Using Thermomechanical Properties to Reassess Particles’ Dispersion in Nanostructured Polymers: Size vs. Content

Nanoparticle-filled polymers (i.e., nanocomposites) can exhibit characteristics unattainable by the unfilled polymer, making them attractive to engineer structural composites. However, the transition of particulate fillers from the micron to the nanoscale requires a comprehensive understanding of how particle downsizing influences molecular interactions and organization across multiple length scales, ranging from chemical bonding to microstructural evolution. This work outlines the advancements described in the literature that have become relevant and have shaped today’s understanding of the processing–structure–property relationships in polymer nanocomposites. The main inorganic and organic particles that have been incorporated into polymers are examined first. The commonly practiced methods for nanoparticle incorporation are then highlighted. The development in mechanical properties—such as tensile strength, storage modulus and glass transition temperature—in the selected epoxy matrix nanocomposites described in the literature was specifically reviewed and discussed. The significant effect of particle content, dispersion, size, and mean free path on thermomechanical properties, commonly expressed as a function of weight percentage (wt.%) of added particles, was found to be better explained as a function of particle crowding (number of particles and distance among them). From this work, it was possible to conclude that the dramatic effect of particle size for the same tiny amount of very small and well-dispersed particles brings evidence that particle size and the particle weight content should be downscaled together.


Introduction
The field of materials science has seen one of the most interesting developments in the emergence of nanostructured polymers.These materials are characterized by their unique properties, which are largely due to the dispersion, size and content of the nanoparticles that they contain.The addition of nanoparticles promotes significant changes to the mechanical, thermal, electrical and optical properties of polymers, making them ideal for a wide range of applications, from biomedical engineering to energy storage [1,2].
Fibrous reinforcements in a polymeric matrix also offer several advantages, making them a popular choice in composite materials.The most commonly used fibers are glass, carbon, aramid and basalt, which are selected to provide the necessary stiffness and strength to the composite, depending on the fiber's orientation, length, volume and interfacial properties.As the polymeric matrix is responsible for the stress transfer at a micromechanical level, and thus for the performance required, its mechanical properties, because they are affected by environmental conditions, such as temperature, humidity, Polymers 2023, 15, 3707 2 of 26 radiation and exposure to chemicals, are of utmost importance and can hinder the final component functionality [2][3][4].
The cost of polymers increases with their degree of engineering.Bearing in mind the importance of the property-to-cost balance to produce viable and scalable composites, the need to finely tune polymers has been looked at from the perspective of tailored-made nanostructured materials by several investigations in the last decades, shedding light on a class of materials known as nanocomposites, disseminated by the Toyota research group and literally opening a new dimension in materials science [5][6][7].
This development has emphasized the importance of factors like particles' specific surface area and mean free path, showing the intimate relationship among the amount of added nanoparticles, faultless dispersions and the resulting properties.This suggests a good economic balance and unveils an encouraging scenario for tailor-made applications and high-performance materials.
Inspired by the above, this article reviews the incorporation of distinct particles in epoxy matrices, analyzing how important parameters such as dispersion methods, particle size and particle content affect the properties of nanostructured polymers.The particles mostly used in particle-filled polymers are described first.Next, widely practiced techniques for incorporating nanomaterials while preventing agglomeration are summarized, emphasizing the processing difficulties.A discussion follows on how particle downsizing influences the particle-epoxy mechanical properties, providing a different perspective on processing-structure-property relationships.

Major Particle Fillers
In light of the distinct features of the numerous types of fillers described in the literature, only those judged more relevant are discussed below, along with advances in the respective field.

Inorganic Nanofillers
Inorganic nanofillers present some significant advantages over most organic ones.Except for cellulose, inorganics tend to be much more scalable in production than carbon nanotubes (CNTs) or graphene, for instance, while still providing interesting property enhancements to polymer composites.Among the inorganic fillers, titanium dioxide (titania, TiO 2 ), silicon dioxide (silica, SiO 2 ) and zinc oxide (ZnO) have been widely used and thoroughly studied [8].
2.1.1.Titanium Dioxide TiO 2 (titania) is a natural oxide found in three allotropic forms: rutile, anatase (both tetragonal) and brookite (orthorhombic).Applications make use of mostly rutile and anatase and are, as such, categorized into two groups.
Rutile is brown-red colored and has the highest density, with antibacterial and UVabsorbent properties.For these reasons, applications include the military industry, cosmetics, soap, sunscreen, toothpaste, cigarette paper and paints to coat electronics, toys, furniture and packages [9].On the other hand, anatase has a lower density and appears in different colors (brown, yellow, orange, green and blue).Its applications include solar cells, catalysts, sensors and battery components.An interesting feature of anatase is its brightness, which enhances the whiteness of paper and removes the greasy-look of resin-coated fibers.It is also used in rubber formulation due to its anti-aging properties [9].
As a filler, nano-TiO 2 is scalable and extensively used in various industries for multiple purposes (mostly photocatalysis, anti-corrosion and chemical stability), ranging from antibacterial materials, anti-fogging mirrors and electronics to sunscreens and cosmetics [9].Particles may be synthesized by numerous methods, with distinct shapes, sizes and properties [10].
When incorporated in polymer matrices, nano-TiO 2 is expected to enhance the mechanical, electrical, optical and rheological properties [11].It may be incorporated either as particles in the matrix [12] or as fiber coating [13].The latter is particularly interesting for natural Polymers 2023, 15, 3707 3 of 26 fibers given the offered barrier properties, thus substantially decreasing the water absorption capability of the fibers and protecting the composite from humidity-related aging.
From the perspective of the influence on the mechanical properties of composites, Hunain et al. [14], for example, demonstrated the beneficial effect of titanium dioxide on the dynamic mechanical properties of fiber-reinforced polymer composites.The increase in fatigue resistance could be attributed to the influence of the filler simultaneously on the matrix, making it tougher and more rigid over the interphase regions, ensuring a higher adhesion among the fibers and matrix.As another illustration, Al-Zubaydi et al. [15] showed that the dispersion of this filler in an epoxy matrix enhanced its resistance to wear, which, allied with the known chemical and impact resistances of the composite, makes it an interesting material for anti-skid flooring applications.

Silicon Dioxide
SiO 2 (silica) may be found in various crystalline and amorphous structures, with low thermal conductivity and high chemical inertness towards most substances.
The compound is abundant throughout the planet and originates from many sources, amongst which sand is the most common.Due to its availability and properties, silica is widely used in industry in fields such as civil construction (mostly in Portland cement concretes), food, pharmaceuticals, cosmetics, paints, inks, rubbers, glass, aerospace, automotive and electronics (semiconductors) [16].
As a filler, nano-SiO 2 is used for its scalability and ability to increase the polymer's mechanical, thermal and electrical properties [17].It has proven to be quite effective in decreasing the frictional wear of polyether-ether-ketone (PEEK) composites [18] by reducing its friction coefficient and increasing the UV resistance of epoxy, both as hydrophilic or hydrophobic nanosilica [19].It has also been found that both DC resistivity and the dielectric breakdown strength improve in epoxy-silica nanocomposites [20], which is supposedly due to the effectiveness of the organic-inorganic interface, opening possibilities for the development of cutting-edge thermoset composites.

