameter of the former is much smaller than that of the latter, so we have modified Equation ( 5) by introducing threshold distances R PP , R CC , and R PC for the present case: Estimation of the size distribution of GDC particles by comparing the scattering intensity of the isolated-particle simulation and observed S-0 data in the q regime from 0.6 to 2 nm −1 .

3D Structural Model
We have introduced two kinds of primary-particles, Pt catalysts, and GDC, for the present catalyst structure of a fuel cell.To create a model structure, it is necessary to determine the particle size and the volume fraction in the simulated cell.GDC particle size distribution parameters were determined by matching the observed S-0 pattern in the high-q regime (0.6 < q < 2 nm −1 ) as shown in Figure 1.The obtained average diameter ( dGDC ) and the coefficient of variation (CV GDC ) of the primary particles were found to be 11.86 nm, and 0.305, respectively, assuming the particle size distribution is log-normal.To determine the particle size distribution of Pt catalysts, we used TEM images at the selected points to extract Pt particles, as shown in Figure 2. The extracted particles size was fitted with a log-normal distribution and determined ( dPt , CV Pt ) as shown in Table 1.It should be noted that the selected points of the TEM images were obtained from locations where the specimens were thin enough for electrons to penetrate, and they may not be representative of the whole structure.The weight ratio of the Pt catalyst (w Pt ) was determined by the inductively coupled plasma-atomic emission spectroscopy (ICP-AES) analysis and the total volume pores (v Pore ) was directly calculated from the volume of adsorbed nitrogen gas at the highest relative pressure, as shown in Table 1.Approximate values of volume fraction, that of Pt catalysts V Pt , GDC supports V GDC , and pores V Pore , could be estimated by the following equations: For the RMC simulation study, we filled GDC and Pt particles based on the above considerations.They are created using a weighted random number generator according to those size distributions until the volume fractions of the particles reach the above-calculated values in a cell that is the size of a 400 nm cubic box.The observed minimum scattering vector (q min ) is 0.02845 nm −1 and the maximum distance (r max ) is 2∕q min ≈221 nm.According to Equation (1), the simulation cell size (L) must be smaller than 2r max to avoid structural overestimation at long distances.As a result, here we use a 400 nm simulation cell size.The resultant total number of GDC particles (N C ) were 4 852, 11 142, 10 084 and 10 961 for S-0, S-20, S-40, and S-50, respectively.From the above volume fractions of the Pt particles V Pt , the total number of particles are calculated as 264 067, 291 161, and 249 957, for S-20, S-40, and S-50, respectively.The initial positions of the GDC particles were selected by sets of uniform random numbers (x C , y C , z C ).The Pt particles were assumed to touch the surface of the randomly selected GDC particles.An additional assumption was to avoid overlapping particles during the next RMC simulation.
We have adopted a goodness-of-fit between the simulation and the experimental data to optimize the constructed structure by using the weight function w i as follows [23] : where, I Exp (q i ) and I Sim (q i ) are the experimental and simulated scattering intensities, respectively, at wavenumber q i .In other conditions for Equation ( 6), the threshold distance R PP , R CC , and R PC are 50, 200, and 200 nm, respectively.The differences in the scattering intensities between R PP = 50 nm and R PP = 200 nm were confirmed to be less than 3% for all samples, which saved a lot of computational time.
Each RMC iteration proceeds as follows.1st step: randomly select particles and modify their positions; 2nd step: evaluate the goodness-of-fit,  ′ , after the movement; 3rd step: compare this value with the value before the movement .If  ′ < , then the model including the modified positions is accepted.Since Shi, et al. have reported that all of the GDC particles were connected due to partial sintering and neck formation with nearest neighbor particles to form a network microstructure, [16] we adopted that an additional assumption when we modify a GDC particle position was to touch a surface of the randomly selected GDC particle.Experimental and simulated scattering intensities with their percent error (ΔI (q) = I Sim (q)∕I Exp (q) − 1) are shown in Figure 3.However, simulated patterns (broken lines) for S-20, S-40, and S-50 did not match the experimental patterns.Specifically, the simulated intensities are higher than the experimental intensities in the q > 1 nm −1 regime, where Pt particle contribution is dominant.Therefore, we tried to find the optimum number of Pt nanoparticles for the RMC simulation by changing their number as shown in Figure 4, and found that  shows a clear minimum at a certain number of Pt particles for all samples.By using the Pt volume fractions that give minimum  values, the simulated SAXS patterns are well in accord with those of the experimental patterns over the entire q regime.However, we noticed that the resultant best fit volume fractions of Pt catalysts are very much less than that of the original values, as 1.6 to 0.8 vol%, 3.9 to 1.0 vol%, and 7.4 to 0.8 vol%, for S-20, S-40, and S-50, respectively.The result suggests that most of the introduced Pt catalysts are not uniformly distributed, especially for the higher volume fraction case.It is emphasized that the SAXS pattern can have information concerning the ratio of the volume fraction of smaller and larger diameter primary particles if the difference of diameters of those particles is sufficient, such as in the current Pt catalysis  Structural models were visualized using the computer program VESTA. [24].and GDC supporter system.The resultant constructed 3D structures obtained by the present SAXS-RMC modeling and slicing images (40 nm thick) are shown in Figure 5.It is noticed that the number of Pt particles located on GDC particles decreases with increasing Pt concentration.This tendency is also seen in the TEM images of Figure 2a.To evaluate a coordination number n i of Pt particles located on the i th GDC particle, the diameter of the i th GDC particle was slightly expanded (specifically, multiply the diameter by 1.01) and counting number of overlapped Pt particles to set the n i value.The resultant histograms P(n) are shown in ∫ nP(n)dn are 12.8, 3.5, and 2.0, for S-20, S-40, and S-50, respectively, clearly indicating decreasing coordination number with increasing introduced Pt amount, quantitatively.In particular, for S-40 and S-50, it should be noted that a number of GDC particles are not coordinated (n = 0) with Pt particles as shown in Figure 6.

