Microstructure reconstruction and structural equation modeling for computational design of nanodielectrics
- First Online:
- Received:
- Accepted:
DOI: 10.1186/s40192-015-0043-y
- Cite this article as:
- Zhang, Y., Zhao, H., Hassinger, I. et al. Integrating Materials (2015) 4: 14. doi:10.1186/s40192-015-0043-y
Abstract
Nanodielectric materials, consisting of nanoparticle-filled polymers, have the potential to become the dielectrics of the future. Although computational design approaches have been proposed for optimizing microstructure, they need to be tailored to suit the special features of nanodielectrics such as low volume fraction, local aggregation, and irregularly shaped large clusters. Furthermore, key independent structural features need to be identified as design variables. To represent the microstructure in a physically meaningful way, we implement a descriptor-based characterization and reconstruction algorithm and propose a new decomposition and reassembly strategy to improve the reconstruction accuracy for microstructures with low volume fraction and uneven distribution of aggregates. In addition, a touching cell splitting algorithm is employed to handle irregularly shaped clusters. To identify key nanodielectric material design variables, we propose a Structural Equation Modeling approach to identify significant microstructure descriptors with the least dependency. The method addresses descriptor redundancy in the existing approach and provides insight into the underlying latent factors for categorizing microstructure. Four descriptors, i.e., volume fraction, cluster size, nearest neighbor distance, and cluster roundness, are identified as important based on the microstructure correlation functions (CF) derived from images. The sufficiency of these four key descriptors is validated through confirmation of the reconstructed images and simulated material properties of the epoxy-nanosilica system. Among the four key descriptors, volume fraction and cluster size are dominant in determining the dielectric constant and dielectric loss.
Keywords
Nanodielectric Material design Descriptor identification Microstructure characterization and reconstruction Structural Equation ModelingBackground
Dielectric materials are widely used in mobile electronics, electrical transmission, and pulsed power applications [1]. There is an increasing demand for new nanodielectric materials, consisting of nanoparticle-filled polymers, for creating future electrical transmission and storage devices. One example is a new capacitor made from nanodielectrics that can store a large amount of energy and discharge it quickly with high-energy density [2]. The design of nanodielectrics is often multi-objective, for example, a tradeoff between dielectric constant and breakdown strength of dielectric materials has been observed [3]. It has been noted that small volume fractions of nanofillers can significantly improve the composites’ dielectric breakdown strength because of their high surface area/internal volume ratio [4]. To achieve design requirements under different application scenarios, a systematic computational design approach is needed to quickly explore the microstructure design space of nanodielectrics. In this work, we are developing characterization, reconstruction, and key microstructure feature identification techniques to support the computational design of nanodielectric systems.
A traditional material design follows a trial-and-error process with the focus on exploring the relationships between processing conditions and material properties. This empirical approach to material design is expensive and time consuming. In integrated computational materials engineering (ICME), a three-link (i.e., processing-structure-property) chain model that enables “microstructure-mediated design” has been proposed to facilitate the design of new materials [5, 6]. The microstructure material design problem can be formulated as an optimization problem, in which the desired material properties drive the design of microstructure first and then the corresponding processing conditions. As pointed out by Xu et al, ICME faces three design-related challenges: design representation, design evaluation, and design optimization [7]. Design representation requires quantitative representation of the design space of heterogeneous microstructures using a small set of design variables. Design evaluation is the process of assessing material properties for a given microstructure morphology, which often involves finite element modeling (FEM) and simulations. Design optimization searches for the optimal microstructure design to achieve the desired material properties. Using the design of nanodielectrics as a focal application, the main focus of this paper is on developing methods to support design representation and identify key microstructural design variables in material design.
