# Towards Sophisticated 3D Interphase Modelling of Advanced Bionanocomposites via Atomic Force Microscopy

Authors: Mohanad Mousa; Yu Dong
Journal: Journal of Nanomaterials
 (2020)

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## Abstract

Nanomechanical properties and interphase dimensions of PVA bionanocomposites reinforced with halloysite nanotubes (HNTs) and Cloisite 30B montmorillonite (MMT) were evaluated by means of peak force quantitative nanomechanical mapping (PFQNM). A three-phase theoretical composite model was established based on hard-core–soft-shell structures consisting of hard mono-/polydispersed anisotropic particles and soft interphase and matrices. Halpin-Tsai model and Mori-Tanaka model were employed to predict experimentally determined tensile moduli of PVA bionanocomposites where effective volume fraction of randomly oriented nanoparticles resulted from the inclusion of interphase properties and volume fractions. Overall, it was suggested that the estimation of elastic modulus according to effective volume fraction of nanoparticles revealed better agreement with experimental data as opposed to that based upon their nominal volume fraction. In particular, the use of polydispersed HNTs and Cloisite 30B MMT clays with Fuller particulate gradation was proven to yield the best prediction when compared with experimental data among all proposed theoretical models. This study overcomes the neglected real interphase characteristics in modelling nanocomposite materials with much more accurate estimation of their mechanical properties.


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## Body

## 1. Introduction

In the last decade, polymer nanocomposites reinforced with a small amount of nanoparticles with different shapes and sizes in order to enhance multifunctional properties, cost effectiveness, and light-weight structures have become the major focus on the innovation of material development in plastics and composite materials. In a nanocomposite system, there are commonly three major nanofiller shapes including nanoplatelets such as MMT clays and graphene nanoplatelets, nanotubes like carbon nanotubes (CNTs) and HNTs, and nanospheroids including zinc oxide nanoparticles or nanosillica.The material merit of using a small amount of nanoparticles can significantly enhance mechanical, thermal, electronic, and antimicrobial properties [1–4] with decreased gas/liquid permeability, while some key features of net polymer systems such as low density and easy processibility [5] can still be well implemented. The applications of polymer nanocomposites have been widely expanded in aerospace and aircrafts, automobile manufacturing, food packaging, and medical devices [6]. It is well recognised that an interphase region in the vicinity of nanofillers or nanoparticles in polymer nanocomposites can have a dramatic property change from those of polymer matrices [5], which in turn plays a prevalent role in the overall enhancement of material performance. Interphase dimensions, volume, and properties can be identified as typical features, which depend primarily on matrix-filler interaction associated with nanoparticle structures and shapes in polymer nanocomposites [7]. Such matrix-filler interaction has been investigated by atomistic and coarse-grained molecular simulations [8] though the limitation of a computational environment may often restrict such simulation work to single-particle or two-particle systems [5, 8]. The PFQNM technique has a powerful characterisation capability to obtain the interphase dimensions and properties of nanocomposites based on various nanomechanical properties including elastic modulus and adhesion of nanocomposite constituents along with corresponding dimensions [7].In simple micromechanical modelling of composite materials using Halpin-Tsai model [9] and Mori-Tanaka model [10], interphase properties and features between fillers and matrices are generally neglected for simplicity. Consequently, the mechanical properties of polymer nanocomposites may be overpredicted based on such conventional composite theoretical models for polymer/CNT composites [11] and silk fibroin/graphene oxide nanocomposites [12], which was consistent with the numerical simulations using the finite element method (FEM) based on the representative volume element (RVE) for polyethylene (PE)/CNT nanocomposites [13, 14]. On the other hand, such properties can also be underestimated, as evidenced by polyamide 66 (PA66)/calcium carbonate (CaCO3) nanocomposites [15]. This drawback can inevitably inhibit accurate micromechanical modelling work in polymer nanocomposite systems owing to the typical lack of interphase properties and features. Despite many recent attempts, the incorporation of interphase regions as the third phase in available Mori-Tanaka model [16] and Maxwell model [17] reveals little success, arising from the simple assumption that the interphase can be only hypothesised as a 1D transitional material phase surrounding nanofillers with uniform interphase thickness and regular shapes. Whereas, it has been well documented that an irregular-shaped interphase with nonuniform thickness usually occurs in real morphological structures of composites [7, 18].In particular, when a real interphase effect is taken into account in the modelling work of nanocomposites, interphase volume fraction is one of the key interphase material parameters to be predetermined. With an additional 5 vol% of monodispersed spherical nanoparticles (particle diameter: 10 nm and interphase thickness: 0.5 nm), the interphase volume fraction can reach as high as 25 vol%, according to Mousa and Dong [7]. When particle diameter is further reduced to be only no more than 5 nm, the interphase volume fraction can be increased by over 50 vol%, as opposed to that of the particles [7]. Amraei et al. [19] established a closed-form interphase model as a function of radial distance using RVE, which was found to have the capability of estimating the effective mechanical properties of polymer nanocomposites reinforced with spherical inclusions. For the other work, Amraei et al. [20] particularly concentrated on a closed-form micromechanical interphase model in terms of CNT diameter and interphase thickness, as well as the mechanical properties of individual constituents to influence overall mechanical properties of polymer nanocomposites, indicating a critical role of the interphase. Moreover, Zare and Rhee [21] developed a new model to predict the tensile modulus of polymer/clay nanocomposites using not only clay size and morphology but also interfacial shear modulus in good accordance with experimental values for the model validation. The dimensions of clay thickness and length were revealed to be key factors for the enhancement level of nanocomposite modulus. Overall, a holistic understanding for the effect of interphase volume fraction on the elastic modulus of bulk nanocomposites still remains challenging when considering the complexity of interphase structures as well as a potential overlap to neighbouring interfacial layers.The aim of this study lies in the systematic development of a novel three-phase composite theoretical model by implementing experimentally measured 3D interphase nanomechanical properties in PVA bionanocomposites reinforced with HNTs and Cloisite 30B MMT clays as the most popular tubular and platelet-like clay nanofillers, respectively. Ultimately, missing real interphase characteristics, often encountered in conventional composite modelling approaches, can be recounted (rather than being completely ignored) for accurate property prediction in advanced nanocomposites.


## 2. Experimental Work

### 2.1. Materials and Fabrication Process

PVA, as a popular water-soluble polymer with a molecular weight of 89000-98000 g/mol and a degree of hydrolysis of 99.0-99.8%, was supplied by Sigma-Aldrich Pty. Ltd., Australia. Cloisite 30B clays modified with methyl, tallow, bis-2-hydroxyethyl, and quaternary ammonium were purchased from Southern Clay Products, Gonzales, TX, USA. HNTs with an outer diameter of 120-140 nm, an inner diameter of 15-100 nm, and a length of 0.3-1.5μm were donated by Imerys Tableware Limited, New Zealand [22].PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays were synthesised using solution casting following the same procedure mentioned in our previous study [1]. Five wt%/v PVA aqueous solution was prepared after mixing PVA powders with deionised water at a magnetic stirring speed of 400 rpm and a processing temperature of 90°C for 3 h to warrant the complete dissolution of PVA. In a similar manner, HNTs and Cloisite 30B clay powders were also mixed in deionised water using a mechanical mixer at 405 rpm and 40°C for 2 h. Subsequently, the ultrasonication process was carried out using an ultrasonicating bath (model: ELMA Ti-H-5, IKA; Germany) at an oscillatory frequency of 25 kHz and a power intensity of 70% at 40°C for 1 h. Aqueous suspensions of HNTs and Cloisite 30B clays were gradually added in a dropwise manner into the PVA solution by mechanical mixing in order to obtain particle contents of 0, 3, 5, and 10 wt%. A further sonication took place for 30 min to ensure a more uniform particle dispersion. In the end, 20 ml mixed solution was poured on a glass petri dish and subsequently dried in an air-circulating oven at 40°C for 48 h. Finally, prepared PVA bionanocomposite films were stored in a desiccator containing silica gels for subsequent material testing and characterisation.


### 2.2. Material Characterisation via Atomic Force Microscopy (AFM)

In this study, the nanomechanical properties of the interphase in PVA bionanocomposites were determined by means of peak force quantitative mapping (PFQM) in a Bruker Dimension FastScan AFM system to acquire a single force-distance curve under ambient conditions. In particular, an RTESPA-525A probe (nominal spring constant: 200 N/m; nominal tip radius: 8 nm; and nominal resonant frequency: 525 kHz) was employed to facilitate such direct measurements. During the calibration process, the deflection sensitivity was warranted with the acquisition of a force curve on a stiff sapphire-12 surface. The spring constant was determined by using a thermal tuning method [23] in which cantilever mechanical response to thermal agitation was measured with the Brownian motion of encompassing fluid molecules. Imaging analysis using atomic force microscopy (AFM) was carried out with the TESPA probe (normal spring constant: 40 N/m; tip radius: 8 nm). The scan rate was set up at 2 Hz with a digital pixel resolution of 256×256. Unwanted features including noise, bow, and tilt were removed when AFM images were first-order flattened using Bruker NanoScope 1.5 software. At least three AFM samples were utilised for each material batch to warrant the test reproducibility.


## 2.1. Materials and Fabrication Process

PVA, as a popular water-soluble polymer with a molecular weight of 89000-98000 g/mol and a degree of hydrolysis of 99.0-99.8%, was supplied by Sigma-Aldrich Pty. Ltd., Australia. Cloisite 30B clays modified with methyl, tallow, bis-2-hydroxyethyl, and quaternary ammonium were purchased from Southern Clay Products, Gonzales, TX, USA. HNTs with an outer diameter of 120-140 nm, an inner diameter of 15-100 nm, and a length of 0.3-1.5μm were donated by Imerys Tableware Limited, New Zealand [22].PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays were synthesised using solution casting following the same procedure mentioned in our previous study [1]. Five wt%/v PVA aqueous solution was prepared after mixing PVA powders with deionised water at a magnetic stirring speed of 400 rpm and a processing temperature of 90°C for 3 h to warrant the complete dissolution of PVA. In a similar manner, HNTs and Cloisite 30B clay powders were also mixed in deionised water using a mechanical mixer at 405 rpm and 40°C for 2 h. Subsequently, the ultrasonication process was carried out using an ultrasonicating bath (model: ELMA Ti-H-5, IKA; Germany) at an oscillatory frequency of 25 kHz and a power intensity of 70% at 40°C for 1 h. Aqueous suspensions of HNTs and Cloisite 30B clays were gradually added in a dropwise manner into the PVA solution by mechanical mixing in order to obtain particle contents of 0, 3, 5, and 10 wt%. A further sonication took place for 30 min to ensure a more uniform particle dispersion. In the end, 20 ml mixed solution was poured on a glass petri dish and subsequently dried in an air-circulating oven at 40°C for 48 h. Finally, prepared PVA bionanocomposite films were stored in a desiccator containing silica gels for subsequent material testing and characterisation.


## 2.2. Material Characterisation via Atomic Force Microscopy (AFM)

In this study, the nanomechanical properties of the interphase in PVA bionanocomposites were determined by means of peak force quantitative mapping (PFQM) in a Bruker Dimension FastScan AFM system to acquire a single force-distance curve under ambient conditions. In particular, an RTESPA-525A probe (nominal spring constant: 200 N/m; nominal tip radius: 8 nm; and nominal resonant frequency: 525 kHz) was employed to facilitate such direct measurements. During the calibration process, the deflection sensitivity was warranted with the acquisition of a force curve on a stiff sapphire-12 surface. The spring constant was determined by using a thermal tuning method [23] in which cantilever mechanical response to thermal agitation was measured with the Brownian motion of encompassing fluid molecules. Imaging analysis using atomic force microscopy (AFM) was carried out with the TESPA probe (normal spring constant: 40 N/m; tip radius: 8 nm). The scan rate was set up at 2 Hz with a digital pixel resolution of 256×256. Unwanted features including noise, bow, and tilt were removed when AFM images were first-order flattened using Bruker NanoScope 1.5 software. At least three AFM samples were utilised for each material batch to warrant the test reproducibility.


## 3. Modelling Approach

### 3.1. 3D Interphase Modulus, Surface Area, and Volume

As mentioned earlier, interphase properties such as interphase modulus and features like interphase surface area and volume were detected via PFQM. The interphase moduli between PVA matrices and dispersed HNTs and Cloisite 30B clays in various shapes and sizes were evaluated on the basis of a dataset associated with elastic moduli collected from PVA/HNT and PVA/Cloisite 30B interphases surrounding 75 different particles at 25 line scan regions (LSRs), respectively.On the other hand, 3D interphase dimensions are classified as interphase widthWInterphase, interphase length LInterphase, and interphase height HInterphase. In particular, individual interphase width WiInterphase, interphase length LjInterphase, and interphase height HkInterphase along the ith transverse plane (i=1,2,3⋯), the jth longitudinal plane (j=1,2,3⋯), and kth height plane (k=1,2,3⋯), respectively, were measured using the same scanning method mentioned in previous literature [7] so that typical features of distinct interphases between nanoparticles and polymer matrices could be characterised according to the great variation in their nanomechanical properties with the aid of PFQM.As illustrated in Figures1(a), 1(b), 1(e), and 1(f) for AFM imaging analysis, it is clearly shown that nanofiller dispersion took place with two typical categories of fully embedded and partially embedded HNTs and Cloisite 30B clays in PVA bionanocomposites. The 3D interphase region can be surrounded between the inner interface area and the outer interface area bound by nanofillers and PVA matrices, respectively, as evidenced in Figures 1(c), 1(d), 1(g), and 1(h). Both surface areas of outer interface SAouterinterface and inner interface SAinnerinterface with respect to a wide range of fully and partially embedded nanoparticles denoted by subscripts “f” and “p” can be estimated in the following by rearranging analytical equations for surface areas of anisotropic particles developed by [24].
(1)SAouterinterfacef=a1+b1LInterphase2+c1WInterphase2+d1HInterphase2,(2)SAouterinterfacep=a2+b2L′Interphase‐effective2+c2W′Interphase‐effective2+d2H′Interphase‐effective2,(3)SAinnerinterfacef=a3+b3Lp2+c3WP2+d3HP2,(4)SAinnerinterfacep=a4+b4L′p‐effective2+c4W′p‐effective2+d4H′p‐effective2,where under the category of partially embedded nanoparticles shown in Figures 1(d) and 1(h), LInterphase‐effective′, WInterphase‐effective′, and HInterphase‐effective′ are the maximum length, width, and thickness for effective interphases, whereas Lp‐effective′, Wp‐effective′, and Hp‐effective′ represent the maximum length, width, and thickness for effective nanoparticles (excluding the nanoparticle portions without an interphase to PVA matrices in bionanocomposites). The rest of the letters ai, bi, ci, and di (i=1,2,3,and4) denote empirical constants determined by curve fitting with corresponding experimental data of the surface areas obtained from AFM measurements (see more details in Supporting Information (available here)).Figure 1
AFM height images of PVA bionanocomposites reinforced with fully embedded and partially embedded HNTs in (a) 2D view and (b) 3D view as well as corresponding proposed schematic diagrams (c and d). AFM height images of PVA bionanocomposites reinforced with fully embedded and partially embedded Cloisite 30B clays in (e) 2D view and (f) 3D view as well as corresponding proposed schematic diagrams (g and h).
(a)(b)(c)(d)(e)(f)(g)(h)Additionally, with the supply of the aforementioned surface area data in equations (1)–(4), nanoparticle/interphase volume Vp/Interphase and nanoparticle volume Vp for fully embedded nanoparticles, as well as effective nanoparticle/interphase volume Vp/Interphase‐effective and effective nanoparticle volume Vp‐effective for partially embedded nanoparticles can be further determined according to the modified equations derived from Behmer and Hawkins [24]:
(5)SAouterinterfacef=ℯ1Vp/Interphase,f1(6)SAouterinterfacep=ℯ2Vp/Interphase‐effective,f2(7)SAinnerinterfacef=ℯ3Vpf3,(8)SAinnerinterfacep=ℯ4Vp‐effective.f4In a similar manner, empirical constantsei and fi (i=1,2,3,and4) can also be determined using a curve-fitting scheme with the experimental data of corresponding volumes via AFM, as given in Supporting Information.Finally, interphase volumeVInterphase in relation to fully and partially embedded nanoparticles in PVA bionanocomposites depicted in Figure 1 is given by
(9)VInterphasef=Vp/Interphase−Vp,(10)VInterphasep=Vp/Interphase‐effective−Vp‐effective.


### 3.2. Interphase Volume Fraction

Interphase volume fraction∅Interphasecan be estimated using a nearest-surface distribution theory [25] where composite media consist of 3D rigid anisotropic particles and voids (see Appendix A for more details). In a typical nanocomposite system, uniformly dispersed nanoparticles with relatively similar sizes are known as monodispersed nanoparticles as opposed to polydispersed nanoparticles where more randomly dispersed nanoparticles occur with different sizes and diameters [26]. With respect to a monodispersed particle system, interfacial volume fraction ∅Interphasecan be estimated as follows (see more details in Appendix A)
(11)∅Interphase=1−∅p1−exp−6∅pλ31−∅p1Sλ2+1λ2+3∅pS21−∅p+43+4∅pS1−∅p+4m∅p23S31−∅p2,where λ=t′/Deq is the geometric size of anisotropic particles in which t′ is the interfacial dimension and Deq is a diameter of a sphere with the same volume of an anisotropic particle. ∅p is the nominal volume fraction of nanofillers. m is considered a parameter used for the theoretical estimation of a radial distribution function in a spherical particle system. In this study, m=2 has been employed according to the Carnahan-Starling approximation [27]. Here, S is a morphological descriptor of anisotropic particles in terms of their aspect ratio α calculated based on the following:
(12)S=1+1.5α2/31+α.As far as a polydispersed particle system is concerned, the impact of particle size distribution (PSD) is critical on both the interphase volume fraction and the reinforcement efficiency of nanocomposite systems. The gradation of irregular aggregates can be experimentally evaluated with the aid of a sieve analyser or laser particle analyser in which each irregular aggregate size is assessed relative to a corresponding spherical particle of equivalent size [28]. When an equivalent diameter is determined, the PSD of spherical particles can be transformed into that of anisotropic particles accordingly [29]. Our study focuses on two particulate gradations, namely, the Fuller gradation [30] and the equal volume fraction (EVF) gradation [29, 30] in order to estimate the ∅Interphase given as follows:
(13)∅Interphase=1−∅p1−exp−6∅pDeq3e′t+d′t3+g′t3,(14)e′=Deq2S1−∅p,(15)d′=2Deq1−∅p+3∅pDeq22S21−∅p2Deq3,(16)g′=431−∅p+4∅pDeqDeq2S1−∅p2Deq3+4m∅pDeq233S31−∅p3Deq32,(17)Deqk=∫DmineqDmaxeqDeqkfNDeqdDeq,(18)fNDeq=−qDmaxeq−q−Dmineq−qDeqq+1q=2.5⟶Fullergradationq=3.0⟶EVFgradation,where fNDeqis a number-based probability function of anisotropic particles. Deqk, Deq‐min, and Deq‐max are the kth moment and the minimum and maximum equivalent diameters of the anisotropic particles. Finally, the aforementioned parameters to calculate ∅Interphase can then be measured for each nanoparticle based on AFM experimental data. For more detailed derivations of ∅Interphase, one can refer to Appendix A.


