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Review

Study of Grain Boundary: From Crystallization Engineering to Machine Learning

1
Electrical and Computer Engineering, Penn State Behrend, Erie, PA 16563, USA
2
Key Laboratory for Precision and Non-Traditional Machining Technology of the Ministry of Education, Dalian University of Technology, Dalian 116024, China
*
Authors to whom correspondence should be addressed.
Coatings 2025, 15(2), 164; https://doi.org/10.3390/coatings15020164
Submission received: 30 December 2024 / Revised: 23 January 2025 / Accepted: 28 January 2025 / Published: 2 February 2025

Abstract

:
Grain boundaries play a vital role in determining the structural, functional, mechanical, and electrical properties of semiconductor materials. Recent studies have yielded great advances in understanding and modulating the grain boundaries via semiconductor crystallization engineering and machine learning. In this article, we first provide a review of the miscellaneous methods and approaches that effectively control the nucleation formation, semiconductor crystallization, and grain boundary of organic semiconductors. Using the benchmark small molecular semiconductor 6,13-bis(triisopropylsilylethynyl) pentacene (TIPS pentacene) as a representative example, the crystallization engineering methods include polymer additive mixing, solvent annealing, gas injection, and substrate temperature control. By studying the grain-width-dependent charge transport, we propose a grain boundary model as a fundamental basis to theoretically understand the intrinsic relation between grain boundary engineering and charge carrier mobility. Furthermore, we discuss the various machine learning algorithms and models used to analyze grain boundaries for the various important traits and properties, such as grain boundary crystallography, energy, mobility, and dislocation density. This work highlights the unique advantages of both crystallization engineering and machine learning methods, demonstrates new insights into discovering the presence of grain boundaries and understanding new properties of materials, and sheds light on the great potential of material application in various fields, such as organic electronics.