A good design representation means an accurate quantitative description of microstructures that is easy to control from the perspective of simulation, design, and processing. In the existing work, methods have been developed to characterize and reconstruct microstructures for different material systems. They can be mainly classified into two categories: one is to use correlation functions (CFs) such as 2-point CF, 2-point cluster CF, and surface correlation [8, 9, 10, 11]; the other is to use physical descriptors, such as volume fraction, particle size, and minimum distance between particles [12]. CF-based reconstruction often involves optimization procedures to minimize the error between the actual CF and the target ones. This approach has been extended for reconstructing multiphase microstructures, for which each phase has its own CF [13, 14]. Although CF-based approaches are flexible and can be adapted for different microstructures, it is computationally expensive and prohibitive for use as a part of the iterative material design procedure. In addition, CFs are infinitely dimensional. While coefficients of the functions can be treated as design variables, they lack physical meaning. The descriptor-based approach, on the other hand, is much more intuitive and offers low dimensionality of design variables with clear physical meaning. Toward this end, Xu et. al. [12, 15] proposed a descriptor-based approach to fully characterize particle-based microstructures by introducing three categories of descriptors: (1) composition: e.g., volume fraction; (2) dispersion: e.g., nearest center distance, interphase area, cluster number, local volume fraction, and orientation; (3) geometry: e.g., cluster area, equivalent radius, aspect ratio, eccentricity, roundness, compactness, tortuosity, pore size, and rectangularity.
Low volume fraction and small number of clusters
Uneven distribution of aggregates (heterogeneity)
Irregularly shaped large clusters that cannot be modeled using simple geometries like a sphere or ellipse (see Fig. 1c)
The volume fraction of the nanodielectric fillers ranges from 0.5 to 3 % over the samples available in our study (collected from several dielectric systems with similar polymer dielectric permittivity). When the filler phase is on the nanoscale, small filler loadings can result in significant property improvement because of the large interfacial area. When the volume fraction is high, aggregation is harder to control and the property enhancements are reduced. As a consequence, the distribution is heterogeneous after processing, which is the reason why, as shown in Fig. 1c, local aggregates (marked by circles) can be observed in these microstructures. In addition, the dispersion (ability to separate primary particles) depends on the particle/particle and particle/polymer attraction [16]. The greater the particle/polymer enthalpic incompatibility, the greater the driving force for agglomeration.
While the descriptor-based method is generally applicable for particle-based nanodielectrics, the original reconstruction algorithm requires the microstructures to be simple and the distribution of filler phase to be even, which is not always satisfied in low volume fraction nanodielectric systems. Therefore, the existing descriptor-based method needs to be tailored to suit the special features of nanodielectrics.
Material informatics [17, 18] is a growing area that exploits information technology and data science to represent, manage, and analyze material data for accelerating new material discovery and design. One of the common challenges associated with material informatics is the high dimensionality of the data and the design space. Recently, efforts have been made in microstructure dimensionality reduction via manifold learning [19] and principal components [20]. However, dimension reduction based only on microstructures does not reflect the influence of microstructure on the properties of interest. To address such limitations, our recent work applied a supervised learning algorithm by using structural information from images and material properties from simulations [21] as supervisory (response) signals. Even though the method can determine the relative importance of descriptors, it introduces subjectivity when determining the final set of design variables. In addition, this learning method is not capable of discriminating redundant features (descriptors); neither it is reliable for cases with a small number of sample images.
In this work, we employ the descriptor-based characterization and reconstruction method to the particular nanodielectric system of interest. To achieve more realistic characterization and more accurate reconstruction, the existing algorithms are modified considering the aforementioned special features of the nanodielectric system. To address the issue of irregularly shaped large clusters, a touching cell splitting algorithm [22] is incorporated to reproduce more realistic structures. To capture the unevenly distributed aggregates, we propose a decomposition and reassembly strategy for reconstruction that preserves local microstructural information. To overcome the limitations of the existing machine learning approach, a Structural Equation Modeling [23] approach is proposed in this work to choose the proper set of independent descriptors using the information learned from images. By introducing latent layers in mapping input and output relations, we are able to identify the relationships and dependencies among descriptors, which allow determination of a small set of key descriptors as design variables. Finally, we illustrate the obtained relationship between key microstructural descriptors and the dielectric properties to support design of a nanoscale silica/epoxy matrix system.
Methods
Descriptor-based characterization and reconstruction
With various microscopic imaging techniques, such as scanning electron microscopy [24] and transmission electron microscopy (TEM) [25], material microstructures can be represented by grayscale digital images. Descriptor characterization is a process for extracting statistical information about the structure descriptors from the images. Due to the heterogeneity of these microstructures, statistical moments are often used as a part of the descriptors. For instance, “cluster area” as a descriptor is best described by a statistical distribution rather than a single value. To reduce dimensionality in representation, the first several orders of statistical moments, such as mean and variance are often used rather than considering the whole distribution.
