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

In this work, the effective mechanical reinforcement of polymeric nanocomposites containing spherical particle fillers is predicted based on a generalized analytical three-phase-series-parallel model, considering the concepts of percolation and the interfacial glassy region. While the concept of percolation is solely taken as a contribution of the filler-network, we herein show that the glassy interphase between filler and matrix, which is often in the nanometers range, is also to be considered while interpreting enhanced mechanical properties of particulate filled polymeric nanocomposites. To demonstrate the relevance of the proposed generalized equation, we have fitted several experimental results which show a good agreement with theoretical predictions. Thus, the approach presented here can be valuable to elucidate new possible conceptual routes for the creation of new materials with fundamental technological applications and can open a new research avenue for future studies.

technologies. The resulting nanocomposites with only 4 wt.% of inclusions of carbon black or fumed silica nanoparticles exhibit 40-50% Young's modulus improvement. [1] Data of Fig 4 (c,b) in manuscript are taken from Ref. [2], corresponds to poly(ether-ether-ketone) (PEEK) composites reinforced by nanosized SiO2 and Al2O3 fillers (PEEK/Al2O3 and PEEK/SiO2). The inclusion of much cheaper (in comparison with carbon nanotubes CNT) nano SiO2 or Al2O3 particles (with diameters ∼15-30 nm) into PEEK is of basic interest for the purposes of processability and mechanical enhancement. The resulting nanocomposites with 10 wt.% SiO2 or Al2O3 nanoparticles exhibit Young's modulus improvement of 30%. [2] Data of fig 4(e) corresponds to polyamide-titania nanocomposites (PTMHMTA/TiO2), taken from Ref. [3]. Polyamides are the first engineering thermoplastic polymers ever commercially produced. These polymers have a lot of applications as fibers, amorphous and crystalline plastics, and adhesives. On the other hand, titania has high melting point, resistance to attack by acids and alkalis and good mechanical properties to reinforce the polyamide matrix. At it is clearly showed when the TiO2 content is 10 wt%, the Young's modulus of PTMHMTA/TiO2 increases by 36% compared to corresponding of PTMHMTA. [3] Data of fig 4(f) corresponds to the composite (P(MMA-co-MTC)/SiO2) formed with SiO2 nanoparticles without surface modification, taken from Ref. [4]. The polymer matrix was intentionally assembled by a promising method to improve the particle dispersion as well as their interfacial adhesion through electrostatic interaction without surface modification. The inclusion of 10 wt% of cationic functional comonomer 2-(methacryloyloxy) ethyltrimethylammonium chloride (MTC) into polymer matrix methyl methacrylate (PMMA) optimize considerable their mechanical reinforcement. As is clearly showed at the figure 4(f) when the MTC content is 10 wt% and SiO2 content is only 1 wt%, the Young's modulus of P(MMA-co-MTC)/SiO2 increases by 35% compared to corresponding P(MMA-co-MTC) which represent an optimal experimental route to produce materials having less costly with higher application. [4] S.2 Derivation of eq.5 On the basis of percolation concepts as discussed in main text, the ratio eff ϕ g ⁄ can be estimated by the following relationship: when → the effective fraction of particles eff → leads to: Substituting s2.2 into s2.1: Figure S2. Schematic illustration of the spherical transformation. The diameter of the hard core, the length of the cylinder and the contact shell are denoted by D, L and respectively. The spherical case is recovered when → 0 which gives rises = ( ⁄ ) being r the particle thickness interface and R the radius of the particle.

S.3 Derivation of eq.6b
The theoretical approach developed by Schilling et al [5] gives rise to the following general equation for the percolation threshold: defines the aspect ratio of the spherocylindrical particles and the connectivity variable is defined as figure S3). results in:

S.4 Derivation of eq.7 and numerical example
Applying Napierian logarithm in both sides of eqs2.3 yields: For each composite we will only have a single exponent value, i.e. the percolation exponent will not depend of the particle concentration ( ⁄ ) → 0 and consequently: S.5 Derivation of eq.8 The eq.8 of the manuscript has been derived mainly based on the X. Ling Ji et al approach [6] however, here, two important improvements have been introduced: (1) Correction to the calculation of the Young modulus of the particles interphase and (2) Introduction of the percolation effect.

Calculation of and :
The moduli of the interface in regions B (block2+block3) and C (block 1) are in quite different forms from each other (shown in Fig. S6). The tensile modulus will be that of block 1 ( 1 ) but the tensile modulus will be calculated as the result of the parallel arrangement between blocks 2 ( 3 ) and block3 ( 3 ) which gives rises: For the calculation of the tensile modulus of each block the following considerations will be taking into account: • The interface region C and region corresponding to block2 will be analysed as parallel and series arrangement of infinite numbers of volume units.
• In the block1 and block 2, a linear gradient distribution of the modulus along the normal direction of the surface will be assumed and will take the form of the function with a linear gradient decreasing along the normal direction of the surface of dispersed phase. ) will be assumed and will take the form of the function with a linear gradient decrease along ⃗⃗⃗ .
• The modulus of the interface at the surface of dispersed rigid phase will be assumed as Here it is important to remark that eq s5.19 has a corrective term, which corrects eq.4 of the paper.
Because the dispersed phase is in spherical form and hence random orientations can be considered deriving the following relationship: