Nanoreinforced polymer composites: 3D FEM modeling with effective interface concept

Composites Science and Technology 71 (2011) 980–988

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Nanoreinforced polymer composites: 3D FEM modeling with effective interface concept H.W. Wang a,b,⇑, H.W. Zhou c, R.D. Peng c, Leon Mishnaevsky Jr. b,⇑ a

Tianjin Key Laboratory of Refrigeration Technology, Tianjin University of Commerce, Tianjin 300134, China Materials Research Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark c State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology (Beijing), Beijing 100083, China b

a r t i c l e

i n f o

Article history: Received 20 September 2010 Received in revised form 18 February 2011 Accepted 6 March 2011 Available online 10 March 2011 Keywords: A. Nanocomposites B. Mechanical properties C. Modeling C. Finite element analysis (FEA) C. Elastic properties

a b s t r a c t A computational study of the effect of structures of nanocomposites on their elastic properties is presented. The special program code for the automatic generation of 3D multiparticle unit cells with/without overlapping, effective interface layers around particles is developed for nanocomposite modeling. The generalized effective interface model, with two layers of different stiffnesses and the option of overlapping layers is developed here. The effects of the effective interface properties, particle sizes, particle shapes (spherical, cylindrical, ellipsoidal and disc-shaped) and volume fraction of nanoreinforcement on the mechanical properties of nanocomposites are studied in numerical experiments. The higher degree of particle clustering leads to lower Young’s modules of the nanocomposites. The shape of nanoparticles has a strong effect on the elastic properties of the nanocomposites. The most effective reinforcement is cylindrical one, followed by ellipsoids, discs, and last, spheres. Ideally random oriented and correlated microstructures lead to the same average Young moduli, yet, the standard deviation of Young modulus for correlated microstructure is nearly 4 times of that for fully random orientation case. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The perspective of development of multiscale composites with improved properties, combining the advantages of polymers, fiber reinforcement or/and nano-reinforced materials present an important step toward the realization of extremely strong, extremely light and extremely tough materials for structural application. Nanostructured polymers, formed by addition of small amount of nanoparticles (e.g. silicates and clay particles with high aspect ratios) have much better mechanical properties than the common polymers, and can be used eventually as matrix in the fiber reinforced composites. In the past few years, a lot of research activities were initiated in the area of modeling of mechanical properties of polymer composites with nanoscale reinforcement, in particular, the effect of structures of nanocomposites on their properties. The important challenges for modeling nanocomposites were to simulate the mechanisms responsible for the strong effect of small amount of nanoreinforcement on the extraordinary properties of nanocom⇑ Corresponding authors. Address: Tianjin Key Laboratory of Refrigeration Technology, Tianjin University of Commerce, Tianjin 300134, China. Tel.: +86 22 2668 6251; fax: +86 22 2668 6268 (H.W. Wang), tel.: +45 75729 5729; fax: +45 4677 5758 (Leon Mishnaevsky). E-mail addresses: [email protected] (H.W. Wang), [email protected] (L. Mishnaevsky). 0266-3538/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2011.03.003

posites, and to generalize the micromechanical methods of modeling common microscale reinforced composites to the case of nanoscale reinforcement. Table 1 shows a review of some research works in the area of micromechanical modeling of nanocomposites. It can be seen in the table many computational models of nanocomposites include the interfacial region as separate phase, sometimes even clusters/ sheets/intercalated regions of nanoparticles as separate phases and seek to take into account the high aspect ratio of the nanoparticles. In this paper, we seek to carry out direct micromechanical analysis of the effect of structures of nanocomposites on their elastic properties. 3D micromechanical FE model of nanocomposites is developed and used to predict the elastic properties of the materials and the effect of the nanoreinforcement on the properties. The results of the investigation are expected to provide some design parameters for the microstructural optimization of the nanocomposites.