1. Introduction

In recent years, major advances have been reported in the study of the grain boundaries of organic semiconductors [1,2,3,4,5,6,7], polycrystalline materials [8,9,10,11,12,13,14], alloys [15,16,17,18,19,20,21,22,23,24], and metals [25,26,27,28,29,30]. The distribution and connectivity properties of grain boundaries exert a great influence on the electronic and mechanical properties of the materials [31,32,33,34,35]. In general, interfaces can be categorized into homophase interfaces and heterophase interfaces [36,37,38]. Homophase interfaces refer to the boundaries among crystals exhibiting identical phases and are also known as grain boundaries [39,40]. In comparison, heterophase interfaces refer to the boundaries among those with distinct phases, which are also known as interphase boundaries [41,42,43]. The complication of characterizing and understanding the grain boundary structure and properties results from the complex dimensionality of crystallographic phase space [44,45,46]. The macroscopic crystallographic parameters of grain boundaries are based on the crystalline misorientation and boundary-plane orientation [47,48,49,50]. Since many important material merits are significantly controlled by the interface-based structures and properties [51,52,53,54], the modulation of grain dimension and boundaries and the quantitative characterization of the boundary structure–property correlation shed light on our understanding of interface-dominated traits and improve many material properties.
Several challenges regarding controlling and characterizing the grain boundaries have caused bottlenecks for the further application of many electronic and functional materials in various fields. First, there has been an absence of crystalline structure–property correlation of grain boundaries, which presents a challenge for understanding crystallography-related characteristics. It can be arduous to fully assess the diverse atomic structures and relate them to the structure–property correlation of grain boundaries [55,56,57,58,59,60]. Dependent on the different parameters of the crystallographic structure, the grain boundary is known to adopt various structures that can range from disordered structures to highly ordered structures [61,62,63,64]. In addition, the grain boundary can transition from one structural phase to another as the temperature changes [65,66,67,68,69,70]. Even for grain boundaries with similar energies, a plethora of metastable states can exist, indicative of glass-like dynamics [71,72,73,74,75]. Based on the diverse structures of grain boundaries, it becomes highly desired to employ descriptors to help better understand the structures, properties, and correlation of the grain boundaries besides the dislocation characterization of atomic structures.
Second, since defects and deformations largely exist at the crystalline grain boundaries and impact charge carrier transport by acting as trap centers of charge [76,77,78,79], it is imperative to control the grain width and grain boundary dimension of the organic semiconductor, since larger grains exhibit less boundaries and defects, which can be beneficial for expediting the transport of charge carriers [80,81,82,83]. The impact on the organic semiconductor charge transport from the grain boundary defects and deformations can be further assessed by using the “grain-width-dependent mobility model.” Consider the channel has a length of L, which can be expressed as follows:
L = n L G + ( n 1 ) L G B
where L G and L G B represent the length of the crystal and grain boundary, respectively, and n represents the number of grain boundaries. Note that L G B has a very small dimension of 1–2 nm and is connected in series. Given that L is much larger than L G B , the total effective mobility μ E can be expressed as [84,85,86] follows:
L μ E = L ( n 1 ) L G B μ G + ( n 1 ) L G B μ G B
where μ G and μ G B represent the mobility of the crystal grain and the mobility of the grain boundary, respectively. After merging Equations (1) and (2), it becomes the following equation:
1 μ E = 1 μ G + n ( L G B L μ G B L G B L μ G )
In the equation above, (n − 1) has been approximated to n, given the very large value of n. The grain width W G and length L G are correlated as follows:
sin θ = W G L G = n W G L
n = L sin θ W G
By assigning A = 1 μ G and B = sin θ L G B ( 1 μ G B 1 μ G ) and merging Equation (5) into (3), it becomes the following equation:
1 μ E = A + B W G
Equation (6) describes a “grain-width-dependent mobility model”, in which the effective total mobility μ E proportionally increases as the grain width increases. It essentially indicates that a fixed channel with a larger grain width can result in higher effective mobility. The charge carrier mobility at the grain boundary μ G B can be estimated using the “grain-width-dependent mobility model.” By measuring the total effective mobility, the crystalline grain mobility, and the grain dimensions, the mobility at the grain boundary can be determined through the provided equations above. This involves experimentally characterizing the material electronic properties and analyzing the grain structure using techniques such as microscopy or modeling. Note that Equations (1)–(6) are specifically applicable to polycrystalline materials, such as organic semiconductors, where charge transport is influenced by grain boundaries and the mobility difference between crystal and grain boundary regions, and are not intended for materials without such features, such as amorphous solids or defect-free single crystals.
As discussed above, it is preferrable to have crystals with a larger grain width in order to reduce the number of crystalline defects and trap centers given a fixed channel dimension. Several factors can significantly impact organic semiconductor crystallization and the dimension of organic crystal grains. We will briefly discuss these factors in the categories of solvent choice, polymer additive, and external alignment and patterning.