Once a microstructure is characterized by descriptors, a typical 2D or 3D reconstruction follows a sequential procedure [12]: (1) dispersion reconstruction: center positions of clusters are adjusted to match dispersion descriptors, e.g., the nearest center distances, using optimization algorithms such as simulated annealing (SA), (2) geometry reconstruction: the geometry is randomly generated for each cluster based on the geometry descriptors and geometry profiles are assigned to each cluster, (3) composition adjustment: the edge of clusters are modified to satisfy the composition descriptors such as the volume fraction.
Because the primary particle size is small in this system, noise in the grayscale TEM images is more likely to be recognized as particles in image processing. In this work, we first employ Gaussian filtering [26] to remove the influence of noise. Gaussian filtering utilizes a Gaussian kernel to smooth the image, which will remove isolated pixels. After the filtering process, the existing descriptor-based characterization and reconstruction algorithm as described earlier is employed.
Touching cell splitting method
- 1)
Polygon approximation of cluster edges
- 2)
Identification of concave points and segmentation of cluster edges
- 3)
Separation with ellipse fitting
- 1)
concavity(P_{c}) ∈ (a_{1}, a_{2}), and
- 2)
Line \( \frac{}{P_{\mathrm{pre}}{P}_{\mathrm{next}}} \) should not cross over the inner region of the aggregates.
Reconstruction using decomposition and reassembly
- 1)
Divide the original microstructure into multiple equal size sub-blocks
- 2)
Apply the 2D descriptor characterization and reconstruction algorithm to each sub-block
- 3)
Randomly assemble the sub-block reconstructions to obtain the fully reconstructed microstructure
Our proposed method is inspired by the Morisita Index approach [27], which is used to analyze local versus global dispersions. The Morisita Index divides the original image using different sizes of small blocks, and then based on the number of particles in each block, a weighted index is calculated to represent the dispersion status of an image. Here we need to choose an appropriate block size based on a specific problem, but the basic idea is similar: let the sub-blocks keep the local information, which would be lost if characterization and reconstruction were directly applied to the global region. The size of the sub-blocks may influence the reconstruction accuracy. In this study, it is found that when the block size is slightly larger than the largest clusters in the microstructure, satisfactory results can be obtained. A decomposition and reassembly strategy was also employed in the evaluation of structure-property relationships and proved to be effective [28]. One additional advantage of the proposed method is that in some sub-blocks, there may be no particles. The block is then maintained as pure matrix (void space) in the reconstruction to ease computation as well as to capture the void space feature which has been used in literature to quantitatively characterize material microstructure dispersion [29, 30] in low volume fraction systems.
Identification of key microstructure descriptors
Each of these equations is a regression equation; factor analysis seeks to find the coefficients λ_{ij} (loadings on factors) that best reproduce the observed variables from the factors. If all coefficients are correlations and factors are uncorrelated, then the sum of the squared loadings for variable X_{i}, e.g., \( {\displaystyle \sum_j}{\lambda}_{ij}^2 \), shows the proportion of the variance of X_{i} explained by these factors. This is called the communality, and the larger the communality for each variable, the more successful the factor solution is.
- Step 1:
Determine the number of latent factors (exogenous variables) n
- Step 2:
Conduct factor analysis with the n factors using proper rotation
- Step 3:
Identify exogenous variables that are poor indicators of latent factors (uniqueness > 0.5) [32]
Equation (5) shows the mathematical relationship of the structural model, the relationship between latent variables, while Eqs. (6) and (7) both represent the measurement model, illustrating the relationship between latent variables and corresponding indicators.
In the context of material structure-property analysis, different categories of microstructure descriptors can be either pre-defined based on experience [12] or identified as the exogenous latent variables ξ_{i} for microstructure descriptors in the proposed Structural Equation Modeling structure after going through the EFA process. Latent variables are often not directly measured. For instance, dispersion can be a latent variable (exogenous variable ξ_{i}) but there is no explicit mathematical definition, while dispersion descriptors, such as nearest center distances and nearest boundary distances [7], can be viewed as indicators, represented by x’s. Different indicators (microstructure descriptors in this case) provide measurements of certain features of the microstructure. The error term, δ_{i}, in the general model can effectively take into account the errors introduced by the approximations in measurement. Considering measurement errors is another strength of the Structural Equation Modeling approach compared to other methods. Rather than assuming the different categories of descriptors are independent, the correlation between them can also be analyzed by studying the relationship among latent factors in the Structural Equation model.
where a is a fitting parameter that describes the shape of CF.
