The ionic conductivity at the grain boundaries (GBs) in oxide ceramics is typically several orders of magnitude lower than that within the grain interior. This detrimental GB effect is the main bottleneck for designing high-performance ceramic electrolytes intended for use in solid-state lithium-ion batteries, fuel cells, and electrolyzer cells. The macroscopic ionic conductivity in oxide ceramics is essentially governed by the underlying polycrystalline microstructures where GBs and grain morphology go hand in hand. This provides the possibility to enhance the ion conductivity by microstructure engineering. To this end, a thorough understanding of microstructure–property correlation is highly desirable. In this work, we investigate numerous polycrystalline microstructure samples with varying grain and grain boundary features. Their macroscopic ionic conductivities are numerically evaluated by the finite element homogenization method, whereby the GB resistance is explicitly regarded. The influence of different microstructural features on the effective ionic conductivity is systematically studied. The microstructure–property relationships are revealed. Additionally, a graph neural network-based machine learning model is constructed and trained. It can accurately predict the effective ionic conductivity for a given polycrystalline microstructure. This work provides crucial quantitative guidelines for optimizing the ionic conducting performance of oxide ceramics by tailoring microstructures.

## I. INTRODUCTION

Polycrystalline ionic conducting oxide ceramics are commonly found in engineering.^{1–3} Depending on the mobile charge carrier therein, they can be categorized into different types, e.g., proton conducting,^{1,3} oxygen ionic conducting,^{2} and alkali-ionic (e.g., lithium, sodium, and potassium ions) conducting^{4} ones. They are promising candidates as electrolytes in solid fuel and electrolyzer cells^{1} and all solid-state batteries (ASSBs). In these energy storage and conversion application scenarios, one of the key properties of concern is the effective (macroscopic) ionic conductivity. Polycrystalline microstructures consist of grains and grain boundaries (GBs). Experimental evidence (e.g., Refs. 1 and 5–12) suggests that many oxide ceramics exhibit high intrinsic ionic conductivity in the grain bulk. However, the ionic conductivity at GBs is typically orders of magnitude lower than the bulk one, resulting in unacceptable low effective or total conductivity. For detailed information regarding the magnitudes of GB, bulk, and total conductivity of common oxide ceramics, one is referred to the above mentioned references.

The low GB ionic conductivity is attributed to different factors, e.g., the GB compositions and structures^{4,6} and space charge layers.^{13,14} Numerous efforts have been devoted to improving the intrinsic GB conductivity.^{1,4,6,7,15} However, this challenging task remains unsolved. On the other hand, the effective properties of a microstructured material depend not only on the composition properties but also on the microstructural features. Thus, apart from the bulk and GB conductivities, the microstructural features relevant to grains and GBs, e.g., grain size, grain orientation, and grain and GB arrangement also strongly affect the effective ionic conductivity.^{7,16,17} Therefore, a thorough understanding of polycrystalline microstructure-effective conductivity correlation is crucial for designing oxide ceramics with enhanced effective ionic conductivity.

In addition to experiment investigations, analytical modeling and numerical simulations (e.g., Refs. 18 and 19) are efficient tools for studying the ionic conducting behavior in oxide ceramics. The brick-layer model^{20} is one of the most widely used analytical models, which provides an explicit expression of the total conductivity as a function of the grain and GB conductivities, grain size, and GB thickness.^{21,22} It can be used in combination with impedance spectroscopy measurements to evaluate the GB conductivity based on the measured total conductivity.^{7} Although it is easy to use, this idealized model ignores some important microstructural characteristics such as the inhomogeneous distributions of grain shape, grain size, and GB conductivities.^{23} To account for these microstructure influences, numerical simulations are desired. Similar to the assumption in the brick-layer model, one can consider that each grain and GB possess a constant ionic conductivity. Then, a steady-state conducting problem governed by the conservation of charge and Ohm’s law in a realistic polycrystalline microstructure is to be solved numerically.^{24} In this case, modeling of GB ionic conduction requires special attentions.