Zinc Oxide
ZnO is a multifunctional filler with a particular set of features, including a high refractive index, high thermal conductivity, antibacterial properties and high UV-shielding, which makes it a strong candidate for reinforcing translucent and UV-resistant composites [21].These are specifically attainable at a filler content lower than usual (0.03 wt.%).Nano-ZnO epoxy-based coatings can promote surface charge dissipation on insulators in DC gasinsulated systems [22] while also being anti-corrosive due to their hydrophobic nature [23].Due to its amenable intrinsic and processing characteristics, ZnO is generally preferred over other possible competitors [24].
Pure ZnO is a white powder with hexagonal wurtzite and the cubic zincblende structure [25,26], naturally occurring as the mineral zincite, usually containing manganese and other impurities that confer a yellow to red color [26].Zinc oxide is an amphoteric oxide and is nearly insoluble in water; however, it will dissolve in the majority of acids [26].
The applications of zinc oxide powder are numerous, and most exploit the reactivity of the oxide as a precursor to other zinc compounds.Currently, it is added to materials and products, including plastics, ceramics, glass, cement [26], rubber, lubricants [27], baby powder and creams against diaper rashes, calamine cream, anti-dandruff shampoos, antiseptic ointments [28], pigments for paints [29], and Li-ion battery and supercapacitors [30] to name some.

Clays
Clays are natural silicate minerals that are divided into several groups according to their composition and particle morphology, such as kaolinite, illite, smectite and vermiculite.Most have a sheet-like morphology that tends to form stacked structures; however, some can roll over and form tubular structures (e.g., halloysite, from the kaolinite group).Those stacked structures can be spread-out into nearly individual high-aspect ratio sheets of nearly 1 nm in thickness and around 100 nm in planar dimensions, as explained by Saba et al. [6] in a very comprehensive review.Similar to CNTs, halloysite nanotubes (HNT) may also be single-or multi-walled [30] but are much cheaper, more abundant and sustainable when compared to CNTs.The layered architecture allows water adsorption/desorption and enables cation exchange, which is vital to define the effectiveness of surface modifications.This can convey an organophilic character to these particles, making them suitable for dispersion in hydrophobic organic polymeric matrices [31].
When incorporated into polymeric matrices, these nanofillers, as others, are only a minor proportion of the composite and still provide enhancements in physical, thermal and mechanical properties, such as permeability, flammability and tensile toughness, respectively [25, [32][33][34].These improvements enable the use of the composites in diversified applications, such as flame-retardant plastics, high-clarity films [30], dental [35] and drug delivery systems [36].

Graphene, Graphene Oxide and Graphene Nanoplatelets
Graphene is a single-layer, two-dimensional sheet of sp 2 -hybridized carbon atoms [37] that has earned a reputation for being capable of enhancing the properties of reinforced polymers, from fracture strength [38] to electric [39] and thermal [40] properties.These improvements are possible thanks to characteristics such as a high surface area and aspect ratio [41], good interfacial adhesion to thermosets, such as epoxy [38], and its high mechanical strength.The applications of this filler are very diversified and include tissue engineering, drug delivery systems [42], magnetic resonance imaging [43] and cancer therapy [44].
The main behavioral difference between graphene oxide (GO) and graphene is the fact that the first is hydrophilic and the latter hydrophobic.This change in polarity can be explained by the presence of oxygen bonded through the sp 2 in GO, and it enables the dispersion of GO in aqueous solutions, enhancing its applicability to polymeric composites, as well as functionalization [45].
Graphene nanoplatelets (GNPs) are small sheets of graphene and, as such, can be effectively embedded in polymeric matrices, enabling many applications due to the increase in thermal and electrical conductivities and a reduction in the porosity of polymers [46].
Nevertheless, it is important to highlight that graphene is not yet applied on a large scale for a few reasons.Firstly, achieving maximum purity in its production process seems to be quite challenging [38].Moreover, the production cost is currently a major barrier to scalability [47].Finally, storage, transportation and health and safety issues are also current concerning topics [48].

Carbon Nanotubes
Carbon nanotubes (CNTs) are basically rolled-up graphene sheets.These CNTs might have one and two or more concentric layers, respectively, called single-walled (SWCNT) or multi-walled (MWCNT).Although the property enhancement from the inherent carbon interatomic bonds is retained, reinforcement by CNTs is unidirectional, unlike the 2D graphene sheets.As such, the reinforcement is most advantageous if the CNTs are aligned in the direction of the applied load.The elastic strain energies of CNTs, however, tend to be affected due to the curvature of the carbon bonds in the tubular shape [46].Biomimetic hierarchical composites using fibers and CNTs have also been investigated [49-52], with inconclusive results.Aside from the mechanical improvements that this filler confers to composites, it is important to underline their influence over thermal and electrical properties [51], enabling applications in the fields of nanoelectronic devices, medicine and defense [41][42][43][44][45][46][47].

Nanocellulose
Cellulosic fillers are amongst the most abundant, sustainable, biodegradable and cheap nanofillers.Its main nanometric forms are cellulose nanocrystals (CNC) and nanofibrillated cellulose (NFC).Cellulose nanocrystals, for instance, are obtained from cellulosic fibers by extracting nanofibrils, removing hemicellulose and lignin through bleaching, and then removing the amorphous regions linking each CNC by acid hydrolysis [53].
Recent research has shown that the addition of CNC to polymeric coatings significantly inhibits corrosion in mild steel, increases the elastic modulus and impact strength in rubber vulcanization and may also be effective in producing tough and resilient polyacrylamidematrix hydrogels [54][55][56].
NFCs are the structures immediately above CNCs, being basically nanometric in diameter and micrometric in length and including the amorphous regions between two CNCs.Because it is longer and more flexible than CNCs, NFCs have some advantages, such as their longer strain to fracture and higher surface area, which enables them to provide a good adhesion with the polymeric matrix.An example of a recent application promoted by this feature is the use of NFC aerogel for developing wearable strain sensors [57].

