Reliability-based robust design optimization of polymer nanocomposites to enhance percolated electrical conductivity considering correlated input variables using multivariate distributions

Polymer 186 (2020) 122060

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Reliability-based robust design optimization of polymer nanocomposites to enhance percolated electrical conductivity considering correlated input variables using multivariate distributions Jaehyeok Doh a, *, Qing Yang b, Nagarajan Raghavan a a b

Engineering Product Development (EPD) Pillar, Singapore University of Technology and Design (SUTD), 487372, Singapore State Key Laboratory of Modern Optical Instrumentation, College of Optical Science and Engineering, Zhejiang University, Hangzhou, 310027, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Polymer nanocomposites (PNCs) Electrical percolation threshold Reliability-based robust design optimization (RBRDO)

In this study, reliability-based robust design optimization (RBRDO) for polymer nanocomposites (PNCs) design is conducted to secure the reliability of design conditions concerning the electrical percolation threshold and carbon nanotube (CNT) aspect ratio as well as the robustness for electrical conductivity. CNT diameter and length are known as following the lognormal and Weibull distributions respectively. To reflect the different probability distributions of CNT geometry parameters and correlations between these random input variables, Nataf transformation is employed. By performing several case studies with the first-order reliability method (FORM)-based RBRDO approaches, the objective function exhibited a noteworthy change according to the correlation coefficients and the reliability and robustness for PNCs were satisfied concurrently. Furthermore, the highlight of this work is to provide a generic framework for practical PNC design and multi-objective optimi­ zation to enhance electrical performance efficiently.

1. Introduction Carbon nanotubes (CNTs) have been reported to possess superior characteristics such as electrical, thermal, and mechanical properties via many research studies in the past. CNTs are utilized as powerful and effective fillers for achieving the multifunctional properties of polymer nanocomposites (PNCs) [1–6]. More so, as CNTs have exceptionally high intrinsic electrical con­ ductivity and high aspect ratio, the electrical conductivity of PNCs can be achieved by compounding only a small amount of CNT fillers into the polymer matrix [7]. The key motivation behind the popularity of PNCs is that electrical properties for polymers can be altered drastically without changing the mechanical properties of the matrix and the electrical properties of these nanocomposites are sensitive to CNT length, their spatial dispersion and functional groups attached to the nanotubes. For this reason, PNCs, which are reinforced by CNT fillers, have been extensively utilized in the field of various engineering applications such as field emission [8], lightning strike protection [9], highly sensitive strain sensors [10–12], and electromagnetic-wave interference mate­ rials [13]. Generally, even though most polymers are insulating, they can be

converted to a conductor by dispersing a small amount of conductive CNT fillers in a polymer matrix. This filler induced conductance phase transition can be modeled by the percolation theory [14]. Based on this theory, the main idea for the percolation phenomenon is the stochastic generation of linkage paths to allow passage of electrical current from the source to drain (two ends of the polymer matrix) by conductive particles or fillers. If the embedded CNT volume fraction in the arbitrary polymer matrix is gradually increased, the electrical current can flow through the formed network path due to percolation occurrence at the specific volume fraction. This phenomenon is known as electrical percolation. At that time, the CNT volume fraction at which the elec­ trical conductivity (σc) increases dramatically is defined as the perco­ lation threshold (φc) and it is very sensitive to the CNT geometry [15]. From the point of view of PNC design, there are two objectives for σ c and φc. The φc should be minimized to reduce material cost but the σ c should be maximized to achieve a high electrical conductivity of PNCs. This can be perceived as a “trade-off” problem in design optimization. Accordingly, the PNCs should be optimized to maximize the electrical conductivity and to minimize the percolation threshold. Furthermore, the design optimization for PNC should be accomplished with the consideration of uncertainty not only in the CNT geometrical parameters

* Corresponding author. E-mail address: [email protected] (J. Doh). https://doi.org/10.1016/j.polymer.2019.122060 Received 22 October 2019; Received in revised form 2 December 2019; Accepted 5 December 2019 Available online 8 December 2019 0032-3861/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Scheme and sequence of the overall research procedure in this study.

employed to consider the correlation between input variables following different underlying probability distributions. For RBRDO of PNCs design, the electrical conductivity has to be insensitive despite pre­ senting variations of PNCs design variables. Accordingly, to secure the objective function robustness, the deviation for electrical conductivity should be decreased under the uncertainty of PNCs design variables. Besides, the probabilistic constraint function has to be satisfied with target reliability such as percolation threshold (i.e. volume fraction) and CNT aspect ratio (i.e. CNT diameter and length). These constraint functions are regarded as PNCs design conditions or criteria in this study. Subsequently, in order to investigate variations in the electrical conductance and percolation threshold robustness of PNCs by taking into account the correlation between these variables, we carried out several case studies according to increasing correlation coefficients and different probability distribution types for each random variable. The procedures of the overall research are illustrated in Fig. 1. 2. Percolation theory

Fig. 2. Classifications of three different states concerning the percolation theory-based electrical conductance transition for CNT-filled polymer nanocomposites.

To improve the electrical property in the design of PNCs, CNTs as conductive fillers are compounded and dispersed into the polymer ma­ trix. When compounding CNTs in the arbitrary polymer matrix, the CNT fillers are randomly embedded without any special orientation in the fabrication process. This process complicates the microstructure of in­ ternal PNCs due to the random spread and spatial distribution and orientation of CNTs. As a result, estimation of a reliable minimum CNT volume fraction to obtain an appropriate conductivity is regarded as a problem given the complexity and variability in the underlying micro­ structure [22]. The states of electrical conductivity for PNCs based on percolation theory can be classified into three different states as shown in Fig. 2 Initially, the percolation network path cannot be physically formed due to the sparse quantity of CNTs, resulting in the PNC remaining in an electrically insulating state. Therefore, the probability of percolation occurrence and electrical conductivity are the lowest in this State 1. Subsequently, if the CNT volume fraction is increased and we approach a specific volume fraction, the electrical conductivity for PNCs abruptly increases due to percolation occurrence of at least one or a few electrical network paths between the electrodes (this phase transition is referred to as State 2). Subsequently, in State 3, many electrically connected paths between the end electrodes are formed due to several CNT filler clusters and the PNC tends to have a high conductivity that is deter­ mined by the high-density of electrical percolation paths [23]. This

but also in the electrical properties of CNTs and the polymer matrix such as intrinsic conductivity and potential barrier height, all of which have their inherent uncertainty (i.e. aleatory uncertainty). There are two techniques to take into account uncertainty in design optimization. These techniques have been widely known as reliability-based design optimization (RBDO) and reliability-based robust design optimization (RBRDO) in the field of computational science and engineering [16–19]. In the reported research concerning PNC design under uncertainty, RBDO for polymer nanocomposite structures via multiscale modeling was performed to enhance mechanical properties by considering applied loading uncertainty [20]. A statistical multiscale homogenization approach of the PNCs elastic property considering the inherent uncer­ tainty of molecular dynamic (MD) simulation was suggested [21]. Unlike conventional research studies that are focused on PNC design for mechanical properties, the highlight of this study involves a design optimization process considering aleatory uncertainty of PNCs to enhance electrical performance. We aim to propose the guidelines for PNC design via RBRDO concurrently satisfying the reliability of design conditions for the percolation threshold and CNT aspect ratio under specific criteria and the electrical performance robustness. Besides, the multivariate distribution based on Nataf transformation is also 2