### 3.3. Micromechanical Model Based on Nominal and Effective Volume Fractions

The prediction of the elastic moduli of a unidirectionally or randomly distributed filler-reinforced composite is generally achieved by using conventional composite theoretical models such as Halpin-Tsai model [9] or Mori-Tanaka model [10]. In particular, the elastic moduli of our PVA biocomposites in Halpin-Tsai model according to different nanofiller orientation states, namely, randomly oriented and parallel (or well-aligned) nanofillers, can be represented by Erandom and Eparallel, respectively, using the following:
(19)Erandom=Em381+ηLξ∅p1−ηL∅p+581+2ηT∅p1−ηT∅p,(20)Eparallel=Em1+ηLξ∅p1−ηL∅p,(21)ηL=Ep/Em−1Ep/Em+ξ,(22)ηT=Ep/Em−1Ep/Em+2,(23)ξ=2α3=2ıp3tp,(24)∅p=WpWp+ρp/ρm1−Wp,where Em and Ep are Young’s moduli of PVA matrices and nanoparticles (i.e., HNTs and Cloisite 30B clays), accordingly. lp and tp refer to the length and thickness of nanoparticles. Øp and Wp are nominal volume fraction and weight fraction of nanofillers in PVA bionanocomposites, respectively, while ρp and ρm are the densities of nanoparticles and PVA.Furthermore, Mori-Tanaka model is an effective theoretical model to predict the elastic moduli of polymer composites, which is also employed in modelling our PVA bionanocomposites. It was originally employed by Tandon and Weng [31] using Mori-Tanaka theory [10] and Eshelby’s solution [32] to derive a complete analytical solution for calculating the elastic moduli of an isotropic matrix containing aligned spheroidal inclusions. Longitudinal and transverse moduli of polymer composites E11 and E22can be experessed as follows:
(25)E11Em=11+ØpΑ1+2νmΑ2/A,(26)E22Em=11+Øp−2νmΑ3+1−νmΑ4+1+νmΑ5Α/2Α,where νm is Poisson’s ratio of polymer matrices and Αi (i=1‐5) and A are the functions of Eshelby’s tensors and properties of the matrices and fillers such as Young’s moduli, Poisson’s ratio, aspect ratio, and volume fraction of fillers. When fillers are inclined to more random orientation, the elastic modulus of composites can be further predicted by using a combination of the laminate theory [33] and Mori-Tanaka model given by
(27)Ec≅0.375E11+0.625E22.In a nanocomposite system, large specific surface areas of nanofillers yield more amounts of polymeric molecular chains to attach to nanofiller surfaces resulting in more effective interphases of nanocomposites. Accordingly, the nominal volume fraction of nanofillers may be altered when reinforced within polymer matrices in nanocomposite systems. It has been proven that the effective volume fraction of nanofillers∅p′ is more applicable instead for accurate property estimation in modelling work using the following [32, 34, 35]:
(28)∅p′=Øp1+kxRgΑΤρp,where Rg is the radius gyration of a polymer, while AT is the specific surface area of nanofillers. The layer thickness of absorbed polymeric chains in nanoparticles is xRg, which means increasing up to x times of Rg. The alteration of kx depends on interfacial interaction levels between polymer matrices and nanoparticles in different nanocomposite systems. Additionally, the dynamic behaviour of polymeric chains near an attractive interface may vary from that of a bulk polymer phase and the effect of attractive surfaces can take place for distances larger than Rg for polymers [36]. The term of kx can be reconsidered based on the following [31, 33]:
(29)kx=EmEptRg,where t is regarded as the thickness of adsorbed layers.The effective volume fraction of nanofillers∅p′ in PVA/HNT bionanocomposites can be determined using equation (28) in a general form. However, in polymer/platelete nanocomposites, ∅p′ should depend on intercalation and exfoliation levels of platelet-like nanofillers taking place within polymer matrices. With respect to clay intercalated structures in nanocomposites [31], ∅p′ can be calculated using the following derived equation:
(30)1∅p′=1+ρp′1−Wp′−Wp′Wp″sd1N−1+hWp′ρmd2N−1+h,where ρp′ is referred to as the clay density, Wp′ is the weight fraction of clays in nanocomposites, and Wp″ is the weight fraction of clay platelets within clays. s is the polymer uptake in a general expression s=1−Wpi/Wpi to represent the mass of polymer matrices swollen into the unit mass of intercalated clays, where Wpi is the weight fraction of intercalated clay structures. d1 and d2 are basal plane spacing values of as-received clay particles and intercalated clays in nanocomposites, respectively. N and h are the average layer number per stack and thickness of a single platelet, respectively.In the case of exfoliated clay structures in nanocomposites where individual clay platelets are dispersed homogeneously in polymer matrices with an adsorption effect,∅p′ can be calculated according to equation (28) in a rewritten form below:
(31)∅p′=Øp1+kxRgΑΤpρp″,where ΑΤp is the total surface area and ρp″ is the density of single clay platelets by replacing AT and ρp accordingly in equation (28).


### 3.4. Volume Fraction of Nanofillers with Interphase Effect

The interphase features and properties are essential in a nanocomposite system, which is often ignored in conventional modelling work. The interphases in PVA bionanocomposites generally have nonuniform dimensions and critical nanomechanical properties to significantly influence nanofiller-matrix interactions. For example, by increasing the interphase volume from 2988 to 4363 nm3 in PVA/nanodiameter bamboo charcoal (NBC) bionanocomposites, their interphase modulus was found to be remarkably enhanced from 34.67 to 49.91 GPa (i.e., an increase by 44%) [7]. Consequently, the application of 3D interphase features such as interphase size and volume fraction to predict the elastic modulus of polymer nanocomposites may yield a more reliable modelling approach to estimate bulk nanocomposite properties according to their nanomechanical behaviour.Apparently, the volume fractions of nanoparticles and interphase∅p and ∅Interphasein a nanocomposite system can be expressed as follows:
(32)∅p=VpVc,(33)∅Interphase=VInterphaseVc,where Vc and VInterphase are the volumes of nanocomposites and interphase, respectively. In this case, Vp and VInterphase can be estimated by using equations (7) and (9) accordingly.By rearranging equations ((32)) and ((33)), they yield a new proposed equation based on the interphase features shown as follows:
(34)∅p=∅InterphaseVpVInterphase,where ∅Interphase can be determined according to the theory of the nearest-surface function [25] elaborated in Section 3.2.


## 3.1. 3D Interphase Modulus, Surface Area, and Volume

As mentioned earlier, interphase properties such as interphase modulus and features like interphase surface area and volume were detected via PFQM. The interphase moduli between PVA matrices and dispersed HNTs and Cloisite 30B clays in various shapes and sizes were evaluated on the basis of a dataset associated with elastic moduli collected from PVA/HNT and PVA/Cloisite 30B interphases surrounding 75 different particles at 25 line scan regions (LSRs), respectively.On the other hand, 3D interphase dimensions are classified as interphase widthWInterphase, interphase length LInterphase, and interphase height HInterphase. In particular, individual interphase width WiInterphase, interphase length LjInterphase, and interphase height HkInterphase along the ith transverse plane (i=1,2,3⋯), the jth longitudinal plane (j=1,2,3⋯), and kth height plane (k=1,2,3⋯), respectively, were measured using the same scanning method mentioned in previous literature [7] so that typical features of distinct interphases between nanoparticles and polymer matrices could be characterised according to the great variation in their nanomechanical properties with the aid of PFQM.As illustrated in Figures1(a), 1(b), 1(e), and 1(f) for AFM imaging analysis, it is clearly shown that nanofiller dispersion took place with two typical categories of fully embedded and partially embedded HNTs and Cloisite 30B clays in PVA bionanocomposites. The 3D interphase region can be surrounded between the inner interface area and the outer interface area bound by nanofillers and PVA matrices, respectively, as evidenced in Figures 1(c), 1(d), 1(g), and 1(h). Both surface areas of outer interface SAouterinterface and inner interface SAinnerinterface with respect to a wide range of fully and partially embedded nanoparticles denoted by subscripts “f” and “p” can be estimated in the following by rearranging analytical equations for surface areas of anisotropic particles developed by [24].
(1)SAouterinterfacef=a1+b1LInterphase2+c1WInterphase2+d1HInterphase2,(2)SAouterinterfacep=a2+b2L′Interphase‐effective2+c2W′Interphase‐effective2+d2H′Interphase‐effective2,(3)SAinnerinterfacef=a3+b3Lp2+c3WP2+d3HP2,(4)SAinnerinterfacep=a4+b4L′p‐effective2+c4W′p‐effective2+d4H′p‐effective2,where under the category of partially embedded nanoparticles shown in Figures 1(d) and 1(h), LInterphase‐effective′, WInterphase‐effective′, and HInterphase‐effective′ are the maximum length, width, and thickness for effective interphases, whereas Lp‐effective′, Wp‐effective′, and Hp‐effective′ represent the maximum length, width, and thickness for effective nanoparticles (excluding the nanoparticle portions without an interphase to PVA matrices in bionanocomposites). The rest of the letters ai, bi, ci, and di (i=1,2,3,and4) denote empirical constants determined by curve fitting with corresponding experimental data of the surface areas obtained from AFM measurements (see more details in Supporting Information (available here)).Figure 1
AFM height images of PVA bionanocomposites reinforced with fully embedded and partially embedded HNTs in (a) 2D view and (b) 3D view as well as corresponding proposed schematic diagrams (c and d). AFM height images of PVA bionanocomposites reinforced with fully embedded and partially embedded Cloisite 30B clays in (e) 2D view and (f) 3D view as well as corresponding proposed schematic diagrams (g and h).
(a)(b)(c)(d)(e)(f)(g)(h)Additionally, with the supply of the aforementioned surface area data in equations (1)–(4), nanoparticle/interphase volume Vp/Interphase and nanoparticle volume Vp for fully embedded nanoparticles, as well as effective nanoparticle/interphase volume Vp/Interphase‐effective and effective nanoparticle volume Vp‐effective for partially embedded nanoparticles can be further determined according to the modified equations derived from Behmer and Hawkins [24]:
(5)SAouterinterfacef=ℯ1Vp/Interphase,f1(6)SAouterinterfacep=ℯ2Vp/Interphase‐effective,f2(7)SAinnerinterfacef=ℯ3Vpf3,(8)SAinnerinterfacep=ℯ4Vp‐effective.f4In a similar manner, empirical constantsei and fi (i=1,2,3,and4) can also be determined using a curve-fitting scheme with the experimental data of corresponding volumes via AFM, as given in Supporting Information.Finally, interphase volumeVInterphase in relation to fully and partially embedded nanoparticles in PVA bionanocomposites depicted in Figure 1 is given by
(9)VInterphasef=Vp/Interphase−Vp,(10)VInterphasep=Vp/Interphase‐effective−Vp‐effective.


