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67c477f081d2151a02bafb24 | 32 | The pressure-area isotherms. The pressure-area isotherms obtained from the DPD simulations at different temperatures are depicted in Figure , along with several experimental measurements, including our data and data from the literature. The experimental data produced by several groups show substantial deviations, as discussed in the introduction. When compared with simulation results, one needs to consider the differences in time and length scales. Our largest monolayer in the simulations is about 56 nm, which is still much smaller than the micrometer-sized kidney-shaped monolayer domains observed through imaging techniques (Figure ). Such large-scale structures may affect surface pressures at high areas per lipid, causing substantial non-uniformity that cannot be captured by a 50 nm film in simulations. As the area per lipid increases, the monolayer becomes more uniform, allowing it to be represented by small patches of films with periodic boundary conditions. On the other hand, at high compressions, the monolayers in experiments tend to collapse, reducing the surface pressure. However, in simulations, the periodic boundary conditions can prevent local deformations and lead to metastable states of high compression. |
67c477f081d2151a02bafb24 | 33 | As depicted in Figure , our simulation data at 293 K shows excellent agreement with several experimental data, including our experimental data, for π πΏ β₯ 53 π΄ 0 2 . At higher temperatures, our results are in fair agreement by being close to the experimental data range. The substantial deviations of the simulation results from experimental data occur at lower areas per lipid. These deviations may be attributed to the effects of small length scales and periodic boundary conditions, which cause simulation systems to exist in metastable states. Given that we used the same force field parameters at all temperatures, such temperature dependence within the range of experimental data is advantageous, which is not achieved by any of the existing coarse-grained force fields. |
67c477f081d2151a02bafb24 | 34 | DPPC bilayers undergo a gel-to-liquid crystalline transition at the main transition temperature π π =314 K. In the liquid crystalline phase, the lipid tails are disordered, in contrast to the gel phase, where the tails are ordered and align parallel to each other. The disordered liquid crystalline phase occurs at higher temperatures and higher equilibrium areas per lipid compared to the gel phase. The gel phase, characterized by highly ordered hydrocarbon chains and restricted molecular mobility, predominates at lower temperatures where entropic contributions to the free energy are minimized relative to cohesive interactions between lipid molecules. These characteristics also result in a higher membrane thickness for the gel phase compared to the liquid crystalline phase. We performed standard DPD simulations of the DPPC membranes in water at different temperatures and at different lateral stretching/compression, determined by the area per lipid in a leaflet. |
67c477f081d2151a02bafb24 | 35 | Figure provides snapshots of the bilayers at different π πΏ at temperatures 293 K, 304 K, and 324 K, while Figure depicts the corresponding membrane hydrophobichydrophilic interfacial tension πΎ π΅πΏ . As shown in Figure , bilayers can be compressed to negative surface tension values up to a point where deformation occurs. Negative surface tension indicates that the bilayer is overcompressed and tends to expand, leading to deformation. At this stage, the membrane area is no longer given by the lateral area of the simulation box. Figures and illustrate that both the deformation point and the extent of deformation are strongly temperaturedependent, and increase with temperature. Above the deformation point, the surface tension increases with area per lipid, nearly linear. However, below the main transition temperature 314 K, the tension area isotherm distinguishes the gel and liquid crystalline phases with different elastic responses, separated by the region of phase transition. In Figure , the bilayer with π πΏ = 52.6 π΄ π 2 at 293 K is in the gel phase, with high lipid order indicated by the red-colored tails. Further stretching leads to the formation of the liquid crystalline phase and eventually to completely disordered, thinner bilayers at high areas per lipid (see snapshot at π πΏ = 69. 7 π΄ π 2 ). Similarly, the bilayer with π πΏ = 55 Γ
2 at 304 K is also in the gel phase, transitioning to the crystalline phase upon stretching. In Figure , the interfacial tension curves at 293 K and 304K consist of a cusp region, which indicates a first order transition. Such a phase transition region is absent in the curve at 324 K, as the bilayer is in the disordered liquid crystalline phase at all values of π πΏ (Figure ). |
67c477f081d2151a02bafb24 | 36 | The equilibrium areas per lipid at which the πΎ π΅πΏ = 0 are found to be βΌ 52 π΄ π 2 at 293 K, βΌ 55 π΄ π 2 at 304 K and 62 π΄ π 2 at 324K. The corresponding experimental values vary within a certain range. In the gel phase, values reported by different groups include 47.9 Γ
2 (293K) , 50Γ
2 (294 K) , 48.6 Γ
2 (298K) , 48.6Γ
2 (300K) and 52.3 Γ
2 (298K). Similarly, the range of values reported for liquid crystalline phase at 323K spans 57 -71 Γ
2 . Our value of 62 Γ
2 at 324 K is in good agreement with the value 62.9 Γ
2 reported by Nagle et al and some of the atomistic and coarse grained force fields. Partial number density profiles π π (π§) along the normal z-direction, of different bead types in bilayer systems of equilibrium area per lipid at various temperatures are given in Figure . The bilayer thickness, estimated as the peak-to peak distance of the choline groups, π‘ ππ = 4.73 nm in the gel phase, is in agreement with the experiments. At 324 K, the thickness of the equilibrium liquid crystalline bilayer is 4.1 nm, also close to experimental result, while the thickness of the bilayer of π πΏ = 69.9 π΄ π 2 is 3.8 nm. In conclusion, our DPPC model reproduces most of the experimentally observed characteristics, with parameters transferable over the temperature range that involves the main phase transition behavior of the lipid. |
67c477f081d2151a02bafb24 | 37 | In this study, we analyzed the temperature-dependent behavior the monolayers of the major lung surfactant phospholipid component, DPPC, at air-water interfaces under lateral compression and expansion using dissipative particle dynamics simulations and complementary experiments. Simulating air-water interface is not standard in DPD and therefore we utilized our DPD gas model developed in a previous work. The gas phase is represented through computationally efficient fictitious beads that capture essential physical characteristics of the air-water interface through carefully parameterized exponential repulsive potentials, enabling accurate simulation of interfacial phenomena. A new specifically parametrized DPD model for DPPC molecules that reproduces interfacial properties was developed. To simulate monolayers at different temperatures, we introduced a temperature scaling approach based on matching surface tension of water, with parameters that are transferable across different temperatures. The effectiveness of our approach was demonstrated by accurately reproducing that surface tension of water at different temperatures. |
67c477f081d2151a02bafb24 | 38 | We demonstrated the temperature-dependent 2D phase behavior of DPPC monolayers simulated at various areas per lipid, spanning high and low lateral compressions. The monolayers exhibited LC and LE phases at high and low compressions, respectively, along with coexistence of the two phases at intermediate compressions. The phases were characterized using parameters representing local monolayer thickness and lipid tail order. At lower temperatures, the LE phase within the coexistence regime tended to form a gas phase, characterized by low-density regions containing lipids oriented parallel to the interface. This low-density regime suggests the potential coexistence of LE, LC, and gas phases at the triple point temperature (~290 K). As the temperature increases, the LE and LC domains become smaller and smaller, and mix upon reaching the critical point. |
67c477f081d2151a02bafb24 | 39 | Our computational model achieves quantitative agreement with experimental surface pressure-area isotherms at 293 K, successfully reproducing key features including the lift-off area, phase transition plateau, and collapse pressure within experimental uncertainty. This is significant, as both experimental and computational approaches have inherent limitations. Given that experimental data in the literature span a range of values, achieving quantitative agreement within experimental uncertainties is desirable. In a field where most computational approaches fail to reproduce DPPC pressure-area isotherms at different temperatures even qualitatively, our approach provides a robust model that captures pressure-area isotherms within the experimental data range, with transferable parameters across temperatures. The quantitative accuracy of the DPPC model is reinforced by its ability to reproduce temperature-dependent phase behaviors and structural characteristics of the DPPC bilayer. Our model can be readily extended to monolayers containing other phospholipids and cholesterol. Future work will explore mixed monolayers and the effects of nanoparticles. This work was supported by the National Science Foundation CBET grant # 2040302 and DMR grant #2138052. |
6511cfbb60c37f4f76705ace | 0 | Protein-protein interactions (PPIs) play a fundamental role in the functioning of biological systems. These interactions govern cellular processes, signal transduction, and the regulation of various biological functions. Understanding PPIs is crucial for unraveling the complexities of living organisms, shedding light on disease mechanisms, and finding potential drug targets. However, the experimental determination of PPIs is still a resource-intensive and timeconsuming endeavor. Consequently, computational approaches that can predict PPIs with high accuracy have gained increasing prominence in the field of bioinformatics. |
6511cfbb60c37f4f76705ace | 1 | Machine learning and deep learning techniques have appeared as powerful tools for predicting PPIs. These approaches use the vast amounts of biological data available, ranging from genomic and proteomic sequences to structural information, to infer and anticipate protein interactions. This research paper embarks on a comprehensive exploration of the application of both traditional machine learning algorithms and advanced deep learning architectures for PPI prediction. |
6511cfbb60c37f4f76705ace | 2 | The motivation behind this research stems from the pressing need to accelerate the discovery of protein-protein interactions. A precise understanding of PPIs can guide biologists and researchers in deciphering complex cellular processes, such as disease pathways, immune responses, and gene regulation. Additionally, accurate PPI predictions hold immense potential in drug discovery, as they enable the identification of novel drug targets and the design of therapeutics tailored to specific protein interactions. |
6511cfbb60c37f4f76705ace | 3 | This study delves into the development and evaluation of predictive models capable of discerning PPIs from a wide array of biological data sources. Through a series of experiments and analyses, we look to assess the performance of these models, find their strengths and limitations, and illuminate the path toward more accurate and efficient PPI prediction. |
6511cfbb60c37f4f76705ace | 4 | 2. To explore the potential of deep learning architectures, such as neural networks, for enhancing PPI prediction accuracy. 3. To evaluate the impact of integrating more biological data, such as gene expression data, on the predictive capabilities of these models. 4. To provide insights into the interpretability of model decisions and the visualization of predicted PPIs. 5. To discuss the applications of accurate PPI prediction in biological research and drug discovery. |
6511cfbb60c37f4f76705ace | 5 | The protein-protein interaction dataset used in this study was collected from the IntAct Molecular Interaction Database [cite: IntAct]. IntAct is a widely recognized and curated resource that supplies comprehensive information on experimentally verified molecular interactions. The dataset forms a diverse array of molecular interactions, encompassing a variety of species, experimental techniques, and interaction types. |
6511cfbb60c37f4f76705ace | 6 | In this section, we delve into the application of traditional machine learning algorithms to predict protein-protein interactions (PPIs). We explore the efficacy of Support Vector Machines (SVM) and Random Forest as representative models for this task. These models are widely used in bioinformatics due to their versatility and capability to handle complex biological datasets. |
6511cfbb60c37f4f76705ace | 7 | Deep learning has appeared as a powerful paradigm for capturing intricate patterns and representations in complex data, making it particularly well-suited for predicting protein-protein interactions (PPIs). In this section, we explore the application of deep learning architectures, specifically neural networks, to enhance the accuracy and predictive capabilities of PPI prediction models. |
6511cfbb60c37f4f76705ace | 8 | In this section, we present the experimental results of our protein-protein interaction (PPI) prediction models, including traditional machine learning models (Support Vector Machines and Random Forest) and deep learning models (Neural Networks). The results highlight the models' performance in distinguishing between interacting and non-interacting protein pairs based on the features extracted from the Intact dataset. |
6511cfbb60c37f4f76705ace | 9 | Deep learning models, while highly effective, are often perceived as black boxes. To enhance interpretability, we employed techniques such as Grad-CAM (Gradient-weighted Class Activation Mapping) to visualize which regions of input feature vectors were most crucial for model predictions. This allowed us to find key features driving the neural network's decisions. |
6511cfbb60c37f4f76705ace | 10 | ROC curves and Precision-Recall curves were generated for the neural network, supplying insights into its classification performance, especially in scenarios where imbalanced classes exist. These visualization techniques helped not only an assessment of model performance but also a deeper understanding of where the models excelled and where improvements could be made. They contributed to the interpretability of our PPI prediction models, making them more accessible and informative to researchers and domain experts. |
6511cfbb60c37f4f76705ace | 11 | One of the key challenges in protein-protein interaction (PPI) prediction is harnessing the wealth of biological data available from diverse sources and integrating it effectively into the modeling process. The integration of multiomics and biological data not only improves predictive accuracy but also provides a holistic view of the underlying biological mechanisms. |
6511cfbb60c37f4f76705ace | 12 | The development and application of PPI prediction models stands for a significant milestone in biology and computational biology. These models have the potential to revolutionize our understanding of biological systems, accelerate drug discovery, and usher in a new era of personalized medicine. As research in this field continues to evolve, its transformative impact on science and healthcare is poised to grow exponentially. |
6511cfbb60c37f4f76705ace | 13 | The future of PPI prediction is marked by exciting opportunities and formidable challenges. Researchers in this field have the potential to make significant contributions to our understanding of biology, drug discovery, and personalized medicine. By addressing the challenges and pursuing innovative directions, we can unlock new insights into the intricate web of protein interactions that govern cellular life. |
6511cfbb60c37f4f76705ace | 14 | The development and application of protein-protein interaction (PPI) prediction models stand for a significant advancement in the field of biology and computational biology. These models have not only improved our understanding of complex biological processes but also have far-reaching implications in drug discovery, personalized medicine, and biotechnology. By integrating diverse biological data sources, using deep learning techniques, and addressing ethical considerations, we have made substantial progress in unraveling the intricacies of PPI networks. |
6493bd4f853d501c004ac918 | 0 | Elastic properties of materials are fundamental physical properties that determine how materials respond to external stress and deformation . Accurately predicting these properties is critical for understanding the behavior of materials and their performance in various practical applications . Although the mechanical behavior of materials is influenced by microstructure, defects, and environmental conditions, single crystal models may offer valuable insights into their fundamental behavior, which can aid in the development of better materials. Nevertheless, the design of materials with exceptional elastic properties at an atomistic scale remains a persistent challenge in data-driven material discovery, given the vast chemical space and numerous possibilities available. |
6493bd4f853d501c004ac918 | 1 | In this context, the application of machine learning (ML) has been applied to many fields , including in materials science, where it has shifted the paradigm of data-driven approaches in predicting new materials with desired properties (inverse design) ML models can learn complex nonlinear relationships between the material composition and its properties, allowing for faster and more accurate predictions. Traditional ML models rely heavily on the feature representations of the chemical environment (e.g., nature of bond, composition, etc.), i.e., descriptors , which can limit their performance. However, choosing appropriate descriptors that accurately capture the underlying properties of materials is challenging and can limit the performance of ML models and leads to significant variability in their performance. Another constraint is the scarcity and heterogeneity of data impose additional challenges in training and validating ML models. |
6493bd4f853d501c004ac918 | 2 | The search for new materials with exceptional elastic properties has mostly relied on high-fidelity albeit computationally costly physics-driven models such as density functional theory (DFT) or molecular dynamics (MD) based structure search techniques followed by experiments in some cases for validating the predictions . ML models have been greatly accelerating this discovery process by predicting materials with target properties (e.g., elastic) by providing a fast and accurate alternative to the physics-based approaches. However, the typical application of ML models in predicting mechanical properties has been limited to only either a selected class of materials (e.g., alloys, glasses, etc.) or a few sets of ML tools. They also do not inform on the reason behind the selection of a particular ML tool across many available options such as kernel ridge regression (KRR) , Decision tree , Gaussian process regression (GPR) , Gradient Boosting (GB) , Random Forest (RF) , and Support Vector Regression (SVR) etc. Additionally, various datasets with variability in computational techniques used to compute the targeted properties have been reported in previous studies . The choice of feature selection thus varies across different studies, with differing emphasis on feature importance. This results in serious nonuniformity and interpretability issues. |
6493bd4f853d501c004ac918 | 3 | Addressing the above challenges require a comprehensive and standardized model benchmarking over a uniform and robust feature space, rigorous validation protocols, and a unified task-specific ranking of feature importance. This involves evaluating and unifying the importance of different features based on their ability to predict the target property across different models, rather than relying on ad hoc or generalpurpose feature selection methods. This can lead to more consistent and reliable results across different studies and datasets, which is important for building trust in the accuracy and reliability of ML models. |
6493bd4f853d501c004ac918 | 4 | While there exist a wide variety of regression models and decision tree-based searches, Graph Neural Networks (GNNs) have shown superior performance in predicting material properties with a large amount of training data required for learning flexible structural representations. GNNs, however, lack interpretability and do not work well with smaller datasets, which may limit their practical applicability . The unified task-specific ranking of features can also be utilized to enhance the performance of GNNs on smaller datasets. One way to enhance the accuracy of these models when dealing with limited data is by training them on features that are weighted based on their significance in contributing to the task. This approach can be particularly beneficial for practical applications where gathering data is costly. |
6493bd4f853d501c004ac918 | 5 | In this study, we present a workflow that constructs a feature space comprising physical properties and structural attributes for standardized benchmarking of ML models in predicting elastic properties, using a dataset of 10570 unique crystal structures from the Materials Project Database 40 , we evaluated the performance of eight different ML models on a subset of the feature space derived based on correlated feature importance using the MRMR algorithm . For the best-performing models in each case, we used SHAP (SHapley Additive exPlanations) derived feature importance to formulate a unified task-specific ranking of features. We next trained a GNN architecture, specifically the CGCNN 7 model, on a weighted feature space based on the overall ranking, which improved the performance of the CGCNN even with lower amounts of data for predicting the Bulk and Shear moduli. |
6493bd4f853d501c004ac918 | 6 | Our workflow comprises of 4 section sections (i) The featurization and Initial feature section, (ii) benchmarking of select popular ML and deep learning models, (iii) evaluation of feature importance, and (iv) Ranking of the features. In the Featurization section, we start by extracting the dataset comprising crystal structures and bulk and shear modulus of the dataset from the materials project database based on section criteria (see dataset section). The extraction of useful features used for the prediction tasks was done primarily using two python-based libraries-Pymatgen 44 and materials data mining package Matminer . A total of 71 feature vectors categorized primarily across two major classes (i.e., local and global), (Fig. ) |
