# Nanoscale Effect Investigation for Effective Bulk Modulus of Particulate Polymer Nanocomposites Using Micromechanical Framework

Authors: Tien-Thinh Le; Minh Vuong Le
Journal: Advances in Materials Science and Engineering
 (2021)

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## Abstract

This paper investigates the nanoscale effect on the effective bulk modulus of nanoparticle-reinforced polymer. An interface-based model is introduced in this work to study the nanoscale effects on the effective properties of heterogeneous materials. That interface model is able to capture discontinuity of mechanical fields across the surface between the nanoparticle and matrix. A generalized self-consistent scheme is then employed to determine the effective bulk modulus. It has been seen from the results that, in a certain range of limits, the influence of nanoscale effects on effective properties of heterogeneous materials is significant and needs to be taken into account. In particular, when the nanoparticle radius is smaller than 10 nm, the value of effective bulk modulus significantly increases when the characteristic size of nanofillers decreases. Besides, it is seen that the harder the inclusion, the smaller the nanoscale influence effects on the overall behaviors of composite materials. Finally, parametric studies in terms of surface strength and filler’s volume fractions are investigated and discussed, together with a comparison between the proposed model and other contributions in the literature.


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## Body

## 1. Introduction

As indicated by various experimental investigations using nuclear magnetic resonance (NMR), there is a disturbed area of the polymer matrix around the nanofillers. Papon et al. [1] observed that the mobility gradient distribution of the polymer matrix is significantly altered by the presence of silica nanoparticles. Using NMR, they found an area of the matrix surrounding the nanoparticles (with a thickness ranging between 2 and 14 nm) with slower dynamics. Similarly, Harton et al. [2] found an immobilized layer of the polymer matrix in the vicinity of the nanoparticle surface when experimentally investigating poly(2-vinylpyridine) reinforced with silica nanofillers. The presence of such a disturbed zone was most likely due to the inclusions and polymer chains interactions at small scales (atomic and molecular) [1, 3–7]. In terms of modeling, the effect of such a “third phase” on the effective properties of the material is no longer negligible [8]. Therefore, it must be taken into account and modelled, especially using micromechanical multiscale techniques [9–12].Currently, two main approaches are employed for modeling the disturbed area of the polymer matrix surrounding nanoparticles. The first approach models the disturbed area as an interphase of finite volume with unknown mechanical properties. For example, in the work of Marcadon et al. [13], the authors have investigated the interphase elastic constants using a micromechanical framework for polyethylene reinforced with silica nanoparticles. The results were compared with Molecular Dynamics simulation assuming that the interphase was isotropic linear elastic. In another study of Kim et al. [14], parameters of Molecular Dynamics such as thickness, interfacial interaction, and interphase structural change have been used to investigate the interphase behavior using a multiscale model. Le et al. [15–17] offered a stochastic multiscale model of the interphase properties, which exhibited random spatial fluctuations from both the experimental and simulation points of view. However, according to experimental results, the relation between the interphase thickness and the nanofiller size is still a difficulty, especially when considering the behaviors at small scales (i.e., atomic and molecular scales) [18, 19]. Moreover, it is difficult to monitor the interface’s mid-plane stretch using an interphase model, especially at small scales such as atomic or molecular scales [20, 21].In the second approach, which is based on continuum mechanics, the force transmission between the nanofillers and matrix has been investigated considering an interface (an interphase with no volume). The main goal of such model is to replace the interphase model, which means that traction or displacement jumps across the interphase are equal to those of the interface [22]. This type of interface model has been effectively applied to determine the influence of nanofiller characteristic size on the effective behaviors of heterogeneous materials. For instance, in the work of Gu and He [23], the interface properties have been investigated using Taylor’s expansion through a thin interphase medium. In another work [24], Benveniste has proposed an interface model to study the behavior of a thin anisotropic interphase. Numerical methods such as Finite Elements have been also used for these interface models. For example, in the work of Yvonnet et al. [25], the authors have used Finite Element Method to implement the interface model for the jumps of mechanical fields. As mentioned above, interphase models are difficult to monitor the mid-plane stretch of the interface; therefore, this is one of the most significant advantages of the proposed interface model [20], which leads to the possibility of dealing with nanoscale effects [26].Thus, this work investigates the nanoscale effect on the effective properties of particulate polymer nanocomposites using an interface model. Section2 describes the microstructure, homogenization scheme, and interface model. Section 3 presents the explicit calculation for obtaining the effective properties (bulk modulus) of the heterogeneous material exhibiting nanoscale effect. Finally, we present some numerical results and discuss them.


## 2. Materials and Methods

### 2.1. Description of Heterogeneous Microstructure

As depicted in Figure1, the thickness of interphase between the matrix and the inclusion is assumed to be zero, which converts it to an interface conveying the mechanical values between the matrix and the nanofillers.Figure 1
Microstructure of heterogeneous material.The microstructure in question comprises three phases: the nanoparticle (denoted by the subscriptp), the interface Г (denoted by subscript s), and the matrix (denoted by subscript m), respectively. The nanoparticle has a radius of rp, whose volume fraction is denoted by fy. The mechanical properties of the matrix and nanoparticle are considered linear, homogeneous, and isotropic. Consequently, the elasticity tensor of the nanoparticle and matrix phases is written as(1)Ci=κi+43μiκi−23μiκi−23μi000κi+43μiκi−23μi000κi+43μi0002μi00Sym.2μi02μi,where i = {p; m} and κ and µ are bulk and shear modulus, respectively. The mechanical property of the interface is considered spherically transversally isotropic (as it is a spherical surface surrounding the nanoparticle, with a unit normal vector n). In spherical coordinates {er, eθ, eφ}, the constitutive law for the zero-volume interface is expressed as follows [27]:(2)σθθsσφφsσθφs=κs+μsκs−μs0κs+μs0Sym.2μsεθθsεφφsεθφs,where κs and µs denote the interface bulk and shear modulus, respectively.


