A study on the synergetic effects of self/induced crystallization and nanoparticles on the mechanical properties of semi-crystalline polymer nanocomposites: experimental and analytical approaches

This work investigates the effects of nanoparticle content, aggregation/agglomeration, polymer/particle interphase, and crystallinity on the mechanical properties of high-density polyethylene (HDPE) nanocomposites. Different samples of HDPE nanocomposites, containing 0.5, 0.75, and 2 wt% of pure and surface-modified silica nanoparticles were prepared by melt-mixing method. The pure silica nanoparticles (PSN) and surface-modified silica nanoparticles with (3-aminopropyl) tri-ethoxy-silane (AMS) were characterized with field emission scanning electron microscopy (FESEM) and Fourier-transform infrared (FTIR) spectra. Differential scanning calorimetry (DSC) and X-ray diffraction (XRD) were used to estimate the crystallinity and crystal size of the nanocomposite samples. Finally, tensile testing was performed on the nanocomposites to establish the relationship between mechanical properties and nanoparticle loading, and surface modification. The results indicate that the crystallinity and elastic modulus of the nanocomposites increased with increasing nanoparticle content. Moreover, the Gutzow–Dobreva theory was applied to approximate the degree of the induced crystallinity in each sample. A mechanical model based on two equivalent box models (EBM) was proposed to determine the crystalline, amorphous phase modulus, thickness and tensile modulus of the polymer/particle interphase region, which showed a decreasing trend with the nanoparticles content and indicated that this region was thicker for the HDPE/AMS relative to HDPE/PSN. Also, it was found that nanoparticles affected both crystalline and amorphous sections, which their effect on the crystals was more significant and presence of the well-dispersed nanoparticles in the amorphous section substantially enhanced their performance against the exerted stress.


Introduction
Poly-olefins, which have a wide range of applications in the polymer industry, are used in many fields, such as agriculture, automotive, packaging, etc., due to their properties, such as good processability and low production costs [1]. Although polymer materials have exceptional unique properties, they have some significant disadvantages such as weak mechanical properties compared to metals which limits their application in some cases [2]. The most important parameter determining the performance and mechanical/physical properties of semi-crystalline polymers is the presence of crystals in their matrix that are mostly formed during crystallization process from molten state [3].
It is well stated that addition of certain amounts of nanoparticles to polymer matrix can drastically enhance its physical/mechanical properties [4][5][6]. Suitable nanoparticles can form a strong polymer/particle interphase with the surrounding matrix. The size and the impact of polymer/particle interphase regions are negligible in polymer composites containing macro/micro-fillers while they can substantially affect the physical/mechanical characteristics of polymer nanocomposites [7]. Accordingly, introducing small amounts of nanoparticles will have a high impact on the composite properties, such as tensile modulus, tensile strength, impact strength, thermal/electrical properties, wear, permeation resistance, and crystallinity percentage [8].
Some of the nanoparticles such as modified montmorillonite (MMT) decorated with silver, copper, and zinc oxide have been used as reinforcing agents in preparing HDPE/modified MMT nanocomposite that was acting as an effective heterogeneous nucleating agent. In this nanocomposite, the enthalpy of crystallization and the crystallization temperature increased with increasing concentration of modified MMT [9]. Also, for polyethylene/silica-silver nanocomposite, the synthesized nanocomposite showed similar or superior thermal properties (melting and crystallization temperatures and crystallinities) compared with the neat polymer [10].
Generally, nanoparticles can act as nucleation agents, in the induced crystallization process that was described by many theories (e.g., fringed micelle, chain folding, and switchboard), but the underlying mechanism has not been still well comprehended [11,12]. The specific arrangement of the polymer chains in crystalline domains can be manipulated by nanoparticles. However, it is very important to investigate, understand, and control the impacts of these particles on both crystalline and amorphous parts of the polymer matrix.
