# A Generalized Approach for Evaluating the Mechanical Properties of Polymer Nanocomposites Reinforced with Spherical Fillers

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## Abstract

**:**

## 1. Introduction

_{2}, Al

_{2}O

_{3}, CaCO

_{3}, carbon black and layered silicates), has been extensively analyzed in the literature based on two-phase models (fillers + polymer matrixes) [17]. Classical models as well as several empirical or semi-empirical equations have been developed to estimate their tensile modulus. For example, based on the consideration that the tensile modulus of the composites under low shear stress would behave similarly to the viscosity of a fluid, the classical Einstein’s equation [18] developed to describe the viscosity increase due to spherical particles in a dispersion was adapted in 1944 by Smallwood [19] to the field of filled elastomers assuming perfect adhesion between fillers and polymer matrixes. Further, the interactions between particle pairs were incorporated by Guth [20] providing the formula known as the Guth-Smallwood-Einstein equation written as:

_{c}and E

_{m}respectively and ϕ is the particle volume fraction.

_{f}− E

_{m})/(E

_{f}+ E

_{m}), E

_{f}and E

_{m}are the tensile moduli of the filler and polymer matrix, respectively.

_{c}(ϕ), written as:

_{c}, E

_{m}, E

_{i}and E

_{f}respectively where the k-parameter is k = E

_{i}/E

_{m}. This parameter takes values between the minimum case, E

_{i}= E

_{m}, i.e., k = 1, meaning no interphase contribution (r = 0, signifying that the volume fraction of the fillers is much greater than that of the interface region) where Equation (4) reduces to the classical Takayanagi two-phase model, and possible maximum values when E

_{i}= E

_{f}, meaning that 1 < k < E

_{f}/E

_{m}.

## 2. Generalized Approach

_{p}/V, where N is the number of particles, the single particle volume is V

_{p}and V is the total system volume. The consideration of the polydispersity effect is an issue, which can be added as a next step into our general approach, but in order to simplify we consider this aspect beyond the scope of this paper. The nanoparticle–interphase regions will be assumed as core–shell assemblies embedded in an infinite polymer matrix. All interfaces between particles and the surrounding matrix will be assumed to be perfectly bonded, thereby removing additional complexities. We will also consider the particles’ interactions as those of hard-spheres, meaning that they cannot interpenetrate.

#### 2.1. Effective Particles Contributing to the Mechanical Reinforcement

_{p}, defined as the percolation volume fraction. This threshold will depend on several variables, such as the sizes, shapes and orientations of particles, with particle interface sizes being a fundamental parameter that cannot be overlooked. Its clarification will be fundamental for the precise understanding of the mechanical reinforcement. On the other hand, it should not be forgotten that as one increases the particle concentration, the system exhibits a dramatic increase in viscosity where upon crossing a critical volume fraction ϕ

_{g}, the particle movements are slow enough that it can be considered essentially frozen, leading to a glass transformation (colloidal glass transition), which was discussed already in 1980’s [55,56,57]. This effect must also be taken into account, especially for a precise characterization of the mechanical properties of the composite.

_{g}≈ 0.65, which is intrinsically correlated with the maximum density of occupancy of the spherical particles [58]. Above this volume fraction, fillers will not diffuse through the sample anymore and, hence, percolation will no longer be possible. This will imply that only a portion of particles ϕ

_{eff}= Aϕ

_{g}will effectively contribute to the formation of the percolating network within a restricted particle volume fraction domain (ϕ

_{p}< ϕ ≤ ϕ

_{g}≈ 0.65), as is illustrated in Figure 1.