In recent years, data-driven or machine learning (ML) methods have been widely applied for surrogate modeling of microstructure–property correlations in microstructured materials including polycrystalline materials. Once trained and validated, data-driven models can efficiently predict the effective properties of a given microstructure. Thus, they can accelerate the process of designing microstructures with optimized properties. There are various ML models. Among them, the convolutional neural networks (CNNs)^{25,26} and graph neural networks (GNNs)^{27–29} are commonly used to model polycrystalline microstructures. CNN models directly take artificial and/or experimental microstructure images or image sequences as inputs, preserving all microstructure details. However, due to the high dimensionality of these inputs, particularly for three-dimensional microstructures, CNN models are computationally expensive. On the other hand, the inputs of GNN models are grain and grain boundary (GB) feature matrices along with an adjacency matrix, which have significantly lower dimensions compared to image-based descriptions. As a result, GNN models are computationally more efficient. Furthermore, thanks to their innovative architecture, GNN models effectively capture inherent grain–grain and grain–GB interactions in polycrystalline microstructures. In fact, recent findings^{29} suggest that GNN models outperform other ML models both in terms of accuracy and efficiency when it comes to modeling polycrystalline microstructures.

Based on the above backgrounds, the contributions of this work are twofold: revealing the quantitative correlations between polycrystalline microstructure and effective ion conductivity in oxide ceramics and encapsulating these correlations into a GNN-based ML model. To this end, we propose a numerical framework to investigate the effective ionic conductivity of polycrystalline oxide ceramics. We conduct numerous simulations on polycrystals with varying grain and GB features to systematically investigate their influence on the effective conductivity. Based on the numerous data generated by simulations, we construct and train a GNN-based surrogate model to correlate the polycrystalline microstructure to its effective ion conductivity. These together provide quantitative guidelines on how to enhance the effective conductivity in ionic conducting ceramics by microstructure engineering.

## II. MODEL AND METHODS

### A. Polycrystalline microstructures: Generation and characterization

As commonly adopted in the literature,^{30} we use two-dimensional (2D) Voronoi diagrams to represent the 2D polycrystalline microstructures. As illustrated in Fig. 1(a), the microstructure generation starts by randomly placing seeds in a 2D rectangular domain. The number of seeds is identical to the number of grains. To achieve periodic microstructures, the seeded domain is duplicated eight times and translated to create a large rectangular domain, on which the Voronoi tessellation is performed. Finally, a unit cell is cropped which acts as the representative volume element (RVE) for the periodic polycrystalline microstructure. In the unit cell, the polygons denote grains and their edges are the GBs. The geometrical features of the microstructure such as the grain size and GB length are determined by the number and distribution of the seeds. As illustrated by Ref. 31, the degree of regularity of the generated polycrystals depends on the minimum distance $\delta $ between any two seeds. A larger $\delta $ corresponds to a more regular microstructure with more uniformly distributed grain sizes. In such a manner, polycrystals with different degrees of regularity are generated by choosing different $\delta $.

Given the geometry of a polycrystal RVE, the crystallographic parameters and material properties are assigned to each grain and GB randomly or in a controlled manner [see Fig. 1(b)]. Numerous RVEs with different microstructural features are generated to study the influence of microstructures on the effective ionic conductivity. The above microstructure generation processes are implemented in the commercial software Matlab R2023a.

### B. Ionic conduction model for grain interior

^{18,24}

^{32}the ionic conductivity depends on the local ion diffusivity and concentration. In this work, we treat the bulk and GB ionic conductivities as known constants. In the steady state, the conservation of charge results in the following governing equation in the bulk:

### C. Ionic conduction model for GB

^{17}The conservation of charge in the steady state yields the following boundary conditions at the GB,

^{6,33}

The GB conductivity may depend on the GB characters such as the misorientation angle, the type, and the local chemical compositions.^{17} Here, we treat the GB conductivity as a known material parameter and focus on its influence on the effective conductivity. Depending on the type of charge carriers and the direction of transportation, the GB effect on the conductivity may be positive or negative (see, e.g., Ref. 34). In this work, we focus on cases with a negative GB effect where GB conductivity is significantly lower than the bulk one.