2. Generalized effective interface model and automatic 3D FE model generation In order to simulate the mechanical behavior of nanocomposites, several peculiar effects should be taken into account, which are not available in common microscale particles reinforced composites. Among them, one should mention the much stronger

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Table 1 An overview of micromechanical models of nanocomposites. Material and shape

Method, predicted properties

Polymer nanocomposites

Takayanagi’s equation ? tensile modulus Halpin–Tsai code LITAC ? buckling, E Halpin–Tsai Mori–Tanaka Experiment ? E

Nano-platelets reinforced composites Nylon 6/layered aluminosilicates (MMT) Nylon 6/glass fibers Nanoclay/P Nanotube/P

Nanoclay/P Montmorillonite silicate/P

Experiment Mori– Tanaka ? E Mori–Tanaka FEM 3D ? E, waviness Shear lag model ? load transfer Mori–Tanaka ? Overall moduli

Nanoclay/P

Halpin–Tsai Mori–Tanaka Experiment FEM 2D ? E

Polymer–clay nanocomposites (PCNs)

FEM ? E

Silica nanoparticle/ polyimide spheres

MD Eshelby ? E

Polymer nanocomposites

FEM ? E

Nanoclay/P

Mori–Tanaka FEM 2D & 3D ? stiffness

Nanocomposites

Stochastic Monte-Carlo approach, ? E

CNT/epoxy

Experiment Micromechanics ? E

Nanotube nanoplatelet

Mori–Tanaka ? E

P nanocomposites sphere P nanocomposites circular

Mean-field (MF) FEM ? E FEM 2D ? viscoelastic properties Experiment Eshelby FEM 2D ? E

Nanoclay/P

SWCNT/PET

Experiment Cox and Krenchel Halpin–Tsai Mori– Tanaka ? E

Influence on

Conclusions/Remarks

F p

D

C

–

–

p

p

p

p

p

p

p

p

p

p

p

– p

–

– –

p

p

p

p

p

–

– p –

–

–

–

–

p

p

p

p

p

p

p

p p

–

–

– –

p

p

p

p

p

p

p p

p

p

–

p

A p

– p

– p p –

–

–

p p

p

I p – –

– –

– p

–

p

p

p –

–

–

–

p

p

– p

p p

p

p

p

–

Three-phase model including the matrix, interfacial region, and fillers [1] Comparison of tensile and compressive elastic moduli, and prediction of effects of incomplete exfoliation and imperfect alignment on modulus [2] The theories of Halpin–Tsai and Mori–Tanaka are used to evaluate the effects of filler geometry, stiffness, and orientation. Model predictions are compared to experimental morphological and mechanical property data [3] Three-phase model including the epoxy matrix, the exfoliated clay nanolayers and the nanolayer clusters was developed [4] The effect of waviness on effective moduli of CNT composites was determined by using FEM and the strain concentration tensor in a composite consisting of wavy carbon nanotubes was evaluated [5,6] A concept of effective length of reinforcement was adopted to represent the load transfer efficiency [7] Formulas for the overall moduli of composite materials reinforced with transversely isotropic spheroids are derived; a hierarchical model is proposed to consider intercalated silicate stacks; a simple model containing constrained regions around the reinforcements is also proposed [8,9] Multiscale modeling was presented taking into account the hierarchical morphology of the nanoclay reinforcement; besides exfoliated nanoclay sheets, an ‘effective particle’ was proposed to represent the nanoclays distributed in the matrix with intercalation structure [10] A locally orthotropic finite element model was developed, and then a number of flake groups with varied orientations were assembled to predict the actual moduli seen in PCNs [11] A micromechanical model was suggested that includes an effective interface between the polyimide and nanoparticle with properties and dimensions that are determined using the results of molecular dynamics simulations; effect of the nanoparticle/Polyimide interface on elastic properties was determined [12] A FEM model with emphasizing on the role of inclusion/matrix interphase was developed with different shapes including spherical, cylindrical, and platelet [13] Both two-dimensional and three-dimensional finite element models are presented for aligned and randomly oriented clay particles which are randomly distributed; the differences between simulation results and Mori–Tanaka model were discussed and owing to the clusters of nearly aligned particles that formed at high volume fractions [14] At the same volume fraction, platelets are generally more efficient than fibers in improving composite modulus; low interfacial adhesion and poor dispersion of inclusions lead to decrease in reinforcement efficiency [15] A new micromechanical model denominated Dilute Suspension of Clusters was developed to consider the heterogeneous dispersion of nanoreinforcement of the composite and the clusters formation [16] The mechanical reinforcing efficiencies of two types of nanoparticles, nanotube and nanoplatelet, are compared. Additionally, the interphase zone in the vicinity of the nanoparticles is addressed [17] Modeling of nanocomposites using the mean field approach [18] The influence of the interphase and its structure was studied [19] A micromechanical analytical approach based on a multiscale framework is presented in which special attention is devoted to the constrained region around reinforcements [20] Elastic constants of SWCNT-reinforced PET composites were determined by tensile tests. The experimental results were compared to some micromechanical models which take into account orientation and aspect ratio of the nanotubes but not curvature of the nanotubes, and as a result it was addressed that the waviness of nanotubes is an important factor that influences the reinforcing efficiency [21]