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Fig. 3. Schematic diagram of the CNT resistor network considering tunneling resistance.

that links the two electrodes in a representative volume element (RVE) in two-dimensions (2-D), the entire network of interconnecting linkage paths for CNTs need to be labeled and accounted for. As shown in Fig. 3, two types of resistance in the CNT network paths can be defined intrinsic resistance of an individual CNT and the tunneling resistance between any pair of adjacent CNTs. The intrinsic resistance of CNTs is the invariant resistance, described by Equation (2, where σCNT is intrinsic electrical conductivity of the CNT and DCNT is the diameter of the CNT. Rint: ¼

Fig. 4. Illustration of the geometrical cases of tunneling distance between CNTs: (a) end-to-end, (b) end-to-body.

(2)

The tunneling effect in quantum mechanics is a stochastic phenom­ enon that accounts for electron transport across an energy barrier height through elastic or inelastic energy transfer. This phenomenon shows a strong dependence on the distance between any arbitrary pair of nearest CNTs. To consider the tunneling resistance, two types of tunneling dis­ tances are geometrically defined: end-to-end and end-to-body, as shown in Fig. 4. The electrical conductivity between any pair of CNTs reduces exponentially with increase in physical distance. Besides, the physical contact between adjacent CNTs is not established perfectly due to the van der Waals distance (dvdw), which gives rise to an energy barrier as well. Electron tunneling can occur if the shortest tunneling distance (dtunnel) between any arbitrary two CNTs is less than the threshold cutoff distance (dcutoff). In our proposed model, the Simmons’ formulationbased tunneling resistance is used, as shown in Equation (3) [22, 24–27]. Here, V is the voltage difference, J is the tunneling current density, A is the cross-sectional area of the tunnel gap (i.e. A¼(mean [DCNTi])2) [25], h is the Planck’s constant, dtunnel is the shortest tunneling distance between CNTs, e is the electron charge, m is the mass of the electron, and λ is the height of the energy barrier. � � V h2 dtunnel 4πdtunnel pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi exp Rtunnel ¼ ¼ 2mλ (3) 2 AJ Ae 2mλ h

Fig. 5. Definition of geometric parameters for straight 1D-CNTs.

phenomenon of dramatic increase in electrical conductivity due to conductive fillers at a specific volume fraction is referred to as the electrical percolation event and it causes an abrupt change in the property of the matrix from an insulator to a conductor. Furthermore, the minimum CNT volume fraction at which this phase transition occurs is defined as the percolation threshold (φc). Based on percolation theory [14], the general electrical conductivity model can be described using the relationship between the electrical conductivity (σc) and the volume fraction (φ) of CNTs as shown in Equation (1). Here, σCNT is the intrinsic electrical conductivity of CNTs and t is the critical exponent.

σc ¼ σCNT (φ –φc)t for φ > φc

4LCNTij

σCNT πD2CNTi

In the proposed model, to obtain the tunneling resistance and con­ nectivity of the tunneling path, the distance between arbitrary pairs of all CNTs is calculated mathematically, and the shortest tunneling dis­ tance is extracted under the condition of dvdw þ mean(DCNTi) � dtunnel � dcutoff þ mean(DCNTi) [26].

(1)

3.2. Geometrical placement of CNTs To predict the percolation probability and to compute the electrical conductivity, the statistical simulation domain is defined as a 2-D RVE problem based on a 1-D line segment for the CNT fillers. In general, the CNTs are embedded in random (without any preferential orientation) in a polymer matrix. For the modeling of arbitrarily spread CNT strands, their geometric parameters are defined as shown in Fig. 5. To model straight-CNT fillers in a 2-D RVE, the orientation angle (θ) and starting point coordinates (xi, yi) of the line segment for CNT fillers

3. Percolation simulation for PNCs considering tunneling effects 3.1. Intrinsic and tunneling resistance for PNCs The CNTs are known as an outstanding conductor because of their one-dimensional (1-D) structure that permits electron flow through ballistic transport [7]. Based on this filler, to form the resistance network 3

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φ¼

N X VCNT where; VCNT ¼ ðDCNTi � LCNTi Þ ; VPolymer ¼ Lx Ly Lz VPolymer þ VCNT i¼1

(6)

Here, VCNT is the total volume of the embedded CNTs; Vpolymer is the volume of the polymer matrix; Lx and Ly are each the lateral length of the RVE; Lz is set as 1 μm because of the 2-D RVE problem and LCNTi is the length of each CNT. 3.3. Simulation procedures The in-house code for percolation simulation of PNCs in the previous study was developed by utilizing MATLAB®, and the essential proced­ ures for percolation simulation to estimate the percolation probability and to compute the electrical conductivity are explained below. 1) After CNT incorporation up to a specific pre-determined volume fraction, the intersection points between the whole pair of CNTs are computed by using the linear equation. Besides, to consider the tunneling resistance, the tunneling coordinates and the minimum distance between the whole pair of line segments are also acquired mathematically. The tunneling paths are labeled by cases where the specific distance criteria are satisfied. 2) To form percolation networks, the number indices of the CNTs are sequentially allocated to the acquired intersection points and tunneling path points and the entire set of distances and connecting points for the percolation evaluation are generated. Afterward, to remove the duplicated connection paths, the slope and path distance between entire CNT lines and connection paths are compared. 3) Once the connectivity of the entire network path is produced in Step 2, the intersection points at both sides of the electrode boundary are collected. Based on these points, the connected shortest distance path is then searched for between the source (electrode 1) and the drain (electrode 2) to generate the circuit network. 4) To estimate the percolation probability (Pp) for the CNT filled-PNCs, statistical simulations based on the Monte Carlo simulation (MCs) are executed. For this estimation, the obtained shortest distance paths from Step 3 are utilized. If there is at least one path where percolation occurs, the number of percolation occurrences is counted and accumulated. The percolation probability is then computed by using Equation (7) [23]. Here, Np is the ratio of the number of percolation occurrences and Ns is the total number of simulations.

Fig. 6. Statistical simulation procedure of the proposed percolation model.

