Dependence of Z Parameter for Tensile Strength of MultiLayered Interphase in Polymer Nanocomposites to Material and Interphase Properties
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Abstract
In this work, the Z interphase parameter which determines the tensile strength of interphase layers in polymer nanocomposites is presented as a function of various material and interphase properties. In this regard, the simple Pukanszky model for tensile strength of polymer nanocomposites is applied and the dependency of Z to different characteristics of constituents and interphase are illustrated by contour plots. The interphase strength (σ _{i}) and B interfacial parameter in Pukanszky model show direct links with Z parameter. Also, it is found that the volume fractions of nanoparticles and interphase reveal dissimilar effects on Z. A high Z is obtained by a low nanoparticle volume fraction and high content of interphase, but the best values of Z are associated with the level of B parameter.
Keywords
Polymer nanocomposites Interphase layer Material properties Tensile strengthAbbreviations
 φ_{f}
Volume fraction of filler
 A_{c}
Specific surface area of filler
 B
Interfacial adhesion parameter
 d
Thickness of nanofiller
 k^{th}
Number of interphase layer
 r
Filler radius
 t
Total thickness of interphase
 x
Distance
 Z
Interphase parameter
 ρ_{f}
Density of filler
 σ_{c}
Tensile strength of composite
 σ_{i}
Strength of interphase
 σ_{m}
Tensile strength of matrix
 σ_{p}
Tensile strength of filler
 σ_{R}
Relative strength
Background
The exceptional improvements of mechanical properties at low nanofiller contents have introduced significant interest in the use of nanoparticles in polymer matrices [1, 2, 3, 4, 5, 6]. The different types of nanofillers have been used to strengthen and toughen the polymers. The most interesting aspect related to nanofiller is that they can improve the properties of polymers at very low filler concentrations compared to microparticles and fibers [7, 8, 9]. This phenomenon can be described as nanoeffect which is the interactions at the atomic scale. When the particle size decreases to the nanoscale, the specific surface area rapidly increases, making the surface properties as the dominant factors and providing unique characteristics with widespread applications in many industrial parts. Additionally, when the filler size is similar to that of polymer chains, molecular interactions between nanoparticles and polymer matrix produce a third phase as interphase which has different properties from both polymer and nanoparticles [10, 11]. The properties of interphase play a main role in the level of dissipated energy by different damaging mechanisms which take place at the nanoscale [12, 13]. As a result, the mechanical properties of the nanocomposites significantly depend on the interphase level.
Many researchers have tried to characterize the interphase properties by modeling of the general properties of nanocomposites because the interphase is affected by many factors, and it cannot be characterized by simple techniques [14, 15]. The theoretical surveys in the recent years provided a large amount of information about interphase and interfacial interactions in polymer nanocomposites.
Several researchers have considered a multilayered interphase in polymer nanocomposites. They assumed that each layer in interphase has different properties from others. The characteristics of interphase layers were hypothetically studied and their influences on the nanocomposite behavior were discussed in many papers [16, 17, 18]. In one study, the thickness of interphase was assumed as a characteristic length scale and the main effects of interphase on stiffness and yield stress of polymer nanocomposites were evaluated [19]. The theoretical results showed a good agreement with the experimental data for polymer/SiO_{2} nanocomposites in that study.
In our previous study [20], it was found that the tensile strength of interphase layers changes by a power function of the distance between nanoparticles and polymer matrix. It was also shown that the calculations of this equation depend on Z parameter which shows the interphase properties. Additionally, it was discussed that the extent of Z determines the level of mechanical properties in the polymer nanocomposites. In this paper, the Z interphase parameter is defined by the material and interphase properties in polymer nanocomposites. The Pukanszky model and many useful equations are applied which only need to tensile strength of polymer nanocomposites and the properties of nanocomposite components. The dependency of Z to different material and interphase characteristics are illustrated by contour plots based on the resultant equations. The obtained results for different types of polymer nanocomposites are also explained by practical views in this area.