### 2.2. Jumps of Mechanical Fields across the Interface Model

The jumps of mechanical fields such as traction or displacement fields across the interface Г are represented below. First, let us denote the stress discontinuity at the interface by Г. Such a jump, denoted byσ, could be expressed as(3)σn=σmn−σpn=−divsσs,where n is the unit normal vector, divs is the divergence operator, and σ(s) is the stress tensor of the interface, respectively [28]. In spherical coordinates, the discontinuity of the stress field is written:(4)σrrer+σrθeθ+σrφeφ=−divsσs.The surface divergence has the following explicit form [29]:(5)divsσs=−σθθs+σφφserr+eθr∂σθθs∂θ+1sinθ∂σθφs∂φ+σθθs−σφφscotθ+eφr∂σθφs∂θ+1sinθ∂σφφs∂φ+2σθφscotθ.Note that, in the interface model, the displacement field across Г is continuous:(6)u=um−up=0.These conditions across the interface Г will be added to the homogenization formulations to determine the effective moduli of the heterogeneous material.


### 2.3. Homogenization Using Generalized Self-Consistent Scheme

Homogenization technique is a two-scale method used to study the behavior of composite materials [30]. The main objective of the method is to determine the behavior of the material at the global scale based on information given from the local scale. The local scale information of the structure is numerically calculated, with the boundary conditions being mostly of the periodic type. It is worth noting that the local problem is not computed on the whole structure, but only on a representative volume element which is sufficient to represent the behavior of the whole structure. A relation between local and global scales is constructed based on the goal of minimizing the potential energy between the two scales. For more advanced developments about homogenization technique, readers are referred to the works [31–38].In this work, the hypothesis of separation of scale was adopted [39, 40], enabling the generalized self-consistent micromechanical scheme to be used to determine the homogenized properties of the heterogeneous material [41–43]. Figure 2 depicts the homogenization scheme including the nanoparticle, the interface, the matrix phase, and the effective medium, respectively.Figure 2
Diagram of generalized self-consistent scheme.As the radius and volume fraction of the nanoparticle are given, the radius of the matrix phase, denoted byRm, is deduced:(7)Rm=Rp3fv1/3.The self-consistent condition for the domainV is written as(8)∫∂Vtu0−t0udS=0,where u and t represent the displacement and traction fields and u0 and t0 are the imposed displacement and stress field on the boundary of the domain, respectively. This condition was established on the basis of the self-consistency of the energy of the system.Finally, Figure3 presents the methodology flowchart of the present work.Figure 3
Flowchart methodology of this work.


## 2.1. Description of Heterogeneous Microstructure

As depicted in Figure1, the thickness of interphase between the matrix and the inclusion is assumed to be zero, which converts it to an interface conveying the mechanical values between the matrix and the nanofillers.Figure 1
Microstructure of heterogeneous material.The microstructure in question comprises three phases: the nanoparticle (denoted by the subscriptp), the interface Г (denoted by subscript s), and the matrix (denoted by subscript m), respectively. The nanoparticle has a radius of rp, whose volume fraction is denoted by fy. The mechanical properties of the matrix and nanoparticle are considered linear, homogeneous, and isotropic. Consequently, the elasticity tensor of the nanoparticle and matrix phases is written as(1)Ci=κi+43μiκi−23μiκi−23μi000κi+43μiκi−23μi000κi+43μi0002μi00Sym.2μi02μi,where i = {p; m} and κ and µ are bulk and shear modulus, respectively. The mechanical property of the interface is considered spherically transversally isotropic (as it is a spherical surface surrounding the nanoparticle, with a unit normal vector n). In spherical coordinates {er, eθ, eφ}, the constitutive law for the zero-volume interface is expressed as follows [27]:(2)σθθsσφφsσθφs=κs+μsκs−μs0κs+μs0Sym.2μsεθθsεφφsεθφs,where κs and µs denote the interface bulk and shear modulus, respectively.


## 2.2. Jumps of Mechanical Fields across the Interface Model

The jumps of mechanical fields such as traction or displacement fields across the interface Г are represented below. First, let us denote the stress discontinuity at the interface by Г. Such a jump, denoted byσ, could be expressed as(3)σn=σmn−σpn=−divsσs,where n is the unit normal vector, divs is the divergence operator, and σ(s) is the stress tensor of the interface, respectively [28]. In spherical coordinates, the discontinuity of the stress field is written:(4)σrrer+σrθeθ+σrφeφ=−divsσs.The surface divergence has the following explicit form [29]:(5)divsσs=−σθθs+σφφserr+eθr∂σθθs∂θ+1sinθ∂σθφs∂φ+σθθs−σφφscotθ+eφr∂σθφs∂θ+1sinθ∂σφφs∂φ+2σθφscotθ.Note that, in the interface model, the displacement field across Г is continuous:(6)u=um−up=0.These conditions across the interface Г will be added to the homogenization formulations to determine the effective moduli of the heterogeneous material.