## 3.2. Interphase Volume Fraction

Interphase volume fraction∅Interphasecan be estimated using a nearest-surface distribution theory [25] where composite media consist of 3D rigid anisotropic particles and voids (see Appendix A for more details). In a typical nanocomposite system, uniformly dispersed nanoparticles with relatively similar sizes are known as monodispersed nanoparticles as opposed to polydispersed nanoparticles where more randomly dispersed nanoparticles occur with different sizes and diameters [26]. With respect to a monodispersed particle system, interfacial volume fraction ∅Interphasecan be estimated as follows (see more details in Appendix A)
(11)∅Interphase=1−∅p1−exp−6∅pλ31−∅p1Sλ2+1λ2+3∅pS21−∅p+43+4∅pS1−∅p+4m∅p23S31−∅p2,where λ=t′/Deq is the geometric size of anisotropic particles in which t′ is the interfacial dimension and Deq is a diameter of a sphere with the same volume of an anisotropic particle. ∅p is the nominal volume fraction of nanofillers. m is considered a parameter used for the theoretical estimation of a radial distribution function in a spherical particle system. In this study, m=2 has been employed according to the Carnahan-Starling approximation [27]. Here, S is a morphological descriptor of anisotropic particles in terms of their aspect ratio α calculated based on the following:
(12)S=1+1.5α2/31+α.As far as a polydispersed particle system is concerned, the impact of particle size distribution (PSD) is critical on both the interphase volume fraction and the reinforcement efficiency of nanocomposite systems. The gradation of irregular aggregates can be experimentally evaluated with the aid of a sieve analyser or laser particle analyser in which each irregular aggregate size is assessed relative to a corresponding spherical particle of equivalent size [28]. When an equivalent diameter is determined, the PSD of spherical particles can be transformed into that of anisotropic particles accordingly [29]. Our study focuses on two particulate gradations, namely, the Fuller gradation [30] and the equal volume fraction (EVF) gradation [29, 30] in order to estimate the ∅Interphase given as follows:
(13)∅Interphase=1−∅p1−exp−6∅pDeq3e′t+d′t3+g′t3,(14)e′=Deq2S1−∅p,(15)d′=2Deq1−∅p+3∅pDeq22S21−∅p2Deq3,(16)g′=431−∅p+4∅pDeqDeq2S1−∅p2Deq3+4m∅pDeq233S31−∅p3Deq32,(17)Deqk=∫DmineqDmaxeqDeqkfNDeqdDeq,(18)fNDeq=−qDmaxeq−q−Dmineq−qDeqq+1q=2.5⟶Fullergradationq=3.0⟶EVFgradation,where fNDeqis a number-based probability function of anisotropic particles. Deqk, Deq‐min, and Deq‐max are the kth moment and the minimum and maximum equivalent diameters of the anisotropic particles. Finally, the aforementioned parameters to calculate ∅Interphase can then be measured for each nanoparticle based on AFM experimental data. For more detailed derivations of ∅Interphase, one can refer to Appendix A.


## 3.3. Micromechanical Model Based on Nominal and Effective Volume Fractions

The prediction of the elastic moduli of a unidirectionally or randomly distributed filler-reinforced composite is generally achieved by using conventional composite theoretical models such as Halpin-Tsai model [9] or Mori-Tanaka model [10]. In particular, the elastic moduli of our PVA biocomposites in Halpin-Tsai model according to different nanofiller orientation states, namely, randomly oriented and parallel (or well-aligned) nanofillers, can be represented by Erandom and Eparallel, respectively, using the following:
(19)Erandom=Em381+ηLξ∅p1−ηL∅p+581+2ηT∅p1−ηT∅p,(20)Eparallel=Em1+ηLξ∅p1−ηL∅p,(21)ηL=Ep/Em−1Ep/Em+ξ,(22)ηT=Ep/Em−1Ep/Em+2,(23)ξ=2α3=2ıp3tp,(24)∅p=WpWp+ρp/ρm1−Wp,where Em and Ep are Young’s moduli of PVA matrices and nanoparticles (i.e., HNTs and Cloisite 30B clays), accordingly. lp and tp refer to the length and thickness of nanoparticles. Øp and Wp are nominal volume fraction and weight fraction of nanofillers in PVA bionanocomposites, respectively, while ρp and ρm are the densities of nanoparticles and PVA.Furthermore, Mori-Tanaka model is an effective theoretical model to predict the elastic moduli of polymer composites, which is also employed in modelling our PVA bionanocomposites. It was originally employed by Tandon and Weng [31] using Mori-Tanaka theory [10] and Eshelby’s solution [32] to derive a complete analytical solution for calculating the elastic moduli of an isotropic matrix containing aligned spheroidal inclusions. Longitudinal and transverse moduli of polymer composites E11 and E22can be experessed as follows:
(25)E11Em=11+ØpΑ1+2νmΑ2/A,(26)E22Em=11+Øp−2νmΑ3+1−νmΑ4+1+νmΑ5Α/2Α,where νm is Poisson’s ratio of polymer matrices and Αi (i=1‐5) and A are the functions of Eshelby’s tensors and properties of the matrices and fillers such as Young’s moduli, Poisson’s ratio, aspect ratio, and volume fraction of fillers. When fillers are inclined to more random orientation, the elastic modulus of composites can be further predicted by using a combination of the laminate theory [33] and Mori-Tanaka model given by
(27)Ec≅0.375E11+0.625E22.In a nanocomposite system, large specific surface areas of nanofillers yield more amounts of polymeric molecular chains to attach to nanofiller surfaces resulting in more effective interphases of nanocomposites. Accordingly, the nominal volume fraction of nanofillers may be altered when reinforced within polymer matrices in nanocomposite systems. It has been proven that the effective volume fraction of nanofillers∅p′ is more applicable instead for accurate property estimation in modelling work using the following [32, 34, 35]:
(28)∅p′=Øp1+kxRgΑΤρp,where Rg is the radius gyration of a polymer, while AT is the specific surface area of nanofillers. The layer thickness of absorbed polymeric chains in nanoparticles is xRg, which means increasing up to x times of Rg. The alteration of kx depends on interfacial interaction levels between polymer matrices and nanoparticles in different nanocomposite systems. Additionally, the dynamic behaviour of polymeric chains near an attractive interface may vary from that of a bulk polymer phase and the effect of attractive surfaces can take place for distances larger than Rg for polymers [36]. The term of kx can be reconsidered based on the following [31, 33]:
(29)kx=EmEptRg,where t is regarded as the thickness of adsorbed layers.The effective volume fraction of nanofillers∅p′ in PVA/HNT bionanocomposites can be determined using equation (28) in a general form. However, in polymer/platelete nanocomposites, ∅p′ should depend on intercalation and exfoliation levels of platelet-like nanofillers taking place within polymer matrices. With respect to clay intercalated structures in nanocomposites [31], ∅p′ can be calculated using the following derived equation:
(30)1∅p′=1+ρp′1−Wp′−Wp′Wp″sd1N−1+hWp′ρmd2N−1+h,where ρp′ is referred to as the clay density, Wp′ is the weight fraction of clays in nanocomposites, and Wp″ is the weight fraction of clay platelets within clays. s is the polymer uptake in a general expression s=1−Wpi/Wpi to represent the mass of polymer matrices swollen into the unit mass of intercalated clays, where Wpi is the weight fraction of intercalated clay structures. d1 and d2 are basal plane spacing values of as-received clay particles and intercalated clays in nanocomposites, respectively. N and h are the average layer number per stack and thickness of a single platelet, respectively.In the case of exfoliated clay structures in nanocomposites where individual clay platelets are dispersed homogeneously in polymer matrices with an adsorption effect,∅p′ can be calculated according to equation (28) in a rewritten form below:
(31)∅p′=Øp1+kxRgΑΤpρp″,where ΑΤp is the total surface area and ρp″ is the density of single clay platelets by replacing AT and ρp accordingly in equation (28).


## 3.4. Volume Fraction of Nanofillers with Interphase Effect

The interphase features and properties are essential in a nanocomposite system, which is often ignored in conventional modelling work. The interphases in PVA bionanocomposites generally have nonuniform dimensions and critical nanomechanical properties to significantly influence nanofiller-matrix interactions. For example, by increasing the interphase volume from 2988 to 4363 nm3 in PVA/nanodiameter bamboo charcoal (NBC) bionanocomposites, their interphase modulus was found to be remarkably enhanced from 34.67 to 49.91 GPa (i.e., an increase by 44%) [7]. Consequently, the application of 3D interphase features such as interphase size and volume fraction to predict the elastic modulus of polymer nanocomposites may yield a more reliable modelling approach to estimate bulk nanocomposite properties according to their nanomechanical behaviour.Apparently, the volume fractions of nanoparticles and interphase∅p and ∅Interphasein a nanocomposite system can be expressed as follows:
(32)∅p=VpVc,(33)∅Interphase=VInterphaseVc,where Vc and VInterphase are the volumes of nanocomposites and interphase, respectively. In this case, Vp and VInterphase can be estimated by using equations (7) and (9) accordingly.By rearranging equations ((32)) and ((33)), they yield a new proposed equation based on the interphase features shown as follows:
(34)∅p=∅InterphaseVpVInterphase,where ∅Interphase can be determined according to the theory of the nearest-surface function [25] elaborated in Section 3.2.


## 4. Results and Discussion

### 4.1. Elastic Properties of Interphase

As seen from Figure1, both HNTs and Cloisite 30B clays when dispersed into PVA matrices in bionanocomposites presented two similar nanoparticle-embedding effects, namely fully embedded and partially embedded formations. When nanoparticles are fully embedded into continuous matrices, a full interphase configuration is manifested in Figures 1(c) and 1(g), while an effective interphase (also known as partial interphase without the consideration of noninteracting nanoparticle portions) is generally applied under the scenario of partial embedding in Figures 1(d) and 1(h) leading to different levels of filler-matrix interaction and reinforcement efficiency to impact the interphase modulus.Furthermore, as a typical case, the AFM images and profiles of the elastic modulus for PVA/HNT bionanocomposites and PVA/Cloisite 30B clay bionanocomposites at a nanofiller content of 3 wt% are depicted in Figure2. The final interphase modulus was determined by using the best curve fitting to discrete modulus datasets based on 25 representative line scan regions (LSRs) illustrated in Figure S9 in Supporting Information. Overall, the interphase modulus demonstrated an almost linearly increasing trend with a modulus-gradient effect in terms of scan distance except for the interphase boundary regions around nanoparticles and PVA matrices regardless of additional HNTs and Cloisite 30B clays in PVA bionanocomposites. In comparison, the interphase modulus of PVA/Cloisite 30B clay bionanocomposites was increased from 20.6±3.4GPa near PVA matrices to 41.6±4.6GPa surrounding nanoclays as opposed to the variation of the interphase modulus in the range from 22.2±2.9 to 43.4±3.1GPa in PVA/HNT bionanocomposites accordingly (Figures 2(b) and 2(d)). However, the interphase moduli measured for PVA bionanocomposites in this study were less than those of silk fibroin (SF)/graphene oxide (GO) nanocomposites in the range of approximately 10-250 GPa [12] as well as those of epoxy/T300 carbon nanocomposites between 23 and 90 GPa [17]. Such variations in the interphase modulus as well as the modulus-gradient effect in the interphase zones from nanoparticles to polymer matrices may be associated with the gradient number of hydrogen bonding from nanofiller surfaces to polymer matrices, which is believed to play an important role in tailoring the mechanical properties of polymer nanocomposites [37, 38]. In a similar manner, Amraei et al. [19, 20] also identified interphase transitional zones with the continuous variations of elastic modulus between the inclusions and matrices with the assumption of a perfect bonding effect in terms of radial distance of RVE in a nanocomposite system.Figure 2
3D AFM mapping images and corresponding typical profile of elastic modulus in PVA/3 wt% HNT bionanocomposites (a and b), as well as PVA/3 wt% Cloisite 30B clay bionanocomposites, respectively. A1B1 and A2B2 are the corresponding typical cut sections for scan distance Sd.
(a)(b)(c)(d)