6493bd4f853d501c004ac918 | 7 | were extracted from the dataset (see supplementary Table. 1). Local features were further expanded using different statistical measures (e.g., mean, and median, etc.) to get an overall representation for each configuration in the dataset leading to a total of 221 feature vectors. These vectors could potentially contain many redundant and irrelevant features for the specific prediction task. To alleviate this issue, an mRMR (Maximum Relevance and Minimum Redundance) algorithm was used to sort features based on their initial relevance score. A certain number of subset features (150 in our case) were selected and passed onto section 2, where a representative set of ML models are used for property prediction. We evaluate 8 different ML models, Linear regression (LR), K-Nearest Neighbor regression (KNN) , Support Vector Regression (SVR) , Random Forest (RF) model, Kernel Ridge Regression (KRR) , Gradient Boosting Method (GBM) , Gaussian Process Regression (GPR) and Multilayer Perceptron (MLP), for the benchmarking task. Additionally, a Graph neural network (GNN) CGCNN 7 was trained on the ranked feature space. All the ML models were implemented using the python-based library Scikit-Learn . Finally for all the trained models, the feature importance of each model was computed using a SHAP workflow in sections 3. |
6493bd4f853d501c004ac918 | 8 | The dataset for benchmarking comprised a total 10570 unique crystals from the Materials Project database with ~6000 ternaries, ~4000 binary, ~500 quaternary, and the rest of unary, and pentanary materials with around ~40% of the structures being cubic (see supplementary Fig. )). All the structures considered are having formation energy within 200meV above the convex hull and configurations with negative Vickers hardness or shear modulus were excluded. The reported elastic moduli were computed using density functional theory as implemented in the Vienna Ab Initio Simulation Package (VASP) software. |
6493bd4f853d501c004ac918 | 9 | Generalized Gradient Approximation (GGA) and GGA+U exchange-correlation functional while the DFT+U (Hubbard parameter) scheme is used for highly correlated systems . The elastic moduli were determined using Voigt-Reuss-Hill approximation . Primarily, two target variables -Bulk Modulus (K) and shear modulus (G), were used for benchmarking the performance of the ML models. |
6493bd4f853d501c004ac918 | 10 | However, materials with super hardness (Vickers hardness Hv >40 GPa) are of particular interest, and we use the criteria (K>300GPa and G> 150GPa) to identify likely super hard structures in our dataset. The geometric attributes of a material, such as atomic positions, interatomic distance, and bond angle, etc., have a direct influence on the bulk and shear modulus, especially for super hard materials. In contrast, chemical attributes alone are not sufficient to distinguish between different phases of a system with similar compositions. While some prior studies have overlooked these geometric attributes , others have included them in various forms . |
6493bd4f853d501c004ac918 | 11 | In this work, we employ an invariant Smooth Overlap of Atomic position (SOAP) as the representation of the geometric attributes, with the features denoted with an 'F' and subscript. To demonstrate, the correlation between these features and the bulk and shear modulus, we use PCA representation of SOAP feature vectors and mark possible super hard structures in 2D PCA space, as shown in Fig. ). These structures seem to form a clustered ridge in the PCA space, an indication to share geometric similarities. To better understand the relationship between the structural attributes (represented by a 24-dimensional SOAP feature vector) and the target variables (shear and bulk moduli), we computed the Pearson correlation coefficient (Fig. ), which ranges from -1 to 1 and indicates both the strength and direction of the relationship. Our analysis reveals a clear correlation between the G and K moduli and the SOAP feature vector, highlighting the importance of these structural attributes in predicting material properties. Interestingly, we found that the correlation between the SOAP vector and the shear modulus was stronger than that for the bulk modulus. This can be explained by the high directional dependence of the shear modulus in contrast to the bulk modulus, which has a more volumetric dependence. |
6493bd4f853d501c004ac918 | 12 | Any physical or chemical attributes of a crystalline system can primarily be categorized into two classesattributes that represent the crystal structure on a macro scale (e.g., Formation Energy, space group.) (i.e.,) global properties, and the attributes that belong to the local environment of an atom (e.g., interatomic distance, and elemental properties, etc.), local properties as shown in Fig. . We split the global attributes into 3 categories, as shown in Fig. ), (i) six property-based attributes (e.g., density, melting point, and bandgap, etc. (ii) six stoichiometry-based attributes (e.g., compositional fractions and their norms) (iii) seven global structural based attributes (e.g., space groups, point group, and Laue symmetry.). A total of 19 global features were selected (see supplementary information Table .1 and supplementary note 1). The local feature set however consists of two primary classes -structural and local elemental attributes. The structural attributes consist of 24 invariant SOAP (smooth overlap of atomic position) feature vectors. |
6493bd4f853d501c004ac918 | 13 | The local elemental attributes can further be subdivided into properties related to the periodic table (e.g., block number, and row number group number, etc.) (8 features), and local elemental properties (e.g., electronegativity, band gap, and magnetic momentum, etc). These local elemental properties can further be subdivided into three major classes (Fig. ), (i) properties related periodic table (8 features) (ii) properties related to electronic structure (10 features) (iii) properties of chemical species (6 features) (see supplementary information Table .1). We note that the aggregation of the local elemental attributes was done using 6 statistical measures such as 'minimum', 'maximum', 'range', 'mean', ' 'avg-dev', and 'mode' to obtain an overall representation for a single configuration (see supplementary note 1). A simple "mean" of the SOAP feature vector (F1, F2, β¦) was used to construct structure-level representation from the atomlevel representations. . Overall, this leads to a total of 221 feature vectors. |
6493bd4f853d501c004ac918 | 14 | The mRMR (Maximum relevance and Minimum redundancy) algorithm ranks the features that are highly correlated to the target having a low correlation between themselves. The F-statistic is used to calculate relevance to the class, and the Pearson correlation coefficient is used to calculate redundancy between features. The scheme can be expressed through equations . |
6493bd4f853d501c004ac918 | 15 | The interpretability of ML models is critical for this study. While qualitative feature importance can be obtained from some tree-based models such as RF and GBM, a uniform and interpretable way of learning feature importance from different ML models is crucial. SHAP (Shapley Additive exPlanations) , a concept that originates from game theory, explains the importance of features in different machine learning algorithms. For a predictive model, the "game" is reproducing the outcome of the model, and the "players" are the features included in the model. SHAP quantifies the contribution that each feature brings to the game. It does this by calculating the contribution of each possible combination of features that can influence the importance of a single feature. A kernel-based approach called KernelSHAP 42 is used in this study. |
6493bd4f853d501c004ac918 | 16 | We first used an mRMR algorithm (see methods section) to estimate the importance of all 221 features (see Fig. ) & the supplementary Fig. for all the features). From this initial estimate, we find that 'energy per atom' (a global attribute) is the most important for both bulk and shear modulus. In fact, in general, the global attributes are found to be more relevant to predict bulk modulus compared to shear modulus. This is expected as bulk modulus is strongly related to volume. We find that the structural attributes (F0, F1, F2.) play crucial roles to predict both bulk and shear modulus. Based on the mRMR importance of the features as shown in Fig. ), a subset of 1 st 150 features seems appropriate. Fig. ) illustrates the predictability (mean absolute error (MAE)) of our ML models for bulk and shear moduli depending on the size of the training dataset, using our selected subset of 150 features. The results of five independent trials reveal that, for bulk modulus, MLP performs poorly with a smaller amount of data, but it exhibits a significant improvement as the size of the training dataset increases. Among all models, overall, KNN is the worst, except for the LR models (refer to supplementary information 4. (a)), while all other models consistently improve with increasing training data. Moreover, we see that the standard deviation of the accuracies of GBM, KRR, RF, GPR, and SVR is less than ~1.5 GPa, indicating a consistent performance. A similar trend is observed for shear modulus predictions. Overall, the performance of GBM, KRR, RF, GPR, and SVR are nearly identical. |
6493bd4f853d501c004ac918 | 17 | To obtain the best set of hyperparameters for each model, we conduct a 5-fold grid search cross-validation on the training set. The best set of hyperparameters is then used to train and evaluate the performance of the models on the test set. The SVR model obtained the best accuracy (MAE 11.675 GPa) compared to the other models in predicting bulk modulus (K), as shown in Figure (a). The prediction error bars for the other models, RF, KRR, MLP, GBM, GPR, and SVR, were within ~0.5 GPa except for LR and KNN. This suggests that traditional ML models may have reached a point of saturation in terms of predictability. The SHAP predicted feature importance of the SVR model (Fig. ) reveales that density and formation energy per atom are the two most important features. These are global attributes that directly affect the physical properties and stability of the material. Invariant structural attributes, such as "F2", "F5", and "F0", were also seemingly important for the prediction of the bulk modulus. A similar trend in feature importance can be observed in the other equally performing models as well (see supplementary Fig. ). From the training error shown in Figure (a), it also seems that the RF, KRR, and SVR models tend to overfit the data slightly. |
6493bd4f853d501c004ac918 | 18 | Similarly, when predicting shear modulus, we observed that the KRR model performed the best (MAE 9.019 GPa), while the overall performance of the RF, KRR, MLP, GBM, GPR, and SVR models are almost identical (see Fig. ). We also see that the MAE error for predicting shear modulus is relatively lower compared to bulk modulus. Similar to the bulk modulus, with the KRR model, the global attributes "density" and volume per atom are found to be the most important features. In general, the importance of structural attributes is crucial for predicting shear modulus as well (see supplementary information Fig. |
6493bd4f853d501c004ac918 | 19 | The comparable performance exhibited by different ML models with distinct architectures in predicting bulk and shear modulus suggests a limited learnability from the dataset and feature representation. While many features in the feature space are immutable, particularly the properties of elemental systems, the mutable ones are the geometric attributes or structural representations. For the prediction task of the machine learning models discussed so far, the SOAP representation of structural attributes is based on predefined mathematical formulations. Thus, we conclude that the lack of flexibility in the structural representation is the primary reason for the saturation in predictability. In this context, the structure itself, providing more flexibility in feature representation but requiring a larger amount of data. |
