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Effective properties of biopolymer composites: A three-phase finite element model
Composites: Part A 40 (2009) 130–136
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Effective properties of biopolymer composites: A three-phase finite element model S. Rjafiallah, S. Guessasma *, D. Lourdin INRA, unite BIA, Rue de la Géraudière, Nantes 44130, France
a r t i c l e
i n f o
Article history: Received 2 July 2008 Received in revised form 13 October 2008 Accepted 28 October 2008 Keywords: B. Elasticity B. Interphase/interface B. Microstructures C. Finite element analysis (FEA)
a b s t r a c t The effective Young’s modulus of starch–zein biopolymer composite is studied using a finite element model. This model handles the fact that starch–zein interface is not perfect in the sense that elasticity properties across the interface are altered. The motivation here is to predict the right modulus of elasticity near the interface for any zein content. In that way, the finite element model requires zein, starch and interphase properties to be implemented for the computation of the composite Young’s modulus as a function of zein content. The main variables are the interphase thickness and modulus. The comparison between the predicted and experimental results shows, firstly, that the effective composite properties are not correctly estimated using standard models with perfect interface hypothesis. Secondly, the threephase model is able to suggest optimal interphase parameters in order to fit the experimental data obtained using three-point bending test. Finally, the optimal interphase parameters (thickness and modulus) vary as a function of zein content. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Biopolymer composites are becoming increasingly employed to replace oil-based components [1]. The knowledge of their mechanical properties is essential to guarantee high performance during their lifetime [2]. While various non-food applications have been developed for biopolymers, their industrial usage is unfortunately limited to the design of low cost sacrificial materials. The main reason explaining the restrictive development of biopolymers is their low mechanical performance and high sensitivity to environment variables (temperature, humidity) [3]. One way to improve the mechanical performance of biopolymer composites is to gain better understanding of the structure–mechanical properties relationship. Central to this understanding is the load transfer in the composite. Indeed, for a two-constituent composite with a perfect bond between phases, the stress in the composite can be written as [4]
re ¼ f r1 þ ð1 f Þr2
ð1Þ
where r1 and r2 are the average stresses in the phases, re is an external stress. f is the fraction of phase 1 in the composite. The key point here is that when a given load is applied to the composite, the reinforcement (the phase with the largest modulus) will carry a large proportion of that load. However, if the interface properties are announced as weak, the load transfer is altered in a way that it can lead to debonding, crack initiation and failure of the material [5]. It is then completely understood that the performance
* Corresponding author. Tel.: +33 (0)2 40 67 50 36; fax: +33 (0)2 40 67 51 67. E-mail address: sofi[email protected] (S. Guessasma). 1359-835X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2008.10.010
of many composites is sensitive to the interface region. It is thus important to study the impact of interface region on the effective properties in order to optimise effective properties. A huge body of work exists in this central question either considering analytical [6,7], experimental or numerical studies [8]. In this study, a finite element model is developed to handle the effect of imperfect contact between phases in starch–zein composite material. A three-phase model is implemented to account for a particular region in the composite around the interface named here the interphase that has different mechanical properties from those of the intrinsic phases. The starting point is to import CLSM (Confocal Laser Scanning Microscopy) images (Fig. 1) of the biopolymer composite into the finite element model. The studied material consists of starch matrix and zein filler. Next, the intrinsic properties are implemented as well as the thickness and elasticity modulus of the interphase. The effective modulus of the composite is computed as a function of zein fraction. The variables of the model (thickness and modulus of the interphase) are adjusted to fit the experimental composite modulus obtained from three-point bending experiments at any zein fraction. By doing so, the model is able to predict the evolution of the interphase properties as a function of zein content. 2. Modelling technique A static linear analysis is used to compute the effective composite properties as a function of zein content using ANSYS software. The method assumes the direct import of confocal microscopy images into a finite element model. The grid size is 512 512 pixels where each pixel is converted into a four-node plane element
S. Rjafiallah et al. / Composites: Part A 40 (2009) 130–136
Fig. 1. CLSM (Confocal Laser Scanning Microscopy) image showing the starch–zein microstructure. Starch is the black phase. Zein percent is 17%. Image dimensions 160 160 lm2.
