Introduction

Machine learning is revolutionizing materials research by enabling data-driven predictive science. In recent years, machine learning-enabled molecular design, with the objective of identifying new materials with specific, desired properties, has made significant strides1,2. The typical workflow consists of forward and inverse predictions3. First, a statistical model is built to predict the target properties of a given material based on its compositional and structural features. Subsequently, inverse mapping of this model is explored to predict materials with the desired properties in the reverse direction. Various machine learning techniques have been actively developed to apply this concept in polymer chemistry, including a wide range of property predictors4,5,6,7 and inverse design methods using virtual polymer generators such as molecular generative artificial intelligence8,9,10. Notably, proof-of-concept examples include the discovery of amorphous polymers with high thermal conductivity4, lithium-ion conducting polymers11,12, gas-separating polymer membranes13, and high-temperature polymer dielectrics14. However, few cases of polymers that were initially predicted by machine learning and subsequently verified experimentally have been reported. In this nascent field, our work advances the proof-of-concept of machine learning-driven polymer material design for real-world applications through the successful discovery of liquid crystalline polymers with high thermal conductivity.

As the demand for the miniaturization and portability of electronic devices increases, researchers have focused on the development of lightweight, highly insulating polymers with high thermal conductivity for efficient heat dissipation. However, the thermal conductivity of amorphous polymers is one to three orders of magnitude lower than those of metals and ceramics15,16, which is a major barrier to their practical application. In general, polymers without free electrons exhibit low thermal conductivity because phonons, the dominant factor in heat transport, tend to scatter in the amorphous region. To overcome this empirical and theoretical limitation, inorganic fillers are commonly added to the polymer matrix17. However, increasing the filler content causes other polymer properties such as adhesion, flowability, processability, and insulation to significantly deteriorate18. Additionally, if the thermal resistance of the matrix polymer is high, the thermal conductivity of the composite material reaches a ceiling; therefore, it is imperative to improve the thermal conductivity of the polymer matrix itself19. Therefore, attempts have been made to suppress phonon scattering by utilizing liquid crystalline phase formation to induce an ordered alignment of polymer chains20. However, the design of liquid crystalline polymers remains purely empirical, and relies extensively on trial and error. While some trends have been identified—for example, the tendency of phenyl benzoate backbones or certain alkyl spacer chains to promote liquid crystallinity21,22,23,24,25,26—polymers with complex molecular interactions, such as charge-transfer interactions in polyimides, often deviate from these trends, resulting in numerous exceptions27.

In this study, we developed a machine learning algorithm to predict the chemical structure of a polymer-repeating unit capable of forming liquid crystalline phases. This is an indirect approach that aims to enhance the thermal conductivity by addressing the challenge posed by the lack of a comprehensive dataset on the thermal conductivity of oriented polymers. We constructed a binary classifier using the compositional and structural features of the polymer-repeating unit as inputs from a labeled dataset of liquid crystalline polymers and other polymers that were previously synthesized. The prediction accuracy of the constructed model in discriminating whether a polymer exhibits a liquid-crystalline phase exceeded 96%. Using this model, we conducted high-throughput virtual screening to search for polyimides that form liquid crystalline states. In liquid crystalline polyimides, rigid mesogens composed of aromatic segments (including conjugated rings and imide groups) promote molecular alignment, while flexible spacer chains (consisting of alkyl or similar groups) enhance the molecular mobility to facilitate the formation of the liquid crystalline state21. In addition, six polymers were selected from a narrowed-down library of candidates for experimental verification by the de novo syntheses of their monomers followed by their polymerization reactions. As a result, all six polymers were successfully synthesized, and the resulting polymers spontaneously formed smectic liquid crystal phases. Moreover, by applying a lock-in photothermal method for measuring the in-plane thermal diffusivity using arrayed temperature sensors on suspended SiNX membranes, their thermal conductivities were experimentally confirmed to reach 0.722−1.26 W m−1 K−1. These are the first liquid crystalline polymers predicted and discovered via machine learning.