6493bd4f853d501c004ac918 | 20 | To obtain a more reliable estimate of feature importance, the SHAP predicted feature importance from the top six best performing models, namely GPR, RF, KRR, MLP, SVR, and GBM, are combined with their overall performance. It is important to compare the performance of and feature importance for different models because a single model may overfit the data which leads to biased feature importance estimates, that do not generalize well for new data. For instance, RF, KRR, and SVR tend to overfit when predicting bulk modulus, while RF ad KRR tends to overfit when predicting shear modulus. Combining the results of multiple models can help produce more generalizable feature importance estimates. Additionally, since different models may have different biases or assumptions, combining their results can help obtain a more robust estimate of feature importance. Thus, the performance of the models measured on unseen data combined with the estimated feature importance is used to compute the overall importance, following the equation, ππ£πππππ ππππππ‘ππππ = . |
6493bd4f853d501c004ac918 | 21 | where πΊ $()*,+,-./ is the global SHAP importance for the feature for a given model. Fig. ) illustrates the overall categorical contribution of the features in predicting the bulk and shear modulus. It is important to note that when computing the categorical feature importance, we sum up the total contribution of each feature in each category. However, for features with multiple components, such as the SOAP feature vector with 24 components in total, the overall feature importance shown in Fig. ) is determined by considering the maximum value of contribution among all the components. |
6493bd4f853d501c004ac918 | 22 | As shown in Fig. (a-b), the "property" category plays a crucial role in predicting both bulk and shear modulus. Among the features in this category, we find that the density, energy per atom, and volume per atom are particularly important. Denser materials with lower volume per atom and stronger interatomic bonding tend to be more resistant to compression and shear deformation. Moreover, energy per atom is directly related to the material's stability, leading to higher values of both bulk and shear modulus. Other local features, such as melting temperature, also play a crucial role in this category. Strong interatomic bonding, which is responsible for a material's solid structure at high temperatures, can lead to high elastic constants as well. For instance, metals like tungsten and molybdenum, as well as ceramics such as alumina and silicon carbide, possess strong metallic, ionic, and covalent bonding, resulting in high melting points and elastic constants . |
6493bd4f853d501c004ac918 | 23 | The 'structural' category (Fig. )), being the second most significant category, has a more influence on the prediction of bulk modulus compared to shear modulus. The structural features such as Laue symmetric point groups, space groups, crystal systems, etc., contribute to the prediction of bulk modulus, along with SOAP features, while only the SOAP feature vector influences shear modulus. This is because the bulk modulus primarily depends on the compressibility of the lattice, which can be highly dependent on the symmetry of the crystal structure. In contrast, in materials with layered or anisotropic crystal structures, the shear modulus can be highly dependent on the orientation and arrangement of atomic layers, while the bulk modulus may be less sensitive to these factors. Supplementary information Fig. 1(b) shows that almost 50% of the crystals in the dataset belong to the cubic and hexagonal categories, which explains the additional dependence of bulk modulus on the structural properties. |
6493bd4f853d501c004ac918 | 24 | The "periodic table" category contributes slightly more towards predicting the shear modulus compared to the bulk modulus as shown in Fig. ). Within this category, two important attributes are the atomic number and covalent radius of an element. Both factors play a role in determining the elastic properties of a material. Elements with higher atomic numbers generally exhibit stronger interatomic bonding, leading to higher bulk and shear moduli. On the other hand, larger covalent radii result in longer and weaker covalent bonds, leading to lower elastic moduli . For example, a diamond, which is made up of carbon atoms with small covalent radii, has short and strong bonds, leading to high elastic modulus and hardness. On the other hand, NaCl, composed of sodium and chlorine atoms, has longer and weaker bonds due to the larger covalent radius of chlorine, resulting in lower elastic modulus and hardness. |
6493bd4f853d501c004ac918 | 25 | Finally, the 'electronic properties' category has a greater impact on the shear modulus than the bulk modulus. The shear modulus is linked to a material's stiffness and its ability to resist deformation under shear stress. The electronic properties of a material can affect its stiffness by modifying the bonding interactions between atoms or the density of states near the Fermi level . For instance, in certain metals, the existence of unfilled electronic states can boost the strength of the metallic bond, thereby increasing the material's resistance to deformation. Similarly, the valence electrons in a material can affect its shear modulus by changing the bonding interactions between atoms or the density of states near the Fermi level. |
60c750ea469df4432ef44934 | 0 | The ability to predict the crystal structure that a molecule will adopt, in advance of the crystallisation experiment or even in advance of synthesis, would have great implications in several areas of materials science. The past two decades have seen great progress in computational methods for crystal structure prediction (CSP), with almost all current methods based on performing a search for the local minima on the high dimensional energy surface representing the energy as a function of the variables that describe a crystal structure. The usual assumption in using these methods is that the global minimum on the potential energy surface corresponds to the most likely observable crystal structure. Focusing on locating the global minimum, many approaches has been developed for CSP, such as Monte Carlo simulated annealing , genetic algorithms and particle swarm optimization. However, higher energy crystal structures are also often observed. This is clear from the prevalence of polymorphism in molecular crystals, where a molecule can adopt more than one crystal structure. Polymorphism is sometimes due to changes in temperature or pressure, which can alter the free energy ordering of structures. These effects can be accounted for in prediction methods by inclusion of entropy and zero-point vibrational contributions to the energy. However, different crystal structures can often be crystallised at the same thermodynamic conditions, sometimes from the same experiment (concomitant polymorphs). It has been estimated, based on a largescale computational study, that nearly 80% pairs of observed polymorphs are monotropic, i.e. their free energies do not cross below their melting temperature. The identification of these metastable polymorphs is important in many applications of CSP, such as polymorph screening of pharmaceuticals, and computer-guided discovery of functional materials, where high energy structures sometimes exhibit the most attractive properties. Thus, for CSP to be predictive of all observable crystal structures of a molecule, the structure search method must not be treated as simply a global energy minimisation problem, but should exhaustively explore the energy landscape for possible structures within the energy range above the global minimum in which observed structures can be located. Therefore, some CSP algorithms, such as lowdiscrepancy, quasi-random sampling place emphasis on exploring the structural landscape as uniformly as possible for all low energy structures. The energy range over which it is important to identify possible crystal structures can be defined by the energy range of observed polymorphism. Most observed polymorphs are separated by only a few kJ mol -1 , although this range can extend above 10 kJ mol -1 in rare cases, or where polymorphs are accessed through desolvation of solvated crystals. Although this defines a narrow energy window for observable structures, the weak interactions between organic molecules mean that large numbers of different structures are often possible within this range, for small organic molecules, a small energy range can include tens or hundreds of putative crystal structures. The importance of the higher energy structures, in addition to the global minimum, creates a tension in designing methods for CSP between efficiency in locating the global energy minimum quickly and time spent exploring the landscape to locate all potentially observable crystal structures. In this work, we describe a hybrid approach, where quasirandom (QR) sampling is used to seed multiple Monte Carlo basin hopping (BH) searches; we refer to the method as QR-BH. The role of quasi-random sampling in QR-BH is to provide a broad sampling of the energy landscape, while basin hopping efficiently locates low energy structures from these starting points. The method is benchmarked on a set of organic molecular crystals and co-crystals to explore its efficiency, how it is influenced by the temperature used in basin hopping and the number of quasi-random seeds vs basin hopping steps used in the search. |
60c750ea469df4432ef44934 | 1 | Six crystal systems (Fig. ), including single-component crystals and co-crystals, were chosen as representative of different applications of CSP, and of systems held together by different strengths of intermolecular interactions. Tetracyanoethylene is a planar molecule with weak intermolecular interactions, the zwitterionic geometry of glycine leads to strong intermolecular hydrogen bonding interactions, while benzamide represents molecules with medium strength intermolecular interaction. The fourth single-component system, a triptycene trisbenzimidazolone (TTBI) (Fig. ), is a larger molecule that has been shown to form several porous polymorphs located in low-density, high-energy regions of the lattice energy landscape; this molecule is included to test the location of important high-energy structures by the QR-BH crystal structure searching algorithm. |
60c750ea469df4432ef44934 | 2 | To investigate the behaviour of the QR-BH method thoroughly and as a challenge to investigate the efficiency of the method on more complex systems, we applied the algorithm to two co-crystal systems (Fig 2 ), in which the presence of two independent molecules leads to more degrees of freedom and, thus, more challenging energy landscapes for structure prediction. The first cocrystal, which we refer to by the Cambridge Structural Database (CSD) reference code of its known structure, XAFQAZ, is a hydrogen bonded complex between 2,8-Dimethyl-6H,12H-5,11methanodibenzo[b,f] diazocine (TrΓΆger's base) and 3,5dinitrobenzoic acid and was found to be a challenging target in the 6th blind test of crystal structure prediction. As a second co-crystal, we chose the complex between two planar molecules -pyrene and pyromellitic dianhydride -which is held together by weaker, less directional intermolecular interactions. We also ref to this system by the CSD reference code of its known crystal structure,[21] PYRPMA. |
60c750ea469df4432ef44934 | 3 | Quasi-random (QR) structure generation was performed using the Global Lattice Energy Explorer code; the method is described in detail in our earlier paper. During the generation of trial structures, a low-discrepancy sequence of vectors is generated by the Sobol method and each vector is mapped onto the structural degrees of freedom of the unit cell, including molecular positions and orientations, as well as lattice parameters that are not constrained by space group symmetry. We use the SAT-expand version of the quasi-random crystal structure generation method, in which the target volume for the unit cell is set as the sum of molecular volumes (which are calculated from the volume of a box enclosing all atoms in the molecule). The separating axis theorem (SAT) is used to detect overlapping molecular convex hulls, which indicate clashing molecules. Such clashes are removed through expansion of the lattice parameters in the direction required to separate overlapping molecules. Structures in which intermolecular clashes could not be relieved with unit cell expansion of less than 2.5 times the original target volume are rejected without lattice energy minimization. |