(PLANE82 using ANSYS nomenclature). Thus, the mesh is regular with a fixed density for all conditions. The element size is about 0.3 lm, which is easily derived from the following expression
element size ¼ physical dimension=number of pixels where the region of interest has a dimension of 160 lm for which the acquisition resolution is 512 pixels per image edge. The calculations are performed under isotropy conditions, that is to say elasticity parameters for the considered element are irrespective of space orientations. Each node of the plane element has two degrees of freedom (translations UX and UY in X- and Y-directions). Plane stress is assumed knowing the fact that the studied material is a thermomoulded film with typical thickness 40 times smaller than its length. Three materials are considered: starch, zein and interphase. Zein and starch elasticity moduli are quite close (EZ = 3.00 GPa and ES = 2.62 GPa). Intrinsic elasticity moduli were measured using three-point bending tests as a part of an earlier study [9]. The interphase modulus is introduced here to allow a varied stress transfer between intrinsic materials. Typical values are in the range 0.1 and 2 GPa. However, for identification purposes described later, the interphase modulus can vary in a wider range. The extent of the interphase region is handled by allowing a varied thickness of this material (Fig. 2). The load conditions are shown in Fig. 3. For a typical loading in Y-direction, all nodes of the bottom line are constrained against displacement in the direction of loading and free to move in the other direction. Nodes of the upper line are displaced by a positive amount in the loading direction and again are not constrained in the transverse direction. To avoid shearing of the structure, periodic boundary conditions are applied to nodes of the lateral lines. Constraint equations are introduced to allow coupling lateral degrees of freedom as illustrated in Fig. 3. The constraint equation illustrated in Fig. 3 assumes that loading is performed in Y-direction. Transverse displacements (UX) of homologue nodes i and j (Xi = Xj + L and Yi = Yj, where L is the microstructure size) are coupled. Such constraints are needed to ensure that periodic boundary conditions are properly defined, which avoid rigid displacements. Analogous conditions are applied for a loading in the X-direction. Effective Young’s modulus is derived from the computation of the reaction forces at the loaded line, giving the sample size. The deformation is calculated from the imposed displacement and knowing the sample size. More than 20 microstructures exhib-
131
iting a zein ratio varied between 5% and 60% are considered. For each microstructure, four interphase thickness and moduli are combined. More than 360 calculations are required with 10 min per run on a PC – INTEL Core 2 CPU 6400 at 2.13 GHz and with 3.37 GBytes of RAM. Convergence tests are performed to ensure that meshing is not too small to cause ill-conditioning or too rough to lead to convergence problem. Ill-conditioning may appear as a consequence of the fine mesh especially when meshing contains elongated elements. We recall here that our meshing is regular for all conditions because of the direct pixel/element conversion. Preliminary accuracy tests are performed on coarser meshes, since we cannot use finer meshes unless to change the main characteristics of the microstructure. All our structural analyses are conducted using the ANSYS software. We use the preconditioned conjugate gradient solver available in the package. The PCG assembles the full global stiffness matrix starting from the element matrix formulation instead of factoring the global matrix. DOF solutions are computed in an iterative way until convergence. The accuracy of the computed primary variables (displacement solutions) is controlled using a tolerance value of 108, which is always reached regardless of the meshing density. Table 1 compares the number of PCG iterations needed to reach a converged solution for different meshing densities. 3. Results and discussion The justification of the use of a three-phase model comes from the fact that analytical approaches giving the bounds of the starch– zein composite fails in fitting the experimental data shown in Fig. 4. Indeed, as starch and zein have similar elasticity moduli, all analytical models predict a linear trend. Three-point bending test results suggest instead a non-linear evolution of the effective modulus as a function of zein content. If we consider the fact that experimental results are obtained with good reproducibility (averages of 10 replicates with a reasonable scatter), the most possible reason attached to the deviation from the analytical approaches is the presence of an interface effect. Moreover, a recent study [10] has shown that in the same material, nanoindentation experiments revealed that near the interface, the maximum penetration force in the bulk material is at least 4.5 times larger than near the interface. Before going any further with the three-phase model, mesh sensitivity is performed in order to check the validity of the computed results at larger scales. Generally speaking, a rough mesh leads to fast computations but at the cost of loosing microstructure details. These details have actually a large importance in our study because we know that interphase effect is dependent on the amount of interface quantity in the microstructure. A large mesh is usually required to obtain accurate results if we accept large computation durations. The balance between the two (i.e., accurate results and run duration) gives the optimal resolution at which the microstructure should be looked at. As pointed out earlier, the finite element mesh is the consequence of the direct import of the composite images at the genuine acquisition resolution. Thus, we cannot perform finer meshes unless to change the main characteristics of the microstructures. Knowing this fact, preliminary tests of mesh sensitivity were performed using coarser meshes. Fig. 5 illustrates this by considering a particular microstructure containing 39% of zein, for which the resolution was lowered thanks to the concept of mesh index. This parameter defines the ratio of the element size (pixel size) at the largest resolution over that at any lower resolution. For example, all original microstructures were acquired at 512 512 pixels corresponding to 160 160 lm2. Thus, the original element size is around
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Fig. 2. Finite element model with interphase properties where t denotes the thickness of the interphase region.