The structural error ζ_{i} describes unmodeled factors that may influence the endogenous variables η_{i}. If the exogenous variables ξ_{i} in the structured model cannot explain the endogenous variables η_{i} well, then a large structural error ζ may exist in the final model, which indicates that additional descriptors may need to be included.
As a result of applying Structural Equation Modeling analysis, key microstructure descriptors are chosen as material design variables. Ideally, for each identified significant latent factor, we want to pick one descriptor as the best indicator (or microstructural design variable). The choice of multiple descriptors within one latent factor will lead to redundancy as multiple descriptors are often correlated.
Constructing structure-property relationship for microstructure design
As a simple illustration of microstructure design of nanodielectrics, the design objectives associated with the mechanical properties can be chosen as maximizing the real part ε ' and minimizing the energy loss, tan δ. Previous work has shown that dielectric permittivity in polymer nanocomposites can be analyzed using a Prony Series approach adapted from viscoelasticity studies. This approach incorporates explicit consideration of microstructure dispersion as well as polymer interphase between the nanofillers and matrix into the Finite Element simulation [44]. Based on experimental results of bimodal brush grafted silica nanoparticles in an epoxy matrix, finite element modeling has been used to accurately capture the dielectric permittivity and loss angle measured in experiments by superposition of frequency-dependent dielectric relaxation constants. Optimization of nanodielectric materials can be achieved following the framework described in Fig. 8 once sufficient data are collected and multiple simulation property models are built.
Results and discussion
Characterization and reconstruction
As shown in Fig. 9b, the irregular clusters are each split into multiple ellipses, providing a better representation compared to the result in Fig. 9a where single ellipses are used to represent the irregular clusters. As a confirmation of improved accuracy using the splitting algorithm, we compare the interfacial area and the nearest center distance using the two methods. The interfacial area (2D boundary between the filler and matrix) in Fig. 9b is 0.0075 after using the splitting algorithm, which matches better with the real surface fraction 0.0078 of the original image, compared to 0.006 of the result in Fig. 9a without using the splitting algorithm. A similar observation is made for the characterized nearest center distance, which becomes 40 pixels after splitting versus 60 pixels before splitting. Nearest center distance reflects the local clustering behavior, and the “true” value is unknown as the evaluation depends on the way the cluster is characterized. The ellipse splitting algorithm implemented in this work is shown to be effective for splitting touching clusters and offering more accurate characterization. This splitting algorithm is utilized for all microstructure characterizations in the following sections of this paper.
Volume fraction (deterministic)
Nearest center distance (mean and variance)
Aspect ratio (mean, variance, normal distribution)
Cluster area (mean, exponential)
Descriptor information of the two sample microstructures (unit of length: nm)
Sample 1 | Sample 2 | |
---|---|---|
Volume fraction | 0.53 % | 0.9 % |
Nearest center distance | 1st mean = 124, 1st var = 15,410 | 1st mean = 79, 1st var = 4151 |
Aspect ratio | Mean = 0.7513, var = 0.0234 | Mean = 0.9701, var = 0.0315 |
Cluster area | Mean = 200 | Mean = 10,120 |
where S_{2}(r) is the target 2-point CF and S_{2} ' (r) is the 2-point CF of reconstruction. The average error from six reconstructions is found to be 5 %, which is acceptable for this low volume fraction material system.
By comparing Fig. 12a, b, it is noted that the 2-point CF obtained using our proposed decomposition and reassembly strategy matches much more closely than using the original algorithm, especially in the short range from r = 50 to 100. The relative error achieved is 14.8 %, which is a big improvement compared to 48.6 % using the original approach. The results show that the uneven local information is maintained through the proposed strategy of using small blocks. It should be noted that when assembling the small blocks, a totally random sequence is applied in our study. Omitting the relationship between the sub-blocks is acceptable here for two reasons: (1) The large clusters are sparse in the microstructure images; therefore, it is difficult to come up with a statistical characterization that is representative for the whole image. (2) For such a low volume fraction system, it has been observed that the main differences among 2-point CF occur in the short distance range, which has been captured by the information obtained from individual small blocks. 2-point CF are usually oscillating with several peaks and valleys. The location of the first deepest drop corresponds to the size of local clusters and the peaks at longer distance relate to certain global periodical patterns. As for this case study, no obvious long distance pattern can be observed, so it is not necessary to consider the higher order block-block relationships. In real implementation, the best results will be achieved by randomly assembling the small blocks multiple times and choosing the best match for the target CF, to compensate for the lack of block-block characterization.