Dispersion Methods of Nanoparticles in Polymeric Matrices
Physical processing methods for dispersing nanoparticles in a polymeric matrix are currently of significant interest.Melt-blending and solvent processing, assisted by highshear mixing or sonication, have been favored in important research contributions due to simplicity and compatibility with standard industrial techniques.Melt blending is one of the most economical methods for fabricating nanocomposites and is the easiest to scale up for a wide range of polymeric composites [47,[58][59][60][61][62][63][64][65][66][67][68][69][70].The method obviously requires temperatures typically higher than the melting point of the polymeric matrix.
The dispersion of the nanoparticle fillers within the polymeric matrix can be achieved by applying high shear forces using, for example, single-or twin-screw extruders or compounders, which help to break apart the filler agglomerates.Its simplicity makes it attractive for industrial-scale processes; however, many parameters (rotation speed of the twin-screw, temperature, high-shear mixers, hydrodynamics, etc.) must still be fine-tuned to optimize the results.However, it has also been observed that adhesion between the matrix and filler is not favored in melt processing, as compared to the very strong interaction between individual nanoparticles [47].While melt blending of hard-to-wet, untreated carbon nanotubes (CNTs) and graphene has shown modest improvement in dispersion, the melt mixing of montmorillonite clay with polymer has met with great success.The chemical functional groups at the silicate surface of the individual clay platelets might explain the observed greater affinity that results in better adhesion with the matrix.
On the other hand, solvent processing coupled with mechanical mixing or sonication is also suitable for obtaining nanocomposites [9,11,47,60,63,[71][72][73][74][75][76][77][78][79][80][81].The technique involves dispersing the nanoparticles in a polymer that is firstly dissolved in a compatible solvent.The breaking up of particle agglomerates is promoted by either a turbulent flow or the formation of cavitation bubbles.The major disadvantage, of course, is that the method is limited to polymeric materials that can be easily dissolved.In addition, the solvents used in the sonication process need to have low viscosity (e.g., acetone, distilled water, ethanol [63]).
Although these physical methods are supposed to be envisioned to benefit industrial processing, it is worth mentioning that most of the studies pointed out that a fully homogeneous dispersion of nanoparticles is difficult to attain unless a rather low content of nanoparticles (typically < 1.0 wt.%) is used.Furthermore, the mixing or processing conditions may need to be tailored by introducing compatibilizers or surfactants to improve the quality of filler dispersion.The recent review by Boon et al. [80] suggested that ionic surfactants should be used for CNTs in aqueous mixtures, whereas nonionic surfactants are preferred for mixtures with organic solvents.Some of the ionic surfactants used with carbon nanotubes include sodium dodecyl sulfate (SDS), dodecyl-benzene Polymers 2023, 15, 3707 6 of 26 sodium sulfonate (NaDDBS) and polyvinyl pyrrolidone (PVP), while nonionic ones include polyoxyethylene-8-lauryl and Tergitol NP-7 [80][81][82].
Further immobilization can be obtained if the particles are dispersed within the monomers prior to polymerization [47,58,60,63,65,68,78,79,[91][92][93][94][95][96].One of the advantages of this technique is that it allows the grafting of polymeric molecules onto the surface of the fillers, which also leads to a better dispersion of hard-to-wet nanomaterials.
In contrast to the physical methods, such an in-situ polymerization route offers the added advantage of higher nanoparticle loads.
The effects of the nanoparticle dispersion method on the physical and chemical structures of composites are widely reported in the literature.Observed changes in glass transition temperatures (T g ), for instance, were attributed to a loss in the flexibility of polymer chain segments resulting from the particle-matrix interaction, which relies on well-dispersed nanoparticles within the matrix.The selected results are shown in Table 1.
Table 1.Summary on the effect of particle dispersion method on the glass transition temperature (T g ) of epoxy nanocomposites relative to that of the matrix.Combining nanoparticles with other nanofillers to make hybrid composites has been shown to help the preparation of polymer-particle mixtures [84,91,98].Coupled with a chemical or a mechanical method, this technique emphasizes the potential of a percolated network of hybrid filling nanomaterials, leading to a significant improvement in nanoparticle dispersion within the polymeric matrix [99].When compared to functionalization, hybridization typically comes at a cost; however, it eases the fabrication process and has become a promising way to counterbalance some of the manufacturing challenges.

Nanoparticles
From a more applied perspective, the applicability of these concepts to the development of structural fiber-reinforced polymeric nanocomposites still needs to be assessed and demonstrated.

Particle Effects on Composite Thermomechanical Properties
Aside from the complexities summarized above for the particle incorporation processes, published research not only shows that the properties of nanostructured polymers may be altered by the addition of nanofillers but also that those changes depend on the particle type, size and content [87,98].The major trends on this issue are described in what follows.
To enable a consistent comparison among the variables involved, the polymer matrix was chosen for the epoxy resin, the four most cited nanoparticles, namely TiO 2 , SiO 2 , GO and GNP, were selected, and three properties, namely tensile strength (TS), storage (elastic) modulus (E') and glass transition temperature (T g ), were studied.
The titania (TiO 2 ) particles are, indeed, the most commonly used particles found in the literature, with more abundant data and several industrial applications.Similarly, epoxy DGEBA is the most common polymeric matrix considered in these studies, given its wide usage in industry, easy access to and ability to disperse particles within low-viscosity media [11].
The shift in the selected properties as a function of particle type, size and content (wt.%) was referred to (linearly normalized by) that of the corresponding epoxy matrix alone and expressed as a percentage (∆%) in order to level the information extracted from the literature.Table 2 summarizes the tensile strength (TS) data collected.In a similar way, Table 3 presents the data collected for the corresponding storage (elastic) modulus (E', related to the loss, or viscous, modulus E" by tan δ = E"/E') and the glass transition temperature (T g ) for the same nanoparticles and the epoxy polymer matrix.* Is for GO and GNP; T is the sheet thickness; W is the width; L is the length.
In order to enable a concise yet representative analysis, the discussion that follows was narrowed down to titanium dioxide (TiO 2 ) particles.
To normalize the information gathered from different authors, the influence of the particle size and weight content on properties was described in terms of the surface area.The specific surface area (SSA), commonly expressed in m 2 /g, can be calculated from the particle size, assuming a spherical shape for the oxide particles (Equation ( 1)) [134].