Pore Size Distribution
The 3D configuration of the particles obtained by SAXS enables us to investigate not only the electrical conductivity of catalysis but also the size and connectivity of the pores, which are the key features in gas mobility.In this section, we introduce a quantitative analysis of the pore size distribution from the structural model based on the experimental SAXS data and compared it with that from gas adsorption analysis.
Many analysis methods, mainly for 3D structures obtained by computed tomography, [25][26][27][28][29] have been proposed for evaluating  quantitative features of such complex structures.Typically, the observed image is transformed to binary data that assigns grid point values (GPV) to their pixels; for example, an inside object (particle) is set to 0, and an outside object (pore) is set to 1.The distance map (DM) can be calculated by the Euclidean distance transformation from the original binary GPV. [30,31]Hildebrand and Rüegsegger developed a spherical volume-weighted distance transformation in order to investigate the thickness of objects [32] which is named "Local Thickness (LT)."An example of an LT transformation for a 2D simple case is shown in Figure 7.This analysis is calculated using the software ImageJ [33] with their plugin for "Local Thickness" analysis. [34]Figure 7b shows a color map of the LT transformation for a ball and stick, and a histogram of the map is shown in Figure 7c.It has maxima at 250 and 400 pixels, which correspond to the width of the stick and the diameter of the ball, respectively.The LT transformation can be applied to extract the size distribution quantitatively.It is applicable not only for such simple structures but also for general, complexshaped objects.Recently, we reported the comparison of the pore size distribution derived from the structural model by matching the experimental SAXS data of silica aerogel, the result of the nitrogen gas absorption experiment, and the result derived from TEM images. [35]These medians of the pore size distributions derived from three experiments were in good agreement.
The actual LT transformations are performed at 200×200×200 grid points on the structural model in Figure 5 and the obtained 3D LT transformed images (color map of pore size) are shown in Figure 8.It should be noted that the observed SAXS pattern is simple and does not have enough information to determine the exact 3D structure.Therefore, the obtained model must be interpreted as one of the possible structures that match the experimental SAXS pattern, and we have to extract proper statistical features from the obtained 3D model. [23]One of the key statistical features of the system can be expressed by the pore size distribution.It has been calculated from RMC simulation runs starting from three initial independent random structures and almost the same results were obtained.The cumulative pore volume (C(d)), obtained by SAXS and Barrett-Joyner-Halenda (BJH) [36]  Even considering this uncertainty in the BJH results, the obtained Q 1 and Q 2 are reasonably consistent with each other, except for the result of the S-0.This may be because that S-0 has a relatively large pore size in the results of BJH, so the observed minimum wavenumber q min of SAXS may not be small enough.

Conclusion
We have developed a SAXS-RMC modeling method applicable to multi-constituent systems for studying the 3D packing structure of Pt/GDC catalysts and successfully constructed 3D models that match the experimental SAXS patterns.It should be noted that the experimental SAXS patterns are sensitive to the ratio   of the number of particles between GDC and Pt.We have performed RMC simulations with varying numbers of Pt particles and found there are clear optimum numbers.The results suggest the uniformly distributed Pt particles are much less than the introduced amount, and also enable us to estimate Pt particle coordination number on GDC support.Pore size distributions of the obtained 3D-structural models were compared with the BJH analysis of the nitrogen gas adsorption data, and the lower quartiles and medians of pore diameters were reasonably consistent with each other.Of course, the obtained structural models are only one possible structure and must be interpreted statistically, but the 3D structural model enables us to evaluate physical properties, such as diffusivity of gasses, electric conductivity, and so on, of those complex systems.