Pp ¼

are randomly generated by using (Equation (4)) and their endpoint co­ ordinates (xiþ1, yiþ1) are obtained by utilizing Equation (5). (4)

xiþ1 ¼ xi þ LCNT cos(θ); yiþ1 ¼ yi þ LCNT sin(θ)

(5)

(7)

5) If percolation occurs, the global conductance matrix (Geff.) is created to calculate the electrical conductivity by employing Kirchhoff’s current law (KCL) and the finite element method (FEM). In order to formulate the global conductance matrix ([G]) by using intersection points (nodes) and connected paths (elements) and to obtain the current at each junction in the percolation network, the analytical finite element method based on the relation of [G]{V} ¼ {I} is employed. Therefore, the nodal voltage at each junction and the total current are obtained by subjecting the finite element model to voltage boundary conditions at the electrode boundaries in the equivalent circuit and the electrical conductivity is calculated by using Equation (8) [29] based on the standard Ohm’s law.

Fig. 7. Validation of our proposed percolation model for PNCs via comparison of the conductance trends with existing experimental and numerical datasets.

xi ¼ Lx � rand; yi ¼ Ly � rand; θ ¼ 2π � rand

Np Ns

σ c ¼ Geff:

Subsequently, periodic boundary conditions (PBC) are applied for the entire RVE, and whole CNT volume fraction is repeatedly calculated by utilizing Equation (6) [28] until the specific volume fraction is attained by sequentially adding CNTs into the RVE.

Lx Itotal where; Geff: ¼ Ly Lz ðVelectrode1 Velectrode2 Þ

(8)

To predict the percolation probability and to calculate the electrical conductivity, these procedures are conducted iteratively many times via MCs. The simulation implementation of the proposed percolation model is shown in Fig. 6. 4

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In this study, to execute design optimizations the following four design variables are established that characterize the CNT’s geometric and electronic properties. These four variables are DCNT - CNT diameter, LCNT - CNT length, λ - Barrier height for a polymer matrix, and σ CNT Intrinsic conductivity of CNT. By referring to property data for the CNT filler and polymer matrix in the above literature, the lower and upper bound for each design variable are determined. These bounds of each design variable are established with DCNT ¼ [0.01, 0.05] (μm), LCNT ¼ [5,9] (μm), λ ¼ [0.5, 5] (eV), and σ CNT ¼ [0.005, 5] (S/μm). Therefore, the initial design variables are set as the mean value between the lower and upper bound of each design variable. According to conventional studies, the percolation threshold is defined at a specific percolation probability that can be reverse calcu­ lated by using the Boltzmann function, as shown in Equation (9) [35]. Here, A1 and A2 are set as 0 and 1 respectively; φ0 and Δφ are acquired by using nonlinear least square (NLS) fit. When obtaining φc (i.e. φc ¼ φ at PP of 50%) using the Boltzmann function based on percolation probability data, the specific percolation probability is defined as 50% [36] since the frequency of the probability density function is usually high at the 50th percentile.

Table 1 Values of physical parameters relating to tunneling resistance and voltage conditions used for the statistical simulation. RVE (Lx � L y)

h (eV⋅s)

m (kg)

e (A⋅s)

dvdw (nm)

dcutoff (nm)

V1

V2

25 μm � 25 μm

4.136 � 10 15

9.11 � 10 31

1.602 � 10 19

0.47

1.00

1V

0V

3.4. Results of verification and initial performance factors via percolation simulation for PNCs In our previous research [30], we have verified the developed in-house code for the percolation simulation of PNCs via comparison to several existing experimental [26,31,32] and numerical datasets [33]. As shown in Fig. 7, our proposed model is in good agreement with experimental datasets both in terms of the quantitative values as well as the shape and sensitivity of the percolative transition trend. In this study, we focus on utilizing the already validated PNC percolation model to acquire performance data of electrical conductivity and percolation threshold based on the design of experiments for executing reliability-based robust design optimization (RBRDO). The percolation simulation considering the tunneling resistance is executed to obtain the initial performance factors. For this work, the parameters for the simulation such as electrical property constant (explained in the next sentence), voltage difference, and RVE size are fixed. To calculate the tunneling resistance approximately, the following electrical property constants are needed: the Planck’s constant (h), the mass of the electron (m), the electron charge (e), the average tunneling distance (dtunnel) between CNTs, and the distance of van der Waals interaction (dvdw). These constants and simulation parameters are rep­ resented in Table 1 [30]. The mean value of diameter and length for single-walled CNTs (SWCNTs) are reported to be anywhere between 1 and 2 nm and 5–20 μm, respectively [34]. In the case of Multi-walled CNTs (MWCNTs), the average diameter and length are 40–80 nm and 5–9 μm, respectively [34]. It is known as that the barrier height of CNTs is 4.95–5.95 eV, whereas typical mean values for epoxy, nylon 66, polystyrene, poly­ carbonate, and polyamide are 0.5–2.5 eV, 3.95 eV, 4.22 eV, 4.26 eV, and 4.36 eV, respectively [22,33]. Accordingly, the λ of a polymer matrix is taken to range widely from 0.5 to 5 eV based on conventional studies [22,33]. Besides, many researchers have reported the experimental intrinsic conductivity of an individual CNT. The intrinsic conductivity of MWCNTs and SWCNTs through a literature review of past reports can be categorized to range between 5 � 103 to 5 � 106 S/m and 17 to 2 � 107 S/m [29,33], respectively. In other studies concerning metallic SWCNTs, their intrinsic conductivity has been shown to range from 104 to 107 S/m, whereas, these CNTs as semiconducting ones have conductivity ranging around 10 S/m [22].

Table 2 Orthogonal array (L934) according to PNC design variable levels and perfor­ mance factors: electrical conductivity and percolation threshold. Level 1 2 3

x1 ¼ DCNT (μm) 0.01 0.03 0.05

x2 ¼ LCNT (μm) 5 7 9

x4 ¼ σCNT (S/μm) 0.005 2.500 5.000

OA #Run

x1 ¼ DCNT (μm)

x2 ¼ LCNT (μm)

x3 ¼ λ (eV)

x4 ¼ σCNT (S/μm)

at the Pp of 50%

σc (S/m)

φc

1

1

1

1

1

0.0109

2

1

2

2

1

3

1

3

3

1

4

2

1

1

3

5

2

2

3

1

6

2

3

2

2

7

2

1

3

2

8

2

2

1

3

9

2

3

2

1

6.07 � 10 3 3.43 � 100 9.97 � 100 7.83 � 101 7.73 � 10 2 3.37 � 101 9.00 � 101 2.07 � 102 2.05 � 10 1

Fig. 8. (a) Initial percolation threshold and (b) electrical conductivity of PNCs at Pp of 50%. 5

x3 ¼ λ (eV) 0.50 2.75 5.00

0.0081 0.0063 0.0315 0.0232 0.0186 0.0504 0.0374 0.0301

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Fig. 9. ANOM for electrical conductivity (a) and percolation threshold (b) according to the design variable levels.