For solvent choice, the difference in solvent choice exerts an important effect on controlling the organic semiconductor crystallization behavior, impacting the grain width and grain boundary dimension [87,88]. Solvents of a high boiling point render greater crystal growth time for the organic semiconductor compared to their counterparts with a low boiling point. Organic semiconductors grown in high boiling point solvent turn out to exhibit larger grain width and reduced grain boundaries. In particular, it is of great interest to employ binary solvents to effectively control semiconductor crystallization and grain dimension. The binary solvents are mainly composed of two solvents, with one being the main solvent and the other being an additive solvent [89,90,91,92,93,94]. After the additive solvent is added into (typically a larger amount of) the main solvent, the molecular structure dissimilarity between the binary solvents, as well as the solvent’s affinity, play a vital role in controlling the intensity of various intermolecular interactions, including solute/solvent, solute/solute, and solvent/solvent interactions. The dielectric constant of the solvents, as well as their boiling point and Hansen’s solubility parameters, will be considered when choosing the specific solvents for the binary solvents. A large dielectric constant gives rise to higher polarity of the solvent and better ability to stabilize the charges in the solvent [95,96]. Similar boiling points between two solvents can maintain a simultaneous evaporation profile and allow more facile control of the crystallization process of organic semiconductors. The Hansen’s solubility parameters [97,98,99,100,101,102] come into play when determining the affinity and solubility of the binary solvent. The polar component, dispersive component, and hydrogen-bonding component, denoted as p , d , and   h , respectivley, coexist as the intermolecular interaction force [103,104]. The mismatch parameter R can be calculated by using p , d , and   h , as follows:
R = P 2 + d 2 + h 2
Equation (7) describes that similar Hansen’s solubility parameters result in a smaller mismatch parameter and better solubility of a given solute in a potential solvent. Conversely, a large mismatch parameter indicates low solubility.
In addition, polymers, including amorphous, conjugated, and semicrystalline polymers, have been reported as additives to modulate the crystallization, morphology, alignment, and grain dimension of the organic semiconductors, which further enhance the charge carrier mobility and performance of the semiconductor devices [105,106,107,108,109,110]. It is important to note that the polymers’ additives can be utilized to independently modulate the semiconductor crystallization in the absence of external alignment methods, which provides a desirable merit to simplify the experimental setup. In particular, amorphous polymers, such as polystyrene (PS) [111,112,113,114,115,116], poly(α-methylstyrene) (PαMS) [117,118,119,120,121], and poly(methylmethacrylate) (PMMA) [122,123,124,125,126,127,128,129,130], have been reported as additives that powerfully improve the semiconductor morphology, reduce crystal misorientation, and enlarge the grain dimension. Conjugated polymers, such as P3HT and regiorandom pentacene-bithiophene polymer (PnBT-RRa), exert similar effects to amorphous polymers on controlling the semiconductor growth, morphological features, and grain dimension [131]. As conjugated polymers promote intermolecular interactions with the organic semiconductor, they provide an additional pathway to strengthen the π-π and hydrophobic interactions and control the morphological features [132,133,134,135]. Dependent on several important material properties of the conjugated polymers, such as chemical structure, molecular weight, polydispersity, and regioregularity, the resultant organic semiconductor/conjugated polymer mixed film can exhibit distinctly different alignment, grain boundary dimension, and charge transport properties. Besides the amorphous and conjugated counterparts, semicrystalline polymers possess combined properties of both an amorphous and crystalline nature and consequently offer a unique playground for fine tuning the crystallization and grain dimension of the organic semiconductors [136,137,138,139,140,141]. It is important to point out that, besides polymer additives, small molecular additive [142,143,144,145,146,147] and nanostructured additives [148,149,150,151,152] have also been reported to exert similar effects on grain dimension and boundary modulation.
Furthermore, external alignment and patterning methods attract abundant attention for their application in improving the organic crystal growth in a solution, morphology uniformity, and grain dimensions [153]. Despite the plethora of studies of miscellaneous external alignment methods, they can be largely categorized into the following methods: substrate patterning, air flow, and blade coating. Substrate-patterning-based methods aim to change the substrate wettability by employing photolithography and/or surfactant treatment to increase or decrease the solution dewetting behavior on the substrate [154,155,156]. The deposited solution can be confined in the wettable regions, leading to crystallization only in these regions. The resultant grain dimension can be easily tuned by controlling the dimension of the wettability region. Air flow techniques involve the use of air or inert gas during the crystallization process of the organic semiconductors [157,158]. The injection of air or inert gas can effectively control solvent evaporation, modulate solute diffusion, reduce the coffee ring effect, improve crystal alignment, tune the grain dimension, and enhance the long-range order of the organic semiconductors. Blade coating entails a movable blade or a moving substrate, as well as a coating head that deposits the solution onto the substrate at a tunable deposition rate [159,160,161,162,163,164,165]. The active layer thickness, morphology, contact resistance, and grain dimension can be controlled by optimizing the deposition speed, blade distance, and viscoelastic properties.

