1. Introduction
Nanocomposites made of polymeric matrices reinforced with intercalated or exfoliated clay nanoparticles have become a focus of research in polymer composites in the last two decades after the seminal research initiated successfully by the Toyota research group in 1980s [
1,
2]. To date, substantial experimental investigations have shown that layered-silicate clay particles can be exfoliated into single nanosized platelets through ion exchange and can be well distributed in polymer melts or solutions as a nanoreinforcing phase in enhancing the mechanical and other physical properties of the resulting polymer nanoclay composites (PNCs). The most attractive evidence as demonstrated by numerous researchers worldwide is that layered silicate nanofillers can tremendously increase the tensile modulus, flexural stiffness and the tensile strength of PNCs at a filler weight content of only a few percent. Such preferable experimental observations are due mainly to the fact that the tensile modulus and strength of well aligned, exfoliated nanoclay platelets are close to the theoretical values of their perfect crystalline counterparts without defects (dislocations) and orders higher than those of the polymeric resins, as well as the high interfacial bonding strength (shear strength) between the clay nano platelets and the polymeric resins that exhaustively exploits the toughening effect of the clay nano platelets. For instance, nylon-6 nanocomposites prepared through intercalative ring-opening polymerization of ε-caprolacetam modified montmorillonite to form fully exfoliated nano platelets can double the tensile modulus of the virgin nylon matrix at the filler weight content at 4.1% [
3]; however, such giant modulus improvement cannot be achieved when the clay particles are dispersed in polymeric matrix to form intercalated polymer microcomposites [
4]. Meanwhile, fully exfoliated clay nanofillers at a weight content of ~5% can also enhance the tensile strength of the nanocomposites up to 50% as demonstrated in nylon nanocomposites that were prepared through in situ intercalative polymerization of ε-ecaprolactam in protonated aminododecanoic acid modified montmorillonite [
4]. Besides, increasing experimental data have indicated that PNCs are capable of having desirable thermal stability, flame retardancy, and gas barrier properties that substantially broaden the applications of conventional polymer composites [
4,
5,
6,
7,
8,
9,
10] in various industrial sectors. Among those, the most recent research investigated the effects of nanoclay morphologies (e.g., particle length, aspect ratio, bird nest structure, aggregation, etc. of halloysite and kaolinite nanoclay particles) on the thermal stability, surface wettability, and mechanical properties of biopolymer nanocomposite films for food packaging and other applications [
11,
12]. The studies showed that the thermal, tensile and surface wetting properties of the pectin resin can be noticeably improved by exfoliated halloysite nanotubes at relatively high weight concentration, which form unique bird nest structures in the pectin matrix of the resulting bionanocomposites. In addition, most recent research on the structural, physical and mechanical properties of nanoclay-reinforced fiber composites can be found in Reference [
13].
