# Percolation Model for Renewable-Carbon Doped Functional Composites in Packaging Application: A Brief Review

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## Abstract

**:**

## 1. Introduction

## 2. Renewable Carbon/Carbon Complex

^{2}/g). It consists of thin graphite layers with exceptional mechanical strength, which highlights its great potential to be used as reinforcement agent in advanced packaging composites.

#### 2.1. Classification of Renewable Carbon

#### 2.1.1. Granular Renewable Carbon

#### 2.1.2. Powdered Renewable Carbon

#### 2.2. Use of Renewable Carbon in the Reinforced Packaging Polymeric Composites

_{c}) and inter-particle distance (τ) [27], to simulate the mechanical behavior of the composites. Moreover, τ

_{c}represents the thickness at which the matrix ductility can be improved drastically, and the expression of τ can be written as Equation (1):

_{c}, a remarkable increase of the toughness is perceivable. Therefore, the smaller the filler, the lower the concentration it would require to realize the brittle-tough transition under sufficient interfacial adhesion. Percolation mechanism was concluded as the leading attribution for the improved strength and toughness as the filler loading increased at τ < τ

_{c}.

## 3. Percolation Theory

_{d}), can be increased as the site occupation rising (ϕ

_{d}). The minimum rate of site occupation for percolation is described as percolation threshold (ϕ

_{c}), which is the critical filler volume fraction needed to obtain an initial percolating path throughout the matrix. As shown in Figure 7b, there is no percolation at ϕ

_{d}< ϕ

_{c}, which means the percolation probability is 0. Meanwhile, at the ϕ

_{d}≥ ϕ

_{c}, the percolation probability gradually increases toward 1.

_{c}is the critical volume fraction and β is a critical exponent which solely related to the system dimension (β is 0.4 for 3D-system).

_{c}, the volume fraction of the infinite cluster ϕ

_{∞}is zero. For ϕ ≥ ϕ

_{c}, ϕ

_{∞}can be defined by using Equation (3). The critical exponent b lies between 0.4 and 1.6. The lower the value of b is, the stronger the interaction between fillers and polymeric matrix will be. If b = 0.4, the reinforcing effect of the dangling bonds is similar as backbone. While b = 1.6 means no reinforcement from dandling bonds. Accordingly, the elastic modulus, E

_{composite}, of the composite can be expressed as Equation (4), in which, E

_{filler-filler}represents the stiffness of the filler–filler clusters. It highly depends on the intrinsic stiffness of filler itself, plus the binding stiffness between fillers. In the literature, the filler–filler stiffness for polymeric composites was usually set at 2.0 or 1.8 GPa [33].

## 4. The Effect of Renewable Carbon Characteristics on Its Percolation Behavior

#### 4.1. Geometry of Renewable Carbon

_{c}), which were estimated by extrapolating the data to zero radius. The circular shape, η

_{c}of 0.9614 ± 0.0005, tends to decrease the interparticle connectivity and has the highest percolation point. The 3D computer realization of 100 circular plates in random distribution is shown in Figure 9d. Similarly, Mathew et al. conducted Monte Carlo simulations on the percolation behavior of hard platelets in 3D continuum systems [35]. Cut-spheres, generated via intersecting a sphere with two parallel planes at equal distance from the equatorial plane, were used in their model, to represent the hard platelets. The simulation results indicated the platelets with lower aspect ratios required a relatively lower percolation threshold. Note that the majority of related works neglected the physical properties of the system, and only a few studies in the technical literature are devoted to the mechanical strength of the filler-doped composites.

#### 4.2. Alignment of Renewable Carbon

_{3}) treated MWCNT with a large amount of oxygenated functional groups were mixed with the epoxy resin in an acetone medium. The curing agent triethylenetetramine was afterward added into the milled degassing mixture and subjected to a DC-electric field for an accelerated curing under elevated temperature (Figure 11a). The preferential orientation of MWCNT was confirmed by the polarized Raman spectroscopy of a nanocomposite containing 0.3 wt% of aligned MWCNT in Figure 11b, which showed the intensity of G-band peak with the incident light applied parallel to the alignment was about 1.7 times of that measured with the incident light perpendicular to it.

#### 4.3. Surface Property of Renewable Carbon

^{2}/g for an individual carbon sheet). The high aspect ratio and surface-to-volume ratio of the modified carbon would significantly facilitate the preparation of graphene oxide (GO) carbon-based composites. Similarly, Nawaz et al. covalently functionalized GO carbon with octadecylamine (ODA) and added into thermoplastic polyurethane (TPU) as model additives [47]. The mechanical properties of the composites were characterized and showed no increase in stiffness or low-strain stress at loading levels below 2.5 vol%. However, above this threshold, the functionalized carbon formed a percolating network and both mechanical quantities increased as a power law with the increase of volume fraction. It was concluded that the formation of this network, other than interfacial stress transfer, would act as a jammed system for dominating the mechanical properties.