According to the De-Gennes self-similar carpet theory, the attractive forces between polymer chains and surface of the functionalized particles led to the formation of polymer/particle interphases, which have had significantly different properties compared to those of the bulk of polymer [13]. On the other hand, the physisorption theory states that the polymer chains with higher molecular weight are more involved in the formation of the interphases, and therefore, it can be concluded that the compatible nanoparticles can alter the characteristics of the crystalline domains as much as improve the physical/mechanical properties of the amorphous sections.
Considering the specific complexities of the crystallization process in highly crystalline polymers, this thermodynamic phenomenon can become even more complicated in the presence of nanoparticles, nucleating agents, conductive materials, etc. [14,15]. Many studies are reporting the variation of the sample crystallization percentage by changing the content of nanoparticles [16]. On the other hand, it is also reported that nanoparticles (e.g., silica) cannot significantly affect the crystallization process of the polymer matrix [17,18]. These contradictory findings clearly showed that the characteristics and the structure of the nanoparticles are very important factors affecting their incorporation with the polymer chains [19]. Also, nanoparticles can act as nucleation agents and simplify the formation of crystals, but their aggregation/agglomeration may prevent this phenomenon. Previous studies have shown that formation of aggregates/ agglomerates in nanocomposites, even at very low content of the nanoparticles, is a determinative parameter in the final physical/mechanical characteristics of the system [20].
Although the effect of nanoparticles as nucleating agents on the crystallization and mechanical properties of polymers is well established, the simultaneous effects of self/induced crystallization and nanoparticles on the mechanical properties of the polymer nanocomposites are not still well understood, but has been precisely investigated experimentally and analytically in this work.
To this end, different HDPE nanocomposite samples containing modified and un-modified silica nanoparticles (i.e., 0.5, 0.75, and 2 wt%) were prepared via melt-mixing and exposed to different tests. The surface modification of silica nanoparticles was performed using (3-aminopropyl) tri-ethoxy-silane to evaluate the impact of these modified nanoparticles on the crystallization and mechanical characteristics of their samples. The structure and the surface chemistry of the nanoparticles were characterized using field emission scanning electron microscopy (FESEM) and Fourier-transform infrared spectroscopy (FTIR), respectively. The aggregation/agglomeration of both types of nanoparticles was approximated using an accurate mechanical model according to which it was possible to better correlate the properties of the system and its constituents. The crystallization characteristics of nanocomposite samples were investigated using differential scanning calorimetry (DSC).
Also, X-ray diffraction (XRD) was used to define the average crystal size and the final mechanical properties of the samples were determined via a tensile test. Furthermore, the Gutzow-Dobreva theory was used to study the induced crystallization in the samples, due to the presence of nanoparticles, and consequently, DSC results at different cooling rates (1, 5, and 10 °C/min) were obtained for all samples. It was revealed that the super-cooling parameter (melting temperature (T m ) − crystallization temperature (T c )) was in direct correlation with the content of nanoparticles and also mechanical properties were substantially affected by the aggregation/agglomeration of nanoparticles and their corporation in the induced crystallization phenomenon.

Synthesis and surface modification of silica nanoparticles
Silica nanoparticles were synthesized using a sol-gel process in an ultra-sonication bath. These uniform-sized-spherical nanoparticles were prepared via hydrolysis of TEOS in ethanol in the presence of ammonium hydroxide. At first, the ethanol/water mixture (1/1.08 vol%) was sonicated for 10 min. While sonicating, a volume ratio of 0.04 of tetraethyl orthosilicate/ethanol was added, and the hydrolysis was carried out for 20 min. Then, ammonium hydroxide, as the catalyst, was added dropwise to facilitate the condensation polymerization and sonication continued for 60 min to obtain a white turbid suspension. Finally, the suspension was centrifuged and dried to get pure silica nanoparticles termed PSN. All of these steps were performed at ambient temperature [21]. The schematic of the procedure of synthesis of pure silica nanoparticles is given in Fig. S1 in the Supplementary Information.