Identification of key descriptors from images using the Structural Equation Modeling approach
Descriptor set for statistical learning. Reference are provided for those descriptors without explicit meaning
Descriptor | Definition | Type |
---|---|---|
Composition | ||
VF | Volume fraction | Deterministic |
Dispersion | ||
ncd | Cluster’s nearest surface distance | Statistical |
nbd | Cluster’s nearest center distance | Statistical |
ornang | Principle axis orientation angle [51] | Statistical |
intph | Surface area of matrix phase | Deterministic |
N | Cluster number | Deterministic |
Loc_VF | Local VF of Voronoi cells [52] | Statistical |
Geometry | ||
pores | Pore sizes (inscribed circle’s radius) [53] | Statistical |
area | Cluster area | Statistical |
rc | Equivalent radius, \( rc=\sqrt{A/\pi } \) | Statistical |
comp | Compactness [54] | Statistical |
rnds | Roundness [55] | Statistical |
eccen | Eccentricity [55] | Statistical |
els | Statistical | |
rectan | Rectangularity [55] | Statistical |
tsst | Tortuosity [55] | Statistical |
The exploratory factor analysis (EFA) approach presented in the “Methods” section and the associated three steps are first followed to identify the proper number of latent factors and group the microstructure descriptors (indicators) based on their associations with the latent factors. When applying the EFA approach, only the data collected on microstructure descriptors from 117 images is used. Methods for choosing the number of latent factors have been well studied and three widely used criteria, K1 method (eigenvalue) [45], Non-graphical Cattell’s Scree Test (optimal coordinates and acceleration factor) [46], and Horn’s Parallel Analysis [47], are employed in Step 1 of EFA to build confidence in the result.
Results of EFA (loading, uniqueness, and complexity). The meaning of descriptors can be found in Table 2. Indices 1 and 2 indicate the mean and variance of the corresponding descriptor, respectively, e.g., area1 stands for the mean of clusters area
F1, cluster size | F2, distribution | F3, composition | F4, geometry1 | F5, geometry2 | Uniqueness^{**} | Complexity | |
---|---|---|---|---|---|---|---|
area1^{*} | 0.99 | −0.03 | −0.08 | −0.04 | −0.09 | 0.0066 | 1 |
area2 | 0.87 | −0.02 | 0.19 | −0.03 | 0.25 | 0.2016 | 1.3 |
comp1 | 0.05 | −0.06 | 0.13 | −0.01 | 0.92 | 0.1244 | 1.1 |
comp2 | 0.06 | 0.15 | 0.13 | 0.23 | −0.28 | 0.7996 | 3.2 |
eccen1 | 0.08 | −0.03 | 0.08 | 0.19 | −0.43 | 0.7363 | 1.6 |
eccen2 | 0.09 | 0.24 | 0.12 | −0.12 | 0.15 | 0.9222 | 3.3 |
els1 | 0.27 | −0.01 | 0.13 | 0.24 | −0.32 | 0.6794 | 3.2 |
els2 | 0.12 | 0.23 | 0.06 | 0.39 | 0.15 | 0.7368 | 2.3 |
locvf1 | 0.37 | −0.02 | 0.57 | −0.1 | −0.22 | 0.397 | 2.1 |
locvf2 | 0.36 | 0.09 | 0.23 | 0.02 | −0.14 | 0.7351 | 2.2 |
nbd1 | 0.11 | 0.69 | −0.38 | −0.01 | 0.23 | 0.0913 | 1.9 |
nbd2 | −0.17 | 1.02 | 0.08 | −0.03 | −0.17 | 0.0544 | 1.1 |
ncd1 | 0.25 | 0.65 | −0.38 | −0.01 | 0.16 | 0.0949 | 2.1 |
ncd2 | −0.13 | 1.01 | 0.08 | −0.04 | −0.18 | 0.0496 | 1.1 |
ornang1 | −0.04 | −0.09 | −0.03 | 0.16 | 0.04 | 0.9673 | 2 |