Loaded line
Exp (bending) Num H&SH&S+
UY=U
3.2
UXi+UXj=0 Yi=Yj
i
j
Young's modulus (GPa)
constraint equation
Mori & Tanaka Reuss Eshelby Voight
3.0
2.8
2.6
Y 2.4 0.0
X
Constrained line UY=0
Fig. 3. Load conditions in the Y-direction applied to the FE three-phase model. For visualisation purposes, the grid resolution is lowered to 128 128 pixels. Interface thickness is 0.63 lm. Phase ratios are 0.56, 0.19 and 0.25, for starch, zein and interphase, respectively.
Table 1 Number of PCG iterations as function of different meshing densities. Mesh factor
Number of elements
DOF
DOF constraints
Total Iterations In PCG
1 2 3 4
262,144 65,536 28,900 16,384
526,338 132,098 58,482 33,282
3591 1028 513 258
79 79 65 56
0.31 lm. It is possible to obtain lower resolutions such as 256 256 using binning operators. Thus, the mesh index in this
0.1
0.2
0.3
0.4
0.5
0.6
Zein content (-) Fig. 4. Comparison between some popular analytical approaches giving the bounds for a two-constituent composite material and three-point bending experimental results of starch–zein composite.
case is 2 and corresponds to an element size twice larger than the original size. When analysing the results depicted in Fig. 5, it appears that the sensitivity of the composite modulus to mesh index increases when the interphase modulus is the lowest possible. For example, the scatter of the composite modulus for Ei = 2 GPa is less than 1.3% whereas for Ei = 0.1 GPa, it increases to more than 21%. When the interphase modulus is near the intrinsic moduli, the sensitivity is quite reduced because the behaviour of the composite compares to that of a two-phase model. The increase of sensitivity to mesh coarsening when the modulus is lower than the intrinsic moduli is mainly related to the loose of interface information when the resolution is changed. Analogous interpretation still holds when the interphase modulus is larger than those of intrinsic phases.
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In the following, the original resolution as obtained from the image acquisition protocol (512 512 pixels) is used for all conditions, knowing the fact that we cannot guarantee the stability of the predicted composite modulus at lower resolutions. Fig. 6a shows the evolution of the effective Young’s modulus as a function of zein content for a varied interphase modulus between 0.1 and 2.0 GPa. At any particular zein content, the increase of Ei has the consequence to increase the effective Young’s modulus, which is an obvious result. The evolution of the effective modulus seems to indicate a fairly linear trend with the zein content, irrespective of the interphase modulus. The exception for this is undoubtedly the distinguished point at a zein content of 15%. We see here that this point is totally predictable by the three-phase model, in the sense that the expected interface modulus lies within the restricted range 0.1–0.5 GPa. This point is related actually to the largest number density of zein particles in the microstructure, where the microstructure can be still described as a matrix/particle composite. At larger ratios, zein particles percolate to form a continuous phase before again the composite switches to a matrix/ particle regime, where the matrix becomes zein. There is another
2.8
Effective Modulus (GPa)
2.6 2.4 2.2 2.0 1.8 1.6
Interphase modulus (GPa) 0.1 0.5 1.0 2.0
1.4 1.2 1.0 1
2
3
4
mesh index (-) Fig. 5. Mesh sensitivity analysis for different interphase moduli. Run conditions: zein fraction = 0.39, interphase thickness 0.31 lm.
a
3.45
Ei = 0.1 GPa Ei = 0.5 GPa Ei = 1.0 GPa Ei = 2.0 GPa E experimental
3.35
Effective modulus (GPa)
3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 2.45 2.35 2.25 0
10
20
30
40
50
60
Zein content (%)
b
3.0
2.9
Ey (GPa)
2.8
2.7
2.6
2.5
Ei =0.1 GPa Ei =0.5 GPa Ei =1.0 GPa Ei =2.0 GPa Ey=1.00Ex (R²=0.90)
2.4
2.3 2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
Ex (GPa) Fig. 6. (a) Predicted composite elasticity modulus as function of zein content for different interphase moduli (interphase thickness = 0.31 lm). (b) Comparison between the modulus in X- and Y-directions for all studied conditions.
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Non-linear dependence of the interphase zone properties on the interphase thickness: this possibility requires a fine description of the interphase region in terms of meshing and physical properties. It should make sense to represent this effect as an elasticmodulus gradient [13,14]. However, it would probably lead to an inconclusive insertion, since we have no idea about the gradient profile that have to be implemented. In addition, Substantial improvement of the modelling technique is required which actually is beyond the scope of the present study. Microstructure anisotropy that affects interphase thickness: Because of the random distribution of the filler, this hypothesis is unlikely to provide a clear say on the large dependence of the composite effective property on zein fraction. Sensitivity of the interphase properties to filler content. At the first sight, it may appear surprising that the fraction of the filler affects the physical properties of the interphase. However, this has to deal with the nature of the biopolymer materials and mainly on different water moisture of individual species. This last hypothesis is hereafter explored. In order to put forward a mechanism explaining the approximated experimental trend, the optimal interphase modulus is plotted against zein content (Fig. 7b). We recall here that the interphase elasticity properties have to be lower than those of the intrinsic materials in order to be in line with the nanoindentation results [10]. A brief examination of Fig. 7b, some points particularly between 7% and 17.5% lead to interphase moduli larger than 3 GPa. The worst case (not shown here), is an interphase modulus beyond 100 GPa predicted for a zein content around 60%. In order to examine more deeply this result, the interphase thickness is considered as a second adjustable parameter. As a matter of fact, the results shown in Fig. 7a, through b are obtained for a constant interphase thickness of 0.31 lm. The objective in the following is
a
3.4 3.3 3.2
Effective modulus (GPa)
unpredictable point at about 45%, which exhibits the largest modulus. However, there is no particular structural feature that can be attached to this point. The orthotropy of the effective modulus is checked giving the result depicted in Fig. 6b. The scatter between moduli increases when the interphase modulus decreases. If we recall the fact that a small interface modulus highlights the interface effect, it appears now that the orthotropy of the composite elasticity is related to the quantity of interfaces parallel and perpendicular to the load direction. If these two quantities are the same, the composite will exhibit the same moduli in X- and Y-directions. In the case of Fig. 6b, the linear trend found between Ex and Ey is predicted with an acceptable correlation factor R2 = 0.90, despite the scatter in the case of Ei = 0.1 GPa. The use of interphase modulus as an adjustable variable makes it possible to derive the optimal Ei value that can explain the experimental trend. For this purpose, the experimental trend is approximated using a fitting curve (solid line in Fig. 6a). For a given zein content, the four values of interphase modulus are plotted against the corresponding composite Young’s moduli. A fitting routine is used to determine the function that best describes the correlation between the effective and interphase moduli. The optimal interphase modulus is then determined based on the knowledge of the experimental effective modulus derived from the curve in solid line. The result of the fitting procedure is shown in Fig. 7a. A fairly good agreement is found between the approximated experimental trend and that obtained using the fitting routine. We have to mention that the non-monotonic dependence of effective Young’s modulus on the zein filler is not a common result, but we do find similar behaviour, for example, in the case of POSS reinforced epoxy [11,12]. We can explore different potential explanations that can be inferred to the observed non-linear trend, among them:
3.1 3.0 2.9 2.8 2.7 2.6
Experimental Numerical
2.5 2.4 0
10
20
30
40
50
60
Zein content (%)
b
14
12
Interphase Modolus (GPa)
134
10
8
6
4
2
0 0
5
10
15
20
25
30
35
40
Zein content (%) Fig. 7. (a) Predicted effective modulus as function of zein content for an optimised interphase modulus. (b) The optimal interphase modulus as function of zein content.
to determine if the approximated experimental trend is a genuine effect or not based on the interphase thickness and modulus. Fig. 8a shows the evolution of the effective modulus for different interphase thickness giving a small interface modulus (Ei = 0.1 GPa). The effective modulus is found to decrease with the increase of zein content up to 40%. At any zein content, the effective modulus decreases with the increase of interphase thickness indicating that the interphase fraction increased in the microstructure. Because the interface modulus is lower than any of the intrinsic phase moduli, it is not surprising to observe such a trend. The same evolution is predicted when the interphase modulus is 2 GPa (Fig. 8b). In this case, the tendency of the effective modulus is, however, towards the increase because the interphase contributes better in the overall rigidity of the composite. When comparing this result with that depicted in Fig. 6a, the rigidity of the composite appears dependent on the ratio Ei/t, where t denotes the interphase thickness. The optimisation of the interphase properties similar to that shown in Fig. 7 will require the building of a complex function of the form Eeff ¼ f ðEi ; C z ; tÞ, where Cz is the zein ratio and Eeff is the effective Young’s modulus. A suitable represen-
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a
into account the increase of the ratio Ei/t and the increase of Cz, announces that the effective modulus spread increases with the increase of zein content (Fig. 7c). It seems that the lowest value of Ei/t controls this spread and thus reveals the effect of interface quantity. The spread profile shown in Fig. 7c encodes the evolution of the interface quantity as a function of zein content.
2.8 2.6
2.2 2.0
a
1.8
1.24
1.6 1.4 1.2 1.0 0.8 0.6
Interphase thickness (µm)
Effective Modulus (GPa)
2.4
160 x160 µm², Ei= 0.1GPa Interphase thickness (µm) 0.31 0.63 0.94 1.25 0
10
20
30
40
50
60
Zein Content (%) 2.85
2.80
Effective Modulus (GPa)
0.62
2.75
0.31
160 x160 µm², Ei = 2GPa Interphase thickness (µm) 0.31 0.63 0.94 1.25
0
2.60
20
15
20
25
30
35
40
2.6
2.65
10
10
b
2.70
0
5
Zein content (%)
Interphase modulus (GPa)
b
0.93
30
40
50
60
2.4
2.2
2.0
1.8
Zein content (%) 1.6
c
3.0 2.8
Increase of Ei/t 2.000
0
10
20
c
2.4
0.025
1.8 1.6 1.4 1.2
57 %
Increase of Zein content
0.8 0
50
50
60
Numerical Experimental
3.1
2.0
1%
40
3.2
2.2
100
150
200
250
300
350
400
FE data (-) Fig. 8. (a and b) Effect of varying the interphase thickness on effective properties of the starchy composite for two particular interphase modulus values. (c) Predicted composite Young’s modulus for all studied conditions.
Effective Modulus (GPa)
Effective modulus (GPa)
2.6
1.0
30
Zein content (%)
3.0
2.9
2.8
2.7
2.6
2.5 0
10
20
30
40
50
60
Zein content (%)
tation as that shown in Fig. 7c of all FE run results can be informative of the global tendency that can be inferred to the result of the three-phase model. Indeed, sorting out all numerical results taking
Fig. 9. (a and b) Optimised interphase parameters as function of zein content. (c) Comparison between FE and experimental Young’s modulus as function of zein content.
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The result of the optimisation process similar to that performed using one adjustable variable is shown in Fig. 9. The optimal interphase thickness exhibits a large scatter between 0.31 and 1.25 lm. Note that because of the binning operations, the interphase thickness evolves as discrete jumps. Despite there is no clear correlation between zein content and interphase thickness as depicted in Fig. 9a, the change in interphase thickness can be inferred to some physical reasons explained hereafter. Because zein and starch have different moisture contents in the composite [15,16], the water content of the composite varies as a function of zein fraction from 12% to 7%, when zein content increases from 0% to 100% [9]. Near the interface, there is undoubtedly a change in the water content that may affect the structure and consequently the mechanical properties at the interphase region. Zein fraction and phase morphology determine together the extent of the region where the water content varies from the equilibrium value in starch to the other one value in zein. The same explanation still holds in the case of the interphase modulus, where Fig. 9b shows that changing zein content in the microstructure predicts the change of Ei in the range 0.79– 2.47 GPa with no evident tendency. Note that all predicted values are lower than Young’s moduli of intrinsic phases, which means that the three-phase model is able to predict the composite effective property evolution under weak interphase conditions. The comparison between the numerical and experimental effective moduli is shown in Fig. 9c. Almost experimental tendency was reproduced using the optimal interphase parameters. The conflicting point is however the underestimation of the effective property (2.82 GPa) at the largest zein fraction (57%). All attempts to fit the experimental value, which is actually above the largest intrinsic phase modulus, were discouraging. The weak adhesion hypothesis for this particular fraction is not realistic if we consider that the experimental data are trustable. 4. Summary Analytical models giving the evolution of the effective Young’s modulus for starch–zein composite as a function of zein content fail to describe the non-linear experimental trend. A three-phase model is considered to explain such evolution taking account interphase parameters (thickness (t) and Young’s modulus (Ei)). The orthotropy of the composite is sensitive to the interface quantity in the direction of loading when the interphase modulus is quite different from that of the intrinsic phases. Combining the effect of interphase parameters shows that the effective composite property depends on the ratio Ei/t. The