### 4.2. Interphase Dimensions

The elastic properties of interphases in composite materials generally vary to a different extent when surrounded with individual particles or fillers in diverse sizes and shapes, thus resulting in the great change of interphase dimensions. Interphase dimensions can be clearly specified according to the variation identified in terms of height and adhesion profiles in PVA bionanocomposites reinforced with 3 wt% HNTs and Cloisite 30B clays, as depicted in Figures3 and 4, respectively. Interphase thickness was determined to be 6.5 and 9.2 nm on both sides of HNTs as opposed to corresponding relatively large dimensions of 8.4 and 10.2 nm on those of Cloisite 30B clays, which appeared to be consistent in view of both height and adhesion profiles. Moreover, the tip adhesion to PVA matrices was detected to be approximately 14.24±2.31nN and became greater than those of HNTs and Cloisite 30B clays at 8.6±1.08 and 8.4±1.1nN, respectively. As clearly seen, the adhesion force of PVA matrices was much greater than those for HNTs and Cloisite 30B clays, which could be explained by more hydroxyl groups identified in PVA when compared to those of HNTs and Cloisite 30B clays.Figure 3
3D AFM height mapping images and typical height profiles of PVA/3 wt% HNT bionancomposites (a and b) and PVA/3 wt% Cloisite 30B clay bionanocomposites (c and d), respectively. A3B3 and A4B4 are the corresponding typical cut sections for scan distance.
(a)(b)(c)(d)Figure 4
3D AFM adhesion mapping images and typical adhesion profiles of PVA/3 wt% HNT bionanocomposites (a and b) and PVA/3 wt% Cloisite 30B clay bionanocomposites (c and d), respectively. A5B5 and A6B6 are the corresponding typical cut sections for scan distance.
(a)(b)(c)(d)Overall, interphase dimensions of PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays revealed very similar tendencies in terms of interphase thicknesstInterphase, maximum interphase length LInterphase‐max, maximum interphase width WInterphase‐max, and maximum interphase height HInterphase‐max from a 3D point of view displayed in Figure 5. In particular, tInterphase appeared to be independent of the HNT diameter (DHNT) and platelet thickness of Cloisite 30B (tCloisite30B), respectively (Figures 5(a) and 5(e)). Nonetheless, both LInterphase‐max and WInterphase‐max increased in a linear manner as a function of HNT length LHNT and HNT transverse diameter DTHNT in PVA/HNT bionanocomposites, as well as Cloisite 30B clay platelet length LCloisite30B and width WCloisite30B accordingly in Figures 5(b), 5(c), 5(f), and 5(g) despite opposite linearly decreasing trends taking place in HInterphase‐max in terms of HNT longitudinal diameter DLHNT and HCloisite30B depicted in Figures 5(d) and 5(h). Such a phenomenon can be ascribed to an increase in the contact surface areas between continuous polymer matrices and more dispersed nanoparticles in a nanocomposite system especially when nanoparticles are fully embedded or covered by surrounding matrices [6]. Increasing DLHNT or HCloisite30B could make it difficult for nanofillers to be fully embedded within polymer matrices, thus resulting in the reduction in HInterphase‐max.Figure 5
Correlation between maximum interphase dimensions and nanofiller dimensions of PVA/HNT bionanocomposites and PVA/Cloisite 30B clay bionanocomposites: (a)tInterphase vs. DHNT, (b) LInterphase−max vs. LHNT, (c) WInterphase−max vs. DTHNT, (d) HInterphase−max vs. DLHNT, (e) tInterphase vs. tCloisite30B, (f) LInterphase−max vs. LCloisite30B, (g) WInterphase−max vs. WCloisite30B, and (h) HInterphase−max vs. HCloisite30B. tInterphase is interphase thickness, and DHNT is HNT diameter. LInterphase−max,WInterphase−max, and HInterphase−max are maximum interphase length, width, and height, respectively. LHNT, DLHNT, and DTHNT are the length, longitudinal diameter, and transverse diameter of HNTs while LCloisite30B, WCloisite30B, and tCloisite30B are the length, width, and thickness of Cloisite 30B clay platelets.
(a)(b)(c)(d)(e)(f)(g)(h)As demonstrated in Figure1 for the proposed mechanism models in terms of nanofiller dispersion and particle-matrix interaction, it is worthwhile investigating these two typical scenarios, namely the fully and partially embedded nanoparticles with PVA matrices in bionanocomposites for the assessment of their interphase modulus. The former apparently yields more effective interphases around each side of nanoparticles (i.e., full coverage of nanoparticles by interphases), which is generally deemed as a good representation of effective load transfer from matrices and nanofillers, while the latter gives rise to less pronounced interphase effectiveness since some remarkable portions of nanoparticles were detected to be free of contact with PVA matrices, further reducing effective interfacial regions. In this study, the interphase surface area SAInterface and interphase volume VInterphase were characterised as two critical interphase dimensional parameters to evaluate the impacts on interphase modulus EInterphase with respect to fully or partially embedded nanoparticles in PVA bionanocomposites (Figure 6). In addition, SAInterface is further divided into SAinnerinterface and SAouterinterface when considering the inner interfaces between nanoparticles and interphase regions, as well as outer interfaces between the interphase regions and PVA matrices, respectively, as illustrated in Figures 1(c), 1(d), 1(g), and 1(h). A very identical behaviour occurred in PVA/HNT bionanocomposites and PVA/Cloisite 30B clay bionanocomposites, which indicated that overall EInterphase tended to be consistently enhanced in a nonlinear manner with increasing SAInterface and VInterphase individually. More consistently, fully embedded HNTs and Cloisite 30B clays yielded higher EInterphase as opposed to those based on partially embedded counterparts. Besides, SAinnerinterface led to relatively high overall EInterphase as opposed to those based on corresponding SAouterinterface. As mentioned earlier by Liu et al. [39], the interphase region with a modulus gradient effect can be divided into two different zones comprising the first zone in contact with nanoparticle areas with relatively high interphase density as opposed to the second zone adjacent to matrix regions. Fan et al. [40] reported that increasing interphase density certainly induced the modulus increase in a nanostructure-specific hierarchy in good accordance with our higher EInterphase in the case of SAinnerinterface in PVA bionanocomposites. The abovementioned results may suggest that SAInterface and VInterphase as typical 2D and 3D interphase dimensional parameters can play a vital role in the improvement of EInterphase leading to the further enhancement in mechanical properties of PVA bionanocomposites. Nevertheless, it is also admittedly mentioned here that partially embedded nanoparticles with the imperfect bonding effect within PVA matrices are inevitably detrimental to their mechanical properties with much less bonded matrix-filler interactions when compared with fully embedded counterparts.Figure 6
Relationships between interphase modulusEInterphase and interphase surface area SAInterphase: (a) PVA/HNT bionanocomposites and (b) PVA/Cloisite 30B clay bionanocomposites. Relationships between interphase modulus EInterphase and interphase volume VInterphase: (c) PVA/HNT bionanocomposites and (d) PVA/Cloisite 30B clay bionanocomposites.
(a)(b)(c)(d)On the other hand, the aspect ratio is used as an important factor to evaluate the reinforcement efficiency of nanofillers within polymer matrices in nanocomposites. Its determination becomes much easier when dealing with nanofillers with regular sizes and shapes. However, irregular nanofillers with different shapes are quite common leading to the complexity of real aspect ratios due to the effect of multistage dispersion processing techniques generally used in the manufacture of nanocomposites [25, 26]. Moreover, in addition to the nanofiller aspect ratio, interphase dimensions and properties can also be affected by nanofiller dispersion [26]. Liu and Brinson [5] as well as Alishahi et al. [41] suggested the use of interphase volume per unit nanofiller volume (VInterphase/Vnanofiller) in theoretical modelling as an effective parameter in promoting the reinforcement efficiency of nanofillers instead of VInterphase, whose relationship as a function of Vnanofiller was presented in Figure 7 in this study. Overall, an increase in Vnanofiller caused a significant decrease in VInterphase/Vnanofiller for both PVA bionanocomposites despite being more pronounced in PVA/Cloisite 30B clay bionanocomposites. This is because Cloisite 30B clay platelets induced more efficient planar reinforcements when compared to predominant longitudinal reinforcement of tubular HNTs.Figure 7
Interphase volume per unit nanofiller volume (VInterphase/Vnanofiller) as a function of Vnanofiller in PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays.