ornang2 | −0.03 | −0.02 | 0.07 | 0.15 | 0.28 | 0.8921 | 1.7 |
pores1 | 0.92 | −0.09 | −0.35 | −0.1 | −0.02 | 0.2054 | 1.3 |
pores2 | 0.92 | 0.01 | −0.04 | 0 | 0.06 | 0.1733 | 1 |
rc1 | 0.88 | −0.01 | −0.22 | −0.06 | −0.28 | 0.0704 | 1.4 |
rc2 | 0.97 | −0.02 | 0.05 | −0.01 | 0.11 | 0.0905 | 1 |
rctan1 | −0.1 | −0.03 | −0.09 | 1 | −0.04 | 0.0075 | 1 |
rctan2 | −0.1 | −0.05 | −0.1 | 1 | −0.04 | 0.015 | 1 |
rnds1 | 0.03 | −0.06 | −0.02 | −0.1 | 0.75 | 0.4154 | 1.1 |
rnds2 | 0.21 | 0.01 | 0.06 | 0.21 | 0.25 | 0.8562 | 3.1 |
ttst1 | 0.05 | −0.08 | 0.15 | 0.51 | 0.44 | 0.5319 | 2.2 |
ttst2 | 0.02 | 0.03 | −0.09 | 0.85 | −0.21 | 0.1797 | 1.2 |
intph0 | −0.07 | −0.02 | 0.94 | −0.09 | 0.01 | 0.0992 | 1 |
n | −0.37 | 0.03 | 0.91 | −0.08 | 0.21 | 0.0726 | 1.5 |
vf | 0.49 | −0.03 | 0.74 | −0.08 | 0.15 | 0.0977 | 1.8 |
In Step 3 of the EFA procedure, based on the rule that at least half of the variance of an independent variable should be explained by a latent factor (uniqueness ≤0.5), we can identify poor factor indicators (underlined) and withdraw them from our data set. Going through steps 1 to 3 in EFA, we reduced the number of characterization parameters from 29 to 19 and associate them to five latent factors. Examining the results, we can relate each factor to a physical interpretation: Factor 1 represents the size of clusters, with 6 descriptors as indicators: cluster area (area1, area2), pore size (pores1, pores2), and equivalent radius (rc1, rc2). Factor 2 represents the distribution status, with the nearest neighbor distance such as nearest boundary distance (nbd) and nearest center distance (ncd) as the indicator. Factor 3 describes the composition information, including volume fraction (vf), interfacial fraction (intph), number of clusters (n), and local volume fraction (locvf1). Volume fraction clearly describes the composition, and the other three describe higher order composition information. Factors 4 and 5 represent two geometric characteristics: Factor 4 is associated with rectangularity (rctan1, rctan2), and tortuosity (ttst2); Factor 5 is associated with compactness (comp1) and roundness (rnds1).
The obtained Structural Equation Model is verified based on the physical meaning of factor coefficients. For example, Factor 1 represents the size of clusters and it is meaningful that it has a negative effect (−0.542) on the surface correlation and a strong positive effect (0.893) on the lineal path correlation. Intuitively, a larger cluster size leads to smaller surface area and larger lineal path. Based on the Structural Equation Model analysis, a potential set of material design variables are chosen by including four descriptor variables, each underlines Factor 1, Factor 2, Factor 3, and Factor 5, respectively. Based on the loadings of each descriptor with respect to each latent factor, the most significant set is identified to be area1 (0.995, average cluster area) for Factor 1, ncd1 (0.959, average nearest center distance) for Factor 2, vf (0.943, volume fraction) for Factor 3, and rnds1 (0.898, average cluster roundness) for Factor 5.