2. Crystallization Engineering in Grain Boundaries

In this section, we will briefly discuss the various important studies dedicated to engineering organic semiconductor crystallization, controlling the grain dimensions, reducing crystalline defects, and assessing grain-boundary-related electrical properties. We will base our discussion on the benchmark example of small-molecular organic semiconductor 6,13-bis(triisopropylsilylethynyl) pentacene (TIPS pentacene) [166,167,168,169,170]. TIPS pentacene was developed based on pentacene [171,172,173,174,175,176,177,178] and exhibits enhanced π-π stacking and improved solubility in various organic solvents [179,180]. Due to these improved merits, TIPS pentacene has been extensively demonstrated in organic electronic applications, including organic thin-film transistors [181,182,183,184,185], organic gas sensors [183,186,187,188,189], and complimentary inverters [190,191,192,193,194,195,196,197,198,199]. While the following discussion will be based on TIPS pentacene, it is important to note that these highly efficient crystallization engineering methods can be easily applied to other organic semiconductors.
He et al. studied the effect of using polyethylene oxide (PEO) as a polymer to modulate the grain dimension and boundary of TIPS pentacene crystals [200]. Without blending the PEO polymer, the in-solution crystallization of TIPS pentacene turned into randomly oriented needle- or ribbon-shaped crystals with various grain widths and grain boundaries. When PEO was blended with TIPS pentacene at a weight ratio of 5% and 10%, the polymer additive resulted in unidirectional growth of crystal needles, while simultaneously leading to long-range alignment and the modulation of the grain boundaries, as shown in the polarized optical images of Figure 1a,b. Based on six crystal measurement, as shown in Figure 1a,b, the average crystal width ( W G ) was quantitatively calculated as the width along the crystal short axis of [1 2 ¯ 0], as presented in Figure 1c. The addition of the PEO polymer at 5% and 10% weight ratios resulted in an average W G of 30.9 ± 12.4 μm and 49.5 ± 23.1 μm, indicating that PEO loading at a higher weight ratio promoted larger grain width and thereby reduced the grain boundaries. A mobility of up to 0.025 cm2/Vs was reported based on the TIPS pentacene/PEO blends. This work demonstrates that, as a semicrystalline polymer, the amorphous nature of PEO improves the organic semiconductor diffusivity and surface energies of semiconductor facets, while the crystalline nature of PEO gives rise to independent nucleation and crystallization events (Figure 1d), offering a previously unexplored pathway to modulating the crystallization and grain boundaries of small molecular organic semiconductors.
He et al. also reported incorporating of poly(butyl acrylate) as a polymer additive with TIPS pentacene to modify the nucleation and grain width of the organic semiconductors [201]. Without the additive, pristine TIPS pentacene showed not only random crystal orientations, but also a small grain width of 36.48 ± 17.59 μm, indicating the abundance of grain boundaries and crystalline defects. The addition of poly(butyl acrylate) into the mixture had threefold benefits, as follows: the polymer improved crystal alignment, enlarged areal coverage, and expanded the grain width, as showcased in the polarized optical images of Figure 2a–d. By using the same method to quantitatively calculate the grain width of the crystals, the TIPS pentacene/poly(butyl acrylate) mixture was found to demonstrate a grain width of 174.11 ± 80.82 μm, which is approximately a 5-fold enlargement compared to the counterpart without the polymer additive. The poly(butyl acrylate) polymer likely improved the TIPS pentacene crystallization and grain width by promoting the uniform deposition of the semiconductor nucleation seeds and facilitating more uniform alignment and crystal width enlargement. The combined effect from crystal alignment and grain boundary reduction can be attributed to an improved mobility of 0.11 cm2/Vs.
Wo et al. studied the influence of the grain boundary on the potential drop and charge transport of the TIPS pentacene semiconductor [202]. TIPS pentacene crystalline films with different grain sizes that range from several micrometers to millimeters were formed by using a hollow capillary writing process with different writing speeds. In particular, well-aligned TIPS pentacene crystal arrays were formed at writing speeds lower than 1 mm/s, whereas randomly oriented crystals were formed at a higher writing speed. Electrical characterization indicated that the extracted mobilities depended on both the crystal grain size and the channel direction, with respect to the domain orientation. Grain boundaries were found to result in a potential drop of more than one volt and impact the charge carrier mobilities. The highest mobility of 0.8 cm2/Vs was obtained from the TIPS pentacene thin-film transistors with a writing speed of 0.1 mm/s and channel direction parallel with crystal orientation.
Hou et al. studied the tuning of grain boundaries in the TIPS pentacene organic semiconductor by using solvent vapor annealing for application in gas sensors [187]. TIPS pentacene was first spin coated on a PMMA-covered substrate and was then exposed to different solvent vapors, including 1,3,5-trimethylbenzene (TMB), o-xylene, and toluene. The thin-film topography of TIPS pentacene after spin coating and after solvent annealing with TMB and toluene is shown in the atomic force microscopy (AFM) images of Figure 3a–c, which indicates that that morphology was dependent on the solvent choice. In particular, TMB, with a high boiling point of 168 °C, allowed sufficient time for TIPS pentacene to reorganize into large crystalline domains. In contrast, toluene, with a lower boiling point of 110 °C, led to the formation of smaller microstructures with a higher density of grain boundaries. The hole mobility was extracted to be 0.036 cm2/Vs for the TIPS pentacene thin-film transistors without solvent vapor annealing, and enhanced mobility of 0.15 cm2/Vs, 0.13 cm2/Vs, and 0.015 cm2/Vs were obtained from the devices based on vapor annealing of TMB, o-xylene, and toluene. Accordingly, a negative shifting of the threshold voltage and an enhanced subthreshold slope were observed, indicating a higher density of charge carrier trapping sites at the charge transport interface. When the TIPS pentacene thin-film transistors were exposed to NO2 gas with various concentrations, increased saturation currents and mobilities were measured as the gas molecules diffused into the grain boundaries of the TIPS pentacene crystalline domains. Thin-film transistors based on toluene vapor annealing showed the most pronounced increase in current and thus the best sensing performance, due to the presence of a high density of grain boundaries.