### 4.3. Particle Debonding

With an increase in nanoparticle loading, interparticle spacing appeared to be smaller and nanoparticles tended to be interconnected with neighbouring counterparts side by side with some nanoparticles free of contact with PVA matrices in bionanocomposites. Moreover, a closely compacted joint particle-to-particle pattern may induce adverse phase separation resulting in scattered load transfer from PVA matrices to nanoparticles. It was evidently shown in Figures8(a) and 8(c) that large HNT aggregates and clustered stack-up Cloisite 30B clays took place at a very high nanofiller content up to 10 wt% in PVA bionanocomposites with the resulting weak interfacial bonding for particle-matrix interactions. Two narrow zones near interphase regions were highlighted with a sharp decline in elastic modulus for both PVA bionanocomposites with excessive amounts of undispersed nanoparticles in Figures 8(b) and 8(d), indicating an existing particle debonding effect with polymer matrices [36]. A lower interphase modulus may also arise from a relatively low density near interphase zones as compared to those of aggregated nanoparticles and polymer matrices [24]. In addition to nanoparticle sizes and dimensions, nanoparticle dispersion within continuous matrices is also among the key factors in controlling the interfacial bonding between fillers and matrices. When excessive amounts of HNTs and Cloisite 30B clays were dispersed within PVA matrices, those nanoparticles tended to agglomerate owing to their weak Van der Waals interactions, while higher nanofiller content inevitably led to a significant reduction in interparticle spacing to hinder uniform nanoparticle dispersion [42, 43]. A nanofiller fine dispersion method such as ultrasonication [41], on the other hand, may give rise to the possible damage of nanoparticle structures especially at the high ultrasonic power intensity resulting instead in declined mechanical properties of bionanocomposites.Figure 8
AFM height mapping images and corresponding typical elastic modulus profiles of (a and b) PVA/10 wt% HNT bionanocomposites, as well as (c and d) PVA/10 wt% Cloisite 30B clay bionanocomposites indicating phase separation and agglomeration of nanoparticles, respectively. A7B7 and A8B8 are corresponding typical cut sections for scan distance.
(a)(b)(c)(d)


### 4.4. Prediction of Elastic Moduli of PVA Bionanocomposites

The prediction of the elastic moduli of PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays as a function of nominal and effective volume fractions of nanoparticles was presented in Figure9 (without predefined particle systems) and Figure 10 (with monodispersed and polydispersed particle systems), respectively, using Halpin-Tsai model, Mori-Tanaka model, and a combination of Mori-Tanaka model and laminate theory [4] along with associated modelling parameters listed in Table 1. In both bionanocomposites (without predefined particle systems), the estimated moduli did not coincide well with corresponding experimental data even at the very low volume fractions of HNTs and Cloisite 30B clays, Figure 9. However, the use of an effective volume fraction and a randomly oriented dispersion state of nanoparticles consistently gave rise to much closer correlation with experimental data regardless of the composite theoretical models implemented, as also confirmed in previous studies [34, 35]. In particular, with respect to PVA/Cloisite 30B clay bionanocomposites, the predicted results appeared to be in better agreement with experimental data at the low volume fraction of 0.061 vol% on the basis of an effective volume fraction and exfoliated clay structures, while better correlation was manifested for those predicted in terms of effective volume fraction and intercalated clay structures at high volume fraction levels. As far as morphological structures are concerned, it was expected that the majority of the Cloisite 30B clay platelets tended to be more exfoliated at low volume fractions as opposed to predominant intercalated structures with an increase in their volume fraction, which further confirmed a mix of exfoliated and intercalated structures taking place in PVA/Cloisite 30B clay bionanocomposites. The major drawback in using conventional Halpin-Tsai model and Mori-Tanaka model lies in their common assumption that both nanofillers and polymer matrices in nanocomposites are linearly elastic and isotropic materials with uniform nanofiller dispersion and the neglect of particle-particle interactions [44]. Generally speaking, isotropic nanoparticles are in possession of very large specific surface areas on which a substantial amount of molecular chains of polymer matrices can be absorbed with the aid of hydrogen bonding or electrostatic forces, resulting in the formation of interphase layers in nanocomposites. Consequently, such interphase layers behave as solid transitional material phases to restrict the mobility of other molecular chains of matrices, thus contributing positively to the total volume of reinforcements [31, 34]. This suggested that composite theoretical models in this study should use effective volume fraction instead of nominal volume fraction for better accuracy of modelling work.Figure 9
Prediction of the elastic moduli of PVA/HNT bionanocomposites: (a) Halpin-Tsai model (H-T) and (b) Mori-Tanaka model (M-T) and the combination of Mori-Tanaka model and laminate theory (M-T-L). Prediction of the elastic moduli of PVA/Cloisite 30B clay bionanocomposites: (c) Halpin-Tsai model (H-T) and (d) Mori-Tanaka model (M-T) and the combination of Mori-Tanaka model and laminate theory (M-T-L).
(a)(b)(c)(d)Figure 10
Prediction of the elastic moduli of PVA/HNT bionanocomposites: (a) Halpin-Tsai model (H-T) and (b) the combination of Mori-Tanaka model and laminate theory (M-T-L). Prediction of the elastic moduli of PVA/Cloisite 30B clay bionanocomposites: (c) Halpin-Tsai model (H-T) and (d) the combination of Mori-Tanaka model and laminate theory (M-T-L).
(a)(b)(c)(d)Table 1
Modelling parameters used for PVA bionanocomposite films.
Modelling parameterPVA/HNTPVA/Cloisite 30B clayRef.Em (GPa)2.082.08[45]Ep (GPa)140a178ba[46]b[32]lp (nm)208 (3 wt% HNTs)109 (3 wt% Cloisite 30B)[45]410 (5 wt% HNTs)206 (5 wt% Cloisite 30B)[45]900.6 (10 wt% HNTs)560 (10 wt% Cloisite 30B)[45]tp (nm)4.6 (3 wt% Cloisite 30B)[45]15.8 (5 wt% Cloisite 30B)[45]41.6 (10 wt% Cloisite 30B)[45]d (nm)35.2 (3 wt% HNTs)[45]66.5 (5 wt% HNTs)[45]85 (10 wt% HNTs)[45]ρm (g cm-3)0.390.39[45]ρp (g cm-3)1.81.98Material data sheetst (nm)~7.3~10.4[45]Rg (nm)1010[32]νm0.40.4[34]W″p0.66[32]d1 (nm)1.87[45]d2 (nm)3.6[45]N5.6[45]h (nm)0.98[45]ρp″ (g cm-3)3.1[45]ΑΤ (m2 g-1)40.3c658dc[47]d[32]Although as mentioned earlier, the implementation of an effective volume fraction can lead to a more accurate estimation of experimental data in PVA bionanocomposites, the layer thickness of polymeric molecules absorbed onto nanofiller surfaces in terms ofkx may vary even in the same grade of polymers with different molecular structures and polarities [32, 34]. Our aforementioned results demonstrated that interphase features actually did not possess a uniform interphase layer thickness, and the interphase dimensions and properties could vary significantly when embedded with heterogeneous nanoparticles with different particle sizes and dimensions, as well as diverse nanomechanical properties depending on matrix-filler interactions. As a result, interphase volume fraction ∅Interphase was incorporated into two different particle systems, namely monodispersed particles and polydispersed particles by considering the Fuller and EVF gradations, respectively. Modelling parameters presented in Table 2 were substituted into equations (11) and (13), respectively, to estimate corresponding ∅Interphase values used in PVA bionanocomposites in this study. Besides, interphase volume can be calculated according to equations (5), (6), (7), and (8) with associated results being used to determine the volume fraction of nanofillers. Such recalculated volume fractions of nanoparticles with respect to the interphase effect were incorporated into Halpin-Tsai model and Mori-Tanaka model accordingly in this study and illustrated in Figure 10 (with monodispersed and polydispersed particle systems). In a consistent manner, all elastic moduli of PVA bionanocomposites with the consideration of the interphase effect in our modelling work revealed a much closer relationship with experimental data as opposed to corresponding theoretical models using both nominal and effective volume fractions without interphase. More evidently, our proposed models in a monodispersed particle system caused better curve fitting with experimental data at low volume fractions up to 0.0604 and 0.061 vol% for HNTs and Cloisite 30B clays, respectively, as opposed to those in a polydispersed particle system with both the Fuller and EVF gradations, but vice versa beyond 0.1109 and 0.1105 vol%, accordingly. It was understandable, since HNTs and Cloisite 30B clays tended to be more uniformly dispersed in PVA matrices with relatively similar nanoparticle sizes (i.e., monodispersed particle system) at their low volume fraction, while typical nanoparticle agglomeration was evidently identified in better accordance with a polydispersed particle system. When a particulate gradation effect was taken into account, those predictions according to the Fuller gradation seemed to be in slightly better agreement with corresponding experimental data in PVA bionanocomposites, which might be ascribed to the theoretical assumption of finer particle dispersion taking place in EVF gradation at the same volume fraction of nanocomposites. Larger specific surface areas of nanoparticles generally took place in the case of EVF gradation, thus resulting in increasing ∅Interphase and volume fractions of nanoparticles. However, it was not always the case in our PVA bionanocomposites especially at very high volume fractions of nanoparticles.Table 2
Modelling parameters measured for the calculation of∅Interphasein PVA bionanocomposites.
Modelling parameterNanoparticle content (wt%)PVA/HNTaPVA/Cloisite 30B clayDeq‐min (nm)364.534.15168.554.310245.7141.6Deq‐max (nm)3836436524601836.21046453141Deq‐mean (nm)3208109541020610900.6560tInterphaseb (nm)58.610.7Note. aDeq‐min, Deq‐max, and Deq‐mean are the maximum, minimum, and mean equivalent diameters of anisotropic particles, which represent the maximum, minimum, and mean equivalent lengths of HNTs, respectively, in PVA/HNT bionanocomposites according to equations (A.24)–(A.34) in Appendix A. btInterphase is the average interphase thickness for the model development based on the assumption that the interphase generated between nanoparticles and polymer matrices was typically considered at the filler content of 5 wt% in PVA bionanocomposites.


## 4.1. Elastic Properties of Interphase

As seen from Figure1, both HNTs and Cloisite 30B clays when dispersed into PVA matrices in bionanocomposites presented two similar nanoparticle-embedding effects, namely fully embedded and partially embedded formations. When nanoparticles are fully embedded into continuous matrices, a full interphase configuration is manifested in Figures 1(c) and 1(g), while an effective interphase (also known as partial interphase without the consideration of noninteracting nanoparticle portions) is generally applied under the scenario of partial embedding in Figures 1(d) and 1(h) leading to different levels of filler-matrix interaction and reinforcement efficiency to impact the interphase modulus.Furthermore, as a typical case, the AFM images and profiles of the elastic modulus for PVA/HNT bionanocomposites and PVA/Cloisite 30B clay bionanocomposites at a nanofiller content of 3 wt% are depicted in Figure2. The final interphase modulus was determined by using the best curve fitting to discrete modulus datasets based on 25 representative line scan regions (LSRs) illustrated in Figure S9 in Supporting Information. Overall, the interphase modulus demonstrated an almost linearly increasing trend with a modulus-gradient effect in terms of scan distance except for the interphase boundary regions around nanoparticles and PVA matrices regardless of additional HNTs and Cloisite 30B clays in PVA bionanocomposites. In comparison, the interphase modulus of PVA/Cloisite 30B clay bionanocomposites was increased from 20.6±3.4GPa near PVA matrices to 41.6±4.6GPa surrounding nanoclays as opposed to the variation of the interphase modulus in the range from 22.2±2.9 to 43.4±3.1GPa in PVA/HNT bionanocomposites accordingly (Figures 2(b) and 2(d)). However, the interphase moduli measured for PVA bionanocomposites in this study were less than those of silk fibroin (SF)/graphene oxide (GO) nanocomposites in the range of approximately 10-250 GPa [12] as well as those of epoxy/T300 carbon nanocomposites between 23 and 90 GPa [17]. Such variations in the interphase modulus as well as the modulus-gradient effect in the interphase zones from nanoparticles to polymer matrices may be associated with the gradient number of hydrogen bonding from nanofiller surfaces to polymer matrices, which is believed to play an important role in tailoring the mechanical properties of polymer nanocomposites [37, 38]. In a similar manner, Amraei et al. [19, 20] also identified interphase transitional zones with the continuous variations of elastic modulus between the inclusions and matrices with the assumption of a perfect bonding effect in terms of radial distance of RVE in a nanocomposite system.Figure 2
3D AFM mapping images and corresponding typical profile of elastic modulus in PVA/3 wt% HNT bionanocomposites (a and b), as well as PVA/3 wt% Cloisite 30B clay bionanocomposites, respectively. A1B1 and A2B2 are the corresponding typical cut sections for scan distance Sd.
(a)(b)(c)(d)


