How Interphase Properties Control the Young’s Modulus and Yield Strength of Polymer Nanocomposites?

In this article, several models are applied to reveal the effects of volume fraction, thickness, strength and modulus of interphase region between polymer matrix and nanofiller on the Young’s modulus and yield strength of polymer nanocomposites. The properties of interphase are calculated for several samples by experimental data of mechanical properties. It is found that the concentration of interphase is higher than that of nanofiller in some samples. The Young’s modulus of nanocomposites largely depends on filler and interphase concentrations. In addition, the highest fraction and strength of interphase region produce the highest yield strength of nanocomposites.


INTRODUCTION
Nanostructures suggest novel applications due to the excellent properties of materials at nanoscale [1][2][3][4]. Therefore, the structure and interaction at nanoscale cause the significant effects on the properties. The dramatic enhancement in the mechanical properties of polymers can be attained by incorporation of a low weight percentage (wt %) of various nanofillers such as layered silicates [5,6]. The large aspect ratio and stiffness of layered silicates may be the main reasons for the highly enhanced mechanical properties of polymer nanocomposites [7].
The Young's modulus of polymer nanocomposites increases by addition of nanoparticles, because they usually have a much higher modulus than polymer matrices [8][9][10]. However, the yield strength of nanocomposites depends on the stress transferal between nanofiller and polymer matrix [11]. The stress applied to nanocomposites can be excellently transferred to nanoparticles in well-bonded nanoparticles to polymer matrix. In this condition, the yield strength of polymer nanocomposites noticeably improves in the tensile test. However, the yield strength reduces by adding of poorly bonded nanoparticles. As a result, the proper-ties of interface/interphase between polymer matrix and nanoparticles cause main effects on the mechanical properties of polymer nanocomposites and discounting of interphase characteristics results in wrong prediction of nanocomposites performances [12,13].
The interphase dimension and stiffness have been determined by micromechanical models for mechanical behavior such as Young's modulus and tensile/ yield strength [14,15]. It was also reported that shape memory polymer nanocomposites with a strong adhesion at polymer-nanofiller interface show pronounced shape memory properties [16]. However, there is not a model which directly expresses the effects of interphase properties such as interphase fraction on the Young's modulus and yield strength.
In this work, Ji and Pukanszky models are used to display the Young's modulus and yield strength of polymer nanocomposites containing different filler geometries as a function of the volume fraction, thickness, tensile strength and modulus of interphase region. The influences of interphase properties on the Young's modulus and yield strength of nanocomposites are discussed. Additionally, the mentioned equations are used to analyze the properties of interphase in various samples.

BACKGROUND
Ji et al. [17] proposed a three-phase model for Young's modulus of nanocomposites taking into account matrix, nanofiller and interphase between polymer and nanoparticles. The Ji model for nanocomposites containing layered (1), spherical (2) and cylindrical (3) nanoparticles is expressed attributed to geometry of nanofillers as , (( ) ) 1 r r r Accordingly, Ji model can be defined by the properties of interphase for polymer nanocomposites containing different nanoparticles as (2 )( 1) ln ) ( 1)ln ) ( 1)ln ) which display the effects of interphase properties on the Young's modulus of polymer nanocomposites.
Pukanszky [18] suggested an equation based on the formation of interphase in composites, where the yield strength is determined as a function of filler content. Pukanszky model is presented as where  r is relative yield strength as  c / m ,  c and  m are yield strengths of composite and matrix, respectively, B is an interfacial parameter which assumes the capability of stress transfer between matrix and filler. This model was well applied for different polymer nanocomposites in the recent studies [19,20]. Therefore, it is applied in this work to analyze the effects of interphase on yield strength of dissimilar nanocomposites. Parameter B depends to interphase characteristics as where A c is the specific surface area of filler,  f is density of filler, and  i is the tensile strength of interphase. To calculate parameter B, Pukanszky model can be rewritten as where the linear plot of ln ( r (1 + 2.5 f )/(1 - f )) against  f shows the slope of B. Using Eqs. (12)- (14), Pukanszky model can be expressed as a function of interphase properties for polymer nanocomposites as Additionally, A c can be defined for layered, spherical and cylindrical nanoparticles as where A, m, v and l are the surface area, mass, volume and length of nanoparticles, respectively. As a result, B can be expressed for polymer nanocomposites as 3. RESULTS AND DISCUSSION