Experimental Section
Synthesis of Catalyst Powder: The GDC support was prepared by the flame oxide-synthesis method; details of the method were described in an earlier report. [16]The Gd dopant concentration was 20 atom% as the nominal value.The as-prepared support powder was annealed at 800 °C for 4 h in air with the use of a rotary kiln furnace.The annealed GDC powder was termed S-0.The Pt nanoparticles were loaded by the colloidal method; details of the method were described in an earlier paper. [16]The as-prepared powder was heat-treated at 400 °C in N 2 atmosphere for 1 h and then reduced to 150 °C in 10% H 2 (N 2 balance) for 1 h, followed by being cooled to room temperature in the same atmosphere.The projected Pt loading on GDC supports were 20, 40, and 50 wt.%.In order to increase the Pt loading value, the samples were termed S-20, S-40, and S-50.
Characterization: The pore size distribution was derived from the nitrogen gas adsorption measurement (Autosorb-iQ, Anton-Paar GmbH, Austria) using the BJH method.The Pt content in the resultant catalyst was quantified by inductively coupled plasma-mass spectrometry (ICP-MS, 7500CX, Agilent Technologies Co. Ltd., USA).Transmission electron microscopy (TEM) images of the synthesized catalyst were taken using a H-9500 microscope (Hitachi High-Tech.Co., Japan).SAXS measurements were performed by a laboratory SAXS measurement system (Rigaku NANOPIX), which employs a high-performance semiconductor detector (Rigaku HyPix-6000), a high-brilliance point-focus X-ray source (Rigaku MicroMax-007 HFMR) focused by a multilayer confocal mirror (Rigaku OptiSAXS), and low parasitic scattering pinhole slits (Rigaku ClearPinhole).The SAXS patterns for samples S-0, S-20, S-40, and S-50 were collected in transmission geometry.The q space resolution of the instrument with the current measurement conditions was Δq = 0.0067 nm −1 .This is used for the construction of 3D structural models based on Equation ( 6) to fit the experimental SAXS profiles a) Particle size distribution parameters, the average diameter ( dPt ) and the coefficient of variation (CV Pt ), of Pt catalysts were determined from TEM images b) Weight ratio of Pt catalysts (w Pt ) was determined by ICP-AES analysis c) Total volume of pores (v Pore ) was directly calculated from the volume of adsorbed nitrogen gas at the highest relative pressure.

Figure 2 .
Figure 2. A typical TEM image (a) with the selected points to measure Pt particle size (b).

Figure 3 .
Figure 3. Experimental (upper solid lines) and simulated (upper dashed lines) scattering intensities with their percent error (lower solid lines).The simulated patterns with introduced Pt contents do not match the experimental patterns.

Figure 4 .
Figure 4. Optimization for Pt particle contents by varying Pt vol%.Experimental (upper black lines) and simulated (upper colored lines) scattering intensities with their percent error (lower solid lines) are shown in (a), (b), and (c) for S-20, S-40, and S-50, respectively.The  values versus V Pt are shown in (d), (e), and (f) for S-20, S-40, and S-50, respectively.

Figure 5 .
Figure 5.The resultant structural models obtained by RMC to match the observed SAXS patterns.Red and gray spheres for Pt and GDC particles, respectively.a) 3D structural models in the parallel projection.b) Slicing images in 40 nm thickness.c) Magnified images of (b).Structural models were visualized using the computer program VESTA.[24].

Figure 6 .
From the definition of P(n), the area of the histogram equals the total number of GDC particles N C , ∫ P(n)dn = N C .The obtained mean numbers n = 1 N c

Figure 7 .
Figure 7.An example of the local thickness (LT) transformation for a) 2D simple binary image consists of a stick and a ball.The LT transformation results in b) a color map and c) a histogram.
methods, with horizontal axis pore diameter (d), are shown in Figure 9.The statistical values (means D ave and the quartiles Q n ) are listed in the Table 2.It should be noted that the statical values of the BJH analysis results are unreliable because of the large influence of a few measurement points in the higher d regime with large volume changes.In particular, the result for S-0 shows relatively large volume changes (C(162) − C(54.6) = 91.8− 8.7 = 83.1 vol%) at the last two measurement points, are 54.6 to 162 nm in d.

Figure 8 .
Figure 8. 3D pore size distributions represented by color maps with a yellow isosurface level of 40 nm in diameter.

Figure 9 .
Figure 9. Pore size distribution curves.The BJH analysis result curves are shown in the black lines with the measurement data points in open circles.The average curves of three RMC simulations run for SAXS patterns are shown in the color lines.

Table 1 .
Structural Properties of Catalyst Powders.

Table 2 .
Means D ave (nm) and the quartiles Q n (nm) of the pore size distribution.