Results

Performance of the machine learning model

The machine learning task is formulated as a supervised learning problem aimed to classify the chemical structure of a polymer repeat unit denoted as \(X\) into two classes with the binary variable \(Y\in \{-1,\,1\}\) indicating a liquid crystalline state (\(Y=1\)) or a non-liquid crystalline state (\(Y=-1\)) (Fig. 1a). The compositional and structural features of a given repeat unit are encoded into a 397-dimensional descriptor vector (\(\phi \left(X\right)\in {{\mathbb{R}}}^{397}\)). Here, only linear homopolymers without additives were considered for the learning and prediction tasks by focusing on their intrinsic properties and avoiding the influence of additives or copolymerization, which can affect liquid crystalline phase formation. The descriptor was formed through the concatenation of two different descriptors: a 207-dimensional vector of RDKit descriptors representing various physical, chemical, and structural features of polymer molecules (https://www.rdkit.org/), and a 190-dimensional quantitative descriptor that encodes empirical force field parameters derived using a general AMBER force field version 2 (GAFF2)28, which is widely used in all-atom classical molecular dynamics simulations29,30. The calculations of these two polymer descriptors were performed using the Python libraries XenonPy5,31 and RadonPy29, which implement wrapper functions for RDKit and the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)32, respectively. To properly describe the periodicity of polymer repeating structures, the descriptor was calculated after linearly linking the head and tail of the repeating unit 10 times to form a decamer30. The descriptor calculation procedure is described in the Supplementary Information (Section 1.2; pages S2 and S3).

Fig. 1: Machine learning process for predicting whether a polymer with a designed repeating unit exhibits liquid crystalline states.
figure 1

a Machine learning workflow. b Prediction accuracy (confusion matrix, precision, recall, and F1 scores) for test datasets; standard deviations of the performance metrics obtained from 100 independent tests are shown in parentheses.

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A machine learning classifier \(Y=f\left(\phi \left(X\right)\right)\) defines a mapping from the vectorized polymer to the binary class label by indicating whether the polymer would exhibit a liquid crystalline state. As positive instances (P) for the binary classification task, we used a list of 951 liquid crystalline polymers compiled from PoLyInfo33, a polymer property database that had been manually compiled based on a literature survey. The polymer list was partially labeled with more detailed annotations representing liquid crystalline states such as the nematic and smectic phases; however, in this study, they were merged into a single label (liquid crystal). A total of 3,597 polymers without any record of forming liquid crystalline states were extracted from the database and used as unlabeled instances (U).

Notably, these polymers are not necessarily true negative cases, and their liquid crystallinity has not been confirmed. No comprehensive database of negative cases has been constructed for liquid crystalline polymers. In machine learning, this problem is known as positive and unlabeled (PU) learning34. In the present study, we applied the classical PU learning algorithm proposed by Elkan et al.35 to calibrate the classification probability. However, the effect of the PU learning calibration was insignificant. The protocols established for data preparation and PU learning are provided in Supplementary Information (Section 1.3; pages S3 and S4).

The binary classifier was modeled as a conventional multilayer perceptron neural network. A total of 85% of the entire dataset was randomly selected and used as the training set, while the remaining 15% was used as a test set for evaluating the model performance. A randomly selected 85% of the training dataset was applied for training, and the remainder was utilized for the validation set. The hyperparameters of the classification model were optimized using the black box optimization software Optuna36, which adjusted the number and width of the hidden layers to minimize the validation F1 score with this random split (see details in Supplementary Information (Section 1.2; page S2 and S3)). To assess the variation in the performance metrics, we independently repeated the training and testing process 100 times using different random data splits.

According to Fig. 1b, the average classification accuracy exceeds 96%. The mean values of recall and precision are 0.92 and 0.90, respectively, suggesting that the trade-off between the false and true positive rates is well balanced. Although the data are not shown herein, we confirmed that other machine learning algorithms, such as ensemble learning, achieve comparative predictive performance.