Along with the enrichment of the database of experimental data of PNCs, remarkable efforts has been devoted to model-based understanding and numerical simulation of the mechanical behavior of PNCs for controllable processing and nanocomposites design. One typical treatment broadly considered by researchers is to consider PNCs as conventional particle-reinforced composites while an effective particle volume (size) is assumed [
14]. Such treatment facilitates the utilization of classic micromechanics and numerical methods (e.g., finite element method-FEM) to analyze the mechanical behavior of PNCs [
15]. With the assumption that the nanofillers are evenly distributed in the matrix, the effective mechanical properties, mainly the effective moduli, of the resulting nanocomposites can be approximately modeled by adopting Eshelby’s equivalent model, self-consistent models of finite-length fillers, Mori-Tanaka type models, bound models, Halpin-Tsai model and its extensions, and modified shear-lag models, among others [
15]. By comparison with these classic micromechanics models plus detailed finite element analysis (FEA), Tuchker and Liang concluded that the simple shear-lag model is capable of giving a good estimate of the longitudinal modulus (
E11) of short-fiber reinforced composites when the fiber aspect ratio is greater than 10 [
15]. Meanwhile, Tsai and Sun [
16] developed a modified shear-lag model to investigate the load transfer efficiency in PNCs reinforced with intercalated and fully exfoliated clay particles, and their model was validated by using FEM. In addition, Weon and Sue [
17] developed an experimental scheme to tailor the clay orientation and aspect ratio in nylon-6 PNCs through controlled shearing in order to study their effects on the mechanical properties. It was found that the effective modulus and tensile strength of the PNCs decreased with decreasing aspect ratio and alignment extent of the reinforcing clay nano platelets; in contrast, the fracture toughness and ductility of the PNCs increase simultaneously. In the study, the Halpin-Tsai and Mori-Tanaka micromechanics models were employed successfully to examine the dependency of composite moduli upon the orientation and aspect ratio of the clay platelets; the effective filler structure was established through mapping the effective structural parameters of fillers to the structural parameters of conventional fillers within the framework of micromechanics [
18]. Similar studies were also conducted by many other researchers. For example, Zhu and Narh [
19] performed detailed FEA of the mechanical behavior of aligned clay platelet-reinforced nanocomposites. In their model, the clay particles and the polymeric matrix were all treated as linearly elastic solids and an interlayer was proposed to model the transition region between the polymeric matrix and the fillers. Numerical simulations based on such a three-phase model showed that either decreasing interlayer modulus or increasing interlayer thickness resulted in decreasing effective moduli of the PNCs. In addition, Dai et al. [
20] recently introduced an improved stagger model to analyze the stiffness and strength of PNCs reinforced with aligned, exfoliated clay nanoparticles; numerical results are close to those obtained in experimental studies. More recently, Dong and Bhattacharyya [
21] performed finite element modeling to determine the effective moduli of polypropylene/organoclay nanocomposites. In the model, representative volume elements (RVEs) of stacking, staggering, and randomly distributed platelets in matrices were considered; their numerical results showed that the interlayer properties had less impact on the effective moduli of the PNCs reinforced with fully exfoliated clay nanoparticles. More detailed reviews on research progress in processing, characterization and modeling of the mechanical properties of PNCs can be found in the recent review papers in this topic, e.g., [
4,
5,
6,
13,
21,
22] and references therein.
In the above, it can be observed that the majority of the modeling studies reported in the literature was focused on the effective moduli of PNCs by adopting the classic micromechanics models and related computational methods within the range of linearly elastic deformation of the composites. Yet, no systematic study has been conducted on the ultimate tensile strength of PNCs though it is an important material parameter in practical applications of PNCs. In view of processing PNCs, the key processing parameters include the clay particle volume (or weight) fraction, exfoliation extent and platelet orientation in polymeric matrices, which dominate the mechanical properties of the resulting PNCs. Thus, in this study, we initiated a computational scaling study on the effects of processing parameters (i.e., the volume fraction and aspect ratio of the clay particles) on the effective modulus, yield strength, and ultimate tensile strength of PNCs by using a nonlinear FEM model. In the model, the typical stack and stagger RVEs of PNCs were adopted, and the polymeric matrix was treated as idealized elastoplastic solid with isotropic hardening behavior. A family of effective stress-strain diagrams was gained at varying volume fraction and aspect ratio of the clay particles through detailed nonlinear FEA. Dependencies of the effective stiffness, yield strength and ultimate tensile strength upon the processing parameters of the PNCs and relevant mechanisms were explored and discussed. Conclusions of the present study and relevant applications were addressed in the last section of the paper.