The obtained nanoparticles were calcined in the presence of oxygen at a temperature of 500 °C for 24 h in the furnace, to increase the surface density of hydroxyl agents [22]. As it is schematically demonstrated in Fig. S2 (in Supplementary Information), the calcinated nanoparticles were then dispersed in 100 mL of ethanol, and about 2-3 mL of APTES was added to it under stirring for 24 h at room temperature. Generally, the surface modification process consisted of the reaction of hydrolyzed ethoxy groups of APTES molecules with silanol groups on the silica surface [23]. Finally, the modified nanoparticles were collected via centrifugation and washed several times with ethanol to separate excess APTES molecules [24]. The obtained superficially modified nanoparticles were termed AMS.
It should be noted that, in the production of HDPE granules used in this work, several types of additives including amino additives had been used as antioxidant agents, which could enhance the polarity of the matrix and increase the dispersion degree of the nanoparticles, lately.

Characterization of silica nanoparticles
Particle sizes of the synthesized silica nanoparticles were measured using field emission scanning electron microscopy (FESEM, MIRA3 Tescan, Czech Republic). Also, investigation and comparison of the chemical structure of the modified and unmodified nanoparticles were performed using A Perkin-Elmer Spectrum One (USA) FTIR spectrometer. The materials were in the form of powder, and the FTIR spectra of these samples were obtained in transmittance mode and the spectral region of 400-4000 cm −1 using a resolution of 4 cm −1 .

Nanocomposite preparation
The nanoparticles were melt-mixed with HDPE at a content of 0.5, 0.75, and 2 wt% of PSN and/or AMS using a Brabender internal mixer [25]. The manufacturing temperature was kept at 190 °C and the screw speed was set at 60 rpm for 10 min. After that, the samples were compression-molded into sheets (100 × 10 mm 2 ) at 180 °C and 100 bar for 5 min in an industrial hot press. Also, HDPE pure sheets were prepared using the same method to be used as reference.

HDPE/SiO 2 nanocomposite samples characterization
Field emission scanning electron microscopy (FESEM, MIRA3, Tescan, Czech Republic) analysis has been used to investigate the dispersion of nanoparticles in the HDPE nanocomposites. The nanocomposite sheets were broken in liquid nitrogen, and the FESEM micrographs were obtained from the fracture surface of the specimens.
The changes in the crystallinity and structural characteristics, induced by nanoparticles, are important to understand the response mechanism of the nanocomposites under different conditions [26]. The nonisothermal crystallization behavior of the pure HDPE and reinforced HDPE nanocomposites was studied using a Mettler-Toledo (DSC3, Switzerland) differential scanning calorimeter under a nitrogen atmosphere. Appropriate amount of each samples (5-10 mg) was heated from room temperature (25 °C) to 160 °C at a rate of 10 K/min and held at high temperature for 5 min to erase thermal history. Then, the cooling process was performed on the sample to room temperature at the rates of 1, 5, and 10 K/min, and a second heating scan was carried out at the same rates. Crystallization and melting phenomena were monitored during the second and third steps, respectively, to obtain the enthalpy of fusion ( ΔH m ), the melting (T m ) and crystallization (T c ) temperatures.
T m was considered to be the temperature at which the maximum of the endothermic peak, from the second heating scan, was obtained and T c was the corresponding temperature to the maximum of the exothermic peak obtained from the cooling scan. The heat of fusion was determined via measuring the surface area under the melting peaks and normalized per gram of the samples. Also, the crystallinity of samples (X c ) was calculated by the following formula [27]: where ΔH m is the melting enthalpy of the sample, ΔH • m is the melting enthalpy of 100% crystalline HDPE (293 J/g) [28] and φ is the mass fraction of the nanoparticles.
X-ray diffraction (XRD) study of HDPE-based nanocomposites was performed on the sheet specimens over the range of 2 = 0 • − 80 • and wavelength of 1.54 Å, using the X'PertPro XRD device (Malvern Panalytical Co., the Netherlands). Accordingly, the Scherrer's equation was used to approximate the apparent crystal size (l) formed in the samples during the cooling process [29]: where β is the breadth of the observed diffraction line at its half-maximum intensity (FWHM), K is the shape factor and λ is the X-ray wavelength.