p F, D, C, A and I represents the effect of fraction, distribution, cluster, aspect ratio and Interface respectively. ‘‘ ’’ denotes the effect is considered and ‘‘–’’ denotes the effect is not considered. ? Studied elastic properties, P – polymer.

effect of interface/interphases on the nanocomposite properties, the importance of the effect of nanoclay distribution and clustering (intercalated versus exfoliated structures), high aspect ratio and clustered reinforcement. In this section, we present the developed computational tools for modeling the nanocomposites. The special program code for the automatic generation of multiparticle unit cells for nanocomposites modeling is developed. The unit cells can contain an effective interface layer around each particle. The thickness of this layer might be varied. The interface layer can consist in turn of several

sublayers, and these sublayers or the interface layers as a whole can overlap with corresponding sublayers of neighboring particle, if they are located close enough. 2.1. Generalized effective interface model One of the main peculiarities of the mechanical behavior of nanoreinforcement as compared with microreinforcement is that the properties of the interface/interphase between the nanoparticles and the matrix play a much bigger role.

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The micromechanical two-phase models of composites, used in many works (see a review in [22]), do not typically take into account the interface/interphase effects. In order to generalize the micromechanical models of composites onto nanocomposites, the effective interface model (EIM) was proposed by Odegard and his colleagues [12]. Here, we develop a computational finite element micromechanical model based on the effective interface model. In this model, the homogeneous region of finite size represents molecular structure of the perturbed polyimide and interfacial molecules with a gradual transition to the bulk molecular structure. The ‘‘effective interface layer’’ is considered here as a region of finite thickness around nanoparticle, which contains interphases, and/or other structural deviations, or has atomistic structures and properties different from those of both matrix and particles. The effective interface model was generalized to be applied to the analysis of the nanoparticles clustering (intercalated microstructures) in the following way. As noted by Odegard [23], the effective interface model in its classical version is not applicable for the case of the high volume fraction of nanoparticles, when the particles might touch one another. In fact, the properties of the interface/interphase layer would change between nanoparticles if they are located very closely. This makes it impossible to apply the effective interface concept to analyze intercalated and clustered microstructures of nanocomposites (what is a very important subject for the nanocomposites structures). In order to overcome this problem, a generalized effective interface model is proposed for the computational modeling of nanocomposites. In the generalized effective interface model, the interface/interphase layer between the matrix and a particle consists of two layers, which have different properties, one of them stiffer (outer layer) and one of them softer (inner layer). While the outer interface layers are allowed to overlap, the inner layers do not overlap with those of neighboring particles. The idea was that the clustering of nanoparticles (which leads to the overlapping of out effective interface layers) should ensure a lower and not a higher stiffness. With the bilayer effect interface model, where only the outer stiffer layers can overlap, the clustering leads to the lower stiffness of the nanocomposites. It was further postulated that after averaging, the elastic properties of the interface layer correspond to those in [12]. The thickness of the effective layer was taken 12 Å (as in [12]), each sublayer having the thickness of 6 Å. The Young’s moduli of the outer Eifout and inner layers Eifinn were calculated by formulas:

Eifout ¼ Emat þ ðEmat  Eif Þ ¼ 2Emat  Eif Eif ¼

Eifout þ Eifinn 2

ð1Þ ð2Þ

or

Eifinn ¼ 2Eif  Eifout

ð3Þ

where Emat and Eif are the Young moduli of matrix and the effective interface [12]. The generalized effective interface model, with two layers of different stiffness (stiffer layer is the outer layer, the softer layer is the inner layer) and only the outer layer can overlap, allows to model the combined effect of the nanoparticles clustering and the interface/interphase properties. 2.2. Automatic generation of 3D multiparticle unit cells with nonoverlapping and overlapping effective interface layers In this section, we present a program code for the automatic generation of 3D unit cell models of nanocomposites. As noted above, the peculiarities of nanocomposites (as compared with

common microscale reinforced composites) include: higher volume content of interface layers and their stronger influence on the material properties, the interface layers are not always absolute stiff and can overlap (if their physical nature is related with the changed atomistic structure and not by interphases), etc. Following the earlier developed program code ‘‘Meso3D’’ and ‘‘Meso3DFiber’’ [24–26], a new program ‘‘Nanocomp3D’’ was developed. This program, written in ABAQUS PDE (Python Development Environment) [27], generates multiparticle unit cells in ABAQUS CAE Environment. The unit cells include many randomly arranged particles of given shapes and orientations. The particles are surrounded by the interface layer (or several layers). The volume content and amount of particles, their sizes and aspect ratios, the thickness of effective interface layers and availability of sublayers, whether the interface layer or sublayers are allowed to overlap, all these parameters are introduced into the program as input data. The program generates the unit cells automatically. The nanoparticles can have different shapes: cylinders, discs, ellipsoids (with a given aspect ratios) and spheres. The elongated particles can be aligned and have random directions. The nanoparticles were placed in random places, according to RSA (random sequential absorption) algorithm [28–31]. In order to model the random and clustered arrangement of high aspect ratio particles with random orientation, the unit cell volume was divided into several subcells (according to the degree of overlapping or the total amount of particles), and clusters with a given amount of particles (which all have the same orientation) are placed in the subcells. This algorithm works only for not very high aspect ratio. The overlapping of effective interfaces or sublayers of the effective interfaces was realized using Boolean operations in ABAQUS. The meshes of different parts are connected by sharing nodes. Several multiparticle models with different shaped filler are shown in Fig. 1. 2.3. Materials properties and boundary conditions The unit cells were subject to a uniaxial tensile displacement loading along Y axis direction (defined as upper face) (applied strain 10%). The simulations were carried out with ABAQUS/STANDARD. Three-dimensional 8-node linear brick, reduced integration with hourglass control elements C3D8R were used in the single particle models. Three-dimensional 4-node linear tetrahedron elements C3D4 were used in the multi-particles models. The materials properties for the considered unit cells are given in Table 2. For interface, the functionalized silica composite properties were adapted [12]. 3. Single particle unit cell models of nanocomposites: effect of effective interface properties 3.1. Comparison with the Mori–Tanaka model Several single particle unit cell models with effective interface layers were tested in the simulations. A 3D single-particle FEM unit cell model is shown in Fig. 2. Using these models, Young’s moduli of the composites with different particle radii (from 11 Å to 1000 Å: 11, 15, 25, 50, 100 and 1000 Å, respectively) are calculated. In our simulations, the Poisson’s ratio of the particle was assumed to be 0.26. The volume content was kept constant at the level of 5% (i.e., when the particle radius increased, the cell size increased proportionally, but the effective interface layer thickness remained constant). The thickness of effective interface layer (in this case, only one layer) was kept constant at the level of 12 Å.

H.W. Wang et al. / Composites Science and Technology 71 (2011) 980–988

(a) Spherical filler (50 fillers)

(c) Aligned disc filler (50 fillers)

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(b) Aligned cylindrical filler (50 fillers)

(d) Aligned ellipsoidal filler (50 fillers)

(e) Ideally random cylindrical filler (64 fillers) (f) Correlated random cylindrical filler (64 fillers) Fig. 1. Multiparticle models with different shaped filler (some interface and particle elements are removed to see the effective interface structure clearly in a–d).

Fig. 3 shows the Young moduli plotted versus the particle radius, calculated using the FEM and the unit cell models with and without the effective interfaces, and compared with the analytical models (Mori–Tanaka, without effective interface, and effective interface model) from [12]. The difference between the analytical and numerical results is of the order of 1.2%.

In Fig. 3, we can directly observe the size effect, i.e., transition from nanocomposite behavior (which is observed for the particle size below 100 Å) to the microcomposite behavior. While the mechanical properties of the nanocomposites improve with increasing the particle size (assuming the soft interface layer) up to the particle size 100 Å, the particle size has almost no influence

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Table 2 Properties of the phases used in the simulations.