### D. Orientation-dependence of ionic conductivity

### E. Effective ionic conductivity: Computational homogenization

The numerical implementation of the above formulation is conducted by the finite element (FE) method. As illustrated in Fig. 1(c), the grains and GBs are discretized into triangular elements and two-node interface elements, respectively. The element size is about 1/10 of the average grain size, which is sufficient to ensure mesh convergence. Periodic mesh is generated in the open-source software Gmsh.^{35} Subsequently, the FE simulations are conducted in the open-source FE software MOOSE.^{36} The interface elements are added by the built-in mesh generator in MOOSE. Python scripts are written to automatize the simulation processes, which enables efficiently simulating numerous microstructures.

The parameters involved in the simulations are the two principal bulk conductivities $ \sigma 1 G$ and $ \sigma 2 G$, the grain orientation angle $\theta $, the GB conductivity $ \sigma G B$, and the GB thickness $ t G B$.

The proposed bulk and GB models for ionic conduction, along with their numerical implementation, enable not only the determination of effective conductivity in an arbitrary direction but also the characterization of microscopic electric fields and current densities within a realistic polycrystalline microstructure. In addition, the influence of grain and GB features on the effective conductivity are well captured. These aspects can not be adequately considered by the brick-layer model^{20} and other existing numerical methods, e.g., Refs. 23 and 37–39.

In the supplementary material, the verification of the numerical implementation of the proposed model by a benchmark problem is presented.

### F. The GNN-based ML model

We construct a GNN-based ML model to predict the effective ionic conductivity for a given polycrystalline microstructure. As illustrated in Fig. 2, a polycrystal is first converted to a graph consisting of nodes (i.e., grains) connected by edges (i.e., GBs). Based on the graph, the grain (node) feature matrix, the GB (edge) feature matrix, and the adjacency matrix characterizing the grain connectivity are extracted, which act as the inputs to the GNN model. For each grain, we select three features, i.e., the normalized grain size $ d G/ t G B$, the grain orientation angle $ \theta G$, and the ratio between the two principal ionic conductivities $ \sigma 2 G/ \sigma 1 G$. Here, the grain size $ d G$ is defined as the square root of the grain area. For each GB, the three features are the normalized length $ L G B/ t G B$, the inclined angle $ \theta G B$ [see Fig. 1(b)], and the normalized conductivity $ \sigma G B/ \sigma 1 G$. Thus, for a polycrystal with a total grain number $N$ and a total GB number $M$, the grain feature matrix, the GB feature matrix, and the adjacency matrix are $N\xd73$, $M\xd73$, and $2\xd72M$ matrices, respectively.

The GNN model is constructed and trained by the deep learning platform Pytorch.^{41} All the relevant model parameters are summarized in Table I.

GNN model architecture and . | |
---|---|

hyperparameters . | Options . |

Embedding layer | A single perceptron layer (3, 10; activation function: relu) |

Graph convolutional layer | 2 CGConv layers (activation function: relu) |

Prediction layer | A feed-forward neural network with 2 hidden layers (10, 128, 128, 3; activation function: relu ) |

Optimization algorithm | Adam |

Learning rate | 0.002 |

Epoch number | 400 |

Batch size | 40 |

GNN model architecture and . | |
---|---|

hyperparameters . | Options . |

Embedding layer | A single perceptron layer (3, 10; activation function: relu) |

Graph convolutional layer | 2 CGConv layers (activation function: relu) |

Prediction layer | A feed-forward neural network with 2 hidden layers (10, 128, 128, 3; activation function: relu ) |

Optimization algorithm | Adam |

Learning rate | 0.002 |

Epoch number | 400 |

Batch size | 40 |

## III. RESULTS AND DISCUSSION

Here, we investigate the influence of grain size, grain orientation, and the bulk and GB conductivities on the effective conductivity of polycrystals. In addition, the training and validation of a GNN-based surrogate model for predicting the effective ionic conductivity of a given polycrystal is presented.

In the following, unless otherwise stated, $ \sigma 1 G= \sigma 2 G= \sigma G$ and $ \sigma \u22c6= \sigma 11 \u22c6$. The total grain number in each RVE is $ n G=100$. The height and width of the RVE are $w=h= n G d \xaf G$ with $ d \xaf G$ being the nominal average grain size.