Mechanical properties
Tensile properties of the prepared HDPE nanocomposite samples containing pure and modified silica nanoparticles were evaluated using a Zwick/Roell testing machine according to the ASTM D638 at a crosshead speed of 10 mm/min. Relatively thin sheets of about 350 ± 25 μm were prepared using an Otto Weber press (Type PW 30 hydraulic) at a temperature of 190 °C and pressure of 50 bar. Then, the molds were cooled to room temperature at the rate of 10 K/ min to provide a similar crystallization condition to what was exerted in the DSC test. To measure the mechanical properties, the sheets were cut into dumbbell specimens and conditioned at 25 °C and 55-60% relative humidity for 48 h. Five specimens for each sample were tested and the average results were reported.

Model background
Besides investigating the effects of PSN and AMS nanoparticles on the mechanical properties of the HDPE nanocomposites, using the tensile test data, it was also significant to evaluate the amount of the aggregated/agglomerated domains for better understanding the effect of nanoparticles on the amorphous sections of the semi-crystalline polymers.
To this end, the volume fraction of the aggregated/agglomerated domains (φ agg ) was defined using Zare's model. It was approximated as follows [30]: where φ n represents the nanoparticle content (vol%), D n and D agg are diameters of the nanoparticle and aggregated/ agglomerated domain, respectively. The nanoparticles content was defined by φ n = (ρ t /ρ n )w n , in which w n and ρ n denote the weight fraction and density of the nanoparticles, respectively, and ρ t represents the density of the nanocomposite samples that was calculated using the rule of mixtures. To define D agg , a combination of theoretical and experimental methods was proposed which resulted in the following equation: where N is a model parameter and A is definable using Young's modulus (E c ), shear yield strength ( ), and Poisson's ratio ( ) of the samples as follows: where λ represents the distance between the fully dispersed nanoparticles, b B is the Burger's vector, τ n and τ m are the shear yield strength of the nanocomposite and pure sample, respectively. Then, Gutzow-Dobreva theory was applied to approximate the degree of the induced crystallinity in each system. The application of the theory required defining the supercooling degree of the samples at different cooling rates and accordingly, all samples were subjected to DSC test at two different cooling rates of 1 and 5 K/min. Consequently, a multi-component model was proposed to define the tensile modulus of the crystals as well as the characteristics of the polymer/particle interphase (thickness and tensile modulus). As mentioned earlier, the formation of the polymer/particle interphase region is the most important parameter affecting the behavior of a nanocomposites system against the exerted stress (e.g., shear, tensile, and pressure) [31]. Therefore, considering the simultaneous effect of crystals and nanoparticles on the enhancement of the mechanical properties of the nanocomposite samples, the tensile modulus of the system was represented as follows: where E t is considered equal E exp , E c and E an are Young's modulus of the crystalline section and the amorphous section containing nanoparticles, respectively, φ c represents the volume fraction of the crystals and η is a model parameter correlating excluded volume model (EVM) to the actual nanocomposite samples and depends on the characteristics of the system, such as content of nanoparticles, aggregation/agglomeration, and X ic . The details of this model are presented in the next section.

Morphological characteristics
The morphology and the size of the synthesized silica nanoparticles were evaluated using FESEM analysis. As demonstrated in Fig. 1, the produced nanoparticles were spherical with 20-40 nm in size, which confirmed their nanoscale (other samples are presented at different magnifications in the Supplementary Information as Fig. S3). Also, the dispersions of PSN and AMS nanoparticles in the HDPE-based nanocomposites are shown in Fig. 2 for the PENC-PSN2 and PENC-AMS2 samples. These FESEM micrographs clearly show that the nanoparticles were well dispersed in the samples and accordingly, they could be categorized as nanocomposites. However, it can be seen that despite the well distribution of nanoparticles, there is a very small amount of aggregation/agglomeration of nanoparticles in the nanocomposite samples. Therefore, FESEM micrographs were provided from specific parts of the samples to only show presence and structure of the aggregates/agglomerates, which are presented in Fig. 2a and b. The agglomeration of nanoparticles in the nanocomposites (although very low) is clearly visible in these micrographs, especially in the PENC-AMS2 sample (Fig. 2b).