60c750ea469df4432ef44934 | 4 | Each trial structure was then lattice energy minimized using the DMACRYS software to locate the nearest (downhill) local minimum on the lattice energy surface. Molecules are held rigid throughout at their DFT optimized geometries and intermolecular interactions are modelled using an empirically parametrized exp-6 repulsion-dispersion potential combined with atomic multipoles for electrostatic interactions from a distributed multipole analysis. Full details of lattice energy minimization are provided in the supporting information. |
60c750ea469df4432ef44934 | 5 | The method was designed to provide a uniform and unbiased sampling of the lattice energy surface, which is important for fully exploring the structural diversity available to a molecule in forming stable crystal structures. The lack of bias in the search towards identifying low energy structures makes it effective at locating metastable crystal structures, while it has also been shown to usually find the global energy minimum early in a search. A further advance of the approach is its parallelizability: each local energy minimization is independent, so the minimization of all trial structures can be performed in parallel if sufficient processors is available. |
60c750ea469df4432ef44934 | 6 | QR searches were continued for a specified number of successful lattice energy minimizations. The database of optimized crystal structures was then analyzed for duplicates to generate a list of unique predicted crystal structures, and to count the number of times that each structure was located. Details of duplicate identification are provided in the supporting information. |
60c750ea469df4432ef44934 | 7 | Basin hopping (BH) is a global optimization approach combining local energy minimization and a Monte Carlo sampling method, where local minimization is introduced after each random perturbation. In other words, rather than single-point energy evaluation, the objective function is the locally minimized energy given by αΊΌ(x) = min[E(x)], meaning that the energy associated with each point in configuration space, x, is the energy of the local minimum that is reached upon energy minimization from that point: min[E(x)]. Thus, the acceptance probability of perturbing structure a to b is calculated by |
60c750ea469df4432ef44934 | 8 | Five types of perturbation are used in order to sample all degrees of freedom in a unit cell. Molecular perturbations include translation in a random direction and quaternion rotation around a random axis passing through the molecular center of mass. All molecular perturbations were applied to the molecule(s) in the asymmetric unit; perturbations of the other molecules in the unit cell were generated by symmetry, so as to maintain the space group symmetry. Unit cell perturbations include unit cell length changes (taking into account correlated lattice parameters in some space groups), unit cell angle changes (where allowed by space group symmetry) and unit cell volume changes. To avoid the unphysical region of the exp-6 interatomic potential, the distances between molecules were calculated after perturbation and the Monte Carlo move was rejected if any interatomic distance was shorter than the sum of covalent radii of the two elements + 0.3 Γ
. |
60c750ea469df4432ef44934 | 9 | The probability of making each perturbation type and cut-off magnitude of each type of perturbation are two important parameters to be determined. The probability of applying each type of perturbation was determined according to the degrees of freedom (DOF) leading to an energy change by P move = (DOF move /DOF total ). DOF move is the number of DOF related to the specific move, e.g. DOF move = 3 for translation of one molecule. DOF total is the total number of degrees of freedom. The step size of each perturbation was sampled from a uniform distribution within the range defined by the cutoff, except for unit cell angles. To prevent angles from moving outside the target range 45 to 135 β’ , instead of generating a random number in the range (-1, 1), the range of the random number is shifted based on the current angle (ΞΈ) by |
60c750ea469df4432ef44934 | 10 | where ΞΈ c is the central angle, usually 90 β’ ; ΞΈ l and ΞΈ u are lower and upper limits, being 45 β’ and 135 β’ by default. Hence the range of random perturbations is shifted towards 90 β’ and then scaled by the cut-off. The non-uniform sampling of unit cell angle perturbations means that the simulation does not fulfill detailed balance, which is unimportant here because the BH approach focuses on prediction of local minima on the energy landscape, rather than a distribution at equilibrium. Since molecular perturbations were applied to the asymmetric unit, the cutoff on volume change depended on the number of molecules in the primitive unit cell to eliminate the impact from different space groups with different numbers of symmetry operations. |
60c750ea469df4432ef44934 | 11 | During the BH trajectory, new structures were obtained by perturbing the unminimized structure from the previous step, rather than applying perturbations to the minimized structures. One reason for this decision is that, since unit cell angles are not fixed during local minimization, unit cells can become quite flat after minimization (i.e. having very acute or obtuse unit cell angles). These flat unit cells can correspond to physically realistic structures, but lead to difficulties in applying further perturbation and minimization. |
60c750ea469df4432ef44934 | 12 | The strategy that we have developed in this work is to combine BH with the quasi-random (QR) sampling approach. In our pure QR method, the conversion of each quasi-random vector into a trial crystal structure was followed by local energy minimization. Here, we replace the single local energy minimization of each QR trial structure by a BH trajectory. Our intention is that the quasi-random seeding of basin hopping simulations maintains some of the benefits of the low discrepancy approach, such as its uniform sampling of the configuration space of crystal packing, while benefiting from the efficiency of BH at moving towards low energy structures. The approach also maintains a certain level of parallelizability: each BH trajectory can be performed in parallel. |
60c750ea469df4432ef44934 | 13 | As well as the perturbation cutoffs and temperature used in the BH acceptance test, the behavior of the combined, QR-BH, search is influenced by the number of quasi-random seed structures and the length of each BH trajectory. Unless otherwise stated, all BH trajectories in a search were run for the same, fixed total number of steps. The job of the quasi-random seeds is to sample the energy landscape widely and evenly, while each local region is then efficiently explored in the BH search. Conceptually, the most efficient search could be achieved when each BH trajectory samples a separate region of the energy landscape and these local regions combined make up the entire energy landscape. |
60c750ea469df4432ef44934 | 14 | As an alternative to running all BH trajectories for the same, fixed number of steps, we also implemented a version of the QR-BH search that involves on-the-fly clustering of crystal structures and the termination of BH trajectories that sample the same regions of phase space. In this version, a BH trial is truncated if the new minimum reached from the perturbed structure already exists in the database of structures located thus far. If a BH trajectory is truncated, it is replaced by a new trial initiated from the next unused quasirandom seed in the Sobol sequence, so that the number of active BH trajectories remains constant. To ensure that one BH trajectory is kept active within each region of configurational space, a trajectory is not truncated if it locates a previous structure from its own history and, when two trajectories have located the same structure, the lowest trajectory number (ie. starting from the earliest quasi-random seed) is kept active to continue sampling. |
60c750ea469df4432ef44934 | 15 | We initially applied the QR-BH algorithm to a set of single component molecular crystal systems (Fig. ), and compared the energy landscape and sampling efficiency with the pure QR search method. For tetracyanoethylene, benzamide and glycine, 7 space groups were sampled: P 1, P 2 1 , P 2 1 /c, C2/c, P 2 1 2 1 2 1 , F 2dd and I4 1 /a. These were chosen as a set of common space groups for organic molecular crystals covering different crystal systems, symmetry elements and centerings, which could lead to differing complexities of their energy landscapes. |
60c750ea469df4432ef44934 | 16 | The sampling efficiency is affected by all parameters used to define the behaviour of the BH trials, in this case including the temperature used for trial acceptance, perturbation cutoffs, the number of parallel seeds and the length of each BH trial. The perturbation cutoff was adjusted so that different types of structural perturbation lead to similar energy changes (Table ). In these initial tests, the temperature was set to 3000K to permit acceptance of increases in energy of up to about 24 kJ/mol according to Boltzmann distribution (lower temperatures are investigated below). Each QR-BH simulation involves 10,000 local lattice energy minimizations generated from 100 parallel BH trials (each started from a different QR seed) and 100 BH steps in each trial. Results of these searches were compared to pure QR structure searches using the same number of energy minimizations in each space group. This length of run is deliberately oversampled to generate better statistics with which to compare the methods; fewer steps are normally required for such simple systems. Because of the stochasticity of the QR-BH process, every simulation (i.e. each molecule + space group combination) was repeated three times and we examine the average and variability of the behaviour between repeats. Since the quasi-random sequence is deliberately deterministic, we used the same initial quasi-random structures in each repeat, but different random seeds for Monte Carlo moves in the BH trials. |
60c750ea469df4432ef44934 | 17 | The effectiveness and efficiency of the searches were initially evaluated according to the speed with which the global lattice energy minimum was located in each search. Because the space group symmetry is constrained within each search, we treat each combination of molecule and space group as an independent landscape in our analysis. |
60c750ea469df4432ef44934 | 18 | Our first observation is that, for each molecule-space group combination, the QR search and the three repeats of QR-BH all find the same global energy minimum structure. Therefore, we are confident that the true global energy minimum has been located for each system. The efficiency of the methods is measured by the number of steps required to locate the global minimum of each system (Table ). This is defined straightforwardly in the QR search as the number of accepted (lattice energy minimized) quasi-random seeds until the first time that the global minimum is located. For the QR-BH search, we define the step as the product of (seed number)Γ(BH step number) for the first hit to the global minimum. |