Fig. 10. Accuracy of the RSM-based surrogate models for (a) electrical conductivity and (b) percolation threshold. Table 3 The empirical formula for Fij according to the combination of different proba­ bility distributions for random variables. PDF Combination

Empirical formulas for Fij

NormalNormal NormalLognormal

Fij ¼ ρij

NormalWeibull LognormalWeibull

Table 5 Random variables and their probability distribution types for the PNC design.

δj Fij ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnð1 þ δ2j Þ Fij ¼ 1:301 0:195δj þ 0:328δ2j Fij ¼ 1:301

0:052ρij þ 0:011δi

0:210δj þ 0:002ρ2ij

​ ​ ​ ​ ​ ​ ​ ​ þ 0:22δ2i þ 0:35δ2j þ 0:005ρij δi þ 0:009δi δj

X1 X2 X3 X4

0:174ρij δj

σ Xk

XLk

μXk

XUk

PDF Set 1

PDF Set 2

0.001 0.100 0.500 0.500

0.010 5.000 0.500 0.005

0.0300 9.0000 5.0000 4.3505

0.05 9.00 5.00 5.00

Normal Normal Normal Normal

Lognormal Weibull Normal Normal

Table 4 The establishment of case studies for PNCs design optimization considering uncertainty. Case

Transformation

Opt.

Correlation coefficient [ρij] (i6¼j)

PDF of input variables DCNT(μm)

1 2 3 4 5 6 7 8 9

– Rosenblatt

Nataf

DDO RBDO RBDO RBRDO RBRDO RBRDO RBRDO RBRDO RBRDO

– – – – – 0 0.3 0.6 0.9

– Normal Lognormal Normal Lognormal

6

LCNT (μm) – Normal Weibull Normal Weibull

λ (eV) – Normal Normal Normal Normal

σCNT (S/m) – Normal Normal Normal Normal

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Table 6 Optimal results for RBDO and RBRDO by using Rosenblatt transformation (Cases 1–5). PDF

DDO

RBDO (βt ¼ 3)

Initial

Case 1

Case 2

Case 3

Case 4

–

–

Set 1

Set 2

Set 1

Set 2

0.0300 7.0000 2.7500 2.5025 0.2139

0.0300 9.0000 5.0000 4.3504 0.0246

0.0268 9.0000 5.0000 4.3272 0.0818

0.0267 9.0000 5.0000 4.3264 0.0861

0.0268 9.0000 5.0000 4.2985 0.0818

0.0267 9.0000 5.0000 4.2977 0.0861

gAR

0.2911

0.0000

GAR

0.0000

0.0000

GAR

0.0000

0.0000

μ*f

1.6379

–

–

–

–

μf

1.7399

1.7345

0.1054

–

–

–

–

σf

0.0527

0.0529

x1 x2 x3 x4 gφc fσc

1.6379

σ*f

Pp ¼

1.8925

X1 X2 X3 X4 Gφc fσc

A1 A2 � � þ A2 1 þ exp φΔφφ0

1.7399

1.7346

fRBRDO

1.5307

1.5258

In many engineering applications, to generate the surrogate model over the design space of interest, the design of experiments (DoE), which is an effective acquisition method for sample data is employed. The use of this method is to reduce computation time and experimental cost in approximate design optimization. Generally, surrogate models of the black-box type are employed in the case of high nonlinearity and these models demand a large number of sample data for training the model. In this study, percolation simu­ lation takes approximately 3 h on a standard workstation to obtain response data (i.e. electrical conductivity and percolation threshold) for every single simulation trial due to the need for executing several sto­ chastic procedures to mimic the spatial randomness of the matrix – fiber network. Thus, it is critical and essential to be able to acquire sample data efficiently for generating a surrogate model based on the DoE. Besides, since the relationship between input and output does not show any complicated nonlinearity in the analysis of means (ANOM), the quadratic polynomial models are employed in this study instead of the black-box types. The surrogate models for electrical conductivity and percolation threshold are generated by using the response surface method (RSM). The composite central design (CCD) [37] has been theoretically known as one of the best DoE methods that are appropriate for generating the quadratic polynomial model since the test combinations (samples) are established taking into consideration of the variation of the responses indicative of a convex or concave trend [38,39]. Based on the CCD, which is known to be suitable for RSM, we establish the DoE to obtain the sample data for the performance factors in the design space. The number of experiments (N) based on CCD is determined by using the equation of N ¼ 2dþ2d þ 1 where d is the number of design variables. Percolation simulations were conducted 25 times by changing the design variables to acquire the performance factors at the percolation probability of 50%. It is worth noting here that the electrical conductivity is converted to a log scale to maintain nu­ merical stability during design optimization. As shown in Fig. 10, we evaluate the accuracy of surrogate models using the standard metric of R2 value. The calculated value of R2 for the surrogate model of the electrical conductivity and percolation threshold is as high as 0.9975 and 0.9999 respectively. The obtained surrogate model is utilized as the objective function and constraint function for design optimization.

4. Analysis of means (ANOM) for design variables For sensitivity analysis of performance factors (i.e. φc and σc) by using the analysis of means (ANOM), an orthogonal array (OA) is established with the L9(34) table. Using the in-house code aforemen­ tioned in Section 3, PNC percolation simulation is executed to acquire the σc and φc at percolation probability value of 50%. The OA and the resulting performance factors from the simulation according to the design variable levels are tabulated in Table 2. We calculate the mean of σ c and φc by using Equation (10), where n is the number of experiments under identical level of a design variable, i is the design parameter level, xk is the design variable and y is the performance factor, mxk i is the mean value of multiple performance factor instances for the same design variable. n 1X yx ij ​ ​ ði; ​ j ¼ 1 � 3; ​ k ¼ 1 � 4Þ n j¼1 k

X1 X2 X3 X4 Gφc

Case 5

5. Surrogate modeling

(9)

Consequently, from the Equations (1) and (9), the initial percolation threshold and electrical conductivity at Pp of 50% for PNCs are computed to be φc ¼ 0.0234 and σc ¼ 43.59 S/m as shown in Fig. 8.

mxk i ¼

RBRDO (βt ¼ 3, w1 & w2 ¼ 0.5)

(10)

The sensitivity of mean performance factors by changing each design variable level is illustrated in Fig. 9. As shown in Fig. 9(a), by increasing levels of DCNT (black line) and σCNT (green line), electrical conductivity is mostly increased proportionally since CNT intrinsic resistance de­ creases. For LCNT (red line) and λ (blue line), there is a trade-off tendency between LCNT and λ. When the LCNT level is high, electrical conductivity decreased relatively owing to the increase in CNT intrinsic resistance. Furthermore, from the perspective of energy barrier difference between the CNT and polymer matrix, when the level of λ is low, electrical conductivity increases relatively since tunneling effects are larger than intrinsic electron transport along CNTs. In the case of the CNT aspect ratio (i.e. LCNT/DCNT), It has an effect on the percolation threshold. The mean percolation threshold indicates the high sensitivity concerning DCNT and LCNT as shown in Fig. 9(b). When the CNT aspect ratio is small, the percolation threshold is raised as the content of CNTs in the polymer matrix increased by short length ‘CNTs, while the σ CNT and λ variables have a greater effect on electrical conductivity than the percolation threshold due to their intertwined relationship with electrical proper­ ties. Hence, through an investigation on performance sensitivity using ANOM, we ascertain that DCNT, LCNT, σ CNT, and λ have a considerable effect on electrical conductivity, and CNT aspect ratio affects the percolation threshold due to changing volume fraction in accordance with DCNT and LCNT.