Effect on Tensile Strength
Figure 1 was constructed from the selected data listed earlier in Table 2, which is plotted in the way usually found in the literature, i.e., as a function of the weight content of the added particles.As there is no known mathematical relationship (i.e., dependency law) between the tensile strength and added particle content, data points were simply linked by smoothed lines to help the discussion.Figure 1 shows that, regardless of the TiO2 content and SSA, the composite's TS was always higher than that of the corresponding epoxy matrix.As a common trend, Figure 1 also shows that there is an optimal particle content that results in a maximum TS gain.Above the optimal content value, the TS gain drops, apparently due to ineffective particle dispersion: the higher the TiO2 content, the harder it is to avoid particle agglomeration.More importantly, it appears that the optimal particle content tends to decrease as the corresponding SSA increases, i.e., higher TS gains might be obtained with lower contents of smaller particles (high SSA).
Figure 1 also highlights several inconsistencies, pointed out as follows.The gain in TS was markedly influenced by the SSA for any given particle weight content.This could be attributed to the influence of the dispersion method on TS, which can be illustrated at a fixed TiO2 content of 1 wt.%.As the SSA increased from 6.4 to 28.4 m 2 /g, TS increased from ~2 to ~7%.On the contrary, when the SSA increased from 28.4 to 56.7 and then to 83.4 m 2 /g, the gain in TS increased from ~7 to ~18 and then fell to ~13%.This drop in TS could suggest the existence of an optimal particle size (or SSA) for the best property gain using a particular added particle weight content.However, it might also be attributed to the structural changes due to differences in particle dispersion, given that, for 56.7 m 2 /g, TiO2 particles were incorporated into the epoxy by sonication [102] whereas, for 83.4 m 2 /g, by using a moderate-speed mixing method [100].As discussed in Section 3, sonication induces the generation of collapsible cavitation bubbles, leading to a more effective dispersion.
The doubtful confidence on the effect of added particle content can be illustrated by referring to the two different authors (Table 2, [100,101]) that explored similar 50 nm TiO2 particles (SSA of 28.4 m 2 /g) seeking to improve the tensile strength performance but used a different added particle content range.As seen in Figure 1, the resulting curves do not seem to match each other, even for the common particle content.
Thus, when specifically observing the particle influence on the structuring process, particle size (expressed as SSA) certainly is an important variable; however, particle crowding (namely expressed as the mean free path between them, g, and the particle number, NP) is not suitably translated by weight content, according to the majority representation shown in works with nanocomposites [136][137][138][139]. Figure 1 shows that, regardless of the TiO 2 content and SSA, the composite's TS was always higher than that of the corresponding epoxy matrix.As a common trend, Figure 1 also shows that there is an optimal particle content that results in a maximum TS gain.Above the optimal content value, the TS gain drops, apparently due to ineffective particle dispersion: the higher the TiO 2 content, the harder it is to avoid particle agglomeration.More importantly, it appears that the optimal particle content tends to decrease as the corresponding SSA increases, i.e., higher TS gains might be obtained with lower contents of smaller particles (high SSA).
Figure 1 also highlights several inconsistencies, pointed out as follows.The gain in TS was markedly influenced by the SSA for any given particle weight content.This could be attributed to the influence of the dispersion method on ∆TS, which can be illustrated at a fixed TiO 2 content of 1 wt.%.As the SSA increased from 6.4 to 28.4 m 2 /g, ∆TS increased from ~2 to ~7%.On the contrary, when the SSA increased from 28.4 to 56.7 and then to 83.4 m 2 /g, the gain in TS increased from ~7 to ~18 and then fell to ~13%.This drop in ∆TS could suggest the existence of an optimal particle size (or SSA) for the best property gain using a particular added particle weight content.However, it might also be attributed to the structural changes due to differences in particle dispersion, given that, for 56.7 m 2 /g, TiO 2 particles were incorporated into the epoxy by sonication [102] whereas, for 83.4 m 2 /g, by using a moderate-speed mixing method [100].As discussed in Section 3, sonication induces the generation of collapsible cavitation bubbles, leading to a more effective dispersion.
The doubtful confidence on the effect of added particle content can be illustrated by referring to the two different authors (Table 2, [100,101]) that explored similar 50 nm TiO 2 particles (SSA of 28.4 m 2 /g) seeking to improve the tensile strength performance but used a different added particle content range.As seen in Figure 1, the resulting curves do not seem to match each other, even for the common particle content.
Thus, when specifically observing the particle influence on the structuring process, particle size (expressed as SSA) certainly is an important variable; however, particle crowding (namely expressed as the mean free path between them, g, and the particle number, Polymers 2023, 15, 3707 11 of 26 NP) is not suitably translated by weight content, according to the majority representation shown in works with nanocomposites [136][137][138][139].
The number of particles, NP, can be calculated (again assuming identical spherical particles) through the volume fraction, V p , from the corresponding weight fraction, as seen in Equations ( 2) and [140].
In Equation ( 2), ρ is the density, w is the weight fraction, and subscripts m and p refer to the matrix and particle, respectively.In Equation ( 3), V sp is the volume of each particle.
Figure 2 was constructed using the commercial software Digimat ® (Version 2021-3 by Hexagon) and the Random Fiber Placement algorithm (the algorithm places the particles at random positions within the cube until the specified volumetric fraction is reached), illustrates the crowding of 0.05 wt.% TiO 2 particles in a 10 × 10 × 10 µm 3 volume as a function of the particle size.As the particle diameter is halved, the SSA duplicates (from Equation ( 1)); however, NP increases sharply (multiplies by 2 3 ).As seen in Figure 2, the same small weight content of well-dispersed TiO 2 particles could result in a homogeneous composite with properties that improve when the particle size decreases (i.e., SSA increases).However, Figure 2 clearly shows that downsizing can result in an excessive number of particles, whose crowding will bring about dispersion difficulties (e.g., agglomeration) and consequently, hindered properties.The number of particles, NP, can be calculated (again assuming identical spherical particles) through the volume fraction, Vp, from the corresponding weight fraction, as seen in Equations ( 2) and [140].
NP  / In Equation ( 2),  is the density, w is the weight fraction, and subscripts m and p refer to the matrix and particle, respectively.In Equation (3), Vsp is the volume of each particle.
Figure 2 was constructed using the commercial software Digimat ® (Version 2021-3 by Hexagon) and the Random Fiber Placement algorithm (the algorithm places the particles at random positions within the cube until the specified volumetric fraction is reached), illustrates the crowding of 0.05 wt.% TiO2 particles in a 10 × 10 × 10 μm 3 volume as a function of the particle size.As the particle diameter is halved, the SSA duplicates (from Equation ( 1)); however, NP increases sharply (multiplies by 2 3 ).As seen in Figure 2, the same small weight content of well-dispersed TiO2 particles could result in a homogeneous composite with properties that improve when the particle size decreases (i.e., SSA increases).However, Figure 2 clearly shows that downsizing can result in an excessive number of particles, whose crowding will bring about dispersion difficulties (e.g., agglomeration) and consequently, hindered properties.Particle crowding can be assessed through the mean free path among particles, i.e., the distance between the surface of the closest particles.To calculate the mean free path, a specific particle arrangement needs to be assumed.To this aim, the concept of atomic packing factor is what most resembles the studied conditions.The densest particle packing, corresponding to the face-centered cubic arrangement (74 vol.%), was chosen (Figure 3), assuming, yet again, spherical particles.This would be the stringiest particle crowding condition, i.e., in actual homogeneous, well-dispersed composites, g would present higher values.Particle crowding can be assessed through the mean free path among particles, i.e., the distance between the surface of the closest particles.To calculate the mean free path, a specific particle arrangement needs to be assumed.To this aim, the concept of atomic packing factor is what most resembles the studied conditions.The densest particle packing, corresponding to the face-centered cubic arrangement (74 vol.%), was chosen (Figure 3), assuming, yet again, spherical particles.This would be the stringiest particle crowding condition, i.e., in actual homogeneous, well-dispersed composites, g would present higher values.
Figure 3 shows the arrangement of particles for the face-centered cubic packing and highlights the mean free path (g) as the distance between the closest spheres, which can be Polymers 2023, 15, 3707 12 of 26 calculated by Equation ( 4) [134].In Equation ( 4), d is the particle diameter, and V p is the volume fraction of particles.Particle crowding can be assessed through the mean free path among particles, i.e., the distance between the surface of the closest particles.To calculate the mean free path, a specific particle arrangement needs to be assumed.To this aim, the concept of atomic packing factor is what most resembles the studied conditions.The densest particle packing, corresponding to the face-centered cubic arrangement (74 vol.%), was chosen (Figure 3), assuming, yet again, spherical particles.This would be the stringiest particle crowding condition, i.e., in actual homogeneous, well-dispersed composites, g would present higher values.The number of particles for a given mass percentage is directly proportional to its diameter, d.The smaller the diameter, the greater the number of particles and, consequently, the smaller the distance g among them.In the case of this review, the values of mass percentages and particle diameters were given by the authors, requiring the transformation of this information to a volumetric percentage and, later, the relative number of particles.
Thus, smaller particles can stand much closer than larger particles (g is directly proportional to the particle size).Hence, there is a negative power law dependency between g and the number of particles, NP, or an inverse linear relationship between log (g) and log (NP).The values calculated for the number of particles, mean free path and specific surface area for the TiO 2 particles used earlier in Table 2 and Figure 1 are presented in Table 4 and expressed graphically in Figure 4. of particles.Thus, smaller particles can stand much closer than larger particles (g is directly proportional to the particle size).Hence, there is a negative power law dependency between g and the number of particles, NP, or an inverse linear relationship between log (g) and log (NP).The values calculated for the number of particles, mean free path and specific surface area for the TiO2 particles used earlier in Table 2 and Figure 1 are presented in Table 4 and expressed graphically in Figure 4.