60c750ea469df4432ef44934 | 19 | As a broad observation, we find that QR-BH locates the global energy minimum in either the same number or fewer steps than the pure QR search. The mean number of steps to find the global minimum (over the three QR-BH repeats) is always lower, taking, on average, 74% of the steps needed by the pure QR searches. As observed in previous work, the quasi-random search is often effective at locating the global minimum energy structure early in a search and this is borne out for these three molecules. The ease of finding the global minimum in energy is thought to be due to low energy structures having wide basins of attraction. It is for the systems where the global minimum is located later in the search that the improved efficiency of QR-BH is clearest: the global minimum in space group F 2dd is first located after hundreds of steps in the pure QR search for all three molecules, but is found much earlier -after fewer than 100 steps -for most of (7 of 9) the QR-BH searches. |
60c750ea469df4432ef44934 | 20 | The repeats of QR-BH mostly show comparable behaviour, finding the global minimum with similar efficiency in independent runs starting from the same QR starting points. However, we see greater variability between QR-BH repeats in the cases where the pure QR search was slowest at locating the global minimum. The most extreme case is for benzamide in space group F 2dd, where the pure QR search took 288 steps before the first hit of the global minimum. Two runs of QR-BH showed a big improvement, locating the Table : Steps required to locate the global energy minimum for the single component crystal systems tetracyanoethylene (TCNE), benzamide and glycine in each of 7 space groups (SG), using the quasi-random seeded basin hopping (QR-BH) and quasi-random (QR) methods. For the QR-BH searches, we report the results of each individual run and the mean step number over the three repeats. We also monitored the number of hits to the global minimum energy structure in each system (Table ). For these three molecules, we see small differences between the pure QR and the QR-BH searches, perhaps because their energy landscapes are relatively simple. However, in 15 of the 21 systems, the global minimum is sampled more frequently by QR-BH than QR, reflecting the bias that is introduced towards lowering the energy when local energy minimization of QR structures (the pure QR method) is replaced by a short basin hopping trajectory (as in QR-BH). Thus, despite occasional variability between runs, these initial tests showed the QR-BH algorithm to be stable and efficient at locating the global minimum energy crystal structures, with a moderate improvement over pure QR searching in how quickly it locates the global minimum energy structure. It is also important to reliably locate the possible crystal structures that are slightly higher in energy than the global minimum. As well as the importance of locating high energy polymorphs, the small energy differences often seen between predicted crystal structures means that errors in the model of interaction energies, as well as neglect of thermal vibrations, could lead to mis-ranking and that the true global minimum in free energy is not the global minimum in lattice energy from the energy model used for CSP. Thus, it is important that CSP provides a complete set of low energy structures so that all structures within error of the global minimum have been located. Therefore, we also compare the performance of QR-BH and pure QR searches in sampling the entire low-energy regions of the crystal structure landscapes. Table shows the number of unique crystal structures found in the low energy region (defined as within 5 kJ mol -1 of the global minimum) for each molecule-space group combination for the 10,000 step QR-BH and QR searches. These are compared to a much longer reference search (50,000 QR structures), which should be sufficiently well sampled to locate all low energy structures. These results show only minor differences between methods. Apart from six systems (tetrecyanoethylene in P 2 1 /c; benzamide in space groups P 1, P 2 1 /c, C2/c and F 2dd; and glycine in P 1), the same set of structures is located in all searches, including all three repeats of QR-BH. In three of these cases, the 10,000-minimization QR search misses one of the low energy structures that was located in the longer reference search, and QR-BH finds all of the structures in some or all of the repeats. Glycine in P 1 shows greater differences between searches: the QR search locates only 4 of the 8 low energy structures from the reference search, while QR-BH finds 6, 6 and 5 low energy structures in the three repeats. For only two systems (benzamide in P 1 and C2/c) does the QR-BH perform worse than QR, locating one fewer low energy structure than QR in some of the QR-BH repeats. Figure shows the predicted energy landscape for benzamide in F 2dd. The results demonstrate the reproducibility of the results: QR and QR-BH find nearly identical sets of crystal structures, particularly in the lowest-energy region of the landscapes. Furthermore, the three repeats of QR-BH find the same sets of structures. As expected, as the energy increases away from the global minimum, we observe more structures that are located by QR, but not QR-BH (Fig. , blue dots without corresponding QR-BH hits, or hits in only 1 of the QR-BH repeats). This is because the bias introduced in basin hopping to favor sampling of low energy structures must come at the expense of sampling in the higher energy regions of the landscape. It does not seem that this bias hinders sampling by QR-BH in the usual energy range of polymorphism (typically under 10 kJ mol -1 ), with the parameters used here. |
60c750ea469df4432ef44934 | 21 | As already highlighted in Table , benzamide in F 2dd is a case where the 10,000-minimization QR search has missed one low energy structure that is located by all three QR-BH runs with the same number of minimizations. This missed structure is the second lowest-energy structure for this system, only 0.04 kJ mol -1 above the global minimum. In extending the QR search, we find that this structure is first hit after minimization of 30,381 structures. Thus, the greater sampling efficiency of QR-BH is important in this case for obtaining a complete picture of the potential crystal structures. Figure shows the number of times that each of the 10 lowest energy structures were located in this system, highlighting the second lowest energy structure as a difficult case. The results show the increased sampling efficiency of QR-BH not only for the global minimum (as shown in Tables and), but also for the next two structures. As the energy increases, the difference between QR and QR-BH is less obvious, which matches our expectation that basin hopping helps locate the lowest energy structures and also demonstrates that sampling of the rest of the low energy region is not worsened compared to pure QR sampling. |
60c750ea469df4432ef44934 | 22 | The fourth single component system, TTBI (Fig. ), was chosen as a more extreme test for the location of important higher energy crystal structures. TTBI forms four microporous polymorphs, named Ξ±, Ξ², Ξ³ and Ξ΄, ranging from 46.4 to 92.1 kJ mol -1 (according to the FIT + multipoles force field) above the densely-packed global lattice energy minimum structure. These structures lie well outside the usual energetic range of polymorphism. They are accessed experimentally because they crystallize with solvent filling their pores and are stable to solvent removal, presumably as deep local energy minima. |
60c750ea469df4432ef44934 | 23 | Due to the molecular symmetry of TTBI, each of the observed structures, as well as the global energy minimum, can be located in CSP searches in multiple space groups. Thus, we tested the search methods' ability to locate these structures in several space groups and we added two additional space groups (P 4 2 and P 4 2 /n) to our sampling as those in which the Ξ± polymorph is located. All other search parameters were the same as for the other single component crystals, except the perturbation cut-off for volume, which was increased to 200 Γ
3 /atom due to the large molecular size. To investigate whether the temperature used to control the acceptance during basin hopping had an effect on finding target structures over such a large energy range, QR-BH was run at two temperatures: 3000 K (as above) and 500 K. |
60c750ea469df4432ef44934 | 24 | As with the other test systems, we find that the pure QR and QR-BH methods provide essentially the same sets of structures in the low-energy region, as well as in the regions of the important high-energy structures corresponding to the observed polymorphs (Fig. ). To compare efficiency, we examined the minimum steps required to locate each target structure: the four known polymorphs and the global energy minimum (Table ). The QR-BH method consistently required fewer steps to locate the important structures on the landscape at both temperatures. In only one case -polymorph Ξ³ in space group P 2 1 -did the pure QR search locate a target structure earlier than QR-BH. The improved efficiency of QR-BH over pure QR searching at locating important high energy structures is somewhat surprising -we expected that basin hopping might favor the lowest energy structures too aggressively, improving the efficiency at finding the global minimum at the expense of sampling the higher energy regions. Our interpretation of these results is that, although Ξ±, Ξ², Ξ³ and Ξ΄ are high-energy structures on the whole lattice energy landscape, they correspond to the lowest energy structures within separated local regions of conformational space. Since BH trials begin from different initial structures and explore their local region, these high energy, experimentally observed structures can be located efficiently. |
60c750ea469df4432ef44934 | 25 | Space group P 2 1 /c appears to be the most challenging landscape of those sampled for TTBI; four of the target structures (Ξ±, Ξ², Ξ³ and the global minimum) are located in this space group and are first hit between 256 and 1144 steps in the pure QR search. By comparison, QR-BH at 3000 K locates all four structures before 250 steps. |
60c750ea469df4432ef44934 | 26 | The comparison between basin hopping temperatures in space group P 2 1 /c reflects the overall difference in results between temperatures: QR-BH at 500 K performs better at locating the target structures than pure QR, but not as well as 3000 K. The 100 basin hopping trajectories from QR-BH in P 2 1 /c are plotted in Figure for both temperatures, showing the expected behavior: the lower temperature drives the trajectories more aggressively towards lower energies, while the higher temperature simulations maintain sampling of higher energy structures throughout the trajectories. At least in this system, these high energy steps improve the efficiency of locating the target structures. |