Methods
Results and Discussion
The effects of material and interphase properties on the Z parameter are explained according to the proposed equations by contour plots which illustrate the Z as functions of different variables.
The different effect of a very high σ _{i} and a very low σ _{p} on Z is not correct, because σ _{i} > σ _{p} cannot be practically occurred in polymer nanocomposites. The calculated Z at σ _{m} = 80 MPa (Fig. 2b) also show similar values to those of σ _{m} = 40 MPa. Accordingly, a high Z is obtained by high interphase strength (σ _{i}) and low level of σ _{p} at different strength of matrix demonstrating the different influences of σ _{i} and σ _{p} on Z parameter.
At σ _{p} = 300 MPa (Fig. 3b), it is found that both the highest B value and the lowest A _{ c } level cause the best value of Z. Also, B < 4 or A _{ c } > 120 m^{2}/g suggests the smallest Z value at all levels of another parameter. In this condition, an increase in A _{ c } and a decrease in B reduce the value of Z. It is obvious that the A _{ c } plays a negative role in the value of Z at different σ _{p}, but the high levels of B parameter show various Z attributed to the value of σ _{p}. Therefore, the value of σ _{p} plays a critical role in the final level of Z at different A _{ c } and B extents. The Z parameter shows a direct link with B parameter which expresses the level of interfacial adhesion in nanocomposite. Additionally, the high levels of Z and B significantly increase the level of σ _{R} in polymer nanocomposites (see Eqs. 4 and 6). So, the expression of Z parameter as an interphase parameter is true.
The effects of r and t on Z parameter are also plotted in Fig. 4b when σ _{p} = 360 MPa such as for TiO_{2}. In this condition, the negative effect of bigger nanoparticles on Z parameter is also illustrated similar to the previous situation, but the t plays different role compared to the former condition. In this state, the best levels of Z are achieved by r = 10 nm and t = 5 nm, r = 20 nm and t = 10 nm, and the bigger nanoparticles at medium level of t (14 < t < 21 nm).
These observations indicate that the size of nanoparticles and interphase differently affect the Z parameter based on the value of σ _{p}. In other words, a high σ _{p} dictates different roles for nanoparticle size and interphase thickness in the level of Z parameter. Accordingly, an optimization should be performed in this case based on the type of nanofiller used in nanocomposite.
At B = 8 (Fig. 5b), the same roles of φ _{f} and φ _{i} in Z value are also shown. However, a high φ _{i} at the lowest level of φ _{f} causes the best Z, while the former condition (B = 5) exhibits the best values of Z at high φ _{i} and medium values of φ _{f}. This occurrence gives the different effects of φ _{i} and φ _{f} on Z parameter attributed to B parameter. Conclusively, the value of B plays a main role in the calculated results of Z by the suggested equations which should be considered in experiments.
Conclusions
The Z interphase parameter for the tensile strength of interphase layers was expressed by material and interphase properties. The Pukanszky model for tensile strength of polymer nanocomposites was applied, and the dependency of Z to characteristics of constituents and interphase were explained by contour plots.

The σ _{i} and σ _{p} show positive and negative roles in Z parameter at all values of σ _{m}, respectively.

The σ _{i} and B reveal direct links with Z parameter.

The A _{ c } plays a negative role in Z value at different σ _{p}, but the dependency of Z to B parameter is associated with the value of σ _{p}. Therefore, σ _{p} affects the final level of Z at different A _{ c } and B.

The d and t affect the Z parameter based on the value of σ _{p} in different manners. A high σ _{p} causes different roles for nanoparticle size and interphase thickness in Z parameter.

The volume fractions of nanofiller and interphase dissimilarly affect the Z parameter. A high Z is obtained by a low nanoparticle volume fraction and a high content of interphase, but the best Z is obtained based on the level of B parameter.
Notes
Funding
No funding.
Authors’ Contributions
Both authors contributed to the calculations and discussion. Both authors read and approved the final manuscript.
Competing Interests
The authors declare that they have no competing interests.
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