6. Nataf transformations In the conventional design optimization under uncertainty such as RBDO and RBRDO, correlated input variables have to transform into independent standard normal variables for reliability analysis. For this transformation, there are representative transformations, which are commonly known as the Rosenblatt [40] and Nataf transformation [41]. 7

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out in two steps. The first step transforms the input variable of any arbitrary distribution into correlated standard normal variables. Sub­ sequently, these correlated standard normal variables transform into uncorrelated standard normal variables [43]. These procedures are described in detail in the next section. 6.1. Nataf distribution for marginal transformation Let the random input variables be denoted by X ¼ ðX1 ;X2 ;⋯Xn Þ. The marginal probability density function (PDF) of each random input var­ iable (Xi) is assumed with fi ðxi Þ; ​ ði ¼ 1; ⋯; nÞ, and its corresponding marginal CDF is available as Fi ðxi Þ; ​ ði ¼ 1; ⋯; nÞ. Based on this rela­ tionship between random variables and CDF, the marginal trans­ formation works as shown by Equation (11) [29]. Here, Φ(⋅) and Φ 1(⋅) are respectively the CDF and inverse CDF of the standard normal variable. 8 > > < Φðzi Þ ¼ Fi ðxi Þ; zi ¼ Φ 1 ½Fi ðxi Þ�; ​ where; ​ i ¼ 1; ​ 2; ⋯n (11) > > : xi ¼ F 1 ½Φðzi Þ�; i The marginal transformation is employed to acquire a standard normal vector, Z ¼ ðZ1 ;Z2 ;⋯Zn Þ. By using the Nataf transformation and rules of implicit differentiation, the joint PDF for the random variable, X, can be indicated as shown by Equation (12), where φ(⋅) is the standard normal PDF. fX ðx1 ; x2 ⋯; xn Þ ¼

∂ðz1 ; z2 ⋯; zn Þ φ ðz; ρ0 Þ ∂ðx1 ; x2 ⋯; xn Þ n �

1 φn ðz; ρ0 Þ ¼ pffiffiffiffiffiffiffiffiffinffiffiffiffiffiffiffiffiffiffiffiffi exp ð2πÞ detρ0

(12)

� 1 T 1 z ρ0 z 2

(13)

Equation (13) is the n-dimensional standard normal PDF with zero mean (μ ¼ 0) and unit standard deviation (σ ¼ 1), and it is generally known as the Nataf distribution model [29,42]. 6.2. Nataf transformation of random input variables Let the correlation matrix of the random variables, X, be ρ. By using the definition of the correlation coefficient in Equations (12) and (13), the component of the correlation matrix can be calculated by Equation (14). �� � Z þ∞ Z þ∞ � � xj μj x i μi fXi Xj xi ;xj dxi dxj ρij ¼ ∞

Z

σi

∞

þ∞ Z þ∞ �

​ ​ ​ ​ ¼ ∞

∞

σj

Fi 1 ðΦðzi ÞÞ

σi

μi

�

Fj

1

Φ zj

σj

��

μj

!

� φ2 zi ;zj ; ρ0ij dzi dzj (14)

Where, ρ0ij is a component of the correlation matrix ρ0 of Z, and φ2 is the bivariate standard normal PDF. If the ρ of X and the marginal PDF for random variables are known, ρ0 A may be obtained using Equation (14). Once all the components for ρ0ij are obtained, the space of random variables can then be transformed from correlated standard normal space Z ~ N(0, I) to the independent standard normal space U ~ N(0, I) by using a linear transformation. The term ρ0 can be decomposed into the lower and upper triangular matrix by using Cholesky decomposition since the covariance matrix of Z is obviously positive (Equation (15)). Here, L0 is the lower triangular matrix.

Fig. 11. Trend of objective and constraint function of RBDO and RBRDO using Rosenblatt transformation (Case 2 & 4: PDF Set 1, Case 3 & 5: PDF Set 2).

Even though both transformations have identical purposes that trans­ form correlated input variables to independent uncorrelated ones, the probability information contained in the random variables is demanded differently in each of these transformations. The Nataf transformation uses a marginal CDF and a correlation matrix is employed. The main idea of this transformation is to generate a correlated random vector with a specified correlation matrix from the independent standard normal CDF [42]. This transformation is carried

ρ0 ¼ L0 LT0

(15)

After the determination of ρ0 , the mutually independent standard normal vector, U, as the symmetric matrix can be calculated by using Equation (16). In addition, by using the inverse Nataf transformation (Equation (17)), the correlated standard normal variables Z can be 8

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Table 7 Comparison of RBRDO results according to correlation coefficients and transformation methods. RBRDO (βt ¼ 3, PDF Set 2, w1 & w2 ¼ 0.5) Rosenblatt transformation Initial [ρij] (i6¼j) x1 x2 x3 x4 gφc

– 0.0300 7.0000 2.7500 2.5025 0.2139

Nataf transformation

Case 5

Case 6

Case 7

Case 8

Case 9

– 0.0267 9.0000 5.0000 4.2977 0.0861

0.0 0.0267 9.0000 5.0000 4.2974 0.0862

0.3 0.0271 9.0000 5.0000 4.3002 0.0461

0.6 0.0273 9.0000 5.0000 4.3018 0.0136

0.9 0.0268 8.7989 4.7656 4.2893 0.0000

GAR

0.0000

0.0000

0.0000

0.0000

0.0000

[ρij] (i6¼j) X1 X2 X3 X4 G φc

gAR

0.2911

μ*f

1.6379

μf

1.7345

1.7343

1.7515

1.7637

1.7179

0.1054

σf

0.0529

0.0528

0.0523

0.0519

0.0517

fσc

σ*f

1.6379

fRBRDO

1.5258

1.5270

obtained. U ¼ L0 1 Z

(16)

Z ¼ L0 U

(17)

(18)

Subsequently, in order to conduct reliability analysis, the Jacobian matrix for transformation from the correlated space (X-space) to un­ correlated space (U-space) can be expressed by using the relationship of fXi ðxi Þdxi ¼ φðzi Þdzi as represented by Equation (19) [43]. � � φðzi Þ Jx→u ¼ diag (19) L0 fi ðxi Þ In summary, the main purpose of the Nataf transformation is to obtain an appropriate correlation matrix ρ0 for Z from the standard normal space under the condition that the correlation matrix ρ for X is guaranteed. This requires the calculation of ρ0ij (i6¼j) of Z for each ρij of X. 6.3. Approximation of correlation coefficients (ρ0ij ) In order to acquire ρ0ij , Equation (14) has to be used. However, this equation is too tedious for direct calculation due to the double inte­ gration. Therefore, Equation (14) can be approximated by using empirical Equation (20), which Liu and Der Kiureghian [44] have proposed.