Virtual screening of liquid crystalline polyimides

Using the PU learning-calibrated classifier, we conducted an exhaustive search for polymers that undergo liquid crystallization. The search space was limited to polyimides, which are usually synthesized via the polycondensation of tetracarboxylic dianhydride and diamine monomers. To create a virtual library, we decomposed the template structure of polyimides into five building blocks labeled A-E in Fig. 2a. According to this decomposition, the acid dianhydride and diamine are composed of symmetric molecules. Fragments matching the structural pattern of each building block were searched among a set of highly available compounds in the small molecule database ZINC37. Considering these combinations, 115,536 virtual polyimides were computationally generated. In polyimides, the rigid chains, which consist of conjugated aromatic rings bonded to imide groups, promote molecular orientation to instill mesogenic behavior. In contrast, the flexible spacer groups tend to form an amorphous structure that can strongly absorb impact energy. Because the rigid and flexible components are arranged symmetrically, the self-assembled higher-order structures are also expected to exhibit structural equivalence (symmetry) and periodicity in the plane of the main chain. Thus, a virtual library was constructed following this design and synthesis strategy.

Fig. 2: Virtual screening of liquid crystalline polyimides.
figure 2

a The rigid and flexible chains of polyimides were divided into five building blocks, and the corresponding fragment set for each block was extracted from the Zinc database. By combining these fragments, 115,536 virtual polyimides were computationally generated. b The chemical structures of the virtual polymers are visualized using the UMAP projection (in gray). The 10,825 candidates predicted to exhibit liquid crystallinity were clustered into 391 groups based on molecular similarity and visualized in different colors. From these clusters, six polyimides were selected and successfully synthesized.

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By applying a median liquid crystal transition probability of unity and a standard deviation of less than 0.2 as thresholds, approximately 91% of the candidate polymers were filtered out, resulting in 10,825 polyimides predicted to exhibit liquid crystallinity. Figure 2b shows a two-dimensional (2D) representation of the selected candidates using the uniform manifold approximation and projection (UMAP) algorithm38, in which the chemical structure of each repeat unit was encoded into the descriptor following the procedure utilized during the construction of the predictive models. We applied hierarchical density-based spatial clustering of applications with noise (known as HDBSCAN)39,40 to the projected polymers, which suggested the presence of 391 different clusters. Subsequently, we selected candidates suitable for experimental synthesis by examining representatives from the 391 clusters and those with closely related structures. Specifically, the repeating unit of a representative polyimide candidate from each cluster was reviewed by a polymer synthesis expert to assess the synthesizability of each dianhydride and diamine monomer, as well as the availability of the necessary ingredients. As a consequence, six polyimides (Fig. 3) were selected from several promising clusters and successfully synthesized; X-ray structural analysis of these polyimide bulk films revealed the formation of smectic liquid crystalline phases (Supplementary Information, Section 3.5 (pages S40–S41)). Further details of the selection procedure are provided in the Supplementary Information (Section 1.5; pages S4–S6).

Fig. 3: Chemical structures of the six polyimides selected for synthesis through virtual screening and their corresponding monomers.
figure 3

The polyimides were derived from combinations of three types of dianhydrides (TAC1, TAC2, TAC3) and two types of diamines (DAC1, DAC2).

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Analysis of higher-order structures in spin-coated films

The higher-order structures within the spin-coated thin films were analyzed by conducting Grazing Incidence Wide-Angle X-ray Diffraction (GIWAXD) measurements. In this study, the in-plane thermal diffusivity of polyimide films—fabricated via spin coating followed by thermal imidization—was measured using SiNX devices. The small size of these devices meant that conducting GIWAXD directly on the polyimide films formed on them was challenging. Therefore, similar films were replicated on silicon wafers under identical spin-coating conditions to facilitate GIWAXD measurements. To quantify the molecular orientation within the spin-coated films, the uniaxial orientational order parameter \(S=\left(3\left\langle {\cos }^{2}\phi \right\rangle -1\right)/2\) was calculated using the azimuthal distribution of the diffraction intensity at \(2\theta \approx {20}^{\circ }\). This calculation method follows the approach described by Ishige et al.41.