## 2.3. Homogenization Using Generalized Self-Consistent Scheme

Homogenization technique is a two-scale method used to study the behavior of composite materials [30]. The main objective of the method is to determine the behavior of the material at the global scale based on information given from the local scale. The local scale information of the structure is numerically calculated, with the boundary conditions being mostly of the periodic type. It is worth noting that the local problem is not computed on the whole structure, but only on a representative volume element which is sufficient to represent the behavior of the whole structure. A relation between local and global scales is constructed based on the goal of minimizing the potential energy between the two scales. For more advanced developments about homogenization technique, readers are referred to the works [31–38].In this work, the hypothesis of separation of scale was adopted [39, 40], enabling the generalized self-consistent micromechanical scheme to be used to determine the homogenized properties of the heterogeneous material [41–43]. Figure 2 depicts the homogenization scheme including the nanoparticle, the interface, the matrix phase, and the effective medium, respectively.Figure 2
Diagram of generalized self-consistent scheme.As the radius and volume fraction of the nanoparticle are given, the radius of the matrix phase, denoted byRm, is deduced:(7)Rm=Rp3fv1/3.The self-consistent condition for the domainV is written as(8)∫∂Vtu0−t0udS=0,where u and t represent the displacement and traction fields and u0 and t0 are the imposed displacement and stress field on the boundary of the domain, respectively. This condition was established on the basis of the self-consistency of the energy of the system.Finally, Figure3 presents the methodology flowchart of the present work.Figure 3
Flowchart methodology of this work.


## 3. Results and Discussion

### 3.1. Determination of Effective Bulk Modulus

The homogeneous isotropic strain boundary condition to determine the effective bulk modulus is represented by(9)u0x=ε0x=ε0xε0yε0zx,y,z,∀x∈∂V,where ε0 is a material constant. In spherical coordinates, this condition reads(10)ur0x=ε0ruθ0x=0uφ0x=0,∀x∈∂V.Applying the above boundary conditions, the displacement solution can be expressed as(11)uri=firuθi=0uφi=0,where i = {p; m; e} (e denotes the effective medium) and f(i) is an unknown function to be solved. The strain field can then be expressed as the first derivative of the displacement field such as(12)εrri=dfirdr,εθθi=firr,εφφi=firr.By virtue of Hooke’s law, the stress field can be written:(13)σrri=λitrεi+2μidfirdr,σθθi=λitrεi+2μifirr,σφφi=λitrεi+2μifirr,where λ is the Lamé elastic constant and(14)trεi=εrri+εθθi+εφφi=dfirdr+2firr.In spherical coordinates, the equilibrium equation is expressed:(15)∂σrri∂r+1r2σrri−σθθi−σφφi=0.The differential equation obtained is of the form(16)r2d2fidr2+2rdfidr+b1ifi=0,where b1(i) are the constants as a function of λi and μi:(17)b1i=−2λi+5μiλi+2μi.The form of the functionf is such that(18)fi=Air3αi−1/2+Bir−3αi+1/2,where αi is a material parameter defined by He and Benveniste [44]. In this case with an isotropic material, αi is equal to 1. On the other hand, Ai and Bi are unknown constants, where i = {p; m; e}. It should be pointed out that, at r = 0, the function f must be finite. Moreover, as r tends toward infinity, the homogeneous strain boundary conditions need to be satisfied. Thus, we obtain Bp = 0 and Ae = ε0. Such simplification allows us to reduce the number of unknown constants to 4, which are solved by applying the conditions across the interface between the nanoparticle and matrix, and those between the matrix and equivalent medium, respectively. At the interface between the nanoparticle and matrix, there is discontinuity of the stress field and continuity of the displacement field, respectively. Such conditions are written:(19)urm−urp=0,σrrm−σrrp−1rσθθs+σφφs=0.At the interface between the matrix and equivalent medium, the mechanical values are continuous across the interface. Therefore,(20)ure−urm=0,σrre−σrrm=0.A linear system of equations is then obtained that allows us to calculateAp, Am, Bm, and Be, respectively. Finally, we obtain the effective bulk modulus of the heterogeneous material, as a function of the mechanical properties of different material phases and geometrical parameters:(21)κeff=12κmκsRm3+12κpμmRp4−12κmμmRp4+16κsμmRp3+9κpκmRpRm3+12κmμmRpRm39κmRp4−9κpRp4−12κsRp3+12κsRm3+9κpRpRm3+12μmRpRm3.