Chen et al. reported that the grain boundary restricted charge transport in TIPS pentacene needle-shaped crystals and isotropic-shaped domains [203]. A solution casting system that allowed controlled nitrogen flushing and deposition temperature was employed in this work to deposit TIPS pentacene crystals with different sizes. When deposited from a toluene solution with a 0.1 wt% concentration, and at a flushing rate lower than 0.5 L/min, TIPS pentacene needle-shaped crystals with a grain width larger than 6 µm were obtained. In contrast, a faster deposition rate resulted in isotropic-shaped TIPS pentacene domains with a grain width smaller than 4 µm. Figure 4a,b shows the optical micrograph image of the TIPS pentacene needles and AFM image of the isotropic-shaped TIPS pentacene domains, respectively. As compared to the smaller TIPS pentacene domains, the long crystal needles allowed the charge carriers to bypass the energy barriers located at the grain boundaries and thereby exhibited improved charge transport. Mobilities of 3 × 10−4 ± 3 × 10−5 cm2/Vs and 0.91 ± 0.08 cm2/Vs were measured from the TIPS pentacene isotropic-shaped domains and long needles, respectively. In addition, the charge carrier mobility at the grain boundary was estimated to be 5 × 10−7 cm2/Vs, which was seven orders of magnitude lower than the intrinsic mobility of TIPS pentacene.
Park et al. reported the tuning of crystal alignment and grain boundaries by applying an Ar gas injection method and studied the correlation between grain boundaries and charge transport [158]. The TIPS pentacene solution was deposited onto a polyvinylphenol substrate and was applied with the Ar gas flow to promote crystallization. Figure 5 shows the different Ar gas flow rates, including (a) at 300 sccm, (b) at 500 sccm, and (c) at 1000 sccm, which resulted in TIPS pentacene thin-film morphology with different crystal alignment and grain boundaries. The thin-film morphology of TIPS pentacene was shown to depend significantly on the Ar gas flow rate during deposition. At a lower flow rate of 300 sccm, there was no clear correlation between crystal alignment and the flow direction. When the flow rate increased to 500 sccm, partial alignment of the crystals along the flow direction was observed. At the highest flow rate of 1000 sccm, the elongated crystal grains became fully aligned with the direction of the Ar gas flow. This demonstrates the effectiveness of controlled gas flow in modulating crystal alignment and improving the uniformity of thin-film morphology. When the Ar gas flow was in parallel with the direction from source to drain contact electrode, the resultant grain boundaries were also parallel with the current direction, yielding a high mobility of 0.53 ± 0.02 cm2/Vs. In contrast, when the grain boundaries were perpendicular to the flow direction, a much lower mobility of 0.06 ± 0.02 cm2/Vs was obtained because the grain boundaries blocked the current flow.
Shao et al. reported the control of grain boundaries of TIPS pentacene crystals for gas sensing applications by tuning the different solvent choice [204]. TIPS pentacene was dissolved in different solvents, including chlorobenzene, 1,2-dichlorobenzene, toluene, and o-xylene, before being spin coated onto a PMMA dielectric layer to form the active layer. The highest hole mobility of 30 ± 6 × 10−3 cm2/Vs was obtained based on the o-xylene solvent. The grain boundaries in the TIPS pentacene crystals enabled the diffusion of NO2 molecules, and thereby the grain boundary density plays a critical role in determining the gas sensor performance. When the gas sensor was exposed to NO2 gas, the gas molecules were absorbed in the grain boundaries, which further lowered the potential barrier, attributed to an increase in the current. In particular, the o-xylene solvent resulted in abundant grain boundaries and, therefore, improved the sensing performance of the gas sensor. In contrast, chlorobenzene and 1,2-dichlorobenzene generated low grain boundary density and consequently low gas sensing performance.
Lee et al. reported the effect of grain boundaries on charge transport in inkjet-printed TIPS pentacene [205]. In this work, polyvinylphenol polymer was first spin coated as the gate dielectric layer, followed by the deposition of gold as the source and drain contact electrodes. Then, TIPS pentacene solution was deposited onto the patterned contact electrodes via inkjet printing, while the substrate was maintained at different temperatures, including room temperature, 36 °C, 46 °C, and 56 °C. The resultant thin-film morphology of the inkjet-printed TIPS pentacene film is shown in Figure 6. Growth at room temperature results in crystal accumulation at the droplet contact line, due to the coffee ring effect. When the substrate was applied with heating, the faster evaporation of the solvent promoted crystal growth from the droplet edge towards the center, giving rise to more uniform morphology and crystal alignment. However, when the temperature further rose up to 56 °C, smaller crystals with higher grain densities were formed as a result of excessively fast solvent evaporation. Average hole mobilities of 0.09 ± 0.04 cm2/Vs, 0.21 ± 0.05 cm2/Vs, and 0.07 ± 0.03 cm2/Vs were obtained from the inkjet-printed TIPS pentacene thin-film transistor based on the temperatures of 36 °C, 46 °C, and 56 °C, respectively. The mobility at 46 °C further increased to 0.44 ± 0.09 cm2/Vs based on multi-drop TIPS pentacene devices, which incorporate grain boundaries in the same direction to minimize the impact from mobility trapping sites.
Table 1 lists a summary of the reviewed works in this section, including the authors, semiconductors, results and mobility.