As a result, according to the reported φ agg , which is very small for all samples, the PSN and AMS are well dispersed and distributed in the HDPE matrix ( Fig. 2c and d) and the desired so-called nanocomposite samples were obtained perfectly.
To understand the variation of nanoparticles and their surface chemical structure, they were subjected to FTIR analysis. According to these results (Fig. 3), the most prominent peaks for both nanoparticles were located between 950 and 1250 cm −1 related to Si-O-Si and Si-O-C bonds, respectively [32]. OH bending and -CH 2 vibration were specified at 800 and 2870 cm −1 , respectively [33]. Also, two peaks at 1480 and 2930 cm −1 belong to C-H vibrations [34]. After modification with APTES, a new characterizing peak appeared at 1450-1600 cm −1 , representing NH 2 vibrations, confirming the presence of the NH 2 group of APTES molecules on the surface of the modified silica [35].

Crystallization and structural characteristics
The study of crystallization assumes particular significance, due to its impact on the mechanical properties. In general, higher crystallinity increases the modulus and yield stress and decreases toughness. Figure 4a and b shows the second heating and cooling scans of pure HDPE and different prepared HDPE/PSN nanocomposite samples at the heating/ cooling rate of 10 K/min, respectively. There was an endothermic melting peak in the heating runs and an exothermic crystallization peak in the entire cooling scans. The data obtained from this analysis are summarized in Table 1, and It is necessary to mention that the output graphs from the DSC software, which are drawn as continuous points without fluctuation, are available in the Supplementary information (Figures S4-S10) and the reason for the oscillations in curves of Fig. 4 is the fact that the output data of the DSC software are separate points drawn with MATLAB software. According to the results, the crystallinity (X c ) of nanocomposites, approximated by Eq. (1), increased with the increase of nanoparticle content. This was attributed to the nucleation effect and induced crystallinity due to the presence of nanoparticles in the polymer matrix.
Gutzow-Dobreva theory (Eq. (7)) was used for better understanding the effect of induced crystallinity [36]. This theory states that the logarithm of the cooling rate (q) is inversely related to the square of the degree of super-cooling (∆T max ). Accordingly, plotting log(q) versus 1/∆T 2 max , at different cooling rates, leads to defining parameter B which is constant and is a function of nanoparticle content as follows: According to this theory, the DSC analysis needs to be performed at different heating/cooling rates, and therefore, besides the obtained results at 10 K/min (Table 1), the complementary tests were performed at two heating/cooling rates of 1 and 5 K/min (Table 2). Consequently, the induced crystallinity parameter (X ic ) was defined as follows [37]: where, parameters B and B′ are obtained using Gutzow-Dobreva theory, for pure and nanocomposite samples, respectively. The related DSC test results can be found in the Supplementary Information (Figs. S11-S17).
As it is demonstrated in Fig. 5, super-cooling degree increased with the cooling rate for all HDPE/PSN samples which was attributed to the nature of the polymer chains and their capability to form crystal domains [38]. On the other hand, the presence of the different amounts of impurities, and nanoparticles, in the system decreased the super-cooling degree at all tested cooling rates as expected.
Indeed, the nanoparticles could act as nucleating agents and facilitate the formation of crystals in the samples.  However, at high nanoparticle loads, i.e., 0.75 and 2 wt%, the aggregation/agglomeration of the nanoparticles increased. It is necessary to mention that although the aggregation in the sample with 2 wt% nanoparticles was high, due to its weight percentage of nanoparticles, the number of dispersed nanoparticles was still more than the sample with 0.75 wt% (this will be discussed later). On the other hand, increasing the number of nucleating agents in the sample having 2 wt% of nanoparticles would lead to the formation of more crystals with less thickness, which caused a lower degree of super-cooling in this sample compared to the one with 0.75 wt% nanoparticles. More interestingly, according to the results, the supercooling degree of PENC-PSN0.5 was higher than the other samples at the cooling rate of 1 K/min. This proved that although nanoparticles can enhance the formation of crystals at high cooling rates, where the polymer chains did not have enough time to settle in crystalline domains, they negatively affect the inherited crystallization process of polymer matrix at low cooling rates.