ρ0ij ¼ Fij ρij

1.5543

1.5446

(PNC) design parameters and their electrical performance does not indicate high nonlinearity. Therefore, an analytical reliability assess­ ment can be suitably accomplished using the FORM method which would suffice in our case. In the case of the Monte Carlo (MC) simulation being employed, the accuracy in the reliability evaluation depends on the number of sample data and demands high computation cost. The MC approach is more suitable for surrogate models of the black-box type (such as Kriging, Artificial neural network, support vector machine and so on) since it is difficult to obtain an analytically differential function of the surrogate model in general. The surrogate models using quadratic RSM are appropriate to conduct FORM-based RBDO and RBRDO for PNC design. When RBDO and RBRDO are carried out, the input variables are usually assumed to follow a normal distribution. However, the optimal value varies depending on the distribution types of the random input variables. In the case of CNT diameter, SWCNT diameters were experimentally measured by Ziegler et al. but they only suggested the diameter histogram data [45]. After that, Esteva [46] statistically quantified a lognormal distri­ bution using that histogram data. In order to consider the correlation between the input variables, each of which follows different probability distribution, the multivariate distribution is applied to random input variables concerning the CNT diameter and length by using the Rose­ nblatt and Nataf transformation. In addition, several cases are estab­ lished to capture the optimal performance in accordance with the correlation coefficients, transformations, and probability distribution type for random input variables as shown in Table 4.

As a result, using the obtained Z and Equation (18), the correlated random variables can be generated from a standard normal distribution xi ¼ Fi 1 ðΦðzi ÞÞ ​ where; ​ i ¼ 1; ​ 2; ⋯n

1.5429

7.1. Deterministic design optimization By conducting deterministic design optimization (DDO), local optimal values (i.e. optimal region search) are obtained previously for use as the initial design variables for RBDO and RBRDO, as mentioned earlier in Section 7. Here, design variables are simply indicated with xk (k ¼ 1–4) wherein x1 ¼ DCNT (μm), x2 ¼ LCNT (μm), x3 ¼ λ (eV), and x4 ¼ σ CNT (S/m). The lower bound (xLk ) and upper bound (xUk ) for each design variable are identically set as mentioned in Section 3.4. The initial variables are set as xk_DDO initial ¼ [0.03, 7, 2.75, 2.5025]. As shown in Equation (21), the formulation of the DDO problem is defined as a minimization problem. Here, a negative sign is appended to the objec­ tive function that deals with maximization of the electrical conductivity. For the constraint functions, percolation threshold ( ​ gφc ) and CNT aspect ratio (gAR ¼ x2 =x1 ) should be satisfied with less than 0.02 (i.e. PNC volume fraction) and more than 300 respectively. � � min: fσc ðXk Þ

(20)

Here,Fij � 1, and it is a correlation coefficient function for ρij . To obtain the reduced correlation coefficient, ρ0ij , Fij is approximated by a polynomial function form. Liu and Der Kiureghian provide 49 empirical formulas for calculating Fij according to the combination of different probability distribution types. In this study, for Fij , we employed the empirical formulas in Table 3 to consider the correlation between random variables. In the table, the symbols δi and δj are the coefficients of variation (i.e. δ ¼ σ /μ) for each random variable. 7. Design optimization under uncertainty for PNC design The approach to reliability analysis relies on the nonlinearity of the surrogate model. According to this dependence, generally, this analysis is accomplished with first-order reliability method (FORM), secondorder reliability method (SORM), or the sampling-based method (i.e. Monte Carlo simulation). In this study, the surrogate models are generated with a quadratic polynomial function. It turns out from our analysis here that the relationship between the polymer nanocomposite

gφc ðxk Þ 0:02 gAR ðx1 ; x2 Þ �0 300

Subject to 1 9

1�0

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Polymer 186 (2020) 122060

Fig. 13. Robustness of optimized electrical conductivity according to PDF Sets 1 and 2 (Rosenblatt transformation).

Fig. 14. Robustness of optimized electrical conductivity according to increasing correlation coefficients (Nataf transformation).

The objective function for maximizing electrical conductivity is iden­ tical to that in the DDO formulation. However, in the case of the constraint functions for the percolation threshold and CNT aspect ratio, these functions should change to probabilistic constraint functions and the probability of the constraint functions being satisfied for negative values should be more than Rt. In Equation (22), XLk and XUk are the lower and upper bound for the random variables and these bounds of random variables are documented in Table 5. Here, Φ(⋅) is the CDF of the stan­ dard normal distribution to achieve a target reliability index (i.e. βt ¼ 3), Rt is the corresponding target reliability level (99.87%). � � min: fσc ðXk Þ � � Gφc ðXk Þ Subject to P 1 � 0 � 1 Φð βt Þ ¼ Rt 0:02 � � GAR ðX1 ; X2 Þ � 0 � 1 Φð βt Þ ¼ Rt P 1 300 � (22) X Lk � Xk � X Uk k ¼ 1 � 4

Fig. 12. Trend of objective and constraint function of RBRDO using Nataf transformation by considering correlation coefficient effects and PDF Set 2.

xLk � xk � xUk

� k¼1�4

(21)

7.2. Reliability-based design optimization

Based on Table 5, the initial mean random variables are set as μXk ¼ [0.03, 9, 5, 4.3505] and the standard deviations for random variables are assumed to be σ Xk ¼ [0.001, 0.1, 0.5, 0.5]. Hence, the distribution type for all random variables is assumed to be a normal distribution of Xk ~ N(μXk ,σXk ) (PDF Set 1). Concurrently, the CNT diameter and length

Since the random input variables that are relevant to geometric and electrical properties for PNC have aleatory uncertainty (i.e. physical uncertainty), RBDO should be carried out to meet the target reliability for PNC design constraints considering the variation of random input variables. The RBDO formulation is defined as shown in Equation (22). 10

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Fig. 15. Optimal performances for (a) percolation probability and (b) electrical conductivity according to each case (Cases 1, 3, 5, 7, 8, and 9).

are considered to comply with the Lognormal and Weibull distribution in PDF Set 2 using the Rosenblatt transformation.

increasing correlation coefficients using the Rosenblatt and Nataf transformation.