Table 4.
Comparison of relative changes in tensile strength (ΔTS) for TiO2-epoxy nanocomposites as a function of particle size and added particle content (particle number (NP), mean free path (g) and specific surface area (SSA) were calculated).Figure 4 allows a clearer analysis of the effect of TiO 2 particles on epoxy compared to that shown in Figure 1.The dependency of the tensile strength variation with the particle number enables a distinct separation of the ∆TS maxima and brings to evidence the influence of the surface area (i.e., particle size and content), which was not apparent in Figure 1.

Size (nm) Content (wt.%) ΔTS
The first point worth mentioning is that when the property is plotted as a function of NP, a better overlap can be seen for the two 28.4 m 2 /g curves, despite the particular processing parameters considered in each study [100,101], which the previous and traditional form of portraying data (in wt.%, as seen in Figure 1) did not show.Although the maximum on the property curves was seen before, the representation in Figure 4 clearly separates such maxima, bringing to evidence their dependence on the specific surface area.In other words, for comparatively large particles (i.e., low SSA), the number of particles required to reach the maximum is low; as the particles are downsized (i.e., increasing SSA), the NP for the maximum also increases.Moreover, Figure 4 shows that the maxima on ∆TS are clearly higher for the lower particle sizes while suggesting that there might be an optimal particle size (or SSA) for the maximum property gain, as envisioned earlier in Figure 1.When Figures 1 and 4 are analyzed together, as the particle size decreases, the weight content needed for the maximum property gain also decreases; however, the number of particles increases, i.e., the maxima move backwards in Figure 1 and forwards in Figure 4.This is a direct consequence of the correlation between volume and weight through density for any given type of material, yet it also hints at the possibility that the property gain might be dependent on the particles' total surface area.In other words, smaller particles would be needed in lower-weight contents to reach a comparable property gain.Figure 5 depicts such an exercise, as it gathers all data in Table 4 by plotting ∆TS as a function of the particles' total surface area (i.e., weight content × SSA).Despite the scatter in the data, the dashed line in Figure 5 represents the general trend among the data points (moving average trend line), suggesting that the location of the maxima observed earlier in Figures 1 and 4 might be just fortuitous.
Thus, it is desirable to find an alternative way to access the property's underlying mechanism.When the property gain is plotted as a function of g (Figure 6), a trend similar to that for NP can be observed, i.e., the distance among larger particles (with a lower SSA) for the best property improvement is higher than that for smaller particles (with a higher SSA).Thus, it is desirable to find an alternative way to access the property's underlying mechanism.When the property gain is plotted as a function of g (Figure 6), a trend similar to that for NP can be observed, i.e., the distance among larger particles (with a lower SSA) for the best property improvement is higher than that for smaller particles (with a higher SSA).The expected relationship between g and NP at the property maximum, i.e., the negative power law relationship (or the linear relationship between log NP and log g), can be observed in Figure 7a.More important, however, is the positive power law relationship between g and d at the property maximum, which can be seen in Figure 7b.If the strengthening of the polymer matrix relies on its texturing around the filler particles, it would have been expected that the maximum benefit would correspond to a particular optimum distance among particles, i.e., to a common g value.That is not what Figure 7b shows.Indeed, Figure 7b suggests that the "affected" matrix volume surrounding each particle depends  Thus, it is desirable to find an alternative way to access the property's underlying mechanism.When the property gain is plotted as a function of g (Figure 6), a trend similar to that for NP can be observed, i.e., the distance among larger particles (with a lower SSA) for the best property improvement is higher than that for smaller particles (with a higher SSA).The expected relationship between g and NP at the property maximum, i.e., the negative power law relationship (or the linear relationship between log NP and log g), can be observed in Figure 7a.More important, however, is the positive power law relationship between g and d at the property maximum, which can be seen in Figure 7b.If the strengthening of the polymer matrix relies on its texturing around the filler particles, it would have been expected that the maximum benefit would correspond to a particular optimum distance among particles, i.e., to a common g value.That is not what Figure 7b shows.Indeed, Figure 7b suggests that the "affected" matrix volume surrounding each particle depends The expected relationship between g and NP at the property maximum, i.e., the negative power law relationship (or the linear relationship between log NP and log g), can be observed in Figure 7a.More important, however, is the positive power law relationship between g and d at the property maximum, which can be seen in Figure 7b.If the strengthening of the polymer matrix relies on its texturing around the filler particles, it would have been expected that the maximum benefit would correspond to a particular optimum distance among particles, i.e., to a common g value.That is not what Figure 7b shows.Indeed, Figure 7b suggests that the "affected" matrix volume surrounding each particle depends on the particle size and increases with it.At the point of maximum gain, larger particles will be surrounded by a thicker affected matrix layer than smaller particles and will then need to stand further apart from each other.
on the particle size and increases with it.At the point of maximum gain, larger particles will be surrounded by a thicker affected matrix layer than smaller particles and will then need to stand further apart from each other.Having no direct experimental access to the values of g, it is important to determine if the enhanced contrast provided by this alternative form of representation also works for other properties.