### A. Grain size-dependence of effective ionic conductivity

^{21,22}Unlike the brick-layer model where the expression is obtained based on an equivalent circuit,

^{20}expression (11) is derived by analytically solving an boundary value problem consisting of the governing equations for both bulk and GB conduction together with the corresponding boundary conditions. This provides a new way to interpret the brick-layer model. Relevant details are found in the supplementary material.

The analytical solution (11) indicates that the effective conductivity is dominated by the grain size-to-GB thickness ratio, i.e., $ d G/ t G B$. Additionally, the GB-to-grain conductivity ratio $ \sigma G B/ \sigma G$ also plays a crucial role. Thus, we simulate a number of RVEs with varying normalized nominal average grain size $ d \xaf G/ t G B$ and $ \sigma G B/ \sigma G$. The normalized effective conductivity $ \sigma \u22c6/ \sigma G$ varying with normalized grain size $ d \xaf G/ t G B$ and the GB conductivity $ \sigma G B/ \sigma G$ are plotted in Fig. 3(a). For comparison, the corresponding results from analytical solution (11) are also displayed.

The numerical and analytical results match well with each other, indicating that simple analytical expression (11) can already well capture the grain size-dependence of the effective ionic conductivity. $ \sigma \u22c6/ \sigma G$ exhibits a monotonic increase with the average grain size $ d \xaf G/ t G B$ and approaches saturation as $ d \xaf G/ t G B$ becomes sufficiently large. This tendency is highly consistent with the experimental measurements.^{7} $ \sigma \u22c6/ \sigma G$ increases with $ \sigma G B/ \sigma G$. Moreover, in the case with a larger $ \sigma G B/ \sigma G$, the grain size-dependence of the effective conductivity is weaker. In other words, the grain size effect can be compensated by increasing the GB conductivity. In turn, increasing the grain size can weaken the detrimental effect on the effective conductivity due to the low GB conductivity.

The macroscopic grain size-dependence of the effective ion conductivity is attributed to the underlying microscopic ionic conducting behaviors. As shown in Figs. 3(b) and 3(c), increasing $ d \xaf G/ t G B$ and/or $ \sigma G B/ \sigma G$ results in a smoother electric potential distribution across the GB and a higher level of current density and hence a higher effective ionic conductivity.

From the above discussion, the effective ionic conductivity can be enhanced by increasing either the average grain size or the GB conductivity. In this regard, the analytical and numerical results above provide quantitative guidelines.

In the above cases with relatively homogeneous distribution of grain and GB features, simple analytical solution (11) agrees well with the numerical simulations. However, only numerical simulations are capable to tackling cases involving an inhomogeneous distribution of grain and GB features. In fact, similar conclusions have been made in some other works, e.g., Refs. 23 and 37–39.

### B. Grain orientation-dependence of effective ionic conductivity

Here, we investigate how the grain orientation affects the effective ionic conductivity. To this end, as shown in Fig. 4, we consider four cases. The grain orientation distributions are plotted in Fig. 4(a). The orientation distribution in case 1 is random. In the other three cases, to consider the influence of the texture,^{42} a preferred orientation is prescribed such that the majority of the grains possess a similar orientation. In addition, to investigate the GB effect, a much lower GB conductivity is taken for case 4. The grain bulk conductivity is considered to be anisotropic, i.e., $ \sigma 1 G/ \sigma 2 G=5$. The polar plots of the effective conductivity are displayed in Fig. 4(b).

Despite the high degree of anisotropy of the bulk conductivity, the effective conductivity in case 1 is almost isotropic, which is due to the random grain orientation. In cases 2 and 3, the effective conductivity is strongly anisotropic with the maximum value appearing along the preferred orientation. The maximum effective conductivity is much larger than the isotropic one in case 1. The effective conductivity in case 4 is much smaller than the other three cases due to the much lower GB conductivity. But, it is almost isotropic, which is explained as follows. As the GB conductivity is much lower than the bulk conductivity, the effective conductivity is governed by the ionic conducting behavior at the GB. Therefore, the random distribution of the GB orientation [see Fig. 4(a)] results in the isotropy of the effective conductivity. In this case, the GB effect compensates the influence of grain orientation.