According to Table 2, the induced crystallinity (X ic ) was maximum for PENC-PSN0.5 which also proved the high interference of PSN nanoparticles in the crystallization process of this sample. This phenomenon was also observed at the cooling rate of 5 K/min comparing the results of nanocomposite samples. It should be also noted that the high nanoparticle content in the PENC-PSN2 caused the formation of higher nucleating agents compared to PENC-PSN0.75 which resulted in a lower supercooling degree at the cooling rate of 10 K/min. Though, the higher degree of aggregation/agglomeration of nanoparticles in PENC-PSN2 seemed to intensify the inherited crystallization which led to a lower supercooling degree at cooling rates of 1 and 5 K/min. This can be also concluded based on the obtained induced crystallinity (Table 2) according to which a significant decrease in X ic was seen by increasing 0.25 wt% to the nanoparticles content in the PENC-PSN0.75 sample while the increment of the nanoparticle content in the PENC-PSN2 sample, compared to PENC-PSN0.75, did not have the same substantial effect.
According to the obtained results for the HDPE/AMS nanocomposite samples (Fig. 6), the presence of APTES molecules increased the tendency of the nanoparticles to form aggregates/agglomerates in the matrix. This was completely obvious in the case of PENC-AMS0.5 and PENC-AMS0.75.
Also, it was revealed that the increment of the nanoparticle's content increased the size of clusters and affected both induced and inherited crystallization. However, it seemed that the exerted shear stress in the blending process efficiently dispersed/distributed higher content of nanoparticles in the PENC-AMS2 sample. Thus, some of the nanoparticles could perform as nucleating agents and reduced super-cooling degree. On the other hand, the reported X ic for PENC-AMS nanocomposite samples also suggested the drastic impact of the nanoparticle's content on their involvement in the induced crystallization. Comparing Figs. 5 and 6, showed that the super-cooling degree was almost independent of the surface chemistry of the nanoparticles at low cooling rates (e.g., 1 K/min).
According to Table 1, by adding PSN or AMS silica nanoparticles to the polymer matrix, there was a specific trend in the crystallinity of the nanocomposite samples. However, the corresponding melting temperatures did not have a similar trend. The reason for this enharmonic was evaluated using XRD analysis.
The XRD patterns of pure HDPE, HDPE/PSN and HDPE/ AMS nanocomposites can be found in the Supplementary  Information (Figs. S18-S24), that showed the characteristic peaks of crystalline phase of HDPE at 21.5 and 23.9 as (110) and (200) crystal lattice respectively. For the nanocomposite samples, diffraction peaks for the nanoparticles are slightly detectable in the XRD patterns and the most obvious diffraction peaks can be assigned to the ordered structure of HDPE in the samples. It implies that the HDPE almost maintains its structure after adding a small amount of nanoparticles. However, the partial shift of diffraction peaks suggested that It should be noted that the d-spacing and apparent crystal size in each sample were calculated using Scherrer's equation [39] from the intense diffraction peak (110) and are listed in Table 3.
The results showed that the increasing/decreasing trend of the melting point was similar to the size of the crystals. Therefore, the decrement of melting temperature, despite the increment of the crystallinity, was due to the decrement of the thickness of crystals and vice versa.

Mechanical characteristics of the nanocomposites
To investigate the effect of adding pure and surface-modified silica nanoparticles in HDPE matrix on the mechanical properties of this polymer, a tensile test was performed on the pure HDPE, and its nanocomposites and their elastic modulus were calculated. These results and the corresponding φ agg (calculated based on Zare's model) are presented in Table 4. Some examples of the graphs obtained in the tensile test of different samples are given in the Supplementary  Information (Figs. S25-S27).
As it is clear, φ agg increased with increasing the nanoparticles content where its effect on the inherited and induced crystallization process was comprehensively discussed earlier.