## 4.2. Interphase Dimensions

The elastic properties of interphases in composite materials generally vary to a different extent when surrounded with individual particles or fillers in diverse sizes and shapes, thus resulting in the great change of interphase dimensions. Interphase dimensions can be clearly specified according to the variation identified in terms of height and adhesion profiles in PVA bionanocomposites reinforced with 3 wt% HNTs and Cloisite 30B clays, as depicted in Figures3 and 4, respectively. Interphase thickness was determined to be 6.5 and 9.2 nm on both sides of HNTs as opposed to corresponding relatively large dimensions of 8.4 and 10.2 nm on those of Cloisite 30B clays, which appeared to be consistent in view of both height and adhesion profiles. Moreover, the tip adhesion to PVA matrices was detected to be approximately 14.24±2.31nN and became greater than those of HNTs and Cloisite 30B clays at 8.6±1.08 and 8.4±1.1nN, respectively. As clearly seen, the adhesion force of PVA matrices was much greater than those for HNTs and Cloisite 30B clays, which could be explained by more hydroxyl groups identified in PVA when compared to those of HNTs and Cloisite 30B clays.Figure 3
3D AFM height mapping images and typical height profiles of PVA/3 wt% HNT bionancomposites (a and b) and PVA/3 wt% Cloisite 30B clay bionanocomposites (c and d), respectively. A3B3 and A4B4 are the corresponding typical cut sections for scan distance.
(a)(b)(c)(d)Figure 4
3D AFM adhesion mapping images and typical adhesion profiles of PVA/3 wt% HNT bionanocomposites (a and b) and PVA/3 wt% Cloisite 30B clay bionanocomposites (c and d), respectively. A5B5 and A6B6 are the corresponding typical cut sections for scan distance.
(a)(b)(c)(d)Overall, interphase dimensions of PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays revealed very similar tendencies in terms of interphase thicknesstInterphase, maximum interphase length LInterphase‐max, maximum interphase width WInterphase‐max, and maximum interphase height HInterphase‐max from a 3D point of view displayed in Figure 5. In particular, tInterphase appeared to be independent of the HNT diameter (DHNT) and platelet thickness of Cloisite 30B (tCloisite30B), respectively (Figures 5(a) and 5(e)). Nonetheless, both LInterphase‐max and WInterphase‐max increased in a linear manner as a function of HNT length LHNT and HNT transverse diameter DTHNT in PVA/HNT bionanocomposites, as well as Cloisite 30B clay platelet length LCloisite30B and width WCloisite30B accordingly in Figures 5(b), 5(c), 5(f), and 5(g) despite opposite linearly decreasing trends taking place in HInterphase‐max in terms of HNT longitudinal diameter DLHNT and HCloisite30B depicted in Figures 5(d) and 5(h). Such a phenomenon can be ascribed to an increase in the contact surface areas between continuous polymer matrices and more dispersed nanoparticles in a nanocomposite system especially when nanoparticles are fully embedded or covered by surrounding matrices [6]. Increasing DLHNT or HCloisite30B could make it difficult for nanofillers to be fully embedded within polymer matrices, thus resulting in the reduction in HInterphase‐max.Figure 5
Correlation between maximum interphase dimensions and nanofiller dimensions of PVA/HNT bionanocomposites and PVA/Cloisite 30B clay bionanocomposites: (a)tInterphase vs. DHNT, (b) LInterphase−max vs. LHNT, (c) WInterphase−max vs. DTHNT, (d) HInterphase−max vs. DLHNT, (e) tInterphase vs. tCloisite30B, (f) LInterphase−max vs. LCloisite30B, (g) WInterphase−max vs. WCloisite30B, and (h) HInterphase−max vs. HCloisite30B. tInterphase is interphase thickness, and DHNT is HNT diameter. LInterphase−max,WInterphase−max, and HInterphase−max are maximum interphase length, width, and height, respectively. LHNT, DLHNT, and DTHNT are the length, longitudinal diameter, and transverse diameter of HNTs while LCloisite30B, WCloisite30B, and tCloisite30B are the length, width, and thickness of Cloisite 30B clay platelets.
(a)(b)(c)(d)(e)(f)(g)(h)As demonstrated in Figure1 for the proposed mechanism models in terms of nanofiller dispersion and particle-matrix interaction, it is worthwhile investigating these two typical scenarios, namely the fully and partially embedded nanoparticles with PVA matrices in bionanocomposites for the assessment of their interphase modulus. The former apparently yields more effective interphases around each side of nanoparticles (i.e., full coverage of nanoparticles by interphases), which is generally deemed as a good representation of effective load transfer from matrices and nanofillers, while the latter gives rise to less pronounced interphase effectiveness since some remarkable portions of nanoparticles were detected to be free of contact with PVA matrices, further reducing effective interfacial regions. In this study, the interphase surface area SAInterface and interphase volume VInterphase were characterised as two critical interphase dimensional parameters to evaluate the impacts on interphase modulus EInterphase with respect to fully or partially embedded nanoparticles in PVA bionanocomposites (Figure 6). In addition, SAInterface is further divided into SAinnerinterface and SAouterinterface when considering the inner interfaces between nanoparticles and interphase regions, as well as outer interfaces between the interphase regions and PVA matrices, respectively, as illustrated in Figures 1(c), 1(d), 1(g), and 1(h). A very identical behaviour occurred in PVA/HNT bionanocomposites and PVA/Cloisite 30B clay bionanocomposites, which indicated that overall EInterphase tended to be consistently enhanced in a nonlinear manner with increasing SAInterface and VInterphase individually. More consistently, fully embedded HNTs and Cloisite 30B clays yielded higher EInterphase as opposed to those based on partially embedded counterparts. Besides, SAinnerinterface led to relatively high overall EInterphase as opposed to those based on corresponding SAouterinterface. As mentioned earlier by Liu et al. [39], the interphase region with a modulus gradient effect can be divided into two different zones comprising the first zone in contact with nanoparticle areas with relatively high interphase density as opposed to the second zone adjacent to matrix regions. Fan et al. [40] reported that increasing interphase density certainly induced the modulus increase in a nanostructure-specific hierarchy in good accordance with our higher EInterphase in the case of SAinnerinterface in PVA bionanocomposites. The abovementioned results may suggest that SAInterface and VInterphase as typical 2D and 3D interphase dimensional parameters can play a vital role in the improvement of EInterphase leading to the further enhancement in mechanical properties of PVA bionanocomposites. Nevertheless, it is also admittedly mentioned here that partially embedded nanoparticles with the imperfect bonding effect within PVA matrices are inevitably detrimental to their mechanical properties with much less bonded matrix-filler interactions when compared with fully embedded counterparts.Figure 6
Relationships between interphase modulusEInterphase and interphase surface area SAInterphase: (a) PVA/HNT bionanocomposites and (b) PVA/Cloisite 30B clay bionanocomposites. Relationships between interphase modulus EInterphase and interphase volume VInterphase: (c) PVA/HNT bionanocomposites and (d) PVA/Cloisite 30B clay bionanocomposites.
(a)(b)(c)(d)On the other hand, the aspect ratio is used as an important factor to evaluate the reinforcement efficiency of nanofillers within polymer matrices in nanocomposites. Its determination becomes much easier when dealing with nanofillers with regular sizes and shapes. However, irregular nanofillers with different shapes are quite common leading to the complexity of real aspect ratios due to the effect of multistage dispersion processing techniques generally used in the manufacture of nanocomposites [25, 26]. Moreover, in addition to the nanofiller aspect ratio, interphase dimensions and properties can also be affected by nanofiller dispersion [26]. Liu and Brinson [5] as well as Alishahi et al. [41] suggested the use of interphase volume per unit nanofiller volume (VInterphase/Vnanofiller) in theoretical modelling as an effective parameter in promoting the reinforcement efficiency of nanofillers instead of VInterphase, whose relationship as a function of Vnanofiller was presented in Figure 7 in this study. Overall, an increase in Vnanofiller caused a significant decrease in VInterphase/Vnanofiller for both PVA bionanocomposites despite being more pronounced in PVA/Cloisite 30B clay bionanocomposites. This is because Cloisite 30B clay platelets induced more efficient planar reinforcements when compared to predominant longitudinal reinforcement of tubular HNTs.Figure 7
Interphase volume per unit nanofiller volume (VInterphase/Vnanofiller) as a function of Vnanofiller in PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays.