Correlation comparison of chosen descriptors. Proposed and prior ranking algorithm, high correlation among area1 (average cluster area), pores2 (variance of pore size), and rc2 (variance of equivalent radius)
Structural Equation Modeling-based analysis | vf | area1 | ncd1 | rnds1 |
Vf—volume fraction | ||||
area1—average cluster area | 0.46 | |||
ncd1—average nearest center distance | −0.32 | 0.33 | ||
rnds1—average cluster roundness | 0.08 | −0.23 | −0.10 | |
Prior ranking algorithm [21] | vf | area1 | pores2 | rc2 |
Vf—volume fraction | ||||
area1—average cluster area | 0.56 | |||
pores2—variance of pore size | 0.43 | 0.90 | ||
rc2—variance of equivalent radius | 0.54 | 0.93 | 0.91 |
Validation using reconstruction and property simulations
Comparison of dielectric properties of the original and reconstructed images. Eight samples are chosen; properties are averaged over five reconstructions for each sample
Dielectric constant ε ' | Loss angle tan δ | |||||
---|---|---|---|---|---|---|
Original | Reconstruction (average) | Error (%) | Original | Reconstruction (average) | Error (%) | |
1 | 4.215 | 4.204 | 0.254 | 0.0167 | 0.0161 | 3.50 |
2 | 4.084 | 4.084 | 0.016 | 0.0173 | 0.0164 | 4.74 |
3 | 3.926 | 3.941 | 0.384 | 0.0190 | 0.0185 | 2.78 |
4 | 3.897 | 3.903 | 0.168 | 0.0192 | 0.0190 | 0.92 |
5 | 4.109 | 4.173 | 1.546 | 0.0167 | 0.0158 | 5.36 |
6 | 3.910 | 3.920 | 0.253 | 0.0191 | 0.0188 | 1.59 |
7 | 3.962 | 3.971 | 0.212 | 0.0185 | 0.0181 | 2.14 |
8 | 3.981 | 4.009 | 0.709 | 0.0181 | 0.0175 | 3.24 |
Average | 0.443 | Average | 3.50 |
Results of linear model fitting
Model | CF (six parameters) | Key descriptors (four) | Standardized coefficients of descriptors | |||||
---|---|---|---|---|---|---|---|---|
Prediction error | R-squared | Prediction error | R-squared | Average cluster area | Volume fraction | Average nearest distance | Average roundness | |
ε ' | 5.53e−04 | 0.95 | 8.36e−04 | 0.93 | −0.419^{a} | 1.060^{a} | 0.0008 | 0.096^{a} |
tan δ | 1.12e−07 | 0.91 | 7.5e−08 | 0.88 | 0.614^{a} | −1.034^{a} | 0.0043 | −0.048 |
Conclusions
In this paper, new characterization, reconstruction, and key microstructure feature identification techniques are developed to support the computational design of nanodielectric systems. For design representation, a descriptor-based characterization and reconstruction method is employed and tailored for a low volume fraction nanodielectric system with uneven local aggregations and irregularly shaped clusters. To handle special microstructures with large local aggregates, we propose a new decomposition and reassembly strategy, based on which the reconstruction accuracy is greatly improved. We also incorporate a touching cell splitting algorithm into the descriptor-based method to deal with irregularly shaped clusters to achieve more realistic characterization. To simplify the material design process and minimize the redundancy among design variables, a new Structural Equation Model-based method is developed to identify key descriptors. To keep the results independent from the material properties of interest, the analysis presented in this paper is based on information from the microstructure images (CF). According to the fitted Structural Equation Model, in which descriptors are classified into five groups based on the identified latent factors, we find volume fraction, cluster size, nearest center distance, and cluster roundness to be a sufficient set of descriptors to represent the structural features of the very low volume fraction nanodielectric system in this research. The relationship between the microstructure and properties are explored based on the epoxy-silica system and a close to linear relationship is observed between dielectric permittivity and the identified key descriptors.
In a future work, more image data of nanodielectric material systems with a wider range of volume fraction will be collected and simulation models for predicting all important dielectric properties will be included. In addition, the design problem will be extended to include both permittivity and breakdown strength as objectives. To make the tradeoff between the two, the Pareto frontier will be used to first identify a set of non-dominated (best achievable) optimal solutions. In addition, the Structural Equation Model-based method can also be applied to find the relations between descriptors and certain properties, which can then be directly used as predictive models in material design. Processing conditions will be taken into consideration to establish the mapping relations across the chain of processing-structure-property to ensure the manufacturability of new nanodielectric materials.
Availability of supporting data
Data presented in this work will be made available upon request.
Acknowledgements
The support from NSF for this collaborative research: CMMI-1334929 (Northwestern University) and CMMI-1333977 (RPI), is greatly appreciated.
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.