Effect on Dynamic Mechanical Properties
By the fact that it is a viscoelastic material, the nanocomposite needs evaluations both in the viscous phase and elastic.Because it allows the evaluation of properties in the temperature domain and because it offers mechanical excitations compatible with the scale of the new material (nanometers), a large number of researchers [141][142][143][144][145] use DMA to determine the storage modulus parameters (E'), loss modulus (E") and glass transition temperature (Tg).Therefore, extending the reasoning just discussed for the tensile strength, the following analysis considers the results for E' and Tg, obtained from dynamic mechanical analysis tests (DMA).Table 5 presents data for the TiO2 particles selected from those listed earlier in Table 3.
Table 5.Comparison of relative change in storage modulus (ΔE') and glass transition temperature (ΔTg) for TiO2-epoxy nanocomposites as a function of particle size and added particle content (particle number (NP), mean free path (g) and specific surface area (SSA) were calculated).Having no direct experimental access to the values of g, it is important to determine if the enhanced contrast provided by this alternative form of representation also works for other properties.

Effect on Dynamic Mechanical Properties
By the fact that it is a viscoelastic material, the nanocomposite needs evaluations both in the viscous phase and elastic.Because it allows the evaluation of properties in the temperature domain and because it offers mechanical excitations compatible with the scale of the new material (nanometers), a large number of researchers [141][142][143][144][145] use DMA to determine the storage modulus parameters (E'), loss modulus (E") and glass transition temperature (T g ).Therefore, extending the reasoning just discussed for the tensile strength, the following analysis considers the results for E' and T g , obtained from dynamic mechanical analysis tests (DMA).Table 5 presents data for the TiO 2 particles selected from those listed earlier in Table 3.
Table 5.Comparison of relative change in storage modulus (∆E') and glass transition temperature (∆T g ) for TiO 2 -epoxy nanocomposites as a function of particle size and added particle content (particle number (NP), mean free path (g) and specific surface area (SSA) were calculated).The first point to note is that there should not be negative gains in the composite modulus.The addition of inorganic (crystalline) particles to a softer polymeric matrix should always result in stiffness gain.Such a discrepancy is frequently attributed to processing difficulties (e.g., bad dispersion, lack of adhesion), meaning that the added particles behave as defects or impurities rather than playing the expected role of strengthening aids.Nevertheless, available data as those listed in Table 5, are scarce and, bearing the above in mind, they were used in the discussion, all the same.

Size (nm
The composites' storage modulus variations (∆E') as a function of TiO 2 weight content and number of particles are presented in Figure 8a,b, respectively.As seen for the tensile strength analysis, both representations show that the storage modulus gain (∆E') for each different SSA increases up to a maximum value and then decreases.The first point to note is that there should not be negative gains in the composite modulus.The addition of inorganic (crystalline) particles to a softer polymeric matrix should always result in stiffness gain.Such a discrepancy is frequently attributed to processing difficulties (e.g., bad dispersion, lack of adhesion), meaning that the added particles behave as defects or impurities rather than playing the expected role of strengthening aids.Nevertheless, available data as those listed in Table 5, are scarce and, bearing the above in mind, they were used in the discussion, all the same.
The composites' storage modulus variations (E') as a function of TiO2 weight content and number of particles are presented in Figure 8 a, b, respectively.As seen for the tensile strength analysis, both representations show that the storage modulus gain (∆E') for each different SSA increases up to a maximum value and then decreases.Figure 8a shows that the highest gain (52%) was reached with small contents of large particles (small SSA, 5.7 m 2 /g).Composites with smaller particles (higher SSA) tend to present their best property at higher particle contents.This effect is clear for the SSA values of 35.5 m 2 /g and 67.5 m 2 /g, where the variation in storage modulus is negative, i.e., despite the awkwardness of this concept, it means that the composite is worse than the epoxy alone.This effect becomes much clearer from the perspective of the number of particles, shown in Figure 8b.For instance, considering the particles with 5.7 m 2 /g SSA, a small number of particles results in a high gain for this composite.The others, prepared with smaller particles (i.e., higher SSA), even if added in the same weight content, need a much higher number of particles, with the dispersion difficulties that entails, to reach their best increment in modulus.
Still, in Figure 8b, another trend can be observed, which is that the peaks of the four curves nearly fall on a straight line, also shown in the graph, suggesting an inverse proportionality between the modulus gain and the log (NP).This is to say that, despite the size gap between the micro and nano size ranges in the data, a high number of particles with high specific surface areas do not provide significant stiffness gains.As it is, the reported low stiffness improvements (and perhaps also the negative gains mentioned above), generally attributed to high SSA, hence poor dispersion, might be the result of particle crowding (particles too close together), which can be evaluated in terms of the mean free path, g, as shown in Figure 9.
Figure 8a shows that the highest gain (52%) was reached with small contents of large particles (small SSA, 5.7 m 2 /g).Composites with smaller particles (higher SSA) tend to present their best property at higher particle contents.This effect is clear for the SSA values of 35.5 m 2 /g and 67.5 m 2 /g, where the variation in storage modulus is negative, i.e., despite the awkwardness of this concept, it means that the composite is worse than the epoxy alone.
This effect becomes much clearer from the perspective of the number of particles, shown in Figure 8b.For instance, considering the particles with 5.7 m 2 /g SSA, a small number of particles results in a high gain for this composite.The others, prepared with smaller particles (i.e., higher SSA), even if added in the same weight content, need a much higher number of particles, with the dispersion difficulties that entails, to reach their best increment in modulus.
Still, in Figure 8b, another trend can be observed, which is that the peaks of the four curves nearly fall on a straight line, also shown in the graph, suggesting an inverse proportionality between the modulus gain and the log (NP).This is to say that, despite the size gap between the micro and nano size ranges in the data, a high number of particles with high specific surface areas do not provide significant stiffness gains.As it is, the reported low stiffness improvements (and perhaps also the negative gains mentioned above), generally attributed to high SSA, hence poor dispersion, might be the result of particle crowding (particles too close together), which can be evaluated in terms of the mean free path, g, as shown in Figure 9. Figure 9 clearly explains the better performance of the 5.7 m 2 /g composite because the average distance among its particles is noticeably higher than in the others.In addition, the height of the peak is also associated with the g value, i.e., higher maxima occur for a lower SSA and higher g.This representation also evidences that when the particle size drops to a few tens of nanometers, the effect of the distance between them seems to overcome the effect of their number, i.e., the individual curves become overlapped.
The relationships between g and NP at the property maximum (i.e., gmax and NPmax) and between gmax and the particle size, d, can be seen in Figure 10.To help the discussion, the data shown earlier in Figure 7 is included again.The expected negative power law relationship between NPmax and gmax (or the linear relationship between log NPmax and log gmax), as well as the positive power law relationship between gmax and d at the property maximum, can also be seen for the storage modulus.Figure 9 clearly explains the better performance of the 5.7 m 2 /g composite because the average distance among its particles is noticeably higher than in the others.In addition, the height of the peak is also associated with the g value, i.e., higher maxima occur for a lower SSA and higher g.This representation also evidences that when the particle size drops to a few tens of nanometers, the effect of the distance between them seems to overcome the effect of their number, i.e., the individual curves become overlapped.
The relationships between g and NP at the property maximum (i.e., g max and NP max ) and between g max and the particle size, d, can be seen in Figure 10.To help the discussion, the data shown earlier in Figure 7 is included again.The expected negative power law relationship between NP max and g max (or the linear relationship between log NP max and log g max ), as well as the positive power law relationship between g max and d at the property maximum, can also be seen for the storage modulus.
It can be seen in Figure 10a,b that the relationships between the interparticle distance at the property maxima (g max ) and the particle size (d), as well as between g max and NP max , are, for all practical purposes, the same for ∆TS and ∆E', supporting the inferred presence of an "affected" matrix volume surrounding each particle, which depends on the particle size and increases with it.However, because the dependence of the stiffness E' on the number of particles was found to be contrary to that of the strength, TS, it appears that, as g decreases (more particles in the polymeric matrix), the polymeric chains in these narrow gaps likely become more oriented and it would be easier for them to slip past each other (easier movement).This apparently results in a more flexible composite (a decrease in E') but one with increased strength, TS (and maximum strain).It can be seen in Figure 10 a, b that the relationships between the interparticle distance at the property maxima (gmax) and the particle size (d), as well as between gmax and NPmax, are, for all practical purposes, the same for TS and E', supporting the inferred presence of an "affected" matrix volume surrounding each particle, which depends on the particle size and increases with it.However, because the dependence of the stiffness E' on the number of particles was found to be contrary to that of the strength, TS, it appears that, as g decreases (more particles in the polymeric matrix), the polymeric chains in these narrow gaps likely become more oriented and it would be easier for them to slip past each other (easier movement).This apparently results in a more flexible composite (a decrease in E') but one with increased strength, TS (and maximum strain).