The Analysis of Experimental Data from the Literature
In this section, the mentioned models are utilized to determine the properties of interphase in several samples from the literature. In addition, the effects of interphase characteristics on the modulus and strength of polymer nanocomposites are plotted. Table 1 shows different samples from the literature as well as the properties of neat polymer and nanofiller. The experimental tensile moduli of samples are applied to Ji model (Eqs. (1)-(6)) and the average levels of t i or r i and E i are calculated. The interphase thickness cannot be more than the gyration radius of polymer chains and E i changes between the moduli of polymer matrix and nanofiller. Therefore, suitable r i or t i and E i are chosen from the explained ranges and finally, the average values of interphase properties are given in Table 1. The presented data show the significant thickness of interphase in the reported samples. Additionally, a high interphase modulus is calculated in all reported samples, which is more than the stiffness of polymer matrix. Accordingly, the interphase can play an important role in the performances of polymer nanocomposites.
Moreover, the experimental yield strength of samples is fitted to Pukanszky model (Eq. (18)) to determine the B parameter. Parameter B is applied into Eq. (19) to measure  i values. Table 1 gives the values The dimension and level of interphase are attributed to some parameters such as interfacial area and compatibility between polymer matrix and nanofiller which control the interfacial interaction. Some procedures such as treatment, modification and functionalization of nanofillers can encourage the compatibility and interfacial adhesion between the components of nanocomposites. Furthermore, A c data are reported in Table 1 using Eqs. (24)- (26). The nanoclay produces the highest level of A c among the nanoparticles. This occurrence causes the highest level of interfacial area between polymer matrix and nanoclay layers, which finally creates the highest level of reinforcement in polymer nanocomposite. In fact, the high value of A c is the significant advantage of nanofiller, which makes the unexpected behavior in polymer nanocomposites. According to Eqs. (24)- (26), A c is inversely related to r and t and the smallest nanoparticles create the highest level of A c as shown in Table 1. Figure 1 illustrates the volume fraction  i in some reported samples by Eqs. (7)- (9). It is observed that the interphase occupies a large volume in polymer nanocomposites which is more than the nanofiller volume in some samples. Volume fraction  i increases with nanofiller content in all samples. The high level of interphase confirms the significant influence of this phase beside matrix and nanofiller phases. As a result, assuming the interphase is compulsory for estimation of mechanical properties in polymer nanocomposites. In addition,  i is directly related to the thickness of interphase (according to Eqs. (7)-(9)), i.e.  i increases when the thickness of interphase enlarges.

The Roles of Interphase Properties According
to the Models Figure 2 demonstrates the effects of  i and E i on modulus of polymer nanocomposites containing spherical nanoparticles by Eq. (16). The Young's modulus more depends to  i than E i .  The low  i generally results in a low modulus at all E i levels. However, the best modulus is obtained by high levels of  i and E i . It means that  i and E i have optimistic effects on the modulus of nanocomposites, where the effect of E i becomes important at high values of  i indicating the important role of  i in Young's modulus of polymer nanocomposites. Figure 3 exhibits the influences of  i and  i on parameter B in polymer nanocomposites containing spherical nanoparticles (Eq. (28)). Parameter B shows comparatively same levels at all  i values depended to the level of  i . In other words,  i more expressively affects the level of parameter B compared with  i .
Parameter B shows negative values when  i is lower than  m at all  i . The best level of B is obtained in the highest values of  i and  i . As a result, B is more depended to  i value in polymer nanocomposites (especially at low  i ), while formation of a high-volume interphase improves the magnitude of B interfacial parameter. Figure 4 also shows the roles of nanoparticle radius and interphase thickness on the interphase fraction  i2 of nanocomposites containing spherical nanoparticles by Eq. (8) in  i2 = 0.02. The high values of nanoparticle size decrease the  i2 , but the small radius of nanoparticles increases it. On the other hand, a thick interphase grows the level of  i2 , while the thinner one produces a smaller  i2 . Therefore, small nanoparticles and thick interphase show beneficial effects on the  i2 .
Since a higher volume fraction of interphase causes a better level for tensile modulus and strength (Figs. 2 and 3), it is concluded that small particles and thick interphase introduce the high mechanical properties in nanocomposites. Likewise, large particles and thin interphase result in low levels for mechanical performances. The small nanoparticles induce the high specific surface area between polymer and nanoparticles. A good interphase is gained by the high level of interfacial interaction/adhesion in nanocomposites [20,26].
A micromechanical model was also proposed by Boutaleb et al. [27] to calculate the modulus and yield stress in polymer/SiO 2 nanocomposites. It considers the interphase as the perturbed region of polymer matrix around the nanoparticles. The predicted effects of nanoparticle radius, interphase thickness and modulus on the modulus and yield strength of nanocomposites in that work are similar to those suggested in the present study. All these remarks confirm the progressive roles of interphase properties in the mechanical behavior of nanocomposites.

CONCLUSIONS
Ji and Pukanszky models were used to show the Young's modulus and yield strength of different polymer nanocomposites as a function of interphase volume fraction, thickness, modulus and strength. The interphase properties of different samples were also studied based on Young's modulus and yield strength. The Young's modulus more depends on  f and  i than E i . A low  i generally results in a low modulus at all E i values. The poorest Young's modulus is found by the thinnest interphase. Besides, small nanoparticles and thick interphase increase the yield strength of nanocomposites. Parameter  i more expressively affects the level of parameter B compared to  i . Nevertheless, the highest values of  i and  i produce the highest level of interfacial adhesion expressed by parameter B. Since a high level of B increases the yield strength of nanocomposites, the high values of  i and  i play positive roles in the yield strength.