optimisation of both parameters, in the case of weak interphase hypothesis, allows the prediction of almost experimental trend. The misaligned points are those where the interphase modulus has to be larger than the moduli of the intrinsic materials to fit the experimental value. It is thought that the interphase properties variation, as predicted by the three-phase model, can be related to the change in the structure of the interphase as a function of zein content. In particular, different equilibrium values of moisture content between starch and zein phases have some implication in the variation of interphase properties. References [1] Gross RA, Kalra B. Biodegradable polymers for the environment. Science 2002;298:803–7. [2] Beg MDH, Pickering KL, Weal SJ. Corn gluten meal as a biodegradable matrix material in wood fibre reinforced composites. Mater Sci Eng A 2005;412(1– 2):7–11. } Z, Dogossy G, Réczey K, Czigány T. Reducing water absorption [3] Gáspár M, Benko in compostable starch-based plastics. Polym Degrad Stabil 2005;90(3):563–9. [4] Hull D, Clyne TW. An introduction to composite materials. Cambridge: Cambridge University press; 1996. [5] Frogley MD, Ravich D, Wagner DH. Mechanical properties of carbon nanoparticle-reinforced elastomers. Compos Sci Technol 2003;63(11):1647–54. [6] Hashin Z. Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J Mech Phys Solids 2002;50(12):2509–37. [7] Sevostianov I, Kachanov M. Effect of interphase layers on the overall elastic and conductive properties of matrix composites. Applications to nanosize inclusion. Int J Solids Struct 2007;44(3–4):1304–15. [8] Esmaeili N, Tomita Y. Micro- to macroscopic responses of a glass particleblended polymer in the presence of an interphase layer. Int J Mech Sci 2006;48(10):1186–95. [9] Chanvrier H, Colonna P, Della Valle G, Lourdin D. Structure and mechanical behaviour of corn flour and starch–zein based materials in the glassy state. Carbohydr Polym 2005;59(1):109–19. [10] Guessasma S, Sehaki M, Lourdin D, Bourmaud A. Viscoelasticity properties of biopolymer composite materials determined using finite element calculation and nanoindentation. Comput Mater Sci 2008;44(2):371–7. [11] Abd Rashid ES, Ariffin K, Choong Kooi C, Md Akil H. Preparation and properties of POSS/epoxy composites for electronic packaging applications. Mater Des 2009;30:1–8. [12] Zhang Z, Gu A, Liang G, Ren P, Xie J, Wang X. Thermo-oxygen degradation mechanisms of POSS/epoxy nanocomposites. Polym Degrad Stabil 2007;92(11):1986–93. [13] Pender DC, Padture NP, Giannakopoulos AE, Suresh S. Gradients in elastic modulus for improved contact-damage resistance. Part I: The silicon nitride– oxynitride glass system. Acta Mater 2001;49(16):3255–62. [14] Askadskii AA, Goleneva LM, Bychko KA, Afonicheva OV. Synthesis and mechanical behavior of functionally gradient polyisocyanurate materials based on hydroxy-terminated butadiene rubber. Polym Sci A 2008;50(7):781–91. [15] Bizot H, Le Bail P, Leroux B, Davy J, Roger P, Buleon A. Calorimetric evaluation of the glass transition in hydrated, linear and branched polyhydro glucose compounds. Carbohydr Polym 1997;32:33–50. [16] Madeka H, Kokini JL. Effect of glass transition and crosslinking on rheological properties of zein: development of a preliminary state diagram. Cereal Chem 1996;73(4):433–8.