3. Machine Learning in Grain Boundaries

Machine learning is a method that uses a set of algorithms to recognize the patterns and trends in large datasets, which can be categorized into supervised learning [206,207,208,209,210,211] and unsupervised learning [212,213,214,215,216,217,218,219,220], depending on training with or without known data labels. Machine learning algorithms have been widely implemented to study and understand the various important traits of grain boundaries. For instance, machine learning models have been developed to successfully generate grain boundary structures [221], study solute segregation [222], and even predict grain boundary energy [223,224], charge transport characteristics [225,226], and shear coupling behavior [227,228]. In particular, it can be critically important to implement quantitative structure symbols to study the atomic structure of grain boundaries. For instance, Kiyohara et al. utilized the descriptors to describe the energy and structure of grain boundaries in a certain document, including the crystallographic parameters of materials at macro and micro levels, atomic density, and bond length based on the nearest neighbor [221]. Huber et al. predicted solute segregating energy to grain boundaries by utilizing Voronoi analysis to understand the various properties of grain boundaries, including the total side length, total surface area, number of faces, number of edges, the ratio of surface area to volume, changes in excess volume and coordination number of grain boundaries, and Q1 and Q8 Steinhardt parameters [222]. Ziatdinov et al. applied deep neural networks to study the atomic species location and defect types based on extraction from atomically resolved scanning transmission electron microscopy (STEM) images of graphene [229]. The authors utilized a “weakly supervised” approach to identity the large variety of defects based on information from all atomic species coordinates. This approach was further applied to understand atomic and defect transformation and interpret the coordination switching between silicon dopants in graphene. Their deep-learning-based approach shows the merit of the logic of a human operator and enlightens the extraction and analysis of raw information from experimental data, such as developing a “self-driving” microscope to study the key information of atomic species and defects based on atomically resolved imaging.
Priedeman et al. reported the correlation between the grain boundary crystallography and the atomic structure in nickel by utilizing the [100] symmetric tilt grain boundaries [230]. By using the structural unit model as a benchmark, the authors evaluated the atomic structure description capacity of the local environment representation, which is a refinement of the Smooth Overlap of Atomic Positions (SOAP) descriptor. The structural unit model can be encoded by the local environment representation, which also reveals additional information, including distortion in the structural units and structural unit arrangement at the interface. The local environment representation gives rise to SPRING, which is a visualization tool representing the structural similarities and the spatial relationship between grain boundaries and projecting high-dimensional LER descriptors into a two-dimensional space for visualization purposes. Using the SPRING representation, objective evidence was produced that enlightens the correlation between crystallography and atomic structure based on the [100] symmetric tilt grain boundaries. Figure 7 shows the [100] symmetric tilt grain boundaries using SPRING visualization. The SOAP descriptor was employed as a superior method to quantify grain boundary atomic structures. The similarity network of grain boundary atomic structures becomes clustered at the grain boundaries, resulting in structural units segregating within the network. The algorithm is in stark contrast with the structural unit model, reduces the requirement of manual input, and simplifies the structural unit identification and descriptions of structural unit ordering. This method meets the requirement for symmetrical configuration and can be easily applicable to the grain boundary structure to assess the varied character boundary plane orientations.
Rosenbrock et al. employed the gradient enhancement tree algorithm to predict the migration trend of grain boundaries and studied the local atomic environments related to grain boundary dislocations and grain boundary mobility in face-centered cubic (FCC) nickel [227]. Two methods were studied to assess the grain boundary SOAP matrices. Firstly, the SOAP vectors of all atoms within a single grain boundary were used in order to acquire one averaged SOAP vector, which serves as a measure of the entire grain boundary, as illustrated in Figure 8. This single averaged vector representation method of the average LAE is named the Averaged SOAP Representation (ASR) method. The second method involves a compilation of a set of unique LAEs based on the comparison of every atom environment in the grain boundary to those by using dissimilarity metrics and a numerical similarity parameter. Based on two LAEs, with both belonging to a unique class of LAEs, a set U with a representative LAE was produced for the grain boundary system. An exponential increase in the number of unique LAEs that characterize a grain boundary is observed at a smaller dissimilar parameter. An LAE that is sufficiently dissimilar to those in the set is then added and is representative for all other similar LAEs. The set size of the U LAEs can increase as a result of discovering additional LAE classes. The results indicated that FCC-like atoms still play an important role in determining the trend of grain boundary mobility, which is in contrast to the previous understanding that atoms located away from the grain boundaries exert little impact on the grain boundary properties. This study provides a new pathway to quantitatively study the correlation between the grain boundary structure and property via machine learning algorithms.
In addition, Rosenbrock et al. reported the combination of ASR and LER to study the grain boundary energies, mobility, and shear coupling in nickel [228]. The ASR is based on the average local environment descriptors, whereas the LER descriptor is based on the globally unique local environment fractions within the entire grain boundary system. First, grain boundary energies were calculated using machine learning. The SOAP descriptor was calculated for each structure, and the result was plotted to form the similarity matrix K. An RMS error of 0.07 with a standard deviation in the grain boundary energy of 0.37 was reported, which was attributed to the small size of data points used for machine learning. This machine learning model was also trained to classify and understand the mobility and shear coupling properties of the grains. Using SVM classification, the prediction of mobility at the grain boundary showed 91.9% and 77.2% accuracy based on the training and validation sets, respectively. These results indicate that the mobility varies dependent on the local environment at the grain boundary and that the mobility prediction at the grain boundary is enabled by the essential atomic environment information captured by the aggregation of the SOAP descriptors. As for shear coupling, a low prediction accuracy of 64% was obtained, which indicated more information, and the local environment conditions were still required in order to further improve the shear coupling prediction result.
Orme et al. employed a machine learning framework to study the maps of orientation image and electron backscatter diffraction in order to understand the factors that lead to the deformation twinning in Mg alloy [231]. A decision tree learning environment was utilized to assess the correlation between the microstructure and twin deformation formation and to analyze the twin nucleation in a specific grain and twin propagation from one grain boundary to another. Each model was found to exhibit various crystallographic attributes that uniquely impact the twinning in Mg. The results showed that the twin nucleation in Mg was due to various factors, including grain dimension and density of bulk dislocation, whereas the twin propagation was impacted by factors such as grain boundary and misorientation, as well as the angle of grain boundary with respect to the RD plane. The prediction accuracy for twin nucleation and twin propagation across grain boundaries was 86.5% and 96.1%, respectively, based on a cross-validated training dataset of 104 grains. In comparison, the accuracy for predicting twin nucleation and propagation was 75.1% and 75.5%, respectively, based on an independent test set of 1127 grain boundaries.
Gomberg et al. reported a “process–structure–property” (PSP) paradigm to study atomistic grain boundary simulations by developing a framework to objectively identify the atoms in the grain boundary in aluminum [223]. The framework was based on local regression, a centro-symmetry parameter, and atomic structure quantification. For asymmetric tilt grain boundaries, the established model was able to correlate the macro degrees of freedom and energy to atomic structure approximation of the grain boundary. The model showed an average prediction error below 13 mJ/m2. This work shows that quantitative linkages are applicable to improve grain boundary engineering, improve global optimization of grain boundary structures, and integrate both computational and experimental results in real time.
Lawrence et al. applied canonical correlation analysis to identify the effect of processing variables on the abnormal enlargement of minority grain growth at the expense of the surrounding grains in both metallic and ceramic materials [232]. By using high-purity aluminum oxide, the authors assessed practical maps and metrics and identified the combination of processing variables most related to abnormality metrics. This work shows the virtue of enabling the selection of processing parameters to achieve more desirable microstructures and optimized properties such as strength, hardness, and thermal conductivity.
Table 2 lists a summary of the reviewed works in this section, including the authors, materials and results.

4. Conclusions

In this review, we systematically examined recent advances in the study of grain boundaries, with a focus on their structural, functional, mechanical, and charge transport properties. We discussed various crystallization engineering strategies, including polymer additive blending, solvent annealing, gas flow modulation, and temperature control, which have demonstrated significant effectiveness in influencing crystal growth, thin-film morphology, and grain dimensions. Using the benchmark organic semiconductor TIPS pentacene as an example, we highlighted the ability of these methods to optimize grain width, minimize crystalline defects and trap centers, and enhance charge carrier mobility. Additionally, we discussed a grain-width-dependent charge transport model, providing a theoretical framework to correlate charge carrier mobility with grain boundary dimensions and offering fundamental insights into charge transport mechanisms.
Furthermore, we reviewed the integration of machine learning approaches to study and predict various grain boundary properties, such as crystallography, energy, mobility, and dislocation density. These methods have shown promise for advancing our understanding of grain boundary structure–property relationships and accelerating material design and optimization. The inclusion of machine learning is particularly innovative, showcasing its potential to drive material discovery; however, the current limitations of machine learning, such as data constraints and computational complexity, remain challenges to address. Similarly, while the review highlights the transformative potential of crystallization engineering strategies, the reliance on TIPS pentacene as the benchmark material narrows the generalizability of the findings, and the reproducibility of these strategies for industrial applications require further validation.
Overall, this review underscores the potential of combining crystallization engineering and machine learning methodologies to refine the structural, mechanical, and electrical properties of not only organic semiconductors, but also other high-performance material systems. Future research should focus on addressing these challenges of crystallization engineering, alongside developing more robust and interpretable machine learning models to explore the correlation between structural properties and electrical performance in high-mobility organic semiconductors. These efforts will pave the way for fabricating next-generation organic electronic devices for real-world applications.