## 5. Percolation Models

#### 5.1. Directed Percolation and Semi-directed Percolation

_{c}. At P < P

_{c}, the system would return to the passive state; P = P

_{c}corresponding to a continuous phase transition; while at P > P

_{c}, the fraction of active states is beyond zero [52].

#### 5.2. Tunneling Percolation

^{2}>, were introduced. <S> stood for the orientational order parameter, and its angular brackets stood for the ensemble averaging of total fillers. 〈S〉 is 1 for the fully aligned fillers and is 0 for the isotropic orientations. The results from the simulations showed that, as the <S> decreased from 1 to 0, the percolation threshold decreased in the alignment direction. This behavior is attributed to the enhanced connectivity between fillers as the alignment diminished. <S

^{2}> is its second moment, which is expected to affect the performance of composites as well.

^{2}〉 can be independently altered. The distribution of azimuthal angle β is chosen between [0, 2π], which is independent from the distribution of θ. The normalization condition for p(θ) requires the following:

^{2}〉 can be expressed as follows:

_{c}at a given aspect ratio and volume fraction of the fillers. The mechanical strength of their samples exhibited a significant increase under S

_{c}[57].

#### 5.3. Dual Percolation

_{2}-g-PS). Their tensile test results confirmed the effect of filler addition on the strength toughening due to the enhanced interfacial stress-transfer efficiency. As shown in Figure 21a, an improved adhesion was obtained as the filler content exceeded the critical value of 0.65 vol%. The microstructure analysis of composites revealed the dual percolation dynamics which simultaneously occurred in both SiO

_{2}-g-PS agglomerates (dispersed phase) and the polypropylene host matrix. This was mainly attributed to the superposition of stress volumes around the single filler and inside of the dispersed agglomerates. Figure 21b illustrated the variation in the toughness of SiO

_{2}-g-PS/PP composites characterized by the area under the stress–strain curves as a function of fraction of stress volume, φ

_{s}(Equation (10)). The critical stress volume fraction φ

_{sc}, which is defined as the peak position of the first derivation of the area with respect to the stress volume fraction, was estimated as being 51%. The percolation behavior at the shear-yielded zones is beneficial for lowering the overall percolation threshold while maintaining or enhancing the performance.

_{c}stands for the critical matrix ligament thickness and V

_{f}stands for the particle volume fraction.

#### 5.4. Other Models

#### 5.4.1. Halpin–Tsai Model

_{c}, E

_{m}and E

_{f}refer to the modulus of composite, matrix and filler (carbon); and v

_{f}refers to the volume fraction of carbon filler. As for α, its value of 1/3 is generally used for the Young’s modulus of composites, in the case that the length of filler is greater than the thickness of specimen (fillers are assumed randomly oriented in two dimensions); however, the value of 1/6 or 1/5 is used if the filler length is much smaller than the thickness of specimen (fillers are assumed randomly oriented in three dimensions) [65]. For carbon doped polymeric composites, α of 1/6 is more widely employed, considering the much shorter length of carbon particles when comparing them with the thickness of specimens [66].

#### 5.4.2. Debonding Model

_{f}, then we get the following:

_{yc}and σ

_{ym}refer to the yield strengths of composite and polymer matrix, respectively, while a and b refer to the constants related to stress concentration, adhesion and geometry of particulates (fillers). As for the spherical particles with no adhesion to the matrix, which fail by the random fracture, Equation (14) becomes the following:

#### 5.4.3. Generalized Method of Cells (GMC) Model

#### 5.4.4. Other Micromechanics Models

_{r}, of carbon fillers in the RVE [81]:

_{bundles}is the bundles volume in the RVE, χ is the volume ratio of the bundles with respect to the total volume of RVE and ζ is the volume ratio of carbon tubes that are dispersed in bundles with respect to the total volume of carbon tubes. A higher χ means a more homogeneous distribution, and at χ = 1, the carbon fillers reach to a fully uniform dispersion. On the contrary, ζ = 1 represents the state in which all the carbon fillers are agglomerated.