3. Results and Discussion
In the present computational micromechanics scaling study, FEM-based nonlinear analysis was conducted to cover a wide range of clay-particle aspect ratios (
ρ = 1, 2, 5, 10, 20, 50 and 100) and volume fractions (
Vf = 0.5%, 1%, 2%, 5%, 7.5%, and 10%) for both the stack and stagger models. The effective moduli predicted by the two idealized clay-particle arrangements (i.e., stacking and staggering) are nearly the same, especially at the low volume fraction of dilute clay particles in the polymeric matrix. This is the trivial conclusion that can be easily drawn in linearly elastic micromechanics as the effective moduli of PNCs of aligned clay particles are independent of either stacking or staggering arrangement in the dilute clay particle state, and the small variation between the two models are possibly induced by the difference of localized plastic deformations and a small deviation of the clay particle interaction in the two micromechanics models. Compared to the effective moduli, the effective ultimate tensile strengths of the PNCs have a little noticeable variation between the stack and stagger models. However, the general varying trends of the effective moduli and ultimate tensile strengths based on the two computational micromechanics models are consistent and will be discussed further. Hereafter, only the results based on the stack model are shown though a comparative study will also be presented later on. In the case of the stack model, variation of the effective elastic modulus
Ee and effective ultimate tensile strength
σue with the varying clay-particle volume fraction
Vf at several aspect ratios are shown in
Figure 4 and
Figure 5, respectively. During the data reduction process, the effective moduli in
Figure 4 were extracted from the slope of the initial linear elastic region of the effective tensile stress-strain diagrams (
σe-ε
e) of all the computational cases as shown in
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10 and
Figure 11. These results indicate a very good linear relationship between the effective tensile stress
σe and the effective tensile strain ε
e as expected. From
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10 and
Figure 11, it can be observed that when the value of effective tensile strain ε
e is low, the PNCs behave as linearly elastic solids until the effective tensile strain ε
e is close to the yield strain of the polymeric matrix. The effective elastic moduli
Ee extracted from the present numerical experiments (see
Figure 4) are close to those available in the literature [
11,
12]. In addition, it can also be found from
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10 and
Figure 11 that the PNCs exhibit slightly softening behavior prior to the arrival of the yield strain of the polymeric matrix. This phenomenon can be attributed to the localized plastic deformation near the ends of clay particles where stress concentration exists due to the sharp edges/corners of the clay particles. In reality, at the nanoscale, the physical/chemical properties of the polymer chains close to the clay platelets are much more complicated than the present material model and are also noticeably different from the polymers in bulk state. Such unusual properties of the interfacial polymers at the nanoscale are beyond the scope of this computational study and are still under intensive investigation.
In addition, at the low clay-particle volume fraction (
Vf < 2%) or low aspect ratio (
ρ = 1, 2, 5, and 10) based on the present computational study, no obvious improvement of the effective ultimate tensile strength
σult is detected as shown in
Figure 5. This characteristic can be also examined from the
σe-ε
e diagrams given in
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10 and
Figure 11. Such an observation is also qualitatively in agreement with the recent experimental results of the mechanical properties of PNCs as reported in the literature [
4,
6]. In these cases, the values of effective ultimate tensile strength
σult are still dominated by the yield strength of the polymeric matrix; the contribution of the load-carrying capacity of clay particles to the nanocomposites is negligible. Such an observation can be understood as follows. Physically, different from the effective stiffness which is governed by the averaging effect of the mechanical properties of the constituents forming the PNCs, the dilute clay nanoparticles in the polymer matrix are unable to form load-transferring bridges and therefore unable to noticeably strengthen the resulting PNCs. Therefore, dilute clay nanoparticles at low volume concentration have less impact on the hardening of the polymeric matrix. However, the effective ultimate tensile strength
σult of the PNCs is significantly enhanced in the cases of
Vf = 5% and 10% and
ρ = 50 as shown in
Figure 10 and
Figure 11. In the case of
Vf = 5% and
ρ = 20 (
Figure 10), the improvement of the effective ultimate tensile strength
σult of the PNC reaches ~60% on the basis of the yield strength of the virgin polymeric matrix; this increment is close to those achieved experimentally in Nylon-6 and silicone rubber nanocomposites reinforced with montmorillonite clay particles [
4]. Moreover, in the case of
Vf = 10% and
ρ = 50, the present simulation predicts the increment of the effective ultimate tensile strength
σult of the PNCs up to ~150%. This value is much higher than the ones reported in the literature. In realistic PNCs, the polymer/clay interphase, interfacial shear failure (sliding/pull-out failure), filler waviness, and orientation will noticeably decrease this value. In the above two cases, the clay particles with an aspect ratio
ρ higher than 10 apparently increase the load-carrying capacity of the resulting PNCs.