### 3.2. Investigation of Nanoscale Effect on Effective Bulk Modulus

The nanoscale effect on the effective bulk modulus is presented in this section. Various works have set out to determine the elastic surface moduli, for instance, by using stress simulation [45], atomistic simulation [46], contrasting between the atomistic simulation and continuum model [47], semi-analytical method [48, 49], and the asymptotic approach [40, 50]. In this work, we adopted the formulation proposed by Quang and He [40] to generate the surface moduli:(22)κs=hEs1−νs2,μs=hEs21−νs,where Es and νs are the surface Young’s modulus and Poisson’s ratio, respectively, and h presents the thickness of the fine interphase. In this work, different values of κs were considered such as 1, 3, 5, and 7 N/m, respectively. In addition, different values of volume fraction fv were investigated such as 0.2, 0.3, 0.4, and 0.5, respectively.On the other hand, for numerical application, the elastic moduli of the matrix and nanoparticle are chosen asEm = 6 GPa, νm = 0.25; Ep = 11.5 GPa, νp = 0.25.Let us denote the effective bulk modulus with and without interface effects such asKeff and κeff0, respectively. It is worth noting that the case without interface effects means that the change of mechanical fields such as traction or displacement between the matrix and inclusion is assumed to be continuous. In this study, we focus on investigating the ratio between κeff and κeff0 to determine the robustness of the proposed scheme.Figure4 presents the results of nanoscale effect, as a function of characteristic size of nanofillers, surface modulus, and volume fraction of nanoparticle, respectively. It can be seen that the effects of characteristic size of nanofillers are mostly detected when the nanoparticle radius is smaller than 10 nm. In particular, the effective bulk modulus increases when the nanoparticle radius decreases. On the other hand, when the characteristic size of nanofillers is greater than 30–40 nm, their effects on the effective bulk modulus are unnoticeable (i.e., κeff/κeff0 ratio is smaller than 1.02, equivalent to a maximum increase of 2% in bulk modulus). This observation shows that when the characteristic size of reinforcement becomes nanometric [9, 10, 51, 52], the interface plays a significant role in the effective behavior of the material. On the other hand, in terms of the volume fraction of the nanoparticle, it did not show any significant impact on the effective bulk modulus [53, 54].Figure 4
Nanoscale effect on bulk modulus as a function of (a)κs and (b)fy.
(a)(b)Moreover, as shown in Figure4(a), the nanoscale effect increases with increasing interface modulus. For instance, at Rp = 2 nm and fv = 0.4, the κeff/κeff0 ratios are 1.029, 1.081, 1.126, and 1.165, when κs are 1, 3, 5, and 7 N/m, respectively. It should be noted that the higher the strength (energy) of the interface, the greater the effect on the effective properties. As observed in Figure 4(b), the size effects are proportional to the inclusion volume fraction, between the solutions with and without the interface. For instance, at Rp = 2 nm and κs = 5 N/m, κeff/κeff0 ratios are 1.059, 1.091, 1.126, and 1.164, when fy are 0.2, 0.3, 0.4, and 0.5, respectively. The degree of influence of the interface on the effective behaviors is also proportional to the volume fraction of the inclusions, as they are dependent on each other.


