Effect of Borophene and Graphene on the Elastic Modulus of PEDOT:PSS Film—A Finite Element Study

A finite element method (FEM) was employed to investigate the interaction of borophene nanoplatelets (BNPs) and graphene nanoplatelets (GNPs) on the mechanical properties of Poly(3,4ethylene dioxythiophene):poly(styrene sulfonate) PEDOT:PSS film. A 3D random distribution of the inclusion into the PEDOT:PSS matrix was constructed by developing a 145 × 145 × 145 representative volume element (RVE) with a 4% volume fraction of BNPs and GNPs. In comparison to the pristine PEDOT:PSS, the calculated effective elastic moduli of the BNP-PEDOT:PSS and GNP-PEDOT:PSS nanocomposites exhibited 9.6% and 10.2% improvement, respectively. The predicted FE results were validated by calculating the elastic moduli of the nanocomposites using a modified Halpine-Tsai (H-T) model. The reinforcing effect of the inclusion into the PEDOT:PSS film offers a promising electrode with improved mechanical stability. Consequently, this intriguing result makes the BNP/PEDOT:PSS nanocomposite highly promising for further investigation and application in cutting-edge devices such as touchscreen, thermoelectric, light-emitting diode, electrochemical, photodiode, sensor, solar cell, and electrostatic devices.


Introduction
The growing market for wearable, flexible, and portable electronics has paved the way for a slew of new wearable applications including smart apparel, stretchable thermoelectric electrodes, biomedical implants, and flexible pseudocapacitive nanocomposites for energy storage [1]. Fabrication of conductive electrodes with desired mechanical characteristics is the key to endowing advanced materials with such capabilities. In the place of rigid metallic oxides, conductive polymers (CPs) such as PEDOT:PSS have emerged as the most promising flexible electrode materials; they play a critical role in groundbreaking devices as transparent electrodes, hole transport layers, interconnectors, electroactive layers, or motion-sensing conductors [2]. Because of its exceptional flexibility, ease of manufacture, high electrical conductivity, and outstanding optical transparency, PEDOT:PSS is widely employed for practical applications such as energy conversion and storage devices [3,4]. However, PEDOT:PSS film has significant drawbacks, such as excessive acidity, inhomogeneous electrical characteristics, hygroscopicity, the low elastic modulus of~2.7 GPa, and the little strain (ε) and break of~5%, all of which contribute to its limited endurance [1,5,6].

Finite Element Model
In this study, the modeling approach for the 2-phase composite uses either BNP or GNP platelets to reinforce the PEDOT:PSS matrix. To reduce the computational cost, BNP is assumed to be transversely isotropic whereas GNP is taken to be isotropic, and both are represented with circular platelets. The shear modulus of BNPs is calculated from Equation (1). The aspect ratio and thickness of the nanosheets chosen for this calculation are 0.01 and 0.29 nm, respectively, which translate to a platelet diameter of 29 nm. Based on this, the density of BNP employed in the modeling was estimated from its mass density of 7.73 × 10 −7 kg/m 2 [18]. A preliminary FE calculation determined a 4% volume fraction as the percolation threshold for the effective reinforcement of PEDOT:PSS with BNP that was in good agreement with the theoretical calculation. Digimat software [24] was then employed to construct the realistic (stochastic) Representative Volume Elements (RVE) of a 4% volume fraction of BNP/PEDOT:PSS and GNP/PEDOT:PSS nanocomposites. An RVE is the minimum volume of microstructure with the general properties of the entire microstructure, such as morphology, dimension, volume fraction, and phase randomization, over which specialized attributes are modeled [25]. It should be large enough to encompass the fundamental microstructural properties, but small enough that the stress and strain levels may be roughly regarded homogenous throughout the RVE [26]. The RVE is considered to be in equilibrium, with a suitable total deformation. Based on this, the software automatically generated an RVE of 145 × 145 × 145 nm 3 with 639 BNP and 639 GNP inclusions for the respective composites. To arrange the inclusions one after another in the RVE, Digimat-FE employs random placement algorithms. Therefore, the inclusions are randomly distributed in 3D within the matrix and voxel-based meshing was used to mesh the matrix and reinforcement components. The interface between the PEDOT:PSS and the inclusions was considered to be fully bonded, resulting in total load transmission. According to the data in Table 1, all relevant variables for an elastic analysis were entered into the models.
where G 23 , E 22 , and v 23 represent the transverse shear modulus, transverse elastic modulus, and transverse Poisson ratio of BNP, respectively.  [19,30] 1000 [31] Shear Modulus (GPa) -287.35 - [19,30] 0.22 [29] Particle size (nm) 30  For validation, we employed a modified version of the H-T theoretical technique (Equations (2)-(4)) to estimate the elastic modulus of our model to assess the FEM findings. This method considered the aspect ratio and volume fraction variables, which are important parameters to consider when evaluating the mechanical characteristics of randomly distributed 2D nanosheet reinforced composites [20].
where, E nc , E m , and E f are the moduli for the nanocomposite, matrix, and fillers, respectively, t /d is the aspect ratio of the BNP and GNP platelets, t is the thickness of the platelet, and d is the diameter.