Furthermore, the FEA results indicate that at the low particle aspect ratios (e.g.,
ρ = 1, 2, 5 and 10), the effective ultimate tensile strengths
σult of the PNCs based on the two models are nearly independent of the clay volume fraction
Vf up to 5%, i.e., the polymeric matrix governs the tensile strength of the PNCs. This observed phenomenon is due to the fact that the relatively dilute clay nanoparticles in polymeric matrix are unable to form load-transferring bridges at the low aspect ratios of the clay particles to strengthen the resulting PNCs. Besides, by examining the results obtained in the cases of the filler aspect ratio
ρ at 10, 20, 50 and 100 and the filler volume fraction
Vf at 0.5%, 1%, 2%, 5%, 10% and 15%, respectively, it can be found from
Figure 12,
Figure 13,
Figure 14 and
Figure 15 that when the effective tensile strain ε
e is low, the PNCs behave as linearly elastic solids until the effective strain ε
e close to the yield strain of the polymeric matrix and the stack and stagger models both predict very similar effective tensile moduli. In addition, the PNCs exert slightly softening behavior prior to the arrival of the yield strain of the polymeric matrix due to stress-concentration induced matrix (resin) yielding near the filler ends. In addition, the percent increase of the effective modulus
Ee for both the stack and stagger models as shown in
Figure 16 indicates that for all the aspect ratios, the computational predictions based on the stack and stagger models are close, especially at the dilute clay particle state, as confirmed by linearly elastic micromechanics. Yet, a small variation of the effective tensile modulus between the stack and stagger models can be detected, which is due to the variation of clay particle interference between the two models. Moreover,
Figure 17 shows the comparative effective ultimate tensile strength
σult of PNCs predicted by the stack and stagger models. It can be observed that the trend of ultimate tensile strength
σult increase is the same for the two models; however, the deviation of the predicted effective tensile strength increases with increasing aspect ratio
ρ for the two models, and the stagger model predicts relatively lower values of the effective tensile strength than the stack model does for all the computational cases under the present consideration. By comparing the clay-particle arrangements in the two models, it can be concluded that the clay nanoparticles in stagger model are arranged more uniform than those in the stack model at a given filler volume fraction. Thus, subjected to the same effective tensile strain, the stack model generated more uneven stress field, or a higher stress gradient, than the stagger model, i.e., a higher ultimate tensile strength by the stack model. Such an observation also indicates that the ultimate tensile strength is more sensitive to the clay nanoparticle arrangement in the model, which can be useful to process strength-controllable PNCs.
It needs to be mentioned that in reality, the aspect ratio of fully exfoliated clay platelets is much larger than the values studied in this work. Yet, not well aligned clay platelets in polymeric matrices will substantially weaken the potential improvement of the effective tensile stiffness Ee and ultimate tensile strength σeu of the realistic PNCs in a specific orientation. This implies the tailorability of the effective tensile moduli and ultimate tensile strength of PNCs via optimal and controlled clay-particle exfoliation and orientation. To date, efficient processing techniques are still desired to effectively achieve controlled platelet orientation in polymeric matrix. Though the present study was based on the simple stack and stagger models, the computational results provide the insight to understand the scaling mechanical properties of PNCs. Such computational models can be further enhanced to involve other processing parameters such as platelet orientation, waviness and extent of intercalated clay particles for predicting the mechanical behavior of PNCs in controlled fabrication.