_{p},the effective particles ϕ

_{eff}go to 0; (ii) when ϕ = ϕ

_{g}, ϕ

_{eff}will reach the maximum value at the glassy phase ϕ

_{g}and (iii) based on considerations pointed out by Ouali [59], the ratio ϕ

_{eff}/ϕ

_{g}can be described by a power law dependence, i.e., A(ϕ − ϕ

_{p})

^{α}where α defines the percolation exponent. Based on that, the effective amount of particles contributing to the mechanical reinforcement can be found by the following equation (see Supplementary Materials):

_{p}. If a glassy phase is immediately formed after the particles percolate, a step function will describe the process (α = 0), corresponding to a hypothetical extreme situation of composites formed with many particle interactions, where all the particles immediately become trapped.

#### 2.2. Percolation Threshold

_{g}to start the mechanical reinforcement, while composites with particles having smaller interphases in comparison with particle radius (r << R) will require a greater number of particles to interconnect with each other, leading to higher percolation volume fractions. The black point of Figure 2a illustrates the last situation, when particle thickness is 10% of particle radius, and the percolation threshold will be 0.26 where the systems will need a considerable amount of particles (40% of ϕ

_{g}) to start mechanical reinforcement. If the system percolates at the limiting case ϕ

_{p}= ϕ

_{g}, a hypothetical and totally unfeasible case will take places at a lower bound of r/R = 0.0151 (Figure 2b). The percolation threshold can never be greater than the glass particle volume fraction (0 ≤ ϕ

_{p}≤ 0.65) meaning that the interphase thickness cannot be smaller than 1.5% of particle radius, for example in composites formed with particles on the micrometers scale (e.g., R = 1000 nm) the thickness will never be smaller than 15 nm. The mentioned situations are extreme cases, but how consistent such boundary predictions are in comparison with real situations must be understood.

_{p}= 0.08, perfectly consistent with the theoretical prediction. On the other hands, for spherocylindrical surfaces of carbon particle (diameters on the order of a nanometer), quantum mechanical treatment gives rise to a representative value of r/R = 0.2 [60], which means ϕ

_{p}= 0.14 which is also perfectly reliable, and this implies that Equation (6) can be a valuable and consistent approach to compute percolation thresholds in polymer composite systems formed with spherical particles and having glassy interphase regions.

#### 2.3. Critical Percolation Exponent

_{eff}to the percentage change in ϕ − ϕ

_{p}of a composite or, equivalently, as the slope of an ln(ϕ

_{eff}) vs. ln(ϕ − ϕ

_{p}) plot (numerical example is plotted in Supporting Information). From the physical point of view, Equation (7) can also be understood as a Grüneisen parameter [68], which for molecular glasses is written in terms of their index of activation energy, which extremely valuable to elucidate the nonlinear thermal behavior of the glass transformation process [69,70,71]. Based on the aforementioned arguments, we can introduce a new interpretation of the percolation exponent as a measure of the aggregation dynamics of the particles, intrinsically related to the speed of the glassy phase formation. This will provide information about how fast or slow the vitreous phase can form, intrinsically correlated with the degree of strength of the interactions of the particles and their coupling/aggregations within the polymer matrix. If the exponent were universal, it would imply that regardless of the nature of the polymer matrix and the type of particles, the particles will always become trapped in the same manner, following a universal pattern curve (Equation (5) with a constant exponent). However, as pointed out in the Introduction, this process depends on several interconnected parameters, such as particle interface, type of particles, particle physical properties (charge values and their sign) and strength of the filler-filler and filler-matrix interactions, all of which will not necessarily take place in the same manner. For the aforementioned reasons, we will not consider the percolation exponent as universal, but as a fitting variable intrinsically coupled with the rest of the parameters.

#### 2.4. Tensile Modulus

_{c}, E

_{m}, E

_{i}and E

_{f}, respectively (Supporting Information Section S4). The modulus of the interfacial region will also be assumed, as in case of the X. Ling Ji et al. approach [1], by a linear gradient change in modulus between the polymer matrix and the surface of the particle, and is quantitatively described in terms of a k-parameter defined as (k = E

_{i/}E

_{m}).