### 3.3. Comparison with Existing Models

In this section, we compare the results of the framework proposed in this study and other frameworks existing in the literature, as in the work of Firooz et al. [52], Zemlyanova and Mogilevskaya [55], Nazarenko et al. [56], Duan et al. [22], and Gu et al. [57]. The formulae, interface models, and homogenization techniques used to compute the effective bulk modulus for each paper are expressed in Table 1.Table 1
Formulae for prediction of effective bulk modulus of nanocomposites taking into account nanoscale effect.
Ref.ExpressionInterface modelHomogenization techniqueFirooz et al. [52]κeff=3Rpκm3κp+4μm+4fvμmκp−κm+4λs+μs3κm+4fvμm/3Rp3κp+4μ2+3fvκm−κp+121−fvλs+μsSimplified general interface model that recovers the elastic (stress-type) interface modelComposite sphere assemblage methodZemlyanova and Mogilevskaya [55]κeff/κm=3κp+4μm+2ηs+4fvμmκp/κm−1+2ηs/3κm/3κp+4μm+2ηs−3fvκp−κm+2ηs/3, where ηs=2λs+μs/RpSteigmann–Ogden interface modelEquivalent particle (particle-plus-interface) + Maxwell’s homogenization approachNazarenko et al. [56]κeff=fvκp+κs+1−fvκm+9fv1−fvκκp−κm+κs2/1−9κfvκm+1−fvκp+κs−κc, where κ=1/9λc+2μc, and λc and µc in this case are the Lamé parameters of the reference medium, which is chosen as the matrix mediumGurtin–Murdoch interface modelEnergy-equivalent inhomogeneityDuan et al. [22]κeff=3κp3κm+4fvμm+2μm4fvμmκsr+3κm2−2fv+κsr/331−fvκp+3fvκm+2μm2+κsr−fvκsr, where κ=1/9λc+2μcGurtin–Murdoch interface modelEquivalent particle (particle-plus-interface) + composite spheres assemblageGu et al. [57]κeff=κm+fvκeq−κm3κm+4μm/31−fvκeq−κm+3κm+4μm, where κeq=κp+4/3κs/RGurtin–Murdoch interface modelEquivalent particle (particle-plus-interface) + generalized self-consistent methodThis workκeff=12κmκsRm3+12κpμmRp4−12κmμmRp4+16κsμmRp3+9κpκmRpRm3+12κmμmRpRm3/9κmRp4−9κpRp4−12κsRp3+12κsRm3+9κpRpRm3+12μmRpRm3Gurtin–Murdoch interface modelGeneralized self-consistent methodMaterial properties used to compute the effective bulk modulus of 6 models are indicated in Table2. Figures 5(a)–5(c) show the comparison of the effective bulk modulus between the framework proposed in this paper and 5 other frameworks in the literature, with the volume fraction fy = 0.2, 0.4, and 0.6, respectively, in function of the radius of the nanoparticle Rp. The contrast of elastic properties (i.e., Young’s modulus) between the nanoparticle and the matrix in this case is fixed at 10. It can be seen that good agreement between all the solutions for the bulk modulus was obtained; i.e., the model used in this work and other models in the literature have a stronger correlation. However, for the case of Duan et al. [22], a slight difference was observed for particle radius smaller than 10 nm. Moreover, for the case of fv = 0.2, the model proposed by Firooz et al. [52] also exhibited a difference compared to others. It is interesting to notice that Duan et al. [22] and Firooz et al. [52] have used composite spheres assemblage and Mori-Tanaka method for the homogenization scheme. Moreover, a general interface model has been developed in Firooz et al. [52]. Besides, Table 3 summarizes the values of κeff/κeff0 for each framework at Rp = 1 nm and Rp = 50 nm. We can see that when the volume fraction increases, the ratio values (especially when Rp = 1 nm) become close to each other.Table 2
Material properties used to compute the effective bulk modulus of 6 models.
EmGPaνmEpGPaνpfvλs=μsN/m60.2511.50.250.41Figure 5
Comparison of effective bulk modulus ratio curves between 6 models, in function of nanoparticle radius, with (a)fy = 0.2, (b)fy = 0.4, and (c) fv = 0.6, and Ep/Em = 10.
(a)(b)(c)Table 3
Values of theκeff/κeff0 ratio between different frameworks in the case where the contrast of elastic properties is fixed and the volume fraction is varied.
Ref.fy = 0.2, Rp = 1fy = 0.2, Rp = 50fy = 0.4, Rp = 1fy = 0.4, Rp = 50fy = 0.6, Rp = 50fy = 0.6, Rp = 50This work1.01511.00041.03431.00091.06361.0016Firooz et al. [52]1.00571.00021.02861.00081.06321.0016Zemlyanova and Mogilevskaya [55]1.01741.00051.03951.00101.07351.0019Nazarenko et al. [56]1.01741.00051.03951.00101.07351.0019Duan et al. [22]1.00851.00021.01931.00041.03531.0008Gu et al. [57]1.01511.00041.03431.00091.06361.0016Figures6(a)–6(d) show the comparison of the effective bulk modulus between the framework proposed in this paper and 5 other frameworks in the literature, with the contrast of elastic properties (i.e., Young’s modulus) between the particle and the matrix Ep/Em = 10, 20, 30, and 1000, respectively, in function of the radius of the nanoparticle Rp. Such a comparison reveals the influence of particle to matrix stiffness ratio on the overall bulk modulus (for very stiff and very soft inclusions, respectively). The volume fraction of the inclusion used in this case is fixed at 0.4. It is seen that the higher the Ep/Em ratio, the smaller the nanoscale effect we have on the overall behavior of the material.Figure 6
Comparison of effective bulk modulus ratio curves between 6 models, in function of nanoparticle radius, withfv = 0.4 and (a) Ep/Em = 10, (b) Ep/Em = 20, (c) Ep/Em = 30, and (d) Ep/Em = 103.
(a)(b)(c)(d)It is seen that the higher theEp/Em ratio, the smaller the nanoscale effect on the effective properties of nanocomposites. From an experimental point of view, the effects of polymer/particle interactions on polymer chain dynamics and on mechanical properties of nanocomposites have been widely investigated, sometimes with non-conclusive evidence. The importance of interactions, like covalent bonds or hydrogen bonds, among the surface of inorganic fillers and the organic matrix has been shown at a molecular level by, for example, Avolio et al. [58] and Nicola et al. [59]. However, the effect of such interactions on experimentally measured macroscopic properties is not easy to rationalize.In several experimental studies, nanoscale effect on the effective properties of heterogeneous materials has been clearly shown, particularly for polymers reinforced with spherical nanofillers. For example, in the work of Douce et al. [60], the inclusion size effect on the effective behavior (tensile modulus) of nanocomposite has been investigated. Three different inclusion diameters have been used for the comparison such as 15, 35, and 60 nm. In particular, it can be noted, for example, that the effective tensile modulus was 4.3 and 3.95 GPa for the inclusions whose diameters are 15 and 35 nm, respectively. However, the effects of 60 nm particles were not observed in this case. A possible explanation is that, for this size of reinforcements (i.e., larger than 50 nm in diameter), the nanoparticles tend to form aggregates, which are responsible for the very significant increase of the modulus. Most recently, Blivi et al. [61] have explored experimental evidence of size effect in silica-reinforced poly(methyl methacrylate). Uniaxial tensile tests have been carried out and showed that Young's modulus has increased in decreasing particle size (particle sizes are 500, 150, 60, and 25 nm and volume fraction of reinforcement is kept constant). Finally, other experimental results on nanoscale effect are available, such as [61–64].For a micromechanics modeling point of view, the proposed model can be used to retrieve information on the interface/interphase formation and properties [51, 53]. More precisely, formation of the interphase is exhibited by its thickness, whereas properties of the interphase are directly related to the thickness and the interface parameters (bulk and shear surface moduli). Thus, if the interface parameters are known, the interphase properties can be computed through an inverse identification. On the other hand, the prediction of nanocomposites overall properties accounting for interphase/interface effect is much more complex in real systems because of various aspects such as the following:(i)
The thickness of interphase in function of particle distribution size, shape of nanoparticle, and temperature [13, 40, 42, 65–67](ii)
Existence of both nanoparticulate agglomeration phenomenon and free nanoparticles in real systems [66, 68](iii)
Existence of overlapping interphase in real systems [69–72](iv)
Stochastic analysis being required in most cases [73, 74, 75, 76, 77, 78, 79]