Glass Transition Temperature
To further clarify the influence of TiO 2 -particle downsizing on the structure-property relationships in TiO 2 -filled epoxy composites, reported data (selected from Table 5) for the glass transition temperature (T g ) was similarly explored.Figure 11a,b depicts the T g variation as a function of the TiO 2 content and the number of particles, respectively.
Figure 11a shows that T g seems to improve for the high SSA (i.e., for particles with smaller sizes), as seen earlier for the tensile strength (Figures 1 and 4) but is contrary to the behavior of the storage modulus seen in Figure 8a.However, the curves appear to be flatter, and more so for the higher SSA, i.e., less sensitivity to the particles' weight content.
From the perspective of the number of particles, shown in Figure 11b, the effect of lower thermal stability for the nanocomposites with bigger particles becomes even more evident.The difference in the size of particles with 44.3 and 35.5 m 2 /g SSA (32 and 40 nm, respectively) appears to be not significant enough to change their number for the highest property improvement despite the different weight contents that were used.This analysis would suggest that the mechanisms for property enhancement of epoxy through the dispersion of nanoparticles are different for storage modulus and glass transition temperature.Thus, it is possible that the increase in T g may be related to a reduction in the mean free path among particles, hence being favored by smaller particles that can stand closer to each other, thus promoting the loss in the flexibility of polymer chains that seems to favor T g .

Glass Transition Temperature
To further clarify the influence of TiO2-particle downsizing on the structure-property relationships in TiO2-filled epoxy composites, reported data (selected from Table 5) for the glass transition temperature (Tg) was similarly explored.Figure 11a  Figure 11a shows that Tg seems to improve for the high SSA (i.e., for particles with smaller sizes), as seen earlier for the tensile strength (Figures 1 and 4) but is contrary to the behavior of the storage modulus seen in Figure 8a.However, the curves appear to be flatter, and more so for the higher SSA, i.e., less sensitivity to the particles' weight content.
From the perspective of the number of particles, shown in Figure 11b, the effect of lower thermal stability for the nanocomposites with bigger particles becomes even more evident.The difference in the size of particles with 44.3 and 35.5 m 2 /g SSA (32 and 40 nm, respectively) appears to be not significant enough to change their number for the highest property improvement despite the different weight contents that were used.

Conclusions and Outlook
Technological advances have allowed the development of new nanoparticles and the improvement of processing techniques for the preparation and design of nanocomposites, and despite the fact that nanocomposites have been studied for decades, several authors have recently devoted their work to providing new findings in various fields, such as ballistics [146], sensors [147], water treatment [148], conducting polymer composites [149], synergetic effects of self/induced crystallization and nanoparticles on the mechanical properties of nanocomposites [150], mechanical properties increment based on carbon nanoparticles, nanosilicon, and cobalt [111,[151][152][153][154], besides thorough, updated, state-ofthe-art reviews [155,156].The chronology of the articles reviewed from the literature clearly depicts this trend, as added nanoparticles with progressively smaller sizes are being investigated, and the consequent exponential gains in the properties of polymeric matrix composites are being reported.This work analyzed the particular case of titania nanoparticles added to epoxy matrices, seeking a better understanding of the observed improvement in important thermomechanical properties, namely the tensile stress (TS), the storage (elastic) modulus (E') and the glass transition temperature (T g ).
The reported composite property improvement due to the simple decrease in particle size, the so-called "scaling effect", is attributed to surface energy effects, as the specific surface area of a given weight content of particles of the same material directly increases as their size decreases.What appears to have been overlooked so far is that the resulting number of smaller particles increases much faster, and what might have seemed a tiny weight of added particles is, in fact, a huge particle number and a very crowded system whose processing difficulties, often directly linked to a good dispersibility, might be tremendous, upsetting the delicate balance between best performance and economic viability.
It should be remembered that real particles may have a variety of sizes (represented by a flatter particle size distribution curve) and may agglomerate before or after addition to the polymeric matrices, all of which hinder homogeneous dispersions.These processing steps are very challenging, even with current technology.It would, therefore, not be hard to accept that the experimental results reported in the literature could be impaired by heterogeneous dispersions and/or agglomerations of uneven particles, resulting in composites that are prone to premature failure.
The significance of considering tiny amounts of smaller and well-dispersed particles within the polymer matrix was highlighted in this work in terms of the number of particles, NP, needed to reach the highest property gain for a given particle size, d (or specific surface area) and the likely mean free path (distance), g, between the closest particles, i.e., particle crowding.For all practical purposes, the same positive power law relationships between g and NP, as well as between g and d, at the property maximum were identified both for TS and E', suggesting that matrix texturing around the particles increases with their size.However, the dependence of the stiffness on the number of particles was found to be contrary to that of the strength, suggesting that, as g decreases (more particles in the polymeric matrix), the likely forced orientation of polymeric chains apparently prompts easier slipping, resulting in a more flexible composite-but one with increased strength (and maximum strain).Surprisingly, the glass transition temperature appears to be less sensitive to particle weight content or crowding, being simply favored by smaller particles that can stand closer to each other, hence promoting the loss in the flexibility of polymer chains that favor the increase in T g .
Nowadays, given the varied techniques and materials used by each author, normalizing the results through particle crowding (the number of particles and distance among them) brings evidence that the particle size and particle content should be downscaled together and has an interesting potential to better compare the laboratory results and further the knowledge on such important processing-structure-property relationships.
Polymers 2023, 15, x FOR PEER REVIEW 10 of 27 law) between the tensile strength and added particle content, data points were simply linked by smoothed lines to help the discussion.