The measured GIWAXD profiles show that the diffraction is somewhat concentrated on the meridional axis, indicating preferential in-plane orientation of mesogens in the thin films (Fig. 4a–f). The \(S\) values calculated for the films ranged from \(-0.33\) to \(-0.45\) (see the rightmost column of Table 1).

Fig. 4: GIWAXD profiles of the spin-coated polyimide films on silicon wafers.
figure 4

a PIC1-1 PI, b PIC1-2 PI, c PIC2-1 PI, d PIC2-2 PI, e PIC3-1 PI, f PIC3-2 PI, and g Intensity vs. q profiles. The films were prepared under the same spin coating conditions as the samples fabricated for μ-TWA to ensure consistency in the film thickness and morphology.

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Table 1 Experimental properties of the six synthesized polyimides
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Mesogens of which the molecular orientation is such that the long axis is oriented perfectly parallel to the substrate surface would display a 2D GIWAXD profile with the diffraction concentrated on the meridional line parallel to the film normal, yielding an \(S\) value of \(-0.5\). The range of \(S\) values obtained in our work suggests that the alignment of the mesogens is not entirely parallel to the substrate surface. The in-plane molecular orientation develops during the imidization process, where a polyimide film fixed onto a silicon wafer undergoes contraction. GIWAXD profiles measured immediately after spin coating and solvent removal, i.e., before the imidization process, showed azimuthally spread diffraction, indicating a lack of preferential molecular orientation (see Fig. S57 in the Supplementary Information, Section 3.5 (page 44)).

Moreover, SAXS and WAXD analyses of the bulk samples suggested that PIC1-1, 1-2, and 2-1 exhibited a smectic B-like phase, whereas PIC3-1 and 3-2 demonstrated a smectic E-like phase. PIC2-2, similar to PIC3-1 and 3-2, formed a close-packed arrangement between neighboring molecules (Fig. S54). In thin films, however, the intensity vs. \(q\) profiles shown in Fig. 4g indicated that the peak shape around \(q\sim 15\,{\text{nm}}^{-1}\) for PIC2-2, 3-1, and 3-2 differed from those observed in bulk analyses. This discrepancy suggests that in thin films, the molecular alignment over long distances may be insufficient to produce the diffraction patterns characteristic of a smectic E-like phase.

According to the \(S\) values listed in Table 1, no significant differences in the degree of in-plane orientation were observed between the PIC1-1 and PIC1-2 samples, nor between the PIC3-1 and PIC3-2 samples. However, a different trend was observed for the PIC2 series, where PIC2-1 exhibited a higher degree of in-plane orientation compared to PIC2-2. Additionally, the PIC3 series displayed a lower degree of in-plane orientation than both the PIC1 and PIC2 series. These findings suggest that differences in the structure of the mesogen significantly affect the \(S\) values, whereas variations in the spacer length do not have a substantial impact.

The presence of a methylene group at the center of the mesogen in the PIC2 series influences the higher-order structure and \(S\) values. The results of subsequent analyses of the bulk samples presented in Fig. S57 reveal the distinct higher-order structural characteristics of the PIC2 samples as compared with the other samples. This suggests that the methylene group of the mesogen in the PIC2 series plays a critical role in determining the degree of molecular orientation and structural evolution during thermal imidization.