It should be noted that the amount/number of fully dispersed nanoparticles in PENC-PSN2 and PENC-AMS2 was significantly more than that of the other samples. This was attributed to the higher impact of the exerted stress on the dispersion/ distribution efficiency of the nanoparticles in the preparation stage of the mentioned samples. Moreover, as expected, the tensile modulus of the nanocomposite samples increased with the nanoparticles content which was due to significant effects on both crystalline and amorphous sections of the samples. Besides evaluating the effects of PSN and AMS nanoparticles on the crystallization process of the polymer matrix, it was also important to study the corresponding effects on their amorphous sections.
A simple equivalent box model (EBM) was used to approximate the tensile modulus of the crystals in each sample. According to Fig. 7a, the geometrical structure of the model, a cube of uniting side length, comprised two amorphous and crystalline components whose volume fractions were defined using the density of the crystals (ρ c ), the density of the pure polymer ( m ) and crystallinity parameter (X c ) as (1 − ρ m /ρ c X c ) and ρ m /ρ c X c , respectively. The amount of ρ c is reported to be 1.003 g/cm 3 for pure PE crystals [40]. Finally, considering a parallel arrangement in the EBM model (Fig. 7b), E c was calculated based on Eq. (9) as follows:  where E m and E a are the tensile modulus of semi-crystalline polymer and pure amorphous matrix, respectively, and k = ρ m /ρ c X c . E a is reported to be 3 MPa in HDPE and accordingly, E c was calculated to be 1.611 GPa [41].
To define the tensile modulus of the amorphous section containing nanoparticles (E an ), an excluded volume model (EVM) was designed based on the impact of a specific number of nanoparticles (φ n − φ agg ) on the amorphous section of the matrix (1 − ρ m /ρ n w n − ρ m /ρ c X c ). Generally, EVM provided the thickness and tensile modulus of the interphase region by interpreting the tensile test results. Figure 8 demonstrates the cubical structure of the EVM which was consisted of cubical excluded volumes (CEVs) each containing a spherical nanoparticle at its center. Parameter N denoted the number of fully dispersed nanoparticles which was defined as follows: According to Eq. (10), it is considered that a part of dispersed nanoparticles is acting as nucleation agents, some are trapped in aggregates/agglomerates and the remaining are dispersed in the amorphous phase. Indeed, it is assumed that nanoparticles are dispersed or aggregated/ agglomerated in both amorphous and crystalline phases; although, they can have different effects on the mechanical/physical characteristics of the system.
Based on the series and parallel arrangement of the CEVs, it was possible to prove that the tensile modulus of the amorphous nanocomposite section (E an ) was equal to the tensile modulus of a CEV (E CEV ). To define the tensile module of a CEV (Fig. 8b), it was possible to use its corresponding EBM (Fig. 8c) which resulted in the following equation: The parameters of Eq. (11) can be defined using Eqs. (12)- (20) as follows: where E i and E N denote Young's modulus of the interphase region and nanoparticles. r n is the radius of the nanoparticles and = 0.25 fraction of different constituents of EBM model is formulated as follows:  where t represents the thickness of polymer/particles interphase region, φ mIII = φ iII , and φ mII = 0.5(1 − φ iII ). The interpreted characteristics of the interphase region, based on the results of EVM are presented in Table 4 for each sample.
According to the results, there was a decreasing trend in the thickness (t) and tensile modulus of the polymer/particle interphase region by increasing the nanoparticles content. We have comprehensively discussed these phenomena in our previous studies according to which the increment of the polymer/ particle interface area with the nanoparticle content facilitated the desirable physisorption of the available high molar mass polymer chains onto the surface of nanoparticles [42,43]. Consequently, it was revealed that the higher polymer/particle interface led to lessen the number of the adsorbed high molar mass chains and therefore thinner the interphase region [31].
Also, this phenomenon directly affected the mechanical characteristics of the region due to the size reduction of the contained polymer chains with the increment of the nanoparticle content. On the other hand, the HDPE/ AMS nanocomposites had thicker interphase, compared to their corresponding samples containing PSN nanoparticles, which ascribed to the nature of APTES molecules. It seemed that the presence of -(CH 2 )-and -CH 3 groups in the structure of these molecules caused good compatibility between PE chains with the surface of the nanoparticles.