Young’s modulus (GPa) Poisson’s ratio

Matrix (polyimide)

Particle (silica)

Interface (functionalized)

Interface (phenoxybenzene)

4.2

88.7

3.5

0.3

0.4

0.4

0.4

0.26

on the elastic properties of the composite as soon as the particles are larger. If the effective interface layer were stiffer than the matrix, the effect would be inverse: the stiffness of the composite would decrease with increasing the particle size up to 100 Å, and remain constant afterward.

(a)

(b)

Fig. 2. Schema and the FE mesh of the single particle unit cell model with effective interface.

3.2. Effect of the particle volume fraction on the elastic properties of the composites

3.3. Effect of the interface properties on the elastic properties of composites In order to analyze the effect of the interface properties on those of the nanocomposites, the following numerical experiments were carried out. A unit cell with side length 52.5 Å, the radius of the particle and the effective interface thickness 12 Å was subject to tensile loading. The Young’s modulus of the effective interface was varied from 0.3 GPa (corresponds to that of phenoxybenzene silica nanoparticle/polyimide interface) to 88.7 GPa (corresponds to that of nanosilica particle). All other materials properties are given in Table 2. The volume fractions of the particle, effective interface and matrix were 5%, 35% and 60%, respectively. The effect of Young’s modulus of interface on that of the composite is shown in Fig. 5. The conclusions from Fig. 5 are similar to those from Fig. 4: with increasing the interface stiffness, the positive effect of the reinforcement increases, which corresponds to the conclusion in [15]. For comparison, the results in [15] are also plotted in Fig. 5. Due to the differences in the material properties and model parameters in our model and in [15], only qualitative comparison was carried out.

Fig. 3. Young’s modulus of nanocomposite versus particle radius for Mori–Tanaka model and effective interface approach (means in [12]).

2.1

1.8

Ec/Em

According to many investigations, several percent of volume fraction of nanoparticles may result in improvement of the composite properties [32]. In this section, we seek to analyze the effect of particle volume fraction on the mechanical properties of the nanocomposites. In our simulations, we considered two types of the interface layers: functionalized silica nanoparticle/polyimide system with the Young’s modulus 3.5 GPa [12] and a model with higher stiffness interface whose Young’s modulus is 8.4 GPa (what is two times that of polymer matrix). We considered the softer interface for the former case and the stiffer interface for the latter case. For each group, the particle volume fraction varied from 0.1% to 6%, while the particle radius and the effective interface thickness were kept constant, both 12 Å. Fig. 4 shows the Young’s modulus of the nanocomposite plotted versus the particle volume fraction, for different interface properties. From Fig. 4, it can be seen that the influence of the volume fraction of nanoreinforcement on the mechanical properties of nanocomposites are determined by the interface properties. The stiffness of the composite increases with increasing the particle volume fraction. The relationship between the composites Young’s modulus and particle volume fraction is nearly linear, which correspond to the results in [13].

1.5

Stiffer interface in this study Rsults in Ref [13] Softer interface in this study

1.2

0.9 0

5

10

15

20

Particle volume fraction (%) Fig. 4. Young’s modulus of the nanocomposite plotted versus the particle volume fraction for different interface properties (Ec: Young’s modulus of composites; Em: Young’s modulus of matrix).

4. Computational analysis of the particle clustering, particle orientation and particle shapes effect In this section, we analyze the effect of the clustering, orientation and shapes of nanoparticles on the elastic properties of the

H.W. Wang et al. / Composites Science and Technology 71 (2011) 980–988

3.2

Ec/Em

2.4

1.6 Rsults in Ref [15] Results in this study

0.8

0.0 0

3

6

9

12

Ei/Em Fig. 5. Effect of Young’s modulus of interface on that of the composite (Ec: Young’s modulus of composites; Em: Young’s modulus of matrix; Ei: Young’s modulus of interface).