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Effective properties of biopolymer composites: A three-phase finite element model S. Rjafiallah, S. Guessasma *, D. Lourdin INRA, unite BIA, Rue de la Géraudière, Nantes 44130, France
a r t i c l e
i n f o
Article history: Received 2 July 2008 Received in revised form 13 October 2008 Accepted 28 October 2008 Keywords: B. Elasticity B. Interphase/interface B. Microstructures C. Finite element analysis (FEA)
a b s t r a c t The effective Young’s modulus of starch–zein biopolymer composite is studied using a finite element model. This model handles the fact that starch–zein interface is not perfect in the sense that elasticity properties across the interface are altered. The motivation here is to predict the right modulus of elasticity near the interface for any zein content. In that way, the finite element model requires zein, starch and interphase properties to be implemented for the computation of the composite Young’s modulus as a function of zein content. The main variables are the interphase thickness and modulus. The comparison between the predicted and experimental results shows, firstly, that the effective composite properties are not correctly estimated using standard models with perfect interface hypothesis. Secondly, the threephase model is able to suggest optimal interphase parameters in order to fit the experimental data obtained using three-point bending test. Finally, the optimal interphase parameters (thickness and modulus) vary as a function of zein content. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Biopolymer composites are becoming increasingly employed to replace oil-based components [1]. The knowledge of their mechanical properties is essential to guarantee high performance during their lifetime [2]. While various non-food applications have been developed for biopolymers, their industrial usage is unfortunately limited to the design of low cost sacrificial materials. The main reason explaining the restrictive development of biopolymers is their low mechanical performance and high sensitivity to environment variables (temperature, humidity) [3]. One way to improve the mechanical performance of biopolymer composites is to gain better understanding of the structure–mechanical properties relationship. Central to this understanding is the load transfer in the composite. Indeed, for a two-constituent composite with a perfect bond between phases, the stress in the composite can be written as [4]
re ¼ f r1 þ ð1 f Þr2
ð1Þ
where r1 and r2 are the average stresses in the phases, re is an external stress. f is the fraction of phase 1 in the composite. The key point here is that when a given load is applied to the composite, the reinforcement (the phase with the largest modulus) will carry a large proportion of that load. However, if the interface properties are announced as weak, the load transfer is altered in a way that it can lead to debonding, crack initiation and failure of the material [5]. It is then completely understood that the performance
* Corresponding author. Tel.: +33 (0)2 40 67 50 36; fax: +33 (0)2 40 67 51 67. E-mail address: sofi[email protected] (S. Guessasma). 1359-835X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2008.10.010
of many composites is sensitive to the interface region. It is thus important to study the impact of interface region on the effective properties in order to optimise effective properties. A huge body of work exists in this central question either considering analytical [6,7], experimental or numerical studies [8]. In this study, a finite element model is developed to handle the effect of imperfect contact between phases in starch–zein composite material. A three-phase model is implemented to account for a particular region in the composite around the interface named here the interphase that has different mechanical properties from those of the intrinsic phases. The starting point is to import CLSM (Confocal Laser Scanning Microscopy) images (Fig. 1) of the biopolymer composite into the finite element model. The studied material consists of starch matrix and zein filler. Next, the intrinsic properties are implemented as well as the thickness and elasticity modulus of the interphase. The effective modulus of the composite is computed as a function of zein fraction. The variables of the model (thickness and modulus of the interphase) are adjusted to fit the experimental composite modulus obtained from three-point bending experiments at any zein fraction. By doing so, the model is able to predict the evolution of the interphase properties as a function of zein content. 2. Modelling technique A static linear analysis is used to compute the effective composite properties as a function of zein content using ANSYS software. The method assumes the direct import of confocal microscopy images into a finite element model. The grid size is 512 512 pixels where each pixel is converted into a four-node plane element
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Fig. 1. CLSM (Confocal Laser Scanning Microscopy) image showing the starch–zein microstructure. Starch is the black phase. Zein percent is 17%. Image dimensions 160 160 lm2.
(PLANE82 using ANSYS nomenclature). Thus, the mesh is regular with a fixed density for all conditions. The element size is about 0.3 lm, which is easily derived from the following expression