Figure 1 .
Figure 1.Tensile strength variation (ΔTS) for TiO2-filled epoxy composites as a function of TiO2 weight content for the particle size and SSA values shown.

Figure 1 .
Figure 1.Tensile strength variation (∆TS) for TiO 2 -filled epoxy composites as a function of TiO 2 weight content for the particle size and SSA values shown.

Figure 2 .
Figure 2. Dispersion of 0.05 wt.% TiO2 within a 10 × 10 × 10 μm 3 volume illustrating particle crowding as a function of TiO2 particle diameter (software: Digimat ® with Random Fiber Placement algorithm).(a)-(e) show the particle number progressively increasing as much as the particle diameter reduces, even though the amount of mass has remained unchanged.

Figure 2 .
Figure 2. Dispersion of 0.05 wt.% TiO 2 within a 10 × 10 × 10 µm 3 volume illustrating particle crowding as a function of TiO 2 particle diameter (software: Digimat ® with Random Fiber Placement algorithm).(a-e) show the particle number progressively increasing as much as the particle diameter reduces, even though the amount of mass has remained unchanged.

Figure 2 .
Figure 2. Dispersion of 0.05 wt.% TiO2 within a 10 × 10 × 10 μm 3 volume illustrating particle crowding as a function of TiO2 particle diameter (software: Digimat ® with Random Fiber Placement algorithm).(a)-(e) show the particle number progressively increasing as much as the particle diameter reduces, even though the amount of mass has remained unchanged.

Figure 3 .
Figure 3. Schematic representation of the FCC particle packing, showing how to calculate the mean free path, g (adapted from LibreTexts TM[33]).

Figure 4 .
Figure 4. Tensile strength variation (ΔTS) for TiO2-filled epoxy composites as a function of the number of TiO2 particles (NP) for the particle size and SSA values shown.

Figure 4 .
Figure 4. Tensile strength variation (∆TS) for TiO 2 -filled epoxy composites as a function of the number of TiO 2 particles (NP) for the particle size and SSA values shown.

Figure 5 .
Figure 5. Tensile strength variation (ΔTS) for TiO2-filled epoxy composites as a function of the TiO2 particles' total surface area.The dashed line represents the general trend among data points.

Figure 6 .
Figure 6.Tensile strength variation (ΔTS) for TiO2-filled epoxy composites as a function of the mean free path among TiO2 particles (g) for the particle size and SSA values shown.

Figure 5 .
Figure 5. Tensile strength variation (∆TS) for TiO 2 -filled epoxy composites as a function of the TiO 2 particles' total surface area.The dashed line represents the general trend among data points.

Figure 5 .
Figure 5. Tensile strength variation (ΔTS) for TiO2-filled epoxy composites as a function of the TiO2 particles' total surface area.The dashed line represents the general trend among data points.

Figure 6 .
Figure 6.Tensile strength variation (ΔTS) for TiO2-filled epoxy composites as a function of the mean free path among TiO2 particles (g) for the particle size and SSA values shown.

Figure 6 .
Figure 6.Tensile strength variation (∆TS) for TiO 2 -filled epoxy composites as a function of the mean free path among TiO 2 particles (g) for the particle size and SSA values shown.

Figure 7 .
Figure 7. Power law relationship observed at the maximum tensile strength gain for TiO2-filled epoxy composites between (a) particle number, NP, and mean free path, g, and (b) particle diameter, d, and mean free path, g.

Figure 7 .
Figure 7. Power law relationship observed at the maximum tensile strength gain for TiO 2 -filled epoxy composites between (a) particle number, NP, and mean free path, g, and (b) particle diameter, d, and mean free path, g.

Figure 8 .
Figure 8. Storage modulus variation (ΔE') for TiO2-filled epoxy composites as a function of (a) TiO2 weight content and (b) the number of TiO2 particles for the particle size and SSA values shown.The dashed line in (b) represents the common straight line among the ΔE' maxima.

Figure 8 .
Figure 8. Storage modulus variation (∆E') for TiO 2 -filled epoxy composites as a function of (a) TiO 2 weight content and (b) the number of TiO 2 particles for the particle size and SSA values shown.The dashed line in (b) represents the common straight line among the ∆E' maxima.

Figure 9 .
Figure 9. Storage modulus variation (ΔE') for TiO2-filled epoxy composites as a function of the mean free path among TiO2 particles (g) for the particle size and SSA values shown.

Figure 9 .
Figure 9. Storage modulus variation (∆E') for TiO 2 -filled epoxy composites as a function of the mean free path among TiO 2 particles (g) for the particle size and SSA values shown.

Figure 10 .
Figure 10.Power law relationship observed at the maximum property gain (TS and E') for TiO2filled epoxy composites, between (a) particle number, NP, and mean free path, g, and (b) particle diameter, d, and mean free path, g.To help the discussion, the data shown earlier in Figure 7 is included again.

Figure 10 .
Figure 10.Power law relationship observed at the maximum property gain (∆TS and ∆E') for TiO 2filled epoxy composites, between (a) particle number, NP, and mean free path, g, and (b) particle diameter, d, and mean free path, g.To help the discussion, the data shown earlier in Figure 7 is included again.
,b depicts the Tg variation as a function of the TiO2 content and the number of particles, respectively.

Figure 11 .
Figure 11.Glass transition temperature variation (ΔTg) for TiO2-filled epoxy composites as a function of (a) TiO2 weight content and (b) number of TiO2 particles for the particle size and SSA values shown.

Figure 11 .
Figure 11.Glass transition temperature variation (∆T g ) for TiO 2 -filled epoxy composites as a function of (a) TiO 2 weight content and (b) number of TiO 2 particles for the particle size and SSA values shown.

Table 2 .
Comparison of relative changes in tensile strength (∆TS) for epoxy matrix nanocomposites as a function of type, size and content of nanoparticles.
* For GO and GNP; T is the sheet thickness.

Table 3 .
Comparison of relative changes in storage modulus (E'), tan δ and glass transition temperature (T g ) for epoxy matrix nanocomposites as a function of type, size and content of nanoparticles.

Table 4 .
Comparison of relative changes in tensile strength (∆TS) for TiO 2 -epoxy nanocomposites as a function of particle size and added particle content (particle number (NP), mean free path (g) and specific surface area (SSA) were calculated).