_{g}≈ 0.65, the radius of the particle will be (R) and the tensile modulus of the polymer matrix E

_{m}and the percolation threshold ϕ

_{p}will be determined from Equation (6). On the other hand, the thickness of the interphase (r), the rate of the interphase modulus k-parameter, the tensile modulus of the particles E

_{f}and the percolation exponent α will be considered as fitting model parameters.

_{c}> E

_{m}) will take place only above the percolation threshold (ϕ

_{p}< ϕ ≤ ϕ

_{g}), manifested by a step-wise change behavior of E

_{c}. Below the mentioned threshold (0 ≤ ϕ < ϕ

_{p}), the tensile modulus of the composites will be the same as that of the polymer matrix E

_{c}= E

_{m}(δ = γ = 0). This trend is numerically visualized in Figure 3 by modelling the hardness (left) and size (right) effects of the particle interphase in the mechanical reinforcement for hypothetical composites. The lines correspond to the plot of Equation (8) with Equation (6). The left figure visualizes the cases of different composites formed with particles of radius R = 50 nm and thickness r = 30 nm. When k changes from 1 to 5, the tensile modulus of the composite E

_{c}will gradually increase, and, especially for k > 3, the slope of the curve (dlogE

_{c}/d

_{ϕ}) will change drastically giving rise to a higher mechanical reinforcement. The right part of Figure 3 shows the case of two different composites formed by the addition of particles with the same radius and different thickness interphases into the same polymer matrix. From this modeled situation, we can clearly see that composites formed with particles with large interfacial thicknesses (brown line r = 70 nm) will need fewer particles in comparison to the particles with smaller thicknesses to initiate mechanical reinforcement and, therefore, will percolate at a lower particle volume fraction. This will give rise to an increase in modulus for the resulting composite, as compared to the smaller thickness case (blue line), which shows that when the particle size is in the nanoscale range, the interfacial region greatly affects E

_{c}. Undoubtedly, the four fundamental discussed effects (particles and interphase size, tensile modulus, ratio of the interface, and percolation exponent) will have a dominant influence on E

_{c}and validation of these parameters with experimental data is the ultimate goal.

## 3. Model Validation and Discussion

_{c}(ϕ). The data correspond to six nanocomposites having unique properties and specific performances for technological applications (see details in data information). They are extracted from dynamic mechanical thermal analysis (DMTA) experiment at temperatures above the glass transition temperature T

_{g}, where a higher modulus is experimentally observed when increasing the filler content. A common trend presented in these types of polymer composites is observed. At lower filler content, the modulus of the composites is only slightly higher than that of the unfilled material. However, a higher modulus is experimentally reached with increasing filler content. In order to explain this, we have fitted the experimental data with five model equations which are plotted in Figure 4. The fitting curves in Figure 4 correspond to: (1) our general approach (Equation (8), solid blue line), (2) the X. Ling Ji et al. model (Equation (4), brown dashed line), (3) the Guth-Smallwood-Einstein equation (Equation (1), pink dashed line), (4) the Kerner equation (Equation (2), green dashed line) and (5) the Halpin and Tsai equation (Equation (3), black dashed line). Table 1 summarizes the parameters (e.g., interphase size, r, modulus ratio of the interface, k, percolation exponent, α, and modulus of the fillers, E

_{f}) corresponding to the fitting of Equation (8) and Equation (4) respectively.

_{c}(ϕ) is reached at a higher volume fraction of fillers, indicating a clear inconsistency of the equations. This implies that, in addition to hydrodynamic reinforcement, both filler−filler interaction and polymer−filler interaction, and the particle interfacial effect, will contribute to the improvement of mechanical properties of the composites and should be considered.

_{c}is manifested above the percolation threshold (ϕ

_{p}< ϕ ≤ ϕ

_{g}), which is true for all six composites.