## 3.1. Determination of Effective Bulk Modulus

The homogeneous isotropic strain boundary condition to determine the effective bulk modulus is represented by(9)u0x=ε0x=ε0xε0yε0zx,y,z,∀x∈∂V,where ε0 is a material constant. In spherical coordinates, this condition reads(10)ur0x=ε0ruθ0x=0uφ0x=0,∀x∈∂V.Applying the above boundary conditions, the displacement solution can be expressed as(11)uri=firuθi=0uφi=0,where i = {p; m; e} (e denotes the effective medium) and f(i) is an unknown function to be solved. The strain field can then be expressed as the first derivative of the displacement field such as(12)εrri=dfirdr,εθθi=firr,εφφi=firr.By virtue of Hooke’s law, the stress field can be written:(13)σrri=λitrεi+2μidfirdr,σθθi=λitrεi+2μifirr,σφφi=λitrεi+2μifirr,where λ is the Lamé elastic constant and(14)trεi=εrri+εθθi+εφφi=dfirdr+2firr.In spherical coordinates, the equilibrium equation is expressed:(15)∂σrri∂r+1r2σrri−σθθi−σφφi=0.The differential equation obtained is of the form(16)r2d2fidr2+2rdfidr+b1ifi=0,where b1(i) are the constants as a function of λi and μi:(17)b1i=−2λi+5μiλi+2μi.The form of the functionf is such that(18)fi=Air3αi−1/2+Bir−3αi+1/2,where αi is a material parameter defined by He and Benveniste [44]. In this case with an isotropic material, αi is equal to 1. On the other hand, Ai and Bi are unknown constants, where i = {p; m; e}. It should be pointed out that, at r = 0, the function f must be finite. Moreover, as r tends toward infinity, the homogeneous strain boundary conditions need to be satisfied. Thus, we obtain Bp = 0 and Ae = ε0. Such simplification allows us to reduce the number of unknown constants to 4, which are solved by applying the conditions across the interface between the nanoparticle and matrix, and those between the matrix and equivalent medium, respectively. At the interface between the nanoparticle and matrix, there is discontinuity of the stress field and continuity of the displacement field, respectively. Such conditions are written:(19)urm−urp=0,σrrm−σrrp−1rσθθs+σφφs=0.At the interface between the matrix and equivalent medium, the mechanical values are continuous across the interface. Therefore,(20)ure−urm=0,σrre−σrrm=0.A linear system of equations is then obtained that allows us to calculateAp, Am, Bm, and Be, respectively. Finally, we obtain the effective bulk modulus of the heterogeneous material, as a function of the mechanical properties of different material phases and geometrical parameters:(21)κeff=12κmκsRm3+12κpμmRp4−12κmμmRp4+16κsμmRp3+9κpκmRpRm3+12κmμmRpRm39κmRp4−9κpRp4−12κsRp3+12κsRm3+9κpRpRm3+12μmRpRm3.


## 3.2. Investigation of Nanoscale Effect on Effective Bulk Modulus

The nanoscale effect on the effective bulk modulus is presented in this section. Various works have set out to determine the elastic surface moduli, for instance, by using stress simulation [45], atomistic simulation [46], contrasting between the atomistic simulation and continuum model [47], semi-analytical method [48, 49], and the asymptotic approach [40, 50]. In this work, we adopted the formulation proposed by Quang and He [40] to generate the surface moduli:(22)κs=hEs1−νs2,μs=hEs21−νs,where Es and νs are the surface Young’s modulus and Poisson’s ratio, respectively, and h presents the thickness of the fine interphase. In this work, different values of κs were considered such as 1, 3, 5, and 7 N/m, respectively. In addition, different values of volume fraction fv were investigated such as 0.2, 0.3, 0.4, and 0.5, respectively.On the other hand, for numerical application, the elastic moduli of the matrix and nanoparticle are chosen asEm = 6 GPa, νm = 0.25; Ep = 11.5 GPa, νp = 0.25.Let us denote the effective bulk modulus with and without interface effects such asKeff and κeff0, respectively. It is worth noting that the case without interface effects means that the change of mechanical fields such as traction or displacement between the matrix and inclusion is assumed to be continuous. In this study, we focus on investigating the ratio between κeff and κeff0 to determine the robustness of the proposed scheme.Figure4 presents the results of nanoscale effect, as a function of characteristic size of nanofillers, surface modulus, and volume fraction of nanoparticle, respectively. It can be seen that the effects of characteristic size of nanofillers are mostly detected when the nanoparticle radius is smaller than 10 nm. In particular, the effective bulk modulus increases when the nanoparticle radius decreases. On the other hand, when the characteristic size of nanofillers is greater than 30–40 nm, their effects on the effective bulk modulus are unnoticeable (i.e., κeff/κeff0 ratio is smaller than 1.02, equivalent to a maximum increase of 2% in bulk modulus). This observation shows that when the characteristic size of reinforcement becomes nanometric [9, 10, 51, 52], the interface plays a significant role in the effective behavior of the material. On the other hand, in terms of the volume fraction of the nanoparticle, it did not show any significant impact on the effective bulk modulus [53, 54].Figure 4
Nanoscale effect on bulk modulus as a function of (a)κs and (b)fy.
(a)(b)Moreover, as shown in Figure4(a), the nanoscale effect increases with increasing interface modulus. For instance, at Rp = 2 nm and fv = 0.4, the κeff/κeff0 ratios are 1.029, 1.081, 1.126, and 1.165, when κs are 1, 3, 5, and 7 N/m, respectively. It should be noted that the higher the strength (energy) of the interface, the greater the effect on the effective properties. As observed in Figure 4(b), the size effects are proportional to the inclusion volume fraction, between the solutions with and without the interface. For instance, at Rp = 2 nm and κs = 5 N/m, κeff/κeff0 ratios are 1.059, 1.091, 1.126, and 1.164, when fy are 0.2, 0.3, 0.4, and 0.5, respectively. The degree of influence of the interface on the effective behaviors is also proportional to the volume fraction of the inclusions, as they are dependent on each other.


