Grain-growth mediated hydrogen sorption kinetics and compensation effect in single Pd nanoparticles

Grains constitute the building blocks of polycrystalline materials and their boundaries determine bulk physical properties like electrical conductivity, diffusivity and ductility. However, the structure and evolution of grains in nanostructured materials and the role of grain boundaries in reaction or phase transformation kinetics are poorly understood, despite likely importance in catalysis, batteries and hydrogen energy technology applications. Here we report an investigation of the kinetics of (de)hydriding phase transformations in individual Pd nanoparticles. We find dramatic evolution of single particle grain morphology upon cyclic exposure to hydrogen, which we identify as the reason for the observed rapidly slowing sorption kinetics, and as the origin of the observed kinetic compensation effect. These results shed light on the impact of grain growth on kinetic processes occurring inside nanoparticles, and provide mechanistic insight in the observed kinetic compensation effect.


Grazing Incidence X-ray Diffraction (GIXRD) of as-deposited Pd disks
Supplementary Figure 1. XRD pattern of as-deposited Pd nanodisks, which reveals average crystallite size of 10 ± 2 nm.

Vacuum setup schematics
Supplementary Figure 2. Schematics of the vacuum setup used in the experiments. A customized Linkam temperature-controlled vacuum chamber was positioned on a motorized stage on the upright optical microscope.

Extraction procedure for t50
In order to extract t 50 of the signal we used a function for mid-reference level crossing for bilevel waveform (Supplementary Figure 3).
Supplementary Figure 3. Example of a H2 absorption trace, where the Matlab "midcross" function is used to extract t 50 (cross outlined with red circle). The vertical line indicates time stamp for introduction of H2.

Measurement scheme for mixed T-sweep
Supplementary Figure 4. Order of measurements of t50 both for H2 absorption and desorption in mixed T-sweep samples. The results of the first three subsequent measurement points at 303 K are shown in Figure 1 in the main text.

t50 vs. T plots fitted with NLLS
Supplementary Figure 5. t50 vs. T plot (crosses) for 24 single Pd particles from 1 st to 5 th T-sweep (left to right, with mixed T-sweep according to Supplementary Figure 4) fitted with non-linear least squares regression (NLLS) for each individual particle (colored lines). The upper and lower panels correspond to absorption and desorption respectively.

Arrhenius plots fitted with LLS
Supplementary

Arrhenius parameters extracted with LLS and NLLS methods
Data were analysed with both linear (LLS) and nonlinear least squares regression (NLLS) methods in order to see whether there are significant differences between the results 2 . In our case, both methods result in similar trends (

Goodness-of-fit statistics for LLS and NLLS methods
The goodness-of-fit of a model describes how well it fits the set of observations. The following are the plots with goodness-of-fit statistics for each T-sweep, which include sum of square errors (SSE), R-square, adjusted R-square and Root Mean Squared Error (RMSE). These values were extracted using the Matlab Curve Fitting Toolbox™ software.
SSE measures the total deviation of the response values from their fit. A value closer to 0 indicates that the model has a smaller random error component, and that the fit will be more useful for prediction. R-square measures how successful the fit is in explaining the variation of the data, or in other words, it is the square of the correlation between the response values and the predicted response values. R-square can take on any value between 0 and 1, with a value closer to 1 indicating that a greater proportion of variance is accounted for by the model. The adjusted R-square uses the R-square statistic defined above, and adjusts it based on the residual degrees of freedom. The residual degrees of freedom is defined as the number of response values n minus the number of fitted coefficients m estimated from the response values (v = rv -fc). v indicates the number of independent pieces of information involving the rv data points that are required to calculate the sum of squares. The adjusted R-square statistic can take on any value less than or equal to 1, with a value closer to 1 indicating a better fit. RMSE is also known as the fit standard error and the standard error of the regression. It is an estimate of the standard deviation of the random component in the data, and just as with SSE, a mean square error value closer to 0 indicates a fit that is more useful for prediction.
Supplementary Figure 10. Goodness-of-fit statistics for the 1 st T-sweep as shown in Supplementary Figure 4. The statistics are presented for both methods that were used to extract Arrhenius parameters, i.e., least square linear regression (LLS -red circles for absorption, blue circles for desorption) and nonlinear least square regression (NLLS -black dots).