^{−1}tendency, meaning that composites formed of particles with large thickness interfaces in comparison with particle radius will need fewer particles to initiate mechanical reinforcement and therefore percolate at a lower volume particle fraction. Fitting the data of Figure 4f corresponds to a special case where composites were intentionally formed with a special polymer matrix where, with only 1 wt.% of particle content, a considerable enhancement of the mechanical reinforcement was achieved, while other composites need 10% particle content to considerably increase their mechanical properties. On the other hand, we can also see from Table 1 that the addition of particles with different physical properties in the same polymeric matrix will yield different percolation threshold values. Data in Figure 4a,b lead to ϕ

_{p}= 0.0117 and 0.0153 for carbon black and fumed silicate, respectively, with a discrepancy of 31%, while data in Figure 4c,d lead to ϕ

_{p}= 0.0127 and 0.0141 for Al

_{2}O

_{3}and SiO

_{2}, respectively, having a discrepancy of 11%. Since the siloxane and silanol groups on the surface of the silica particles are hydrophilic in nature, attractive filler−filler interactions are strong due to the hydrogen bonds between silica particles. Thus, silica particles often form larger agglomerates that will lead to inhomogeneous filler distributions, making the dispersion of silica particles more difficult than the dispersion of other particles, such as carbon black and Al

_{2}O

_{3}. This implies that composites formed with silicate particles will have a tendency to reach higher percolation threshold values. The lower compatibility of spherical Al

_{2}O

_{3}and SiO

_{2}particles in PEEK could be the reason for the lower discrepancy of the percolation threshold. On the other hand, the dispersion of silica particles in polyolefins offers more resistance than the case of carbon black particles, leading to a higher percolation threshold difference. These data show that the formation of some percolating filler structures will affect the modulus of the composites and their effect should not be omitted.

_{g}will contribute. This will imply that the area of the total surface formed as a result of adding the individual interface thicknesses of each particle in the effective group of particles will be smaller than the area of the group formed by the total amount of particles. An ineffective extra interface thickness is hidden behind the k-values obtained from Equation (4), which gives rise to higher values of the k-parameter.

_{2}O

_{3}and SiO

_{2}particles in PEEK could be the reason why both composites have the same power law exponent of 0.74, irrespective of other differences [12]. Conversely, for the case of data in Figure 4a,b, the power law exponents are different, with the smallest values being the case of fume silicate. This could be justified due to the strong hydrogen bond interactions between silica particles, which lead to a faster interphase formation, resulting in a lower percolation exponent. In regard to the values of the tensile modulus of the fillers E

_{f}, we can also see in Table 1 that for each model equation, the obtained values are different, and the reason is because X. Ling Ji et al.’s approach (Equation (4)) considers an ineffective excess of material which does not contribute to the mechanical reinforcement below the percolation threshold (0 ≤ ϕ < ϕ

_{p}) and above the maximum density of occupancy of the spherical particles (ϕ

_{g}< ϕ ≤ 1). The obtained values are also considerably higher than those of the matrix E

_{m}, although variations of E

_{f}/E

_{m}will have only slight effects on the modulus of the composite E

_{c}.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Percolation threshold: (

**a**) numerical illustration of Equation (6) as the function of thickness and radius size effect. When the thickness of the particles is 1.5% of their radius, a hypothetical and extremely R-bound case will take place ϕp = ϕg. (

**b**) A modeled situation for different composites formed with particles of R = 50 nm having different thicknesses, ranging from 5 nm (black point) to 100 nm (blue point) where the dotted line is the plot of Equation (6) in the entire r/R domain. The inset part of the right figure displays the corresponding values of the effective number of particles determined from Equation (5) where the power law exponent is assumed to be 0.4.

**Figure 3.**Numerical evaluation of the (

**a**) hardness and (

**b**) size effects of the interphase in the mechanical reinforcement for hypothetical composites. The lines are the plot of Equation (8) with Equation (5). The percolation exponent for both figures is assumed as 0.4, k parameters in (

**b**) as 1.5, the tensile modulus of the filler and the matrix as E

_{f}= 1 × 10

^{11}Pa and E

_{m}= 2.4 MPa respectively.