## 3.3. Comparison with Existing Models

In this section, we compare the results of the framework proposed in this study and other frameworks existing in the literature, as in the work of Firooz et al. [52], Zemlyanova and Mogilevskaya [55], Nazarenko et al. [56], Duan et al. [22], and Gu et al. [57]. The formulae, interface models, and homogenization techniques used to compute the effective bulk modulus for each paper are expressed in Table 1.Table 1
Formulae for prediction of effective bulk modulus of nanocomposites taking into account nanoscale effect.
Ref.ExpressionInterface modelHomogenization techniqueFirooz et al. [52]κeff=3Rpκm3κp+4μm+4fvμmκp−κm+4λs+μs3κm+4fvμm/3Rp3κp+4μ2+3fvκm−κp+121−fvλs+μsSimplified general interface model that recovers the elastic (stress-type) interface modelComposite sphere assemblage methodZemlyanova and Mogilevskaya [55]κeff/κm=3κp+4μm+2ηs+4fvμmκp/κm−1+2ηs/3κm/3κp+4μm+2ηs−3fvκp−κm+2ηs/3, where ηs=2λs+μs/RpSteigmann–Ogden interface modelEquivalent particle (particle-plus-interface) + Maxwell’s homogenization approachNazarenko et al. [56]κeff=fvκp+κs+1−fvκm+9fv1−fvκκp−κm+κs2/1−9κfvκm+1−fvκp+κs−κc, where κ=1/9λc+2μc, and λc and µc in this case are the Lamé parameters of the reference medium, which is chosen as the matrix mediumGurtin–Murdoch interface modelEnergy-equivalent inhomogeneityDuan et al. [22]κeff=3κp3κm+4fvμm+2μm4fvμmκsr+3κm2−2fv+κsr/331−fvκp+3fvκm+2μm2+κsr−fvκsr, where κ=1/9λc+2μcGurtin–Murdoch interface modelEquivalent particle (particle-plus-interface) + composite spheres assemblageGu et al. [57]κeff=κm+fvκeq−κm3κm+4μm/31−fvκeq−κm+3κm+4μm, where κeq=κp+4/3κs/RGurtin–Murdoch interface modelEquivalent particle (particle-plus-interface) + generalized self-consistent methodThis workκeff=12κmκsRm3+12κpμmRp4−12κmμmRp4+16κsμmRp3+9κpκmRpRm3+12κmμmRpRm3/9κmRp4−9κpRp4−12κsRp3+12κsRm3+9κpRpRm3+12μmRpRm3Gurtin–Murdoch interface modelGeneralized self-consistent methodMaterial properties used to compute the effective bulk modulus of 6 models are indicated in Table2. Figures 5(a)–5(c) show the comparison of the effective bulk modulus between the framework proposed in this paper and 5 other frameworks in the literature, with the volume fraction fy = 0.2, 0.4, and 0.6, respectively, in function of the radius of the nanoparticle Rp. The contrast of elastic properties (i.e., Young’s modulus) between the nanoparticle and the matrix in this case is fixed at 10. It can be seen that good agreement between all the solutions for the bulk modulus was obtained; i.e., the model used in this work and other models in the literature have a stronger correlation. However, for the case of Duan et al. [22], a slight difference was observed for particle radius smaller than 10 nm. Moreover, for the case of fv = 0.2, the model proposed by Firooz et al. [52] also exhibited a difference compared to others. It is interesting to notice that Duan et al. [22] and Firooz et al. [52] have used composite spheres assemblage and Mori-Tanaka method for the homogenization scheme. Moreover, a general interface model has been developed in Firooz et al. [52]. Besides, Table 3 summarizes the values of κeff/κeff0 for each framework at Rp = 1 nm and Rp = 50 nm. We can see that when the volume fraction increases, the ratio values (especially when Rp = 1 nm) become close to each other.Table 2
Material properties used to compute the effective bulk modulus of 6 models.
EmGPaνmEpGPaνpfvλs=μsN/m60.2511.50.250.41Figure 5
Comparison of effective bulk modulus ratio curves between 6 models, in function of nanoparticle radius, with (a)fy = 0.2, (b)fy = 0.4, and (c) fv = 0.6, and Ep/Em = 10.
(a)(b)(c)Table 3
Values of theκeff/κeff0 ratio between different frameworks in the case where the contrast of elastic properties is fixed and the volume fraction is varied.
Ref.fy = 0.2, Rp = 1fy = 0.2, Rp = 50fy = 0.4, Rp = 1fy = 0.4, Rp = 50fy = 0.6, Rp = 50fy = 0.6, Rp = 50This work1.01511.00041.03431.00091.06361.0016Firooz et al. [52]1.00571.00021.02861.00081.06321.0016Zemlyanova and Mogilevskaya [55]1.01741.00051.03951.00101.07351.0019Nazarenko et al. [56]1.01741.00051.03951.00101.07351.0019Duan et al. [22]1.00851.00021.01931.00041.03531.0008Gu et al. [57]1.01511.00041.03431.00091.06361.0016Figures6(a)–6(d) show the comparison of the effective bulk modulus between the framework proposed in this paper and 5 other frameworks in the literature, with the contrast of elastic properties (i.e., Young’s modulus) between the particle and the matrix Ep/Em = 10, 20, 30, and 1000, respectively, in function of the radius of the nanoparticle Rp. Such a comparison reveals the influence of particle to matrix stiffness ratio on the overall bulk modulus (for very stiff and very soft inclusions, respectively). The volume fraction of the inclusion used in this case is fixed at 0.4. It is seen that the higher the Ep/Em ratio, the smaller the nanoscale effect we have on the overall behavior of the material.Figure 6