According to the above investigations, in the case with an anisotropic bulk conductivity, the degree of anisotropy of the effective conductivity can be tuned by controlling the grain orientation distribution. In particular, introducing a preferred orientation (i.e., a texture) is beneficial for achieving a high conductivity along a specific direction, e.g., the current flow direction. Thus, the results presented here can guide the engineering of grain orientations in practice.^{42}

It is worth mentioning that the influence of grain orientations on the effective conductivity in polycrystals has been investigated by analytical methods in some works (e.g., Refs. 43 and 44). However, these analytical methods ignore the influence of GBs and only applicable to idealized distribution of grain orientations. In contrast, the current numerical framework with consideration of GB conduction is general and capable of analyzing both artificially generated and experimental measured (e.g., Refs. 45 and 46) grain orientations.

### C. GB conductivity-dependence of effective ionic conductivity

Depending on the GB characters and the local chemical compositions, the ionic conductivity may differ among GBs in a certain polycrystal.^{17,23,33} Thus, the distribution of GBs with low and high conductivities is also of great importance. In fact, as suggested by Ref. 17, the percolation of GBs with high conductivities can significantly enhance the effective conductivity. Here, the influence of the GB conductivity distribution is considered. Without loss of generality, we classify the GBs into low-conducting and high-conducting GBs. Their normalized GB conductivities $ \sigma G B/ \sigma G$ are $ 10 \u2212 3$ and $ 10 \u2212 1$, respectively. In real materials, the distribution of GB conductivity could be more complicated.^{17} The present model and numerical methods are capable of analyzing cases with a general distribution of GB conductivities such as the normal distribution considered in Ref. 33.

In Fig. 5(a), the normalized effective conductivities $ \sigma 11 \u22c6/ \sigma G$ and $ \sigma 22 \u22c6/ \sigma G$ in the $x$- and $y$-directions varying with the percentage of high-conducting GBs are plotted. For all the cases except the one with a prescribed percolation path (i.e., the two markers), the high-conducting GBs are randomly distributed. The effective conductivities tend to increase with the percentage of high-conducting GBs. The fluctuations on the curves indicate that a smaller percentage but a better arrangement of high-conducting GBs can result in a larger effective conductivity. This phenomenon is more evident when comparing case A with case B. Both cases possess about 18% high-conducting GBs, while the effective conductivity in the $x$-direction in case A is about three times that in case B. This is attributed to the prescribed percolation path along the $x$-direction in case A [see Fig. 5(b)]. The effective conductivities in the $x$- and $y$-directions are not always the same. This anisotropic behavior is induced by the uneven distribution of high-conducting GBs such that the conducting behavior is different in the two directions as suggested by the distribution of current density in Figs. 5(c) and 5(d).

There exists a threshold value of the percentage of high-conducting GBs, by which the effective conductivity vs percentage curves in Fig. 5(a) are divided into two ranges with distinctive increasing rates (i.e., the nominal slope). Seen from the curves, the threshold value is around 40% for the current problem. For cases B and C in the first range with a percentage smaller than the threshold value, the percolation does not occur. For case D in the second range with a percentage larger than the threshold value, the percolation takes place. The existence of percolation magnifies the contribution of high-conducting GBs, which leads to a much higher increasing rate of the effective conductivity in the second range. The influence of the percolation is also reflected by the distribution of current density in Figs. 5(c) and 5(d). The current density along the percolation path is much higher, indicating that it is a highway for ionic conducting (see cases A and D).

In a short conclusion, to achieve a higher effective conductivity, it is necessary to have at least a certain percentage of high-conducting GBs. Moreover, to optimize the effective conductivity, it is crucial to engineer their distribution to form percolation paths.