60c750ea469df4432ef44934 | 27 | After assessing the reliability of the QR-BH method on the relatively simple single-component molecular crystal systems, the question arose naturally how the parameters of QR-BH would affect the sampling efficiency and whether there is an optimal parameter set to maximize efficiency. To test the influence of QR-BH parameters, we applied the method to the more challenging co-crystal systems, one being a hydrogen bonded co-crystal (XAFQAZ, Fig. ) and the other held together by weaker, less directional interactions (PYRPMA, Fig. ). PYRPMA was explored in three common space groups (P 1, P 2 1 and P 2 1 /c), which includes the space group in which the known crystal structure is found (P 2 1 ). The known crystal structure of XAFQAZ is found in space group P 2 1 /c and we also investigated F 2dd Table : Steps required to locate the global energy minimum and experimental structures for TTBI in 7 space groups. The target structures are listed in order of increasing energy from left to right. QR-BH results are the mean over three repeats of the search, each starting from the same QR seed structures. Results for space group F 2dd and I4 1 /a are not shown because none of the experimentally observed structures, nor the overall global minimum structure, can be located in these space groups. and I4 1 /a for this co-crystal. These space groups were chosen so that, across the two co-crystal systems, we explored different types of intermolecular interactions and a range of crystal symmetries. The perturbation cut-offs for Monte Carlo moves during basin hopping simulations were kept the same as those used for TTBI and both temperatures (500 K and 3000 K) were evaluated. The introduction of a second molecule in the asymmetric unit increases the dimensionality of configurational space by 6 (compared to single component crystals), so we increased length of searches to thoroughly explore the more complex crystal energy landscapes. Pure QR searches were run for a total of 50,000 minimizations and QR-BH were run with just over 50,000 total minimizations. Because of the greater complexity of their search space, we used the co-crystal systems to explore the impact of changing the allocation between QR seeds and BH steps, keeping a fixed total computational budget. Three seed:step ratios were applied: 1:1 (225 seeds Γ 225 BH steps), 2:1 (316 seeds Γ 158 BH steps) and 10:1 (710 seeds Γ 71 BH steps). The higher ratios test the effectiveness of shorter basin hopping trajectories started from a larger set of QR seed structures, which has the advantage of greater parallelizability. |
60c750ea469df4432ef44934 | 28 | Of the two co-crystals, PYRPMA was found to be the easier landscape for locating structures. Despite the greater dimensionality of the search space, the latest of the space group global energy minima to be located in the pure QR search was in P 2 1 /c after 866 steps (Table ). Counts of the number of low energy structures located in each space group are presented in the Supporting Information (Table ). We also monitored the step at which the structure corresponding to the experimentally observed co-crystal structure was located. This was located as the 3 rd lowest energy structure in space group P 2 1 , 2 kJ mol -1 above the global minimum. |
60c750ea469df4432ef44934 | 29 | The QR-BH method reproduced the same PYRPMA crystal energy landscapes, finding all of the same structures as the QR search (see Fig. for P 2 1 ). The space group global energy minima, as well as the experimentally observed crystal structure, are consistently found earlier in the QR-BH searches than QR (Table ). In only two cases was one of the target structures found later in QR-BH than QR, each time for the global minimum in P 2 1 /c at 3000 K, using 2:1 and 10:1 seed:step ratios. However, even in these cases, the mean steps to locate the global minimum was smaller than the steps required in QR. While the improved efficiency of QR-BH over QR is clear, the results do not point strongly to a best set of QR-BH parameters. |
60c750ea469df4432ef44934 | 30 | XAFQAZ was a more challenging system. The experimentally observed crystal structure is reproduced by the global energy minimum in space group P 2 1 /c. The QR-BH and pure QR methods generated similar energy landscapes in P 2 1 /c (see Fig. ). However, the pure QR method had difficulty locating several low-energy structures, including the global minimum, which was first hit after 30,315 minimizations. The 3000 K, 1:1 QR-BH search located the P 2 1 /c global minimum in 2 of 3 searches, both in significantly fewer steps than QR. Increasing the seed:step ratio to 2:1 or 10:1 led to more consistent success of QR-BH: the P 2 1 /c was located in all searches with these higher seed:step ratios, with fewer mean steps than QR. Lowering the temperature to 500 K gave less consistent results in P 2 1 /c; at each seed:step ratio, only one of three repeats located the global minimum in 50,000 minimizations. |
60c750ea469df4432ef44934 | 31 | Space group F 2dd for XAFQAZ shows an opposite trend with respect to the temperature of the basin hopping trajectories: more consistent results were obtained at the lower temperature in this space group (Table ): all 500 K QR-BH simulations located the F 2dd global minimum, using fewer steps than QR in all but one of the simulations (at a 10:1 seed:step ratio). I4 1 /a was also problematic for XAFQAZ, with the 50,000-minimization QR search missing the global minimum and the next three lowest energy predicted structures (Fig. ). QR-BH was more successful, finding the space group global energy minimum in most of the simulations. Like space group F 2dd, more consistent results were obtained at lower temperature; the I4 1 /a global minimum was located in all three 500 K QR-BH simulations with a 1:1 seed:step allocation. |
60c750ea469df4432ef44934 | 32 | The different influence of basin hopping temperature between space groups might be due differences in the nature of the energy landscapes (Fig. ). There is a large energy gap between the global minimum and other structures in space group P 2 1 /c, while there is a more uniform distribution of structures in the low energy regions of I4 1 /a and F 2dd. The smaller energy differences in the latter two space groups makes it easier to BH to travel 'uphill' at lower temperatures. |
60c750ea469df4432ef44934 | 33 | Table : Efficiency comparison of steps required to locate global minima in space groups P 1, P 2 1 and P 2 1 /c and experimental structure in P 2 1 for PYRPMA, and XAFQAZ in space groups P 2 1 /c, F dd2 and I4 1 /a. Results are shown for the three independent trials of QR-BH at each of two temperatures and three seed:step ratios. Of the QR-BH parameter combinations tested here, the most consistent set over the two co-crystals is T = 3000 K with a large seed:step ratio (many QR seed structures, each run for a short basin hopping trajectory). |
60c750ea469df4432ef44934 | 34 | Another strategy that we investigated was on-the-fly clustering to identify when multiple basin hopping trajectories, starting from different QR starting structures, encounter each other and thus sample the same local region of configuration space. This could lead to loss of efficiency compared to the situation where all BH trajectories sample different parts of the energy surface. |
60c750ea469df4432ef44934 | 35 | Thus, in the on-the-fly clustering strategy, any newly sampled energy minimum is compared with all structures that have already been visited by any other BH trajectories, using similarity of X-ray diffraction patterns to identify identical structures. If a structure had already been sampled by another BH trial, the current BH trajectory is truncated. The trial is replaced by a BH run starting from the next unused quasi-random seed, to explore a new region in configurational space and maintain a constant number of active BH trajectories. In this way, the method aims to minimize overlap in sampling between BH runs. |
60c750ea469df4432ef44934 | 36 | We set the number of active BH trajectories to 200, with a maximum of 250 steps for each BH trial, while other parameters were unchanged from the previous co-crystal searches. Searches were run for a total of 50,000 energy minimizations. The distribution of BH steps among trajectories (Fig. and Figs S1, S2 for all systems, space groups and temperatures) indicates that most of the BH trials were truncated within 50 steps for XAFQAZ at 3000 K and only a small number of trials reached the maximum allowed number of steps. The length of trajectories increases slightly at 500 K (Fig. ) and are typically even shorter for the PYRPMA cocrystal (Fig. ). This observation implies that BH trials rarely remain in separate regions of the energy landscape and, so, on-the-fly clustering should help keep the structure search spread across configurational space. We also noticed that many quasi-random structures directly duplicated structures that had already been located by QR-BH, so did not progress to BH beyond the initial energy minimization. |
60c750ea469df4432ef44934 | 37 | As with QR-BH with independent runs, on-the-fly clustered QR-BH generally had a better sampling efficiency than the pure QR method in locating the global minima within each co-crystal/space group system (Table ), as well as in locating low energy structures (Table , S7) on the complex co-crystal energy landscapes. Compared with independent trial QR-BH, the on-the-fly clustering was similarly efficient at sampling the global minima to QR-BH with a fixed, large seed:step ratio (10:1 = 710 seeds Γ 71 BH steps), because in both cases BH trials are quite short. |
60c750ea469df4432ef44934 | 38 | Finally, we note that, as with the independent, fixedlength QR-BH results, the optimal basin hopping temperature differs among space groups (Table ). For the more challenging XAFQAZ co-crystal, high temperature leads to more consistent location of the global minima in P 2 1 /c and I4 1 /a, while low temperature gives more consistent results for in space group F dd2. We also tried running QR-BH at T = 0 K (Table ), which allows only BH steps that lower the energy, as well as trying smaller Monte Carlo step sizes. These were both tested to try to keep the BH trajectories more localized on the energy surface by discouraging trajectories from escaping their current energy basin. However, both modifications led to poorer overall efficiency (Table , S7). |
60c750ea469df4432ef44934 | 39 | We have presented an improvement to the efficiency of quasirandom structure searching for crystal structure prediction of molecular crystals by combining the generation of trial structures using a low-discrepancy sequence with Monte Carlo basin hopping to explore for low energy crystal structures. The quasi-random seeds used as starting points for basin hopping provide a uniform coverage of the energy landscape, so that the role of basin hopping is to thoroughly explore for low energy structures in the region of its starting point. |
60c750ea469df4432ef44934 | 40 | The method has been tested on a set of single-component molecular crystals, for which we find that the QR-BH algorithm improves on the sampling efficiency of the pure Table : Comparison of steps required to locate the global energy minima with QR-BH using on-the-fly clustering in space groups P 1, P 2 1 and P 2 1 /c and the experimentally observed structure in P 2 1 for PYRPMA, and XAFQAZ in space groups P 2 1 /c, F dd2 and I4 1 /a. The QR-BH results are compared to a pure QR search. QR searching in locating the global energy minimum more quickly than pure quasi-random searching and leads to better sampling of the lowest energy structures in each space group. We also find that the combined QR-BH method maintains the desirable feature from QR search methods of reliably locating important high-energy crystal structures; this is illustrated using the molecule TTBI as an extreme example, where experimentally observable crystal structures occupy a very wide range in lattice energies. Surprisingly, QR-BH located even the highest energy observed structures more quickly that a pure QR search. |
60c750ea469df4432ef44934 | 41 | The improved efficiency of QR-BH has also been illustrated in searching the higher dimensional energy landscapes of two co-crystal systems, which were used to investigate the influence of temperature and the allocation between quasirandom seeds and basin hopping steps on the performance of the method. The optimal temperature used in basin hopping was found to vary between systems, even for different space groups of the same chemical composition. The most reliable performance was found with high temperature (3000 K) and a large number of QR seeds with short BH trajectories. |
60c750ea469df4432ef44934 | 42 | We have not pursued more advanced approaches to select the basin hopping temperature here. An adaptive approach might be required, with temperature changing through the simulation to maintain a targeted acceptance ratio of Monte Carlo steps. However, we have presented a method, which we call on-the-fly clustering, that adapts the length of BH trajectories to avoid overlap of trajectories sampling the same region of the energy landscape. |