As reported in the literature [44], APTES molecules can attach to the surface of silica particles by forming bonds with hydroxyl groups at different states. Almost all states resulted in molecular structures containing a considerable number of ethoxy (-EtO) and -CH 2 -groups. Considering the presence of all mentioned molecular states, on the surface of the modified nanoparticles, it can be concluded that APTES molecules had a direct corporation in increasing the tendency of silica nanoparticles toward the HDPE polymer phase which indeed increased the dispersion and strengthening the polymer/particle interphase region, for example.
Though, as mentioned before, APTES also enhanced the aggregation/agglomeration volume fraction of the nanoparticles (φ agg ) in the samples containing modified nanoparticles. Based on the reported results in Tables 2 and 4, it was clear that nanoparticles affected both crystalline and amorphous sections of the semi-crystalline matrix. However, their impact on the crystals was more significant.
Indeed, the obtained results for the tensile modulus of the amorphous section of the nanocomposite samples were substantially lower than the crystalline section which was attributed to the very low tensile modulus of the amorphous (20) iII = 4 r n + t 2 − r n 2 1 2 3 D n + 2 + 2t matrix (3 MPa). Accordingly, the simultaneous evaluation of both crystalline and amorphous sections of the semi-crystalline nanocomposites seemed to be crucial for defining the mechanical characteristics of these systems.

Conclusion
The influence of adding different amounts of pure and surface-modified silica nanoparticles on the induced crystallinity at different cooling rates and subsequently, mechanical properties of the high-density polyethylene-based nanocomposites were systematically investigated. The results showed that by increasing the pure and modified silica nanoparticles content, crystallinity of their HDPE nanocomposites increased. According to Gutzow-Dobreva theory, the nanoparticles acted as nucleating agents and facilitated the formation of crystals in the samples, which had increased the induced crystallinity that was less affected by the cooling rate than the inherited crystallinity.
On the other hand, in some samples with 0.5 and 2 wt% of PSN where the melting temperature had decreased despite the increase in the crystallinity, a decrease in the thickness of the crystals had been seen according to the XRD results. This means that the increase in crystallinity was due to an increase in the number of crystals and a decrease in their thickness, and therefore with decreasing the size of crystals, the melting temperature of these samples had a decreasing trend. The elastic modulus of HDPE/PSN and the HDPE/AMS nanocomposites were higher than that of the pure HDPE which were compatible with the increase in their crystallinity. To simultaneously investigate the effect of nanoparticles presence and crystallinity on the mechanical properties of nanocomposites, the effects of PSN and AMS nanoparticles were evaluated on the crystallization process of both of the polymer matrix and amorphous sections. For this purpose, a mechanical model based on two EBMs was considered to calculate the crystalline and amorphous phase modulus of the nanocomposites considering the polymer/particle interphase effect. According to the results, these nanoparticles affected both crystalline and amorphous sections of the semi-crystalline matrix. Indeed, the dispersion of the nanoparticles was increased with increasing their content which resulted in the higher corporation of nanoparticles to both amorphous and crystalline phases. This was proved based on the variation of the characteristics of the polymer/particle interphase region as well as the value of X ic . However, their effect on the crystals was more significant and the presence of the well-dispersed nanoparticles in the amorphous section substantially enhanced its performance against the exerted stress. Also, the thickness and tensile modulus of the polymer/particle interphase region decreased by increasing the nanoparticle content which may be due to the lower number of the adsorbed high molar mass chains onto the surface of the nanoparticles. It is necessary to mention that the HDPE/AMS nanocomposites had thicker polymer/particle interphase, compared to their corresponding samples containing PSN nanoparticles, due to the nature of APTES molecules and their additive effects on the aggregation/ agglomeration volume fractions of the nanoparticles.
Author contributions SH: Performed the experiments, analyzed the data, wrote the draft version of the manuscript. FR: Developed and supervised the research, performed the experimental preparation, analyzed all characterization data; revised the manuscript. ES: Developed the research, performed the experimental preparation, analyzed all characterization data; revised the manuscript. All authors discussed the results and contributed to the final manuscript.
Data availability All data, models, and code generated or used during the study appear in the submitted article.