Author Contributions

Z.H. wrote the manuscript. Z.H., S.B. and K.A.-Y. revised the manuscript together. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Polarized optical images showing the effect of the PEO polymer additive on modulating the grain boundary of TIPS pentacene organic crystals, including different weight ratios of (a) 5% and (b) 10%. The triangle in (a,b) marks the uncovered substrate. (c) A plot showing the change in TIPS pentacene crystal grain boundary with different loading ratios of PEO, based on six measurements. The red arrows in (a,b) represent the long axis [210] of TIPS pentacene crystals, which is also shown in (c). Crystal width, or W G , is measured as the width along the short axis of [1 2 ¯ 0]. (d) A cartoon showing the improvement of TIPS pentacene (TP) morphology and highlighting the improved crystal alignment and grain boundary enlargement as a result of PEO polymer addition. Reproduced from reference [200], with permission from Springer.
Figure 1. Polarized optical images showing the effect of the PEO polymer additive on modulating the grain boundary of TIPS pentacene organic crystals, including different weight ratios of (a) 5% and (b) 10%. The triangle in (a,b) marks the uncovered substrate. (c) A plot showing the change in TIPS pentacene crystal grain boundary with different loading ratios of PEO, based on six measurements. The red arrows in (a,b) represent the long axis [210] of TIPS pentacene crystals, which is also shown in (c). Crystal width, or W G , is measured as the width along the short axis of [1 2 ¯ 0]. (d) A cartoon showing the improvement of TIPS pentacene (TP) morphology and highlighting the improved crystal alignment and grain boundary enlargement as a result of PEO polymer addition. Reproduced from reference [200], with permission from Springer.
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Figure 2. Polarized optical images in subfigures (ad) showing the enlarged grain width of TIPS pentacene crystals with the blending of poly(butyl acrylate) as a polymer additive. All images have the same scale bar as in (a). The crystal long axis [210] is marked with arrows. Reproduced from reference [201], with permission from Springer.
Figure 2. Polarized optical images in subfigures (ad) showing the enlarged grain width of TIPS pentacene crystals with the blending of poly(butyl acrylate) as a polymer additive. All images have the same scale bar as in (a). The crystal long axis [210] is marked with arrows. Reproduced from reference [201], with permission from Springer.
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Figure 3. AFM height images of (a) as-cast TIPS pentacene film, (b) TIPS pentacene film with TMB solvent vapor annealing, and (c) TIPS pentacene film with toluene solvent vapor annealing. Reproduced from reference [187], with permission from MDPI.
Figure 3. AFM height images of (a) as-cast TIPS pentacene film, (b) TIPS pentacene film with TMB solvent vapor annealing, and (c) TIPS pentacene film with toluene solvent vapor annealing. Reproduced from reference [187], with permission from MDPI.
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Figure 4. (a) An optical micrograph image of TIPS pentacene needles with grain widths larger than 6 µm. (b) An AFM image of rounded and isotropic-shaped TIPS pentacene film with grain widths smaller than 4 µm. The gold contact electrode is shown in the bottom right corner. Reproduced from reference [203], with permission from AIP Publishing.
Figure 4. (a) An optical micrograph image of TIPS pentacene needles with grain widths larger than 6 µm. (b) An AFM image of rounded and isotropic-shaped TIPS pentacene film with grain widths smaller than 4 µm. The gold contact electrode is shown in the bottom right corner. Reproduced from reference [203], with permission from AIP Publishing.
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Figure 5. Thin-film morphology of TIPS pentacene grains depending on different Ar gas flow rates, including (a) at 300 sccm: no clear correlation between alignment and flow rate, (b) at 500 sccm: crystal alignment started to be correlated with flow direction, (c) at 1000 sccm: the elongated crystal grains were aligned along the flow direction. Reproduced from reference [158], with permission from Elsevier.
Figure 5. Thin-film morphology of TIPS pentacene grains depending on different Ar gas flow rates, including (a) at 300 sccm: no clear correlation between alignment and flow rate, (b) at 500 sccm: crystal alignment started to be correlated with flow direction, (c) at 1000 sccm: the elongated crystal grains were aligned along the flow direction. Reproduced from reference [158], with permission from Elsevier.
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Figure 6. Optical microscope images showing the morphologies of TIPS pentacene single droplet based on different drying temperatures, including room temperature (RT), 36 °C, 46 °C, and 56 °C, respectively. Reproduced from reference [205], with permission from Elsevier.
Figure 6. Optical microscope images showing the morphologies of TIPS pentacene single droplet based on different drying temperatures, including room temperature (RT), 36 °C, 46 °C, and 56 °C, respectively. Reproduced from reference [205], with permission from Elsevier.