Evaluation of thermophysical properties

The in-plane thermal conductivity (\(\lambda\)) and phase transition temperatures of the synthesized liquid crystalline polyimides were investigated. The results, including the in-plane thermal diffusivity (\(\alpha\)), specific heat capacities (\({C}_{p}\)), densities (\(\rho\)), and \(\lambda\) (calculated using \(\alpha\), \({C}_{p}\), \(\rho\)), as well as the phase transition temperatures, are summarized in Table 1. Measurement of the thermal conductivity and diffusivity of microscopically ordered liquid crystalline polymer thin films requires unique setups and equipment with high precision and fine spatial resolution. Several techniques, such as a thermal bridge method42, atomic force microscopy cantilever-assisted measurements43, and time-domain thermoreflectance44, have been employed for measuring the microscale thermal conductivity of polymers. In this study, the thermophysical properties were evaluated by a microscale temperature wave analysis method (μ-TWA) using a device consisting of a line-shaped or spiral-shaped temperature sensor array on a 2D silicon nitride (SiNx) membrane with a thickness of 50 or 150 nm45 and ultrafast scanning calorimetry (FSC)46. The method employing the array-type sensors (μ-TWA) enabled rapid measurements of the in-plane thermal diffusivity of the aligned polyimide nanofilms to obtain thermal diffusivity distributions at different positions. The thermal conductivities of liquid crystalline polymers were obatind by measured thermal diffusivities, densities and specific heat capacities according to the following equation \(\lambda =\alpha \rho {C}_{{\rm{p}}}\). Detailed descriptions of the measurement techniques and the corresponding results are provided in the Supplementary Information (Sections 3.3 and 3.4; pages S36–S39).

Using this method, the in-plane thermal conductivity of the newly synthesized liquid crystalline aromatic polyimide nanofilm (PIC2-1) prepared via spin coating was computed from the measured thermal diffusivity, which reached 1.26 W m−1 K−1. This value was significantly higher than those of commercial polyimides in their nearly amorphous states47 and of previously reported non-crosslinked liquid crystalline polyimides48. The PICX-2 samples exhibited higher thermal conductivities compared to the PICX-1 samples. This result aligns with previous studies showing that increased molecular rigidity promotes thermal conductivity49. The PICX-2 samples, with their shorter alkyl spacer chains, have more rigid molecular structures than the PICX-1 samples, which likely contributes to their superior thermal conductivities. Furthermore, among the PICX-2 samples, those with \(S\) values closer to \(-0.5\), indicative of a higher degree of in-plane molecular orientation, exhibited higher thermal conductivities. Specifically, PIC1-2 demonstrated the highest thermal conductivity, whereas the thermal conductivity of PIC3-2, which had the largest \(S\) value, was the lowest. These findings clearly indicate that the molecular chain orientation exerts a significant effect on the thermal conductivity. Figure 5 illustrates the relationship between the structural characteristics of the molecule (chain rigidity), degree of in-plane molecular orientation (evaluated on the basis of the order parameter \(S\)), and thermal conductivity.

Fig. 5
figure 5

Relationship between rigidity of the molecular chain, in-plane molecular orientation, and in-plane thermal conductivity (\(\lambda\)).

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The FSC measurements revealed first-order phase transitions at temperatures higher than 350 °C in the course of an ultra-fast temperature scan (10,000 K/s using FSC) during the heating and cooling processes. These observations were made possible by the extremely short residence times at high temperatures, which significantly suppressed the thermal decomposition of aromatic polyimides. To the best of our knowledge, this is the first study in which a liquid crystal transition at temperatures above 350 °C has been actually observed as a transient transition with an enthalpy change.

Discussion

This study demonstrates that machine learning can predict the formation of liquid crystalline phases for various polymers based on their chemical structures. The trained predictor achieved an accuracy of over 96% within the chemical space of the tested polymer sets. Six polyimides were successfully synthesized according to the molecular design predicted via machine learning. They formed higher-order structures with smectic liquid crystalline phases. These are the first liquid crystalline polymers predicted and discovered by machine learning in the history of polymer materials research. In this proof-of-concept study, we aimed to suppress phonon scattering and enhance the thermal conductivity of the polymers by increasing the molecular orientation order through liquid crystalline phase formation. As anticipated, the thermal conductivity of the synthesized liquid crystalline polyimides exceeded 1.2 W m−1 K−1, which is significantly higher than those of commercial polyimides in their nearly amorphous states.