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overlapping, V – interface volume when some interfaces overlap. When V/V0 is equal 1, there is no cluster. A small value means that the particles are densely clustered. Using the newly developed software ‘‘Nanocomp3D’’, we generated the multiparticle unit cells with the effective interface layers around the particles, and varied degrees of the interface overlapping. The FEM model is shown in Fig. 1a and f. The relation between Young’s modulus of the composite and the degree of the effective interface overlapping is shown in Fig. 6. Fig. 6 indicates that the degree of the interface overlapping has an effect on the Young’s modulus of the nanocomposite. Higher degree of particle clustering (and, therefore, higher degree of the effective interface overlapping) leads to lower Young’s modules of the nanocomposites. In the experimental works of Lam et al. [34,35], the similar trend was observed: small clusters of nanoclays influence the mechanical properties of the nanoclay/polymer nanocomposites. The increase of the size and amount of the clusters will results in the decrease of the hardness of nanocomposite. A similar conclusion was drawn by Termonia [15]: in the Young modulus of the nanocomposites decreases linearly with the increasing the degree of clustering. 4.2. Effect of the nanoparticles shapes on the elastic properties of the nanocomposites: Aligned elongated particles

Fig. 6. The relation between Young’s modulus of the composite and the degree of interphase intersection.

In this part, we analyze the effect of the particle shapes on the elastic properties on nanocomposites. The multiparticle unit cell models with 50 aligned particles (cylindrical, discs or ellipsoids), volume content of particles 5%, and aspect ratios 5 were generated and studied. The effective interfaces contained two layers as described above. The materials properties of phases were calculated by Eqs. (1)–(3). The elongated particles were aligned vertically, and the tensile load was applied along the particle axis. The elastic properties of the materials are shown in Fig. 7. For comparison, the Young’s moduli for the material with spherical reinforcements are also shown. One can see from Fig. 7 that the shape of the particles in nanocomposites has a strong effect on the elastic properties of the nanocomposites. The most effective reinforcement is cylindrical one, followed by ellipsoids, discs, and last, spheres. In order to analyze the effect of the orientation of elongated particles, the unit cells were subject to the horizontal tensile loading. Fig. 8 gives the comparisons of the vertical and horizontal loading data for all the types of reinforcement. As expected, the

nanocomposites, using the generalized effective interface model and the multiparticle unit cell models of nanocomposites described above. 4.1. Effect of particle clustering The clustering of particles occurs naturally if particles are randomly arranged. Further, the particles are arranged in groups or in clusters due to the specific technologies of the material production. Materials with intercalated microstructures represent an important group of the nanocomposites. In the clustered structure, the neighboring particles can touch and be located fairly close one to another. If the peculiar properties of the interface regions around nanoparticles are caused by changed local atomistic structured, molecular structures or diffusion processes, such interfaces can overlap. In our multiparticle unit cell models, we allow the overlapping of effective interface layers or outer sublayers. In order to characterize the degree of effective interface overlapping, we introduced a coefficient V/V0, where V0 is the total effective interface volume in a unit cell when there is no interface

Fig. 7. Elastic properties and typical overlapping coefficients for the composites with nanoparticles of different shapes.

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4.3. Random versus aligned orientation of elongated nanoparticles In this section, we study the effect of the angle between the elongated nanoparticles and the loading direction and the random orientation of elongated nanoparticles on the elastic properties of nanocomposite. We considered unit cells with 64 cylindrical particles, oriented at some angle to the vertical axis (i.e., to the loading direction). The angle varies from 0° to 90°. The aspect ratio of the cylindrical particles was 2. The results are given in Fig. 9a. For comparison, the results of simulation for the unit cell models with the random orientation of the cylindrical nanoparticles are presented in Fig. 9a as well. From Fig. 9a, it can be seen that the Young’s modulus of the composites strongly depends on the angle between the particles (aligned, elongated ones) and the loading direction. When the angle is 0° (vertical particles), the stiffness of the composites reaches its highest value. The lowest stiffness is when this angle around 55°. A similar observation (i.e., U-shaped dependency of the Young modulus of composites on the angle between the tensile loading and direction of elongated reinforcement) was made in the framework of experiments carried out on polymer composites, reinforced by long glass fibers [36], as shown in Fig. 9b. In these experiments, the Young modulus was at the level of 15 GPa, if the angle was 0, reduced to 6. . .7 GPa at the angles 45. . .65° and increased to 10 GPa at the angle 90°. The fully random orientation of nanoparticles gives the elastic properties lying between the highest and lowest stiffness. In [20], similar conclusion, i.e., nanocomposites with unidirectionally oriented particles is stiffer than that with randomly oriented particles, was also drawn. 4.4. Correlated versus random orientation of elongated nanoparticles

Fig. 8. Effect of the vertical versus horizontal loading of aligned elongated particles: (a) cylindrical, (b) ellipsoidal and (c) disc-shaped nanoparticles.