## 4.3. Particle Debonding

With an increase in nanoparticle loading, interparticle spacing appeared to be smaller and nanoparticles tended to be interconnected with neighbouring counterparts side by side with some nanoparticles free of contact with PVA matrices in bionanocomposites. Moreover, a closely compacted joint particle-to-particle pattern may induce adverse phase separation resulting in scattered load transfer from PVA matrices to nanoparticles. It was evidently shown in Figures8(a) and 8(c) that large HNT aggregates and clustered stack-up Cloisite 30B clays took place at a very high nanofiller content up to 10 wt% in PVA bionanocomposites with the resulting weak interfacial bonding for particle-matrix interactions. Two narrow zones near interphase regions were highlighted with a sharp decline in elastic modulus for both PVA bionanocomposites with excessive amounts of undispersed nanoparticles in Figures 8(b) and 8(d), indicating an existing particle debonding effect with polymer matrices [36]. A lower interphase modulus may also arise from a relatively low density near interphase zones as compared to those of aggregated nanoparticles and polymer matrices [24]. In addition to nanoparticle sizes and dimensions, nanoparticle dispersion within continuous matrices is also among the key factors in controlling the interfacial bonding between fillers and matrices. When excessive amounts of HNTs and Cloisite 30B clays were dispersed within PVA matrices, those nanoparticles tended to agglomerate owing to their weak Van der Waals interactions, while higher nanofiller content inevitably led to a significant reduction in interparticle spacing to hinder uniform nanoparticle dispersion [42, 43]. A nanofiller fine dispersion method such as ultrasonication [41], on the other hand, may give rise to the possible damage of nanoparticle structures especially at the high ultrasonic power intensity resulting instead in declined mechanical properties of bionanocomposites.Figure 8
AFM height mapping images and corresponding typical elastic modulus profiles of (a and b) PVA/10 wt% HNT bionanocomposites, as well as (c and d) PVA/10 wt% Cloisite 30B clay bionanocomposites indicating phase separation and agglomeration of nanoparticles, respectively. A7B7 and A8B8 are corresponding typical cut sections for scan distance.
(a)(b)(c)(d)


## 4.4. Prediction of Elastic Moduli of PVA Bionanocomposites

The prediction of the elastic moduli of PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays as a function of nominal and effective volume fractions of nanoparticles was presented in Figure9 (without predefined particle systems) and Figure 10 (with monodispersed and polydispersed particle systems), respectively, using Halpin-Tsai model, Mori-Tanaka model, and a combination of Mori-Tanaka model and laminate theory [4] along with associated modelling parameters listed in Table 1. In both bionanocomposites (without predefined particle systems), the estimated moduli did not coincide well with corresponding experimental data even at the very low volume fractions of HNTs and Cloisite 30B clays, Figure 9. However, the use of an effective volume fraction and a randomly oriented dispersion state of nanoparticles consistently gave rise to much closer correlation with experimental data regardless of the composite theoretical models implemented, as also confirmed in previous studies [34, 35]. In particular, with respect to PVA/Cloisite 30B clay bionanocomposites, the predicted results appeared to be in better agreement with experimental data at the low volume fraction of 0.061 vol% on the basis of an effective volume fraction and exfoliated clay structures, while better correlation was manifested for those predicted in terms of effective volume fraction and intercalated clay structures at high volume fraction levels. As far as morphological structures are concerned, it was expected that the majority of the Cloisite 30B clay platelets tended to be more exfoliated at low volume fractions as opposed to predominant intercalated structures with an increase in their volume fraction, which further confirmed a mix of exfoliated and intercalated structures taking place in PVA/Cloisite 30B clay bionanocomposites. The major drawback in using conventional Halpin-Tsai model and Mori-Tanaka model lies in their common assumption that both nanofillers and polymer matrices in nanocomposites are linearly elastic and isotropic materials with uniform nanofiller dispersion and the neglect of particle-particle interactions [44]. Generally speaking, isotropic nanoparticles are in possession of very large specific surface areas on which a substantial amount of molecular chains of polymer matrices can be absorbed with the aid of hydrogen bonding or electrostatic forces, resulting in the formation of interphase layers in nanocomposites. Consequently, such interphase layers behave as solid transitional material phases to restrict the mobility of other molecular chains of matrices, thus contributing positively to the total volume of reinforcements [31, 34]. This suggested that composite theoretical models in this study should use effective volume fraction instead of nominal volume fraction for better accuracy of modelling work.Figure 9
Prediction of the elastic moduli of PVA/HNT bionanocomposites: (a) Halpin-Tsai model (H-T) and (b) Mori-Tanaka model (M-T) and the combination of Mori-Tanaka model and laminate theory (M-T-L). Prediction of the elastic moduli of PVA/Cloisite 30B clay bionanocomposites: (c) Halpin-Tsai model (H-T) and (d) Mori-Tanaka model (M-T) and the combination of Mori-Tanaka model and laminate theory (M-T-L).
(a)(b)(c)(d)Figure 10
Prediction of the elastic moduli of PVA/HNT bionanocomposites: (a) Halpin-Tsai model (H-T) and (b) the combination of Mori-Tanaka model and laminate theory (M-T-L). Prediction of the elastic moduli of PVA/Cloisite 30B clay bionanocomposites: (c) Halpin-Tsai model (H-T) and (d) the combination of Mori-Tanaka model and laminate theory (M-T-L).
(a)(b)(c)(d)Table 1
Modelling parameters used for PVA bionanocomposite films.
Modelling parameterPVA/HNTPVA/Cloisite 30B clayRef.Em (GPa)2.082.08[45]Ep (GPa)140a178ba[46]b[32]lp (nm)208 (3 wt% HNTs)109 (3 wt% Cloisite 30B)[45]410 (5 wt% HNTs)206 (5 wt% Cloisite 30B)[45]900.6 (10 wt% HNTs)560 (10 wt% Cloisite 30B)[45]tp (nm)4.6 (3 wt% Cloisite 30B)[45]15.8 (5 wt% Cloisite 30B)[45]41.6 (10 wt% Cloisite 30B)[45]d (nm)35.2 (3 wt% HNTs)[45]66.5 (5 wt% HNTs)[45]85 (10 wt% HNTs)[45]ρm (g cm-3)0.390.39[45]ρp (g cm-3)1.81.98Material data sheetst (nm)~7.3~10.4[45]Rg (nm)1010[32]νm0.40.4[34]W″p0.66[32]d1 (nm)1.87[45]d2 (nm)3.6[45]N5.6[45]h (nm)0.98[45]ρp″ (g cm-3)3.1[45]ΑΤ (m2 g-1)40.3c658dc[47]d[32]Although as mentioned earlier, the implementation of an effective volume fraction can lead to a more accurate estimation of experimental data in PVA bionanocomposites, the layer thickness of polymeric molecules absorbed onto nanofiller surfaces in terms ofkx may vary even in the same grade of polymers with different molecular structures and polarities [32, 34]. Our aforementioned results demonstrated that interphase features actually did not possess a uniform interphase layer thickness, and the interphase dimensions and properties could vary significantly when embedded with heterogeneous nanoparticles with different particle sizes and dimensions, as well as diverse nanomechanical properties depending on matrix-filler interactions. As a result, interphase volume fraction ∅Interphase was incorporated into two different particle systems, namely monodispersed particles and polydispersed particles by considering the Fuller and EVF gradations, respectively. Modelling parameters presented in Table 2 were substituted into equations (11) and (13), respectively, to estimate corresponding ∅Interphase values used in PVA bionanocomposites in this study. Besides, interphase volume can be calculated according to equations (5), (6), (7), and (8) with associated results being used to determine the volume fraction of nanofillers. Such recalculated volume fractions of nanoparticles with respect to the interphase effect were incorporated into Halpin-Tsai model and Mori-Tanaka model accordingly in this study and illustrated in Figure 10 (with monodispersed and polydispersed particle systems). In a consistent manner, all elastic moduli of PVA bionanocomposites with the consideration of the interphase effect in our modelling work revealed a much closer relationship with experimental data as opposed to corresponding theoretical models using both nominal and effective volume fractions without interphase. More evidently, our proposed models in a monodispersed particle system caused better curve fitting with experimental data at low volume fractions up to 0.0604 and 0.061 vol% for HNTs and Cloisite 30B clays, respectively, as opposed to those in a polydispersed particle system with both the Fuller and EVF gradations, but vice versa beyond 0.1109 and 0.1105 vol%, accordingly. It was understandable, since HNTs and Cloisite 30B clays tended to be more uniformly dispersed in PVA matrices with relatively similar nanoparticle sizes (i.e., monodispersed particle system) at their low volume fraction, while typical nanoparticle agglomeration was evidently identified in better accordance with a polydispersed particle system. When a particulate gradation effect was taken into account, those predictions according to the Fuller gradation seemed to be in slightly better agreement with corresponding experimental data in PVA bionanocomposites, which might be ascribed to the theoretical assumption of finer particle dispersion taking place in EVF gradation at the same volume fraction of nanocomposites. Larger specific surface areas of nanoparticles generally took place in the case of EVF gradation, thus resulting in increasing ∅Interphase and volume fractions of nanoparticles. However, it was not always the case in our PVA bionanocomposites especially at very high volume fractions of nanoparticles.Table 2
Modelling parameters measured for the calculation of∅Interphasein PVA bionanocomposites.
Modelling parameterNanoparticle content (wt%)PVA/HNTaPVA/Cloisite 30B clayDeq‐min (nm)364.534.15168.554.310245.7141.6Deq‐max (nm)3836436524601836.21046453141Deq‐mean (nm)3208109541020610900.6560tInterphaseb (nm)58.610.7Note. aDeq‐min, Deq‐max, and Deq‐mean are the maximum, minimum, and mean equivalent diameters of anisotropic particles, which represent the maximum, minimum, and mean equivalent lengths of HNTs, respectively, in PVA/HNT bionanocomposites according to equations (A.24)–(A.34) in Appendix A. btInterphase is the average interphase thickness for the model development based on the assumption that the interphase generated between nanoparticles and polymer matrices was typically considered at the filler content of 5 wt% in PVA bionanocomposites.


## 5. Conclusions

A pioneering modelling approach was successfully developed to determine the nanomechanical properties of PVA bionanocomposites reinforced with HNTs and Cloisite 30B clays, as well as 3D interphase dimensions and modulus in terms of interphase surface areaSAInterphase and volume VInterphase. It was clearly shown that the interphase modulus was consistently enhanced with increasing SAInterphase and VInterphase, respectively, regardless of the embedding status of nanoparticles and proposed interfaces. The interphase moduli of PVA/Cloisite 30B clay bionanocomposites appeared to be greater than those of PVA/HNT bionanocomposites based on VInterphase/Vnanofiller. The nanomechanical properties and dimensions of the interphases were found to depend greatly on nanofiller dispersion states. As expected, increasing the nanofiller content inevitably decreased particle-particle spacing resulting in nanoparticle agglomeration with the visible interfacial debonding effect between nanoparticles and PVA matrices. The incorporation of an effective volume fraction and monodispersed and polydispersed particle systems with the interphase effect in conventional Halpin-Tsai model and Mori-Tanaka model was validated to be in better agreement with experimental data as opposed to those based on a nominal volume fraction of nanoparticles with the neglect of interphase features.



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