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High-Accuracy Prediction of Complex Permittivity in Ternary Composites Using Parameters Provided by Binary Systems

This study demonstrates that a binary-informed Maxwell–Garnett mixing-law framework, utilizing fixed effective constituent parameters derived from binary PANI/PU and PANI/paraffin systems, enables high-accuracy, calibration-free prediction of the complex permittivity for ternary PANI/CIP/PU composites across the 2–18 GHz frequency range.

Original authors: Jia Li, Zihan Yang, Qiqi Li, Xiaoji Liu, Hao Wei, Qiang Wang, Lin Wang, Teng Ma

Published 2026-10-23

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CC BY 4.0

Original authors: Jia Li, Zihan Yang, Qiqi Li, Xiaoji Liu, Hao Wei, Qiang Wang, Lin Wang, Teng Ma

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the invisible world of radio waves and radar, materials are not just passive blocks; they are active participants that can store energy, absorb signals, or let them pass through. The ability of a material to do this is measured by a property called complex permittivity. Think of this as a two-part scorecard: one number tells us how much electrical energy the material can hold, like a sponge holding water, while the other number reveals how much of that energy is lost as heat or friction. Engineers need to know these numbers precisely to design everything from stealth aircraft to faster communication networks. However, predicting these numbers for new materials is notoriously difficult. When scientists mix different powders into a plastic to create a new composite, the resulting electrical behavior is not a simple average of the ingredients. The way the particles touch, their shapes, and how they interact with each other create a complex web of effects that are hard to calculate without building and testing the material first. This trial-and-error approach is slow, expensive, and often leaves researchers guessing which combination of ingredients will work best.

A team of researchers at Harbin Engineering University has developed a new way to predict these electrical properties without needing to build every possible version of a material first. They focused on a specific mixture of three components: a conductive plastic called polyaniline, a magnetic metal powder known as carbonyl iron, and a flexible plastic called polyurethane. Instead of trying to guess the behavior of this three-part mixture from scratch, the researchers used a clever strategy of learning from simpler, two-part mixtures. They began by studying mixtures of just the conductive plastic and the flexible plastic, and then mixtures of just the metal powder and the flexible plastic. By carefully analyzing how these simpler combinations behaved across a wide range of frequencies, from 2 to 18 gigahertz, they were able to extract a reliable "fingerprint" for each ingredient. They tested several different mathematical models to see which one best described the behavior of the simple mixtures, eventually choosing a model that treated the plastic as a continuous background with the other materials scattered inside it. This model proved to be the most stable and accurate for describing how the ingredients responded to electrical fields.

The true test of their method came when they tried to predict the behavior of the complex three-part mixture using only the data they had gathered from the simpler two-part mixtures. They did not tweak their calculations or adjust any numbers to make the prediction fit the new three-part results; they simply plugged the pre-determined properties of the ingredients into a single, unified equation. The results were strikingly accurate. When they compared their predictions to the actual measurements of the three-part mixtures, the numbers matched with a high degree of precision. For the energy storage part of the measurement, the predictions were almost perfect, matching the real data with a statistical confidence level above 98 percent. For the energy loss part, the match was also very strong, with only a few minor exceptions where the signal was very weak. Even more impressively, they tested whether the "fingerprint" of the conductive plastic they found in the first experiment could be used in a completely different plastic mixture. They swapped the flexible plastic for a wax-like substance called paraffin and used the exact same conductive plastic properties they had found earlier. The model successfully predicted the behavior of this new mixture without any re-calibration, proving that the properties they extracted were truly inherent to the material itself and not just a result of the specific mixture they were studying.

This work demonstrates that it is possible to move beyond guessing and trial-and-error in material design. By establishing a reliable way to describe how individual ingredients behave in simple mixtures, scientists can now use those descriptions to forecast how those same ingredients will perform in more complex combinations. The researchers showed that once the fundamental electrical character of a component is understood, it can be transferred to new recipes with high accuracy. This approach offers a practical path forward for designing advanced materials for radar and communication systems, allowing engineers to screen and select the best combinations on a computer before ever mixing a single gram of powder. It turns the process of material discovery from a game of chance into a more predictable, guided journey, saving time and resources while opening the door to more sophisticated electromagnetic designs.