In the ideal model, the random orientations of elongated particles mean that there is no correlation between the orientations of even closely located neighboring particles. In reality, some experimental results indicate that adjacent elongated particles in nanocomposites have roughly same orientation, as shown in Fig. 10 [3]. We will call such distribution of particles ‘‘correlated’’. In order to generate the unit cell model for the ‘‘correlated distribution’’, the unit cell was divided into 8 subcells (2  2  2). Eight nanocylinders were placed into every sub-model. While the orientation of particles inside each subcell was constant, it was randomly determined for all the 8 subcells. For comparison, we carried out also a simulation for ideally random (uncorrelated) particle orientations. 16 realizations of the unit cells with both the random and correlated orientations of the nanoparticles were generated and tested. The results are plotted in Fig. 11. The averaged Young’s modulus of the composites with uncorrelated microstructures is 4.394 GPa with standard deviation 0.012. The averaged Young’s modulus of the composites with correlated microstructures is 4.404 GPa with standard deviation 0.045. Comparing the results for the ideally random and correlated microstructures, we can see that while the average values of Young modulus are almost the same, the standard deviation of Young modulus for correlated microstructure is nearly 4 times of that for fully random orientation case, as shown in Fig. 11. 5. Conclusions

stiffness of composites with aligned elongated particles is higher if the load is applied along the bigger axis, and is lower if the load is applied normal to the bigger axis. Also, as expected, the results for discs are fully opposite. The simulation results correspond to the results and conclusions from [33].

A computational study of the effect of structures of nanocomposites on their elastic properties is presented. A special program code for the automatic generation of multiparticle unit cells with effective interface layers around particles

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Fig. 9. Relationship between composites Young’s modulus and loading direction. (a) Composites reinforced by cylindrical nanoparticle in this study, (b) composites reinforced by long fiber.

Fig. 10. A typical TEM photomicrograph of a nanocomposite [3]. Reprinted with permission from Elsevier. Fig. 11. Comparison between the cases of ideally and correlated random orientation of nanoparticles.

is developed for nanocomposite modeling. The interface layer can consist in turn of several sublayers, and these sublayers or the interface layers as a whole can overlap with corresponding sublayers of neighboring particle, if they are located close enough. The generalized effective interface model, with two layers of different stiffness (stiffer layer is the outer layer, the softer layer is the inner layer) and the option of overlapping layers is developed here. This model allows simulating the combined effect of the nanoparticles clustering and the interface/interphase properties. A series of computational simulations has been carried out, using the developed numerical tools. In the simulations, the transition from size-dependent nanocomposite behavior (which is observed for the particle size below 100 Å) to the size-independent microcomposite behavior was observed. While the mechanical properties of the nanocomposite improve with increasing the particle size (assuming the soft interface layer) up to the particle size 100 Å, the particle size has almost no influence on the elastic properties of the composite as soon as the particles are larger. If the effective interface layer were stiffer than the matrix, the effect would be inverse: the stiffness of the composite would decrease with increasing the particle size. The effect of the volume fraction of nanoreinforcement on the mechanical properties of nanocomposites is determined by the interface properties. While the stiffness of the composite increases

with increasing particle volume fraction for the stiff interfaces, the effect is inverse for the soft interface. The higher degree of particle clustering leads to lower Young’s modules of the nanocomposites. Ideally random oriented and correlated microstructures lead to the same average Young modulus, yet, the standard deviation of Young modulus for correlated microstructure is nearly 4 times of that for fully random orientation case.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 11002100), Program of International S&T Cooperation, MOST (Grant No. 2010DFA64560) and the Commission of the European Communities through the Sixth Framework Program Grant UpWind.TTC (Contract No. SES6-019945). The support by Tianjin Natural Science Foundation (No. 08JCZDJC24400) is also greatly acknowledged. H.W. Wang appreciates DANIDA for supporting his stay in Denmark under the framework of SinoDanish Committee of Scientific and Technological Collaboration. He would like to express his special thanks to Drs. Bent F. Sørensen, Xiaoxu Huang and Hai Qing for their help and fruitful discussions.

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