Currently, fundamental methodology is not available for computationally predicting the likelihood of liquid crystalline phase formation in any given polymer, and the molecular design of these materials remains highly empirical. The proposed method offers potential as a powerful tool for facilitating the study of not only liquid crystalline polyimides but also other liquid crystalline polymers; however, several challenges persist. The reported predictive accuracy is an estimate based on a limited set of polymers, which necessitates the determination of the true generalization performance outside the current data distribution. Additionally, this study focused on binary classification to predict whether a polymer would exhibit liquid crystalline behavior, yet the prediction of specific phase types, such as nematic, smectic, and crystalline phases, should also be addressed to enable the synthesis of polymers with targeted liquid crystalline phases. Another matter is the predictability of phase transition temperatures. Furthermore, predicting the physical properties of liquid crystalline polymers represents another important challenge. Constructing a comprehensive database and establishing a methodological basis for data-driven liquid crystalline polymer chemistry are desirable goals for future studies.

Methods

Synthesis of predicted polyimides

Detailed reaction conditions and nuclear molecular resonance (NMR) and Fourier-transform infrared (FT-IR) spectra of the resulting compounds are provided in the Supplementary Information (Sections 2.1–2.5; pages S6–S34). Examples of the synthesis procedures are provided in Fig. 6.

Fig. 6: Synthesis procedure for the designed polyimides.
figure 6

a Synthesis of acid dianhydride TAC3; TAC1 and TAC2 were synthesized via the same procedure after replacing the starting compounds. b Synthesis of diamine DAC2; DAC1 was synthesized via the same procedure after replacing the starting compounds. c Synthesis of polyimides using a drop casting method.

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Synthesis of tetracarboxylic dianhydrides

The three types of tetracarboxylic dianhydrides (TAC1-3) were synthesized through a three-step process using different aromatic diamines as starting materials. In the first step, the aromatic diamines reacted with 4-methoxybenzoic acid chloride to form amides, using N-methyl-2-pyrrolidone (NMP) as the solvent and pyridine as the base. The resultant compounds were purified via recrystallization from dimethylformamide. The second step involved a demethylation reaction using thiophenol, following the method developed by Chakraborti et al.50. The demethylated products were purified through reprecipitation in methanol. In the third step, the hydroxyl-terminated compounds were esterified with trimellitic anhydride chloride, followed by recrystallization using γ-butyrolactone.

Synthesis of diamines

The two diamines were synthesized from dibromoalkanes in a two-step process. The first step involved Williamson ether synthesis with 4-nitrophenol, followed by reduction of the nitro groups to amino groups via hydrogenation using a Pd/C catalyst. The resulting compounds were purified by recrystallization in ethanol for subsequent polymerization.

Preparation of polyimide samples

The three tetracarboxylic dianhydrides and two diamines were polymerized in NMP to prepare six poly(amic acid) solutions. Portions of these solutions were drop-cast onto glass plates and thermally imidized for wide-angle X-ray analyses. Additional samples were prepared for FSC measurements. The remaining solutions were reprecipitated, dried, and redissolved for further analysis. For thermal diffusivity measurements, a 10 wt.% solution of redissolved poly(amic acid) was spin-coated onto the sensor device and silicon wafer (2 cm × 2 cm) for GIWAXD measurements, followed by thermal imidization. Inherent viscosities were measured using an Ostwald viscometer after dissolving 0.5 g/dL of each poly(amic acid) in NMP.

Analysis of higher-order structures in spin-coated films

The molecular orientation and higher-order structures of polyimide films were analyzed by conducting GIWAXD measurements (Bruker D8 DISCOVER instrument equipped with a VANTEC-500 detector and CuKα radiation). The incident angle was consistently set to 0.25°.

Measurement of thermophysical properties

Detailed procedures for the measurement of the polymer properties are provided in the Supplementary Information (Sections 3.2–3.4; pages S36–S39).