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Figure 7. Visualization of the [100] symmetric tilt grain boundaries based on SPRING. The colored nodes represent the grain boundaries dependent on the structural unit content in the grain boundaries, as indicated in the color scale. White nodes indicate an absence of structural unit in the grain boundary. Reproduced from reference [230], with permission from Elsevier.
Figure 7. Visualization of the [100] symmetric tilt grain boundaries based on SPRING. The colored nodes represent the grain boundaries dependent on the structural unit content in the grain boundaries, as indicated in the color scale. White nodes indicate an absence of structural unit in the grain boundary. Reproduced from reference [230], with permission from Elsevier.
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Figure 8. Illustration of the construction process for the ASR and LER. ASR: after forming a SOAP matrix, an averaged SOAP vector is produced based on the sum of Q columns in the matrix, which represents all grain boundaries. LER: the grouped SOAP vectors are reduced to unique vectors (a set of U) represented by unique LAE based on the SOAP similarity metric. A histogram was generated based on each grain boundary that counts the number of unique vector examples existing in the grain boundary. The LER matrix is a collection of histograms showing LEA unique for the M grain boundaries in the M × U collection. Reproduced from reference [227], with permission from Springer Nature.
Figure 8. Illustration of the construction process for the ASR and LER. ASR: after forming a SOAP matrix, an averaged SOAP vector is produced based on the sum of Q columns in the matrix, which represents all grain boundaries. LER: the grouped SOAP vectors are reduced to unique vectors (a set of U) represented by unique LAE based on the SOAP similarity metric. A histogram was generated based on each grain boundary that counts the number of unique vector examples existing in the grain boundary. The LER matrix is a collection of histograms showing LEA unique for the M grain boundaries in the M × U collection. Reproduced from reference [227], with permission from Springer Nature.
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Table 1. Summary of the reviewed works, including the authors, semiconductors, results, and mobility.
Table 1. Summary of the reviewed works, including the authors, semiconductors, results, and mobility.
AuthorSemiconductor Result SummaryMobilityReferences
He et al.TIPS pentaceneSemicrystalline polymer PEO modulated semiconductor diffusivity, crystallization, and grain boundary 0.025 cm2/Vs[200]
He et al.TIPS pentacenePoly(butyl acrylate) polymer promoted semiconductor nucleation, alignment, and grain width 0.11 cm2/Vs[201]
Wo et al.TIPS pentaceneDifferent speeds of hollow capillary writing process impacted domain orientation and grain boundary0.8 cm2/Vs[202]
Hou et al.TIPS pentaceneVapor annealing using different solvents impacted the topography, grain boundary density, and further gas sensing properties0.15 cm2/Vs[187]
Chen et al.TIPS pentaceneSolution casting system enabled good control of grain dimension and boundary in needle-shaped and isotropic-shaped domains0.91 ± 0.08 cm2/Vs[203]
Park et al.TIPS pentaceneAr gas injection resulted in crystal and grain boundary alignment in parallel with the current flow direction0.53 ± 0.02 cm2/Vs[158]
Shao et al.TIPS pentaceneSolvent choice impacted grain boundary density and gas sensing performance30 ± 6 × 10−3 cm2/Vs[204]
Lee et al.TIPS pentaceneDifferent substrate temperatures modulated grain boundary and charge transport0.44 ± 0.09 cm2/Vs[205]
Table 2. Summary of the reviewed works, including the authors, materials, and results.
Table 2. Summary of the reviewed works, including the authors, materials, and results.
AuthorMaterial Result SummaryReference
Priedeman et al.NickleGrain boundary crystallography is correlated with the atomic structural unit content [230]
Rosenbrock et al.NickleGrain boundary mobility is impacted by the local atomic environments and grain boundary dislocations[227]
Rosenbrock et al.NickleGrain boundary energies, mobility, and shear coupling are assessed [228]
Orme et al.Mg alloyGrain dimension and bulk dislocation density impact twin nucleation, while grain boundary and misorientation impact twin propagation[231]
Gomberg et al.AluminumAtomistic grain boundary simulations are studied using a “process–structure–property” (PSP) paradigm[223]
Lawrence et al.Aluminum oxideAbnormal enlargement of minority grain growth is impacted by processing variables[232]
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He, Z.; Bi, S.; Asare-Yeboah, K. Study of Grain Boundary: From Crystallization Engineering to Machine Learning. Coatings 2025, 15, 164. https://doi.org/10.3390/coatings15020164

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He Z, Bi S, Asare-Yeboah K. Study of Grain Boundary: From Crystallization Engineering to Machine Learning. Coatings. 2025; 15(2):164. https://doi.org/10.3390/coatings15020164

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He, Zhengran, Sheng Bi, and Kyeiwaa Asare-Yeboah. 2025. "Study of Grain Boundary: From Crystallization Engineering to Machine Learning" Coatings 15, no. 2: 164. https://doi.org/10.3390/coatings15020164

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He, Z., Bi, S., & Asare-Yeboah, K. (2025). Study of Grain Boundary: From Crystallization Engineering to Machine Learning. Coatings, 15(2), 164. https://doi.org/10.3390/coatings15020164

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