Distribution of plateau pressures at absorption and desorption
In addition to the kinetic measurements, it is instructive to assess the evolution of the thermodynamics of hydrogen absorption and desorption in our samples. Following this line, we have measured sorption isotherms at 303 K (Supplementary Figure 11) for a sample comprised of an array of 24 single Pd nanodisks of the same size as in kinetic measurements, which before the measurement were not exposed to any (de)hydrogenation cycles. The obtained data were then compared with a corresponding isotherm measurement on a sample that had been cycled 41 times in kinetic measurements prior to the isotherm measurement (i.e., sample Tmix after kinetics measurements). We use separate samples for this purpose, i.e., we cannot use asdeposited sample that underwent isotherm measurement to cycle it in kinetics measurementsthe kinetics results of such sample will be different, since exposure to H2 during an isotherm measurement of an as-deposited sample will inevitably change it, and therefore it cannot be used for comparison with a sample that was deposited with Pd and then cycled directly (without prior isotherm measurement). The samples are then measured not in vacuum, but in a gas flow mode (i.e., at atmospheric pressure), with step-wise increase/decrease in hydrogen partial pressure, and at each pressure step there is dwelling time in order to allow the particles reach stable state at this pressure step. From the isotherm measurements we extracted the plateau pressures for absorption and desorption (Pabs and Pdes) for each particle and observe sizable increase in hysteresis for the cycled sample, as well as a larger spread in Pabs and Pdes values for the individual particles. This distinct increase in hysteresis further supports the idea of grain growth in the particles upon cycling 6 .
Supplementary Figure 11. Individual normalized intensity (Inorm) and hydrogen partial pressure isotherms at absorption (left) and desorption (right) for (a) as-deposited sample and (b) sample cycled 41 times prior to isotherm measurements. (c) Distribution of (left) desorption and (right) absorption plateau pressures measured at 303 K on a sample in as-deposited condition (darkgrey) and after 41 (de)hydrogenation cycles (light-orange). Cycling with hydrogen clearly increases the hysteresis and the spread in plateau pressure values, which confirms grain growth.

CQF analysis
This section describes the analysis developed by Griessen et al. 1 (Eq. 28-35 therein), which we applied to characterize the compensation effect observed between Arrhenius parameters in our data. The analysis requires calculation of parameters such as Compensation Quality Factor (CQF), which depends on the number of samples in the measurement (in our case, number of particles measured in one T-sweep (N = 24 or 180), coefficient of determination value (R square ), as well as variance in and covariance between Arrhenius parameters. Analytically calculated variance in ln(t 50 ) for the set of the N investigated particles allows to determine the temperature T min at which the variance of ln(t 50 ) reaches a minimum. The variance of ln(t 50 ) at T min is a direct measure of the degree of coalescence of the Arrhenius plots. The ratio of the variance of ln(t 50 ) at T min normalized to the largest experimentally measured ln(t 50 ) variance defines a CQF that characterizes quantitatively the extent of the crossing region of Arrhenius lines. The CQF is by definition unity for perfect compensation (T min = T isokin , where T isokin is the isokinetic temperature that corresponds to the slope of the Constable plot (i.e., the temperature at which all the particles in the specific measurement set have the same rate of reaction 1 ) and tends towards zero when the Arrhenius lines do not come close to a single crossing. The calculated value of CQF is also compared to a threshold value , which depends on N and the choice of the confidence level (i.e., level of certainty, %). According to this analysis, if CQF < at the chosen confidence level, an observed compensation effect is a statistical artefact. This is the case for our absorption and desorption data if we analyse each T-sweep separately at a 99.5% confidence level (see Supplementary  Figures 6-8 where there is no well-defined crossing of Arrhenius plots for each of the independent T-sweeps). Accordingly, when we apply this analysis to the data presented in Fig.  2 in the main text, i.e., the 1 st decreasing T-sweep for 24 single particles, CQF is lower than (at 99.5 % confidence level) in both cases. This implies that if analysed for this specific data set, the observed compensation effect is of statistical origin, despite Cremer-Constable plots showing high level of correlation, with relatively high R square ≥ 0.98, both for absorption and desorption data (Supplementary Figure 14). However, as we discuss in the main text by invoking a larger set of particles to increase N in this analysis (180 vs. 24), in fact, the compensation effect can be traced back to particle-specific grain structure prior to the very first hydrogenation and therefore has non-statistical origin (SI Section 13). Similarly, we have also applied this CQF analysis to the scenario where we include whole series of T-sweeps with specific T-sweep directions (i.e., not just a single T-sweep) to maximize the grain growth effect (see stars in Supplementary Figures 15-17 d, e). Also, this analysis then reveals a non-statistical origin of the compensation effect for desorption with 99.5% confidence level for "T-sweeps up and down", as well as for absorption with "T-sweep down", thereby identifying grain growth as the physical mechanism behind the observed compensation effect. We attribute the failure of the analysis to identify the compensation effect as non-statistical for the case of absorption for "T-sweep up" to the larger experimental error in absorption measurements, since the absorption process tends to be much faster than desorption, which leads to less accurate t 50 data. This is especially pronounced for measurements at higher T, where we are close to the time the resolution limit of our instrument. For the mixed T-sweep case, the CQF is low also for desorption because the Arrhenius parameters tend to oscillate between different T-sweep directions, which also implies a higher ratio between minimum and maximum of t 50 variance, and therefore by definition leads to a low CQF value.