_{0}/r

_{1})

^{m}refers to the volume fraction of fillers in the two-phase system, with r

_{0}and r

_{1}as the inner and outer radii of the interphase region, and m of 2 or 3 for fillers; the superscripts “eff”, “i” and “f” refer to the effective property, interphase and fiber/fillers, respectively. The ${\mathsf{\alpha}}_{\mathrm{C}}^{\mathrm{i}}$ relates to the Poisson ratio of the interphase and boundary conditions of the far-field loading. Feng et al. simplified the carbon nanotubes as straight cylindrical tubes, and the carbon particles together with their surrounding interphases were simulated as equivalent solid fillers, due to the electron hopping, thus turning the composite into a two-phase system for micromechanical modeling [80]. A good agreement between the modeling results and the experimental data was achieved for both single-wall and multi-wall carbon-nanotube-based nanocomposites. The simulation results indicated that the size of carbon nanotubes had significant effects on the percolation threshold.

## 6. Nanocellulose: Carbon-Based Reinforcing Filler

_{t}(g/g) refers to the swelling capacity at time t (s), K

_{is}(g/g) refers to the swelling rate constant and Q

_{∞}(g/g) refers to the equilibrium swelling capacity. Figure 26a presents the swelling kinetics for both BE film and co-blended film, and a significant decrease in the swelling capacity (Q

_{t}, P < 0.05) was confirmed (swelling rate constant K

_{is}decreased from 621 to 500 g/g; equilibrium swelling capacity Q

_{∞}, decreased from 454.55 to 434.78 g/g).

_{g}) of the matrix. In contrast, carbon nanotube presented with a negative “turning point” in its reinforcing behavior at the temperature below the T

_{g}of the matrix. The evaluation results of these two models confirmed their suitability in prediction of the mechanical properties over a wide range of testing conditions.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Chemical modification of CNT or graphene via four typical schemes [12].

**Figure 2.**SEM images of carbonized bamboo (

**a**), and bamboo-derived granular renewable carbon with magnification of ×100 (

**b**), ×1500 (

**c**) and ×5000 (

**d**) [14].

**Figure 3.**Mechanism for granular carbon in enriching the hydrogen-utilizing methanogens, Geobacter and other methanogens capable of direct interspecies electron transfer, to accelerate the methane generation and sludge consumption [16].

**Figure 4.**SEM images of a typical powdered renewable carbon (

**a**,

**b**) and transmission electron microscopy (TEM) images corresponding to its pore structure (

**c**,

**d**). Insets in (

**c**,

**d**) show the FFT pattern which confirms the amorphous structure of renewable carbon [17].

**Figure 5.**Rod-like CNT/graphene doped composites with anticipated enhanced properties [12].

**Figure 6.**Incorporation of CNT into PHBV: PLA blends and their corresponding properties [24].

**Figure 8.**Schematic of finite and infinite clusters with a percolation concept (

**a**), and series/parallel springs model for mechanical response of the percolated composite (

**b**) [29].

**Figure 9.**Three fundamental geometries of plates (

**a**–

**c**), and computer realization of one hundred circular plates in 3D space (

**d**) [34].

**Figure 10.**Scaled particle concentration c

_{p}= πDL

^{2}ρ

_{p}/4 at the percolation threshold as a function of the dimensionless field strength, βK. Where D refers to the diameter (particles were presumed mutually impenetrable rigid cylinders), L refers to the length and ρ

_{p}is percolation threshold). Solid lines— connectivity percolation; dashed curves—contact-volume approach for λ/D = 0.3 (top), 0.6 (middle) and 1 (bottom). The percolating network only existed in the enclosed areas. The shaded area is the region of coexisting isotropic (paranematic) and nematic phases. Inset (

**a**): calculated percolation thresholds scaled to the zero-field value c

^{0}

_{p}. Inset (

**b**): order parameter S

_{2}with, from steepest to flattest, λ/D = 0.3, 0.6 and 1. The dots indicate the largest value of |βK| that allows for a percolation threshold [38].

**Figure 11.**Alignment of MWCNT in epoxy nanocomposites (

**a**), and Raman spectra of aligned MWCNT/epoxy composites, showing different G-band peak intensities as the change of the direction of applied polarized light against the CNT alignment (

**b**) [41].

**Figure 12.**Quasi-static fracture surface morphologies of neat epoxy (

**a**), 0.3 wt% of randomly distributed MWCNT-nanocomposites (

**b**), 0.3 wt% of aligned MWCNT-nanocomposites (

**c**and

**d**), crack bridged by aligned single and bundle MWCNT (

**e**) and after MWCNT pull-out (

**f**) [41].

**Figure 13.**Schematic representation of nanocomposite cylinder with functionally graded (FG) distribution along with CNT coordinates (