Comparison of effective bulk modulus ratio curves between 6 models, in function of nanoparticle radius, withfv = 0.4 and (a) Ep/Em = 10, (b) Ep/Em = 20, (c) Ep/Em = 30, and (d) Ep/Em = 103.
(a)(b)(c)(d)It is seen that the higher theEp/Em ratio, the smaller the nanoscale effect on the effective properties of nanocomposites. From an experimental point of view, the effects of polymer/particle interactions on polymer chain dynamics and on mechanical properties of nanocomposites have been widely investigated, sometimes with non-conclusive evidence. The importance of interactions, like covalent bonds or hydrogen bonds, among the surface of inorganic fillers and the organic matrix has been shown at a molecular level by, for example, Avolio et al. [58] and Nicola et al. [59]. However, the effect of such interactions on experimentally measured macroscopic properties is not easy to rationalize.In several experimental studies, nanoscale effect on the effective properties of heterogeneous materials has been clearly shown, particularly for polymers reinforced with spherical nanofillers. For example, in the work of Douce et al. [60], the inclusion size effect on the effective behavior (tensile modulus) of nanocomposite has been investigated. Three different inclusion diameters have been used for the comparison such as 15, 35, and 60 nm. In particular, it can be noted, for example, that the effective tensile modulus was 4.3 and 3.95 GPa for the inclusions whose diameters are 15 and 35 nm, respectively. However, the effects of 60 nm particles were not observed in this case. A possible explanation is that, for this size of reinforcements (i.e., larger than 50 nm in diameter), the nanoparticles tend to form aggregates, which are responsible for the very significant increase of the modulus. Most recently, Blivi et al. [61] have explored experimental evidence of size effect in silica-reinforced poly(methyl methacrylate). Uniaxial tensile tests have been carried out and showed that Young's modulus has increased in decreasing particle size (particle sizes are 500, 150, 60, and 25 nm and volume fraction of reinforcement is kept constant). Finally, other experimental results on nanoscale effect are available, such as [61–64].For a micromechanics modeling point of view, the proposed model can be used to retrieve information on the interface/interphase formation and properties [51, 53]. More precisely, formation of the interphase is exhibited by its thickness, whereas properties of the interphase are directly related to the thickness and the interface parameters (bulk and shear surface moduli). Thus, if the interface parameters are known, the interphase properties can be computed through an inverse identification. On the other hand, the prediction of nanocomposites overall properties accounting for interphase/interface effect is much more complex in real systems because of various aspects such as the following:(i)
The thickness of interphase in function of particle distribution size, shape of nanoparticle, and temperature [13, 40, 42, 65–67](ii)
Existence of both nanoparticulate agglomeration phenomenon and free nanoparticles in real systems [66, 68](iii)
Existence of overlapping interphase in real systems [69–72](iv)
Stochastic analysis being required in most cases [73, 74, 75, 76, 77, 78, 79]


## 4. Conclusions

In this work, an interface model was integrated into the classical homogenization scheme to explore the nanoscale effect of materials reinforced with nanometric fillers. To this end, the mechanical values across the surface between the nanofiller and matrix exhibited discontinuities, which were captured by the interface model. The effective modulus of the material was then derived based on the generalized self-consistent micromechanical scheme. It was shown that the effective bulk modulus depended on the characteristic size of fillers, especially when they are nanometric in scale. Moreover, the effective modulus depended on the surface strength and the volume fraction of the fillers.The comparison with experimental data (mostly for measurement of Young’s modulus of nanocomposites) will be considered in our next research, where effective bulk, shear, and Young’s moduli are expressed as a function of surface parameters. In this work, we mainly focus on the nanoscale effect on the effective behaviour of heterogeneous materials in the elastic regime. In future works, other regimes such as elastoplastic or viscoplastic can be taken into account for this type of study. In addition, besides using numerical method such as Finite Element Method, Molecular Dynamics can also be applied to study the atomic and molecular scales. At present, there is a shortfall in such simulations coupling continuum mechanics and atomistic simulations to model the interface region between the matrix and nanofillers.



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