**Figure 4.**Comparison between experimentally obtained (red squares) Young’s modulus vs theoretical prediction using different approaches: our generalized approach (Equation (8), solid blue line), X. Ling Ji model (Equation (4), brown dashed line), Guth-Smallwood-Einstein equation (Equation (1), pink dashed line), Kerner equation (Equation (2), green dashed line) and Halpin and Tsai equation (Equation (3), black dashed line) for (

**a**) polyolefin/carbon black (

**b**) polyolefin/fumed silica (

**c**) PEEK/Al

_{2}O

_{3}(

**d**) PEEK/SiO

_{2}(

**e**) PTMHMTA/TiO

_{2}and (

**f**) P(MMA-MTC)/SiO

_{2}.

Our Model | X. Ling Ji et al. Model [1] | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

No. | Composite [ref.] | R (nm) | R (nm) | E_{m} (GPa) | K = E_{i}/E_{m} | E_{f} (GPa) | $\mathit{\alpha}$ | ${\mathit{\varphi}}_{\mathit{p}}$ | R (nm) | K = E_{i}/E_{m} | E_{f} (GPa) |

1 | Polyolefin ^{1}/CB [31] | 50 | 51 | 2.4 × 10^{−3} | 1.66 | 364 | 0.75 | 0.0117 | 59 | 2.76 | 579 |

2 | Polyolefin ^{1}/fumed silica [31] | 7.5 | 8 | 2.4 × 10^{−3} | 1.43 | 4.3 | 0.63 | 0.0153 | 8 | 4.09 | 4.1 |

3 | PEEK ^{2}/Al_{2}O_{3} [72] | 15 | 15 | 3.9 | 4.37 | 19.2 | 0.74 | $0.0127$ | 14 | 7.22 | 15.8 |

4 | PEEK ^{2}/SiO_{2} [72] | 15 | 17 | 3.9 | 4.76 | 16 | 0.72 | 0.0141 | 15 | 7.72 | 17 |

5 | PTMHMTA ^{3}/TiO_{2} [73] | 4.5 | 4 | 1.82 | 2.04 | 9 | 0.64 | 0.0212 | 4 | 2.22 | 24.9 |

6 | P(MMA-MTC) ^{4}/SiO_{2} [74] | 10 | 23 | 1.91 | 2.35 | 428 | 0.72 | 0.0036 | 21 | 5.59 | 271 |

^{1}: Carboxy-telechelic polyolefin prepolymers.

^{2}: poly(ether ether ketone).

^{3}: poly(trimethyl hexamethylene terephthalamide).

^{4}: methyl methacrylate copolymerized with 2-(methacryloyloxy)ethyl trimethyl ammonium chloride comonomer.

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## Share and Cite

**MDPI and ACS Style**

Martinez-Garcia, J.C.; Serraïma-Ferrer, A.; Lopeandía-Fernández, A.; Lattuada, M.; Sapkota, J.; Rodríguez-Viejo, J. A Generalized Approach for Evaluating the Mechanical Properties of Polymer Nanocomposites Reinforced with Spherical Fillers. *Nanomaterials* **2021**, *11*, 830.
https://doi.org/10.3390/nano11040830

**AMA Style**

Martinez-Garcia JC, Serraïma-Ferrer A, Lopeandía-Fernández A, Lattuada M, Sapkota J, Rodríguez-Viejo J. A Generalized Approach for Evaluating the Mechanical Properties of Polymer Nanocomposites Reinforced with Spherical Fillers. *Nanomaterials*. 2021; 11(4):830.
https://doi.org/10.3390/nano11040830

**Chicago/Turabian Style**

Martinez-Garcia, Julio Cesar, Alexandre Serraïma-Ferrer, Aitor Lopeandía-Fernández, Marco Lattuada, Janak Sapkota, and Javier Rodríguez-Viejo. 2021. "A Generalized Approach for Evaluating the Mechanical Properties of Polymer Nanocomposites Reinforced with Spherical Fillers" *Nanomaterials* 11, no. 4: 830.
https://doi.org/10.3390/nano11040830