**A**), and representative volume element (RVE) with Eshelby cluster model of agglomerated CNT (

**B**) [42].

**Figure 14.**Polymeric composite which consists of both continuous carbon nanotube (CNT) and chemically modified CNT or graphene carbon as matrix modifiers (

**A**), and loss modulus and electrical conductivity with respect to interfacial volume fraction in CNT or graphene carbon nanocomposites (

**B**) [12].

**Figure 15.**The preparation procedure of polyimide composite film with MWNT carbon (

**a**), and Young’s modulus of PI-MWNT composites films against the MWNT content (

**b**) [48].

**Figure 16.**SEM pattern of the cross-section of PI-MWNT (MWNT = 1.14 vol%) composite film (

**a**), and the enlargement of its selected area (

**b**) [48].

**Figure 17.**A semi-directed site percolation cluster on the square lattice, containing N = 17 sites (filled circles) on a strip of width n = 5, due to periodic boundary conditions the sites denoted by 1 and 1′, are identified. The strip is infinite along the horizontal axis (the preferred direction). In contrast to the case of fully directed percolation, vertical lines of the lattice are not directed, which means that the set of all semi-directed percolation clusters includes the set of all possible fully directed clusters [54].

**Figure 18.**Increased alignment of prolate ellipsoids at various nematic order parameter <S> of (

**a**) 0, (

**b**) 0.25, (

**c**) 0.5 and (

**d**) 0.9. The green fillers are percolating clusters, while red fillers are other isolated clusters. The m value is equal to m

_{min}(m≥m

_{min}=3〈S〉/(1−〈S〉)) [40].

**Figure 19.**Schematic diagram of dual percolation network constructed by the cooperation of EG and MWCNT in the matrix. Follow the direction of the arrows, MWCNT content inside of the PP/EG-MWCNT ternary composites is increasing [58].

**Figure 20.**Infrared camera images of samples that represent the effect of dual percolation [59].

**Figure 21.**Tensile strength of SiO

_{2}-g-PS/PP composites as a function of the volume fraction of SiO

_{2}(

**a**), and area under the tensile stress–strain curve of SiO

_{2}-g-PS/PP composites as a function of the stress volume fraction of φ

_{s}(

**b**) [60].

**Figure 22.**Interfacial debonding model [67].

**Figure 23.**Representative microscale volume element (RVE) containing straight carbon fillers [79].

**Figure 24.**Sketch of volume element (RVE) with ellipsoidal bundles of carbon fillers [79].

**Figure 26.**Swelling kinetics (Q

_{t}) and linear plotting (t/Q

_{t}) of BE film and co-blended film at different submerse time (t) (tests were conducted inside of distilled water at room temperature) (

**A**), gas diffusion dynamic as the relative humidity (RH) increases (

**B**), and effect of relative humidity (RH, %) on the gas permeability and gas selectivity of the BE film (Bottom Side, test was conducted at the temperature of 25 °C) (

**C**) [9].

**Figure 27.**Interaction between oxidized nanocellulose and pulp fibrils (

**a**), and tensile strength of composites made from unmodified nanocellulose-fibrils, modified nanocellulose-fibrils and PEI-fibrils (

**b**) [89].

Geometry | Circular Plates | Square Plates | Triangular Plates |
---|---|---|---|

η_{c} | 0.9614 | 0.8647 | 0.7295 |

Error | ± 0.0005 | ±0.0006 | ±0.0006 |

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## Share and Cite

**MDPI and ACS Style**

Sun, B.; Kong, F.; Zhang, M.; Wang, W.; KC, B.S.; Tjong, J.; Sain, M. Percolation Model for Renewable-Carbon Doped Functional Composites in Packaging Application: A Brief Review. *Coatings* **2020**, *10*, 193.
https://doi.org/10.3390/coatings10020193

**AMA Style**

Sun B, Kong F, Zhang M, Wang W, KC BS, Tjong J, Sain M. Percolation Model for Renewable-Carbon Doped Functional Composites in Packaging Application: A Brief Review. *Coatings*. 2020; 10(2):193.
https://doi.org/10.3390/coatings10020193

**Chicago/Turabian Style**

Sun, Bo, Fangong Kong, Min Zhang, Weijun Wang, Birat Singh KC, Jimi Tjong, and Mohini Sain. 2020. "Percolation Model for Renewable-Carbon Doped Functional Composites in Packaging Application: A Brief Review" *Coatings* 10, no. 2: 193.
https://doi.org/10.3390/coatings10020193