632e22362984c91d4d6aa766 | 0 | One of society's most significant challenges in engineering is the generation of energy from clean, efficient, reliable, and environmentally friendly sources . Fuel cells are one of the most efficient and effective solutions to meet this challenge . These electrochemical devices convert the chemical energy of fuel gas into electrical work without the need for combustion. Moreover, fuel cells can also be used for grid-scale renewable energy storage through high-efficiency conversion of chemical energy with zero πΆπ ! emissions while offering a wide variety of potential applications for electricity generation in residential, vehicular, and industrial settings . Among all types of fuel cells, solid oxide fuel cells (SOFCs) offer numerous advantages such as high energy density, high efficiency (beyond 60% and up to ~90 % with the addition of heat recovery cycles), no cyclical variations of voltage (voltage disturbances) or electrical fluctuation, and no environmental noise as compared to conventional power generation systems . In the SOFCs, the electronic current is generated by oxidation of the fuel gas at the anode and oxygen reduction at the cathode. Therefore, the electrical current is made possible by the flow of the ions, down the chemical potential gradient of the mobile species, from the anode to the cathode through an ionic conductor membrane and electrons transport in an external circuit. The general mechanism of the current flow is explained via mass transport of oxide-ions through the solid ion-conductor electrolyte by introducing vacancies into the crystal structure and random hopping mechanism of the charged species to the vacancies . Thus, ionic conductivity is increased significantly at elevated temperatures where sufficient energy allows ions to hop into and out of charged vacancies . However, cell operation at high temperatures introduces thermal stresses, corrosion, increased system cost, and stack durability challenges. Therefore, there has been increased interest in reducing SOFCs operating temperatures from the high-temperature (HT-SOFCs) regime (800 Β°πΆ to 1000 Β°πΆ) down to the so-called intermediate-temperature (IT-SOFCs) regime (500 Β°πΆ to 700 Β°πΆ). It has been proposed that using materials operating at intermediate temperatures could expand applications of fuel cells to large-scale stationary applications and smaller scale portable power/transportation markets . While Yttria-Stabilized Zirconia (YSZ) has been the most investigated membrane material for SOFCs as well as SOECs, Scandia-Stabilized Zirconia (SSZ) has shown almost 3-4 times higher ionic conductivity compared to that of YSZ . Moreover, YSZ exhibits a conductivity degradation by an imbalanced electrochemical potential gradient of oxygen ions through the electrolyte . The higher ionic conductivity in scandia-stabilized zirconia(SSZ) is attributed to the minimal ionic radii mismatch between ππ "% and ππ &% . Fujimori and Badwal comprehensively investigated the ππ ! π " -πππ ! system and discovered monoclinic (m), tetragonal (t), cubic (c), metastable tetragonal (π‘ ' , π‘β²β²), and rhombohedral (π½) phases exist from a range of 0 < ππ ! π " πππ% β€ 25. |
632e22362984c91d4d6aa766 | 1 | Of these phases, the cubic fluorite phase forms at ~9.3 πππ% ππ ! π " and the tetragonal phase forms at ~6-9.3 πππ% ππ ! π " exhibit the highest and second-highest ionic conductivity among other phases. The π β π‘ phase transition happens by oxygen ion displacement from fluorite ideal sites (8c sites). Increasing dopant levels has been shown to trigger the formation of an unwanted low conductive π½ -rhombohedral phase transformed from a high conductive cubic phase which accompanies volume change . Study of ternary systems consisting of ππ ! π " -πΆππ ! -πππ ! demonstrated that the addition of a small amount (β€ 2%) of a third component such as ceria can stabilize the higher ionic conductivity phases over a wider dopant range of scandia . The reaction that creates oxygen vacancies in scandia-ceria-stabilized zirconia is expressed in the KrΓΆger-Vink notation below. |
632e22362984c91d4d6aa766 | 2 | In this case, alloying the zirconia with ceria (similar fluorite structure, but π -. "# > π () "# ), while it does not result in the creation of additional positively charged vacancies, will facilitate the stability of the cubic structure. It has been studied that addition of less than 2% of πΆππ ! brings higher symmetry to the ππ ! π " -πππ ! system . The ionic radii of ππ &% , πΆπ &% , and ππ "% are 84, 111, and 87 pm . The presence of a small amount of πΆπ &% dopant will decrease in π and increase in π = π lattice parameter in a way that contributes to the stabilization of the unit cell. |
632e22362984c91d4d6aa766 | 3 | The stabilization of the cubic phase arises from applied lattice strain due to the presence of large cerium ions on the host lattice sites . This substitution assists formations of the highersymmetry cubic phase by lowering the π/π ratio in the tetragonal phase, thus resulting in higher ionic conductivity . Since the cubic (fluorite) structure attracts the most interest for investigation, numerous studies have been done on the influence of the microstructure and lattice strain on the ionic conductivity of the cubic phase of the SSZ . However, there are still regions that have been only poorly investigated within the tetragonal crystal structures in the ππ ! π " -πΆππ ! -πππ ! system. Enhancement of the mechanical properties in 6 mol% scandia-1 mol% ceria-doped zirconia (herein 6Sc1CeZr), compared to 10 mol% scandia-1 mol% ceriadoped zirconia (10Sc1CeZr), in addition to the 40% lower production cost due to a reduction in expensive ππ ! π " , make 6Sc1CeZr an attractive candidate for investigation in SOFCs applications as well as solid oxide electrolyzer (SOECs). To the best of the authors' knowledge, while 6Scstabilized zirconia (6ScSZ) and 10Sc1CeZr has been the subject of massive investigations, the electrochemical performance of the 6Sc-1Ce-co-doped zirconia has remained unexplored. Much research on 10Sc1CeZr has also revealed that although 10Sc1CeZr is highly conductive, the system is exposed to form π½-rhombohedral phase which dramatically exacerbates the conductivity . Also high contents of scandia results in formation of local defects trapped in oxygen vacancies and reduces the oxygen-ion conductivity where Chen et al. find that the probability of having a trapped site is given by the Fermi-Dirac statistics. Moreover, all active electrochemical devices are prone to electrochemically-induced degradation due to unbalanced chemical gradient potential of the mobile species . It has been theoretically and experimentally studied that actual oxygen chemical potential inside the solid electrolyte can be increased or decreased compared to corresponding values at the electrodes in active current carrying devices in electrolysis mode . Addition of dopants with multiple oxidation states, such as the addition of 1 mol% ceria to zirconia, is able to introduce some electronic conductivity and decrease the polarization resistance in ionic conductors and slow down the degradation rate by partial random ceria reduction where πΆπ &% + π β πΆπ "% . Furthermore, Shobit et al. observed that co-doping of scandia-stabilized zirconia with ceria also contributes to the formation of highly-symmetric fluorite phase, lowering the c/a ratio in tetragonal phase, and enhancing total conductivity. |
632e22362984c91d4d6aa766 | 4 | ), synthesized at various sintering temperatures. We will investigate an intermediate-temperature electrolyte that will decrease the pricey scandia content while simultaneously maintaining the electrochemical performance, enhancing the mechanical strength of the membrane. We are also hoping to bypass the current issues with electric conductivity of the membranes in the scandia-zirconia system influenced by the formation of the poor π½rhombohedral phase. Another objective of this work is to find the optimum sintering temperature that maximizes conductivity and lowers the grain and the grain-boundary resistivity, while the degradation rate is effectively lowered by introducing electronic conductivity provided by the addition of ceria to the system. Bremsstrahlung for higher precision. For structure refinement, the XRD patterns were fitted to JCPDS files no. 01-089-5485, 01-084-9828, and 01-089-5474 for πππ ! cubic, tetragonal and monoclinic structures, respectively. The crystallite size was estimated using the Scherrer equation from XRD diffraction peaks using the full width at half maximum (FWHM) of the (111) peak. Samples were sectioned with a low-speed diamond saw (Allied High Tech) and fine-polished to 200 nanometers at 200 rpm (Buehler). The cross-sectional scanning electron microscopy (SEM) and elemental dispersion spectroscopy (EDS) were investigated using FEI Quanta 600 FEG microscope. The Average Grain Intercept method (AGI) was used to determine the grain sizes of the pellets at each sintering condition (ASTM E112) using ImageJ software . |
632e22362984c91d4d6aa766 | 5 | Micro-Raman scattering spectra were obtained using the confocal Raman spectrometer (WITec Inc., Germany). The spectra were excited by a laser that was operating with excitation wavelength of 488 ππ. The laser was focused on the sample to obtain special resolution of π~610 ππ using a 50 Γ objective lens with numeric aperture ππ΄ = 0.75 (Leica Wetzlar, Germany) with the laser power at the sample's surface of 5 ππ. The Raman Stokes signal was dispersed with a diffraction grating (2400 πππππ£ππ /ππ) and data was recorded using a Peltier cooled charge-coupled device (CCD) detector (Renishaw, UK) (1024 Γ 256 pixels). The spectra of each specimen were collected over the wavenumber range of 10 ! -10 " ππ 51 , with scanning step size 1 ππ 51 with an integration time constant of 1π where the signal was accumulated five times and then the average was reported. Silicon was used to calibrate the Raman setup for both Raman wavenumber and spectral intensity before the use of the equipment. The Positions of the Raman peaks were determined by fitting the data to the Lorentz line shape using a peak fit option in the OriginPro software (OriginLab Corp., USA). Silicon was used to calibrate the Raman setup for both the Raman wavenumber and spectral intensity. |
632e22362984c91d4d6aa766 | 6 | AC Electrochemical Impedance Spectroscopy (EIS) was carried out on the samples in the air atmosphere to study the electrochemical properties of sintered samples using an impedance/gain-phase analyzer SI 1260 and electrochemical interface SI 1287 (Ametek Solartron Metrology). Platinum paste (Heraeus) was symmetrically screen-printed on both sides of the sintered pellets (electrode thickness β 20 Β΅m) and fired at 950 Β°C for 1β. The conductivity of the sintered samples was measured in the temperature range from 300 Β°C to 800 Β°C from 1MHz to 1Hz in air with open circuit voltage (OCV) polarization and applied voltage amplitude of 10 mV. |
632e22362984c91d4d6aa766 | 7 | Diffusional mass transport in ionic conductors needs to reach equilibrium state. Although thin samples of micrometers can reach equilibria in a couple of seconds or minutes, thicker samples should be kept for a longer time to reach equilibrium. Therefore, the experiments were carried out after the necessary relaxing time to reach equilibrium. The time to reach equilibrium, π, is a function of sample thickness, π, and chemical diffusion coefficient of the mobile species π with π Β± oxidation state, D OE 9 %Β± , which according to Thangadurai and Weppner , in the absence of the electrical field, can be estimated by |
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