Technical Summary: High-Accuracy Prediction of Complex Permittivity in Ternary Composites Using Parameters Provided by Binary Systems

Problem Statement
Accurately predicting the frequency-dependent complex permittivity of ternary composites remains a significant challenge. Current approaches often rely on inverse modeling to retrieve constituent parameters, which are typically model-dependent and require re-estimation for every new composite system. This reliance on system-specific fitting limits the transferability of parameters and hinders the ability to independently predict ternary properties without recalibration. Furthermore, sequential mixing approaches or refitting to target data can introduce dependencies on mixing order, parameter non-uniqueness, and overfitting, particularly for fillers exhibiting frequency dispersion and conduction loss.

Methodology
The study focuses on Polyaniline (PANI)/Carbonyl Iron Powder (CIP)/Polyurethane (PU) composites over the 2–18 GHz frequency range. The research employs a "binary-informed" mixing-law framework structured in four distinct stages:

Model Selection via Binary Systems: Three mixing laws—Maxwell–Garnett (MG), Coherent-Potential (CP), and Bruggeman (BR)—were evaluated using identical experimental datasets from binary PANI/PU composites. The models were compared based on goodness-of-fit ( $R^2$ ), frequency-dependent residuals, and residual distributions.
Parameter Retrieval and Physical Validation: The optimal model was used to retrieve the frequency-dependent effective complex permittivity of PANI. The physical consistency of these retrieved parameters was verified using a modified Debye–conductivity model to ensure the data reflected realistic dipolar relaxation and conduction loss mechanisms.
Cross-Matrix Transferability Test: To test parameter independence, the PANI parameters retrieved from the PANI/PU system were applied to PANI/Paraffin composites without any recalibration. This served as an independent validation of the parameters' transferability across different matrix environments.
Ternary Prediction: The validated, binary-derived effective constituent responses of PANI and CIP (retrieved independently from CIP/PU data) were fixed as inputs into a unified MG-based ternary mixing expression. This expression incorporated the morphology-dependent depolarization factors for needle-like PANI and platelet-like CIP simultaneously within a single local-field framework, avoiding sequential binary mixing. The model then predicted the complex permittivity of five ternary PANI/CIP/PU formulations (fixed at 30 wt.% total filler) and one binary endpoint without fitting any ternary experimental spectra.

Key Results

Model Performance: The Maxwell–Garnett (MG) model demonstrated the best balance between fitting accuracy and residual stability for the binary PANI/PU system, outperforming both CP and BR models. The MG model achieved $R^2$ values of 0.9892–0.9977 for the real part ( $\varepsilon'$ ) and 0.9232–0.9956 for the imaginary part ( $\varepsilon''$ ).
Parameter Transferability: The MG-retrieved PANI parameters successfully predicted the dielectric response of PANI/Paraffin composites without recalibration. For medium-to-high loading samples, $R^2$ values reached 0.9961–0.9966 for $\varepsilon'$ and 0.9797–0.9995 for $\varepsilon''$ , confirming the parameters' utility across different matrices.
Ternary Prediction Accuracy: The binary-informed ternary model reproduced the measured complex permittivity of the PANI/CIP/PU system with high fidelity.
- Real Permittivity ( $\varepsilon'$ ): $R^2$ values ranged from 0.9891 to 0.9953, with Normalized Root Mean Square Error (NRMSE) between 5.62% and 11.71%.
- Imaginary Permittivity ( $\varepsilon''$ ): $R^2$ values ranged from 0.9375 to 0.9963. NRMSE values were between 12.74% and 18.95% for five of the six datasets. One dataset (P-CIP-2) exhibited a higher NRMSE of 35.12%, attributed to its narrow response range which amplifies the effect of local fluctuations, though absolute errors (RMSE/MAE) remained low.
Mechanism: The model successfully captured the frequency dispersion, magnitude, and composition dependence of the ternary system, demonstrating that explicit three-phase interfacial correction terms were not required to reproduce dominant permittivity trends within the investigated range.

Significance and Claims
The paper claims to establish a practical, integrated modeling route that transitions from retrospective curve fitting to prospective composition screening. By demonstrating that binary-derived effective constituent parameters can be transferred to ternary systems without ternary-specific recalibration, the study provides a quantitative basis for reducing experimental screening costs. The authors assert that this approach allows for the forward prediction of complex permittivity spectra for candidate ternary formulations by varying volume fractions prior to sample preparation, thereby guiding the composition design of multicomponent dielectric composites. The work validates that model-dependent effective constituent responses, when rigorously selected and physically verified, possess sufficient transferability to support accurate prediction in complex multiphase systems.

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