Correlation of a kinetics slowing factor with Ea
The slowing factor (SF) is defined as the ratio of the latest t 50 measured at 303 K to the first t 50 measured at 303 K. All three samples used for sweeps Tup, Tdown and Tmix were pre-cycled 3 times with H2 at 303 K before sets of T-sweeps for extraction of Arrhenius parameters were performed. Similar to Fig. 6 in the main text, where SF versus Ea values at the first T-sweep is plotted, in Supplementary Figure 18 we plot SF vs. Ea for all the other corresponding sweeps not shown in the main text.
Supplementary Figure 18. Slowing factor (t50 (last) / t50 (first) at 303 K) versus activation energy Ea at the 1 st sweep for samples measured according to scheme Tup (left panel, red data) and Tdown (right panel, blue data) at absorption (upward pointing triangles) and desorption (downward pointing triangles). For Tup SF = t50 (53)/ t50 (1) and Tdown SF = t50 (43)/ t50 (1), where numbers in parentheses indicate the cycle number at which the corresponding last and first measurement at 303 K was performed for each sample.

Data set size and CQF values
To illustrate the influence of sample size on the calculated CQF value within a single T-sweep, we use the dataset with 180 particles measured using a Tdown sweep, where the correlation between slowing factor and activation energy was the most pronounced ( Fig. 6b in the main  text). For this purpose, we randomly divide the dataset of 180 particles into 7 subsets each consisting of 25 particles, where the 1 st set includes particles 1 to 25, the 2 d set -particles 26 to 50 and so on, while the 7 th set includes particles 150 to 175. Particle numbers indicate their position on the sample. Then we plot corresponding SF vs. Ea (Supplementary Figure 19a, b) for each of the particle subsets. We see that with fewer particles included in the analysis it becomes difficult to see the correlation between the two parameters. We also calculate the CQF value for each of the 7 subsets and compare it to CQF value of the entire set of 180 particles (Supplementary Figure 19c). The comparison shows that depending on the constituent nanoparticles in the subset, the CQF value can be greatly different than the value of the entire 180-particle set both for absorption and desorption, highlighting the importance of large data sets (N-value) if this analysis is to be reliably applied.
Supplementary Figure 19. Slowing factor, SF = (t50 (43) / t50 (1) at 303 K vs. Ea, obtained from the 1 st T-sweep for a sample of 180 particles measured according to Tdown scheme, divided into 7 sets of 25 particles, at (a) absorption and (b) desorption. Numbers in parentheses for SF indicate the cycle number at which the corresponding last and first measurement at 303 K was performed for each sample. (c) CQF value calculated for each of the 7 subsets (triangles) and for the entire 180-particle set (stars) in relation to the threshold level at 95, 99 and 99.5 % confidence levels (dotted, dashed and solid lines, respectively and according to number of particles in the set, i.e., 25 and 180) according to Ref. 1 (d) CCD image of the sample with 180 particles with black and white boxes indicating corresponding particle subsets.