Result and Discussion
The mechanical characteristics of BNP and GNP-reinforced composites are studied using a finite element model in this work. The features of the RVE model of the PEDOT:PSS matrix and the inclusions are shown in Figure 1, having the dimension of 145 × 145 × 145 nm 3 . In addition, the elastic result of the nanocomposites is summarized in Table 2. Whereas Table 3 tabulates the effective moduli of the nanocomposites calculated using FEM and H-T methods. As a function of the characteristics of the inclusions and matrix, as well as the filler content of the inclusions, the effective elastic moduli (Equations (5) and (6)) of the filled nanocomposite material (i.e., effective bulk moduli K and effective shear moduli G), are calculated: where G is the shear moduli of the nanocomposites, K is the bulk moduli of the nanocomposites, f is the volume fraction of inclusions, and subscript m and f are the matrix and inclusions, respectively [26]. The average Young's modulus, Poisson's ratio, and shear modulus of BNP/PEDOT:PSS nanocomposites are predicted to be~3 GPa,~0.32, and~1.1 GPa, respectively. For the GNP/PEDOT:PSS nanocomposite, the average Young's modulus, Poisson's ratio, and shear modulus are~3 GPa,~0.32, and~1.1 GPa, respectively. The global densities of the composites are 1.12 g/cm 3 and 0.88 g/cm 3 for the BNP/PEDOT:PSS and GNP/PEDOT:PSS nanocomposites, respectively.   In comparison to pristine PEDOT:PSS with a maximum elastic modulus of 2.7 GPa, the incorporation of BNPs and GNPs translates to a 9.6% and 10.2% enhancement in moduli. The improvement in mechanical property is attributable to the homogeneous random dispersion of the BNPs and GNPs in the PEDOT:PSS matrix. Moreover, the platelet morphology of the inclusions is another noteworthy structural property that improves the composites' Young's modulus. The calculated elastic moduli for the nanocomposites from the H-T model are~3 GPa and~3 GPa for BNP/PEDOT:PSS and GNP/PEDOT:PSS nanocomposites, respectively, which are in excellent agreement with the FEM prediction. The experimental result of Young's modulus for GNP/PEDOT:PSS reported in the literature is 4.6 GPa [16]. This result showed a significant difference from the FEM and H-T findings. The discrepancy between the experimental and theoretical results is mainly due to the incorporation of additional reinforcement with PAA which resulted in~54% enhancement to the modulus of pristine PEDOT:PSS matrix. Moreover, PEDOT:PSS and PAA benefit from hydrogen bonding which promotes strong bonding between the matrix and the reinforcer.
The equivalent von Mises stress and maximum primary total strain distribution of the nanocomposites are shown in Figure 2. The portion of the PEDOT:PSS film where two inclusions come near to one another experiences the most strain, resulting in greater stiffness. Meanwhile, the stiff inclusions embedded in the softer matrix carry the most stress. Figure 3 shows the findings of the Digimat-FE stress-strain prediction in the RVE of the BNP/PEDOT:PSS and GNP/PEDOT:PSS nanocomposites. The nanocomposites exhibit similar responses which are linear and conform to the elastic range of material's behavior.

Conclusions
We introduce borophene and graphene sheets as reinforcing materials for PEDOT:PSS in this work. Both resulting nanocomposites have an increased elastic modulus of 9.6% and 10.2% in their mechanical properties, respectively. BNP and GNP are feasible additives to PEDOT:PSS to improve its mechanical durability. The modeling method used in this study might be useful for the FEM investigation of BNP and GNP fillers in PEDOT:PSS composites. Furthermore, the FEM and H-T conclusions were shown to be in great agreement with theoretical approaches. The stress-strain curves predicted from the RVE models of the nanocomposites show elastic response based on the assumed critical strain of 0.03.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.