7.3. Reliability-based robust design optimization In the RBRDO problem, the objective function (fσc ) of the electrical conductivity is revised using the mean (μf) and deviation (σf) with Equations (23) and (24). The objective function deviation (σ f) considers both the sensitivity and deviation (σ Xi ) with regards to each random variable. By using μf and σf, fRBRDO is expressed by Equation (25), where μ*f and σ*f are defined as the mean and deviation of the objective function

μf ¼ fσc ðXk Þ

(23)

vffiffiffiffiffiffiffi( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) ffiffiffi u n � ��2 uX ∂fσc Xk t 2 σf � ⋅σXk ∂Xk μ k¼1

(24)

min: fRBRDO ¼

w1

σ*f μf þ w2 σf μ*f

! where; w2 ¼ 1

w1

� � Gφc ðXk Þ 1 � 0 � 1 Φð βt Þ ¼ Rt Subject to P 0:02 � � GAR ðX1 ; X2 Þ � 0 � 1 Φð βt Þ ¼ Rt P 1 300 � L X k � Xk � X Uk k ¼ 1 � 4

for the initial random variables (μXk ) in Table 5 and w1 and w2 are the weighting factor for the mean and deviation of objective function respectively. In RBRDO for PNCs design, it assumed that both the mean and deviation of electrical conductivity are important. Therefore, their weighting factors are identically set as 0.5. Accordingly, the RBRDO objective function (fRBRDO) for the electrical conductivity is calculated by the summation of the initial mean ratio (μf =μ*f ) and initial deviation

(25)

8. Results and discussion

ratio (σ*f =σ f ).

8.1. Results of optimization for PNC design

In this study, in order for the RBRDO formulation to be defined as a minimization problem for fRBRDO, the negative sign is assigned to the objective function, as discussed earlier. The mean and deviation for the objective function are generally minimized in an RBRDO problem. However, in this work, there is a trade-off problem between mean and variation. Under identical constraint conditions of RBDO, the mean electrical conductivity (μf) has to be maximized, whereas the variation in electrical conductivity (σf) has to be minimized. Thus, the reciprocal of the variation term is considered for fRBRDO. The initial parameters to conduct RBRDO of the PNC are consistently established with RBDO as aforementioned in Section 7.2 (i.e. initial random variables, mean and deviation for a normal distribution, and target reliability index). More­ over, the optimal results for RBRDO are investigated in accordance with

The result in DDO (Case 1) shows that the electrical conductivity increased by 15.54% compared to the initial conductivity by minimizing the objective function (fσc ). The values of the design variables x2, x3, and x4, obtained via DDO have increased. This is physically reasonable because the percolation threshold is reduced by longer CNT length (x2), and electrical conductivity increases when the barrier height difference between the polymer matrix (x3) and CNT is low and also when the CNT intrinsic conductivity (x4) is high. The optimal results for Cases 1–5 are summarized in Table 6. For Case 2–5, based on the optimum point of DDO, RBDO and RBRDO are accomplished by using the Rosenblatt transformation. All random input variables in Cases 2 and 4 are assumed to follow the 11

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Polymer 186 (2020) 122060

normal distribution, while X1 and X2 for Cases 3 and 5 are set as Lognormal and Weibull PDF respectively. The percolation threshold is an inactive constraint as seen in Fig. 11(a), whereas the CNT aspect ratio is the active constraint (i.e. reliability index boundary is very close to the failure surface) as in Fig. 11(b). In addition, the number of iterations for the objective and constraint functions in RBDO and RBRDO is lower than that for DDO relatively because the feasible design space for the optimal solution is reduced by DDO. Thereby, the optimal value of RBDO and RBRDO can be obtained rapidly with a smaller number of iterations. For the increasing and decreasing rates for the objective function of RBDO and RBRDO, Case 2 and 4 showed an identical increase by 6.227% and Cases 3 and 5 showed a rise of 5.904% and 5.898%, in comparison to the initial electrical conductivity performance, respectively. Howev­ er, as compared with the electrical conductivity of DDO, Cases 2 and 4 showed a reduction by 8.063% and Cases 3 and 5 also showed a reduction by 8.343% and 8.348% respectively. In a conservative design scenario, to meet target reliability under inherent uncertainties, the decrease of performance achievement for the objective function than DDO cannot be avoided, as shown in Fig. 11(c). In random variables, since CNT intrinsic conductivity (X4) is a random variable, which is relevant to the electrical conductivity per­ formance of PNCs, the values of X4 is reduced slightly to secure the objective function robustness in the RBRDO result (Case 5). For PDF Sets 1 and 2, particularly, the percolation threshold of Cases 3 and 5 show a constraint that is more inactive than Cases 2 and 4, while the CNT aspect ratio is still an active constraint. For the electrical conductivity robustness of Cases 4 and 5, for which PDF Sets 1 and 2 are applied respectively, as seen in Fig. 13, the standard deviations of Cases 4 and 5 are minimized in comparison to the initial performance by 50% and 49.8%. The mean values of the objective functions are enhanced from the initial performance by 6.23% and 5.90%. From these results, it is clear that Cases 4 and 5 obtain almost identical results since the Rosenblatt transformation takes into account independent random variables based on a standard normal PDF without considering the correlation between the variables. Subsequently, when RBRDO is accomplished with the Nataf trans­ formation, the change in optimal results is examined as a function of the increase in the correlation coefficient (ρ), which in general ranges all the way from 1 to 1 [47]. In this study, this coefficient is set to range from 0 to 0.9 at interval steps of 0.3. The off-diagonal components of the correlation matrix are set at 0.3 step intervals and the diagonal component is of unit value. The optimal results of RBRDO in accordance with the correlation coefficients (i.e. Cases 6–9) are summarized in Table 7. For Case 5 and Case 6, the optimal results are identical. The reason is that Rosenblatt transformation does not consider the correlation coefficient here. For Case 6, since the correlation coefficient is set as zero despite using the Nataf transformation, the results of the two cases prove the consistency in the analysis being carried out. The constraint functions for the percolation threshold in Fig. 12(a) indicate that is gradually close to the failure surface (i.e. an active constraint) for increasing correlation coefficients, whereas the CNT aspect ratio remains an active constraint (Fig. 12(b)). The objective functions for Cases 7 and 8 in Fig. 12(c) are minimized as compared to Case 6 by the higher correlation coefficients, while the value for Case 9 is higher than for Case 8. This is because the probabilistic constraint for percolation threshold approaches the failure surface for enhanced cor­ relation effects of the random variables, in order to ensure that reli­ ability and robustness are satisfied concurrently. As shown in Fig. 14, when increasing the correlation coefficients, the electrical conductivity robustness of Cases 6–9 is secured by minimizing the standard deviation (σ f) from the initial deviation (σ *f ) by 49.9%,