### D. GNN-based surrogate modeling

In the above sections, we have systematically investigated the influence of various grain and GB-relevant features on the effective conductivity by FE simulations, which results in important guidelines for optimizing the effective conductivity. Here, we construct and train a GNN-based ML model to encapsulate these important microstructure–property correlations. The details on the GNN model are found in Sec. II F. The data generation is discussed first. We generate 2000 RVEs by varying the grain and GB features including the number of grains in an RVE (25–196), the nominal average grain size $ d \xaf G/ t G B$ (10–1000), the distribution of grain orientation (fully random to fully specified), the degree of regularity (almost regular to fully irregular), the GB-to-grain conductivity ratio $ \sigma G B/ \sigma G$ ( $ 10 \u2212 4$– $ 10 \u2212 1$), and the ratio between the two principal bulk conductivities $ \sigma 2 G/ \sigma 1 G$ (0.1–1). The effective conductivities of each RVE are evaluated by FE simulations. The resulting data set consists of 2000 RVEs paired with the corresponding effective conductivities. 80% of them are used for training the model and the rest 20% are for the model validation. As they vary greatly in magnitude, both inputs and outputs are scaled to the range $ [ 0 , 1 ]$ by the min-max scaling. The effective conductivities from both data sets are shown in Figs. 6(a) and 6(b). The four exemplary RVEs with the corresponding grain orientation distribution are displayed in Fig. 6(c). As will be demonstrated below, this data set is representative and sufficient for training the GNN model to achieve good performance.

The performance of the trained GNN model is illustrated in Fig. 7. A good agreement between the GNN model prediction and the ground truth from FE simulations is observed. The coefficients of determination (i.e., $ R 2$) for the two axial components $ \sigma 11 \u22c6/ \sigma 1 G$ and $ \sigma 22 \u22c6/ \sigma 1 G$ exceed 0.99 in both the training and test cases. That for $ \sigma 12 \u22c6/ \sigma 1 G$ is smaller but still larger than 0.93. The good match between the test loss-epoch curve and the training one suggests that the GNN model is well-trained without underfitting or overfitting. These results together indicate that the trained GNN model performs well.

The GNN model can act as an efficient property predictor. It takes the defined grain and GB features of a polycrystalline microstructure as inputs and predicts the corresponding effective ionic conductivity. To determine the effective conductivity of a microstructure, it takes about $10s$ by FE simulations. With the same computing resource, the time required for the GNN model is about $0.0015s$. Thus, the GNN model is much more efficient than FE simulations. This efficient surrogate model can be combined with traditional optimization methods (e.g., Ref. 47) to optimize the performance of ionic conducting polycrystalline materials. The data set and trained GNN model are found at https://github.com/XiangLongPeng/GNN_polycrystal_ion_conducting_ceramics.

## IV. CONCLUSION

In this work, we investigate the effective ionic conductivity of polycrystalline oxide ceramics. An ionic conducting model with consideration of the GB effect is introduced. Its numerical implementation is realized by the FE method, which is verified by a benchmark problem with an analytical solution. The computational homogenization method is adopted to calculate the effective ionic conductivity by conducting FE simulations at the RVE level. The proposed model and numerical methods allow for the calculation of effective conductivity in an arbitrary direction, as well as the characterization of microscopic electric fields and current densities within a realistic polycrystalline microstructure.

The influence of grain and GB features including the grain size, grain orientation, and GB conductivity on the effective ionic conductivity is systematically studied. These investigations provide the following guidelines on improving the effective conductivity:

Increasing the average grain size and/or the GB conductivity leads to the increase of effective conductivity. The detrimental effect of the low GB conductivity can be compensated by increasing the average grain size and vice versa. This is well captured by analytical expression (11).

The degree of anisotropy of the effective conductivity is tunable by engineering the grain orientation distribution. Introducing a preferred orientation (or texture) results in a high conductivity along a specific direction, which can act as the current flow direction.

A certain percentage of GBs with a high conductivity is necessary for attaining a high effective conductivity. Engineering their distribution to form percolation paths is crucial for achieving optimized effective conductivity.

In this work, we only consider artificially generated microstructures. The model and numerical method proposed here is also applicable to real microstructures with experimentally measured features. Here, the forward prediction task, i.e., evaluating the effective properties of a known microstructure is tackled. The inverse design and optimization of microstructures are also of great importance. These aspects are left for future investigations.

## SUPPLEMENTARY MATERIALS

The supplementary material presents a benchmark problem to verify the numerical implementation of the bulk and GB models. In addition, the analytical expression of the effective conductivity for an idealized polycrystal system is derived.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge the computing time granted on the Hessian High-Performance Computer “Lichtenberg”.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Xiang-Long Peng:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead). **Bai-Xiang Xu:** Funding acquisition (lead); Resources (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The data required to reproduce findings of this work are found at https://github.com/XiangLongPeng/GNN_polycrystal_ion_conducting_ceramics.

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