tendency of the increasing mean values (μf) for electrical conductivity when increasing correlation coefficients (ρij) in Cases 6–8, in Case 9, μf is decreased even when the correlation coefficient is relatively the highest, while the robustness of electrical conductivity is remarkably guaranteed by minimizing the deviation (σf) as shown in Table 7. 8.2. Performance verification of optimal solution for PNCs Using the optimal variables of PNC design obtained via DDO, RBDO, and RBRDO, the optimal performance is now verified through the percolation simulation routine again. The percolation probability and electrical conductivity corresponding to each case are illustrated in Fig. 15. In Fig. 15(a) (left), the percolation for Cases 1, 3, 5, 7, 8, and 9 occurred at a lower volume fraction than the initial design. It implies that the φc is minimized significantly, and one can see that the slope of percolation probability by increasing the correlation coefficients in Cases 7 and 8 dwindled relatively. Particularly, the percolation threshold of Case 9 is lower than that of Case 8 in spite of the higher correlation coefficients, as shown in Fig. 15(a) (right). The reason is that the values of the design variables are altered too much due to the high correlation effect to satisfy the target reliability. Furthermore, the electrical conductivity for all cases showed an improvement when compared to the initial σc as following Fig. 15(b) (left), which gradually converged at the high volume fraction. In Fig. 15(b) (right), even if the electrical conductivity for Case 1 is the highest amongst all cases, this case does not secure the reliability of the constraints. Case 4 shows a slightly higher conductivity than Case 5 due to the absence of consid­ eration of the robustness of the objective function. In Cases 7 and 8, electrical conductivity is increased by the correlation coefficient effect, while for Case 9, it reduced in value to meet the target reliability of the percolation threshold despite higher correlation coefficients than in Cases 7 and 8. As a result, from the point of view of the percolation threshold under physical uncertainty, the best design is Case 5. However, Case 5 does not consider the correlation between the input variables even though the target reliability and robustness are satisfied. For electrical conductivity, Case 1 is the best design but the constraint functions are not satisfied with target reliability. In this study, although the percolation threshold is increased by a conservative design, the optimal design of Case 8 is recommended since the most important thing being attained is the simultaneous fulfillment of the PNC design condition reliability and robustness in electrical conductivity in the event of high uncertainty in the values of the design variables. Thereby, referring to each optimal parameter of Case 8, the polymer matrix and CNT type are proposed based on literature data, as mentioned in Section 3.4. The optimal CNT diameter and length are respectively obtained as DCNT ¼ 0.0273 μm and LCNT ¼ 9 μm due to reliance on the constraint condition of the CNT aspect ratio, optimal CNT intrinsic conductivity is σ CNT ¼ 4.3018 S/μm, and optimal barrier height for the polymer matrix 5eV. In conclusion, it may be recommended that the optimum CNT type is the long MWCNT with high intrinsic conductivity and the material for the polymer matrix could be either polystyrene (4.22 eV), polycarbonate (4.26 eV), or polyamide (4.36eV). The reason for the suggested polymer matrix is that the difference of the barrier height between CNT (4.95–5.95 eV) and polymer matrix should be minimized to achieve ease of tunneling through the quantum effect. 9. Concluding remarks In this study, RBRDO for polymer nanocomposites is performed to improve electrical percolation robustness and to achieve the target reliability of the percolation threshold and CNT aspect ratio. The changing tendency of the objective and constraint functions is investi­ gated via several case studies accounting for correlation coefficient ef­ fects and different probability distribution types for the random

50.4%, 50.8%, and 51%, respectively. The mean of electrical conduc­ tivity (μf) in Cases 6–9 has improved by 5.9%, 6.94%, 7.68%, and 4.88%, respectively in comparison to the initial mean (μ*f ). Especially, unlike the

12

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variables using suitable transformation methods. The optimal results for each case study are verified by using the in-house code that was developed for the percolation simulation of PNCs in the previous study [30]. The local optimum design point is determined via DDO, RBDO, and RBRDO for PNC design and then subjected to further design optimiza­ tion considering aleatory uncertainty in the CNT geometry and electrical properties. As a result, although the electrical conductivity in DDO dramatically increased as compared to the initial conductivity, the reliability of the constraint functions and the robustness of electrical conductivity were still not guaranteed. Thus, RBDO and RBRDO are conducted based on DDO results as the initial value. One can see that minimization of the objective function for electrical conductivity by considering inherent uncertainty is unavoidable for the case of conser­ vative PNC design. Furthermore, with increasing correlation between the random variables, concurrently satisfying the target reliability of probabilistic constraints for the CNT aspect ratio and percolation threshold, the robustness of the objective function for electrical con­ ductivity was improved by minimizing deviation and the mean value of electrical conductivity was enhanced in comparison to the initial per­ formance. Based on the results of the optimization, it is suggested that the best design in terms of reliability and robustness for PNCs would be to consider a long MWCNT with high intrinsic conductivity and choose polymer matrices with a high barrier height such as polystyrene, poly­ carbonate, or polyamide. The use of these polymer matrices ensures a lower barrier height between the CNT and polymer matrix so that the tunneling effect can be enhanced. The key contribution of this study is to recommend an organized guideline for design of PNCs to attain improved design condition reliability and performance robustness at the same time. In addition, the relevance of the Nataf transformation for reliability analysis in RBRDO is exemplified considering practical engi­ neering and science problems with correlated design variables. In the future work, the percolation simulation model will be extended using a 3-D RVE by taking into account, CNT waviness and the configuration of the percolation network so as to investigate the impact on the electrical performance, percolation threshold and conductance transition by considering the mechanical loading on the matrix in an operating environment.

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Author contribution Dr. Jaehyeok Doh conceived the idea for the entire design-based model framework and executed the model and analysis. Dr. Qing Yang provided the experimental support for the validation of the model. Dr. Nagarajan Raghavan is the principal investigator of the project and provided his mentorship to the team in the idea generation, execution and manuscript writing procedure. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is funded by the Ministry of Education (MOE), Singapore – Academic Research Fund Tier-1 under Project No. SUTDT12017005 and the SUTD-ZJU Research Collaboration Office under Project No. ZJURP1500103. References [1] T.W. Ebbesen, H.J. Lezec, H. Hiura, J.W. Bennett, H.F. Ghaemi, T. Thio, Electrical conductivity of individual carbon nanotubes, Nature 382 (6586) (1996), 54 %@ 1476-4687.

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