element size ¼ physical dimension=number of pixels where the region of interest has a dimension of 160 lm for which the acquisition resolution is 512 pixels per image edge. The calculations are performed under isotropy conditions, that is to say elasticity parameters for the considered element are irrespective of space orientations. Each node of the plane element has two degrees of freedom (translations UX and UY in X- and Y-directions). Plane stress is assumed knowing the fact that the studied material is a thermomoulded film with typical thickness 40 times smaller than its length. Three materials are considered: starch, zein and interphase. Zein and starch elasticity moduli are quite close (EZ = 3.00 GPa and ES = 2.62 GPa). Intrinsic elasticity moduli were measured using three-point bending tests as a part of an earlier study [9]. The interphase modulus is introduced here to allow a varied stress transfer between intrinsic materials. Typical values are in the range 0.1 and 2 GPa. However, for identification purposes described later, the interphase modulus can vary in a wider range. The extent of the interphase region is handled by allowing a varied thickness of this material (Fig. 2). The load conditions are shown in Fig. 3. For a typical loading in Y-direction, all nodes of the bottom line are constrained against displacement in the direction of loading and free to move in the other direction. Nodes of the upper line are displaced by a positive amount in the loading direction and again are not constrained in the transverse direction. To avoid shearing of the structure, periodic boundary conditions are applied to nodes of the lateral lines. Constraint equations are introduced to allow coupling lateral degrees of freedom as illustrated in Fig. 3. The constraint equation illustrated in Fig. 3 assumes that loading is performed in Y-direction. Transverse displacements (UX) of homologue nodes i and j (Xi = Xj + L and Yi = Yj, where L is the microstructure size) are coupled. Such constraints are needed to ensure that periodic boundary conditions are properly defined, which avoid rigid displacements. Analogous conditions are applied for a loading in the X-direction. Effective Young’s modulus is derived from the computation of the reaction forces at the loaded line, giving the sample size. The deformation is calculated from the imposed displacement and knowing the sample size. More than 20 microstructures exhib-
131
iting a zein ratio varied between 5% and 60% are considered. For each microstructure, four interphase thickness and moduli are combined. More than 360 calculations are required with 10 min per run on a PC – INTEL Core 2 CPU 6400 at 2.13 GHz and with 3.37 GBytes of RAM. Convergence tests are performed to ensure that meshing is not too small to cause ill-conditioning or too rough to lead to convergence problem. Ill-conditioning may appear as a consequence of the fine mesh especially when meshing contains elongated elements. We recall here that our meshing is regular for all conditions because of the direct pixel/element conversion. Preliminary accuracy tests are performed on coarser meshes, since we cannot use finer meshes unless to change the main characteristics of the microstructure. All our structural analyses are conducted using the ANSYS software. We use the preconditioned conjugate gradient solver available in the package. The PCG assembles the full global stiffness matrix starting from the element matrix formulation instead of factoring the global matrix. DOF solutions are computed in an iterative way until convergence. The accuracy of the computed primary variables (displacement solutions) is controlled using a tolerance value of 108, which is always reached regardless of the meshing density. Table 1 compares the number of PCG iterations needed to reach a converged solution for different meshing densities. 3. Results and discussion The justification of the use of a three-phase model comes from the fact that analytical approaches giving the bounds of the starch– zein composite fails in fitting the experimental data shown in Fig. 4. Indeed, as starch and zein have similar elasticity moduli, all analytical models predict a linear trend. Three-point bending test results suggest instead a non-linear evolution of the effective modulus as a function of zein content. If we consider the fact that experimental results are obtained with good reproducibility (averages of 10 replicates with a reasonable scatter), the most possible reason attached to the deviation from the analytical approaches is the presence of an interface effect. Moreover, a recent study [10] has shown that in the same material, nanoindentation experiments revealed that near the interface, the maximum penetration force in the bulk material is at least 4.5 times larger than near the interface. Before going any further with the three-phase model, mesh sensitivity is performed in order to check the validity of the computed results at larger scales. Generally speaking, a rough mesh leads to fast computations but at the cost of loosing microstructure details. These details have actually a large importance in our study because we know that interphase effect is dependent on the amount of interface quantity in the microstructure. A large mesh is usually required to obtain accurate results if we accept large computation durations. The balance between the two (i.e., accurate results and run duration) gives the optimal resolution at which the microstructure should be looked at. As pointed out earlier, the finite element mesh is the consequence of the direct import of the composite images at the genuine acquisition resolution. Thus, we cannot perform finer meshes unless to change the main characteristics of the microstructures. Knowing this fact, preliminary tests of mesh sensitivity were performed using coarser meshes. Fig. 5 illustrates this by considering a particular microstructure containing 39% of zein, for which the resolution was lowered thanks to the concept of mesh index. This parameter defines the ratio of the element size (pixel size) at the largest resolution over that at any lower resolution. For example, all original microstructures were acquired at 512 512 pixels corresponding to 160 160 lm2. Thus, the original element size is around
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Fig. 2. Finite element model with interphase properties where t denotes the thickness of the interphase region.
Loaded line
Exp (bending) Num H&SH&S+
UY=U
3.2
UXi+UXj=0 Yi=Yj
i
j
Young's modulus (GPa)
constraint equation
Mori & Tanaka Reuss Eshelby Voight
3.0
2.8
2.6
Y 2.4 0.0
X
Constrained line UY=0
Fig. 3. Load conditions in the Y-direction applied to the FE three-phase model. For visualisation purposes, the grid resolution is lowered to 128 128 pixels. Interface thickness is 0.63 lm. Phase ratios are 0.56, 0.19 and 0.25, for starch, zein and interphase, respectively.
Table 1 Number of PCG iterations as function of different meshing densities. Mesh factor
Number of elements
DOF
DOF constraints
Total Iterations In PCG
1 2 3 4
262,144 65,536 28,900 16,384
526,338 132,098 58,482 33,282
3591 1028 513 258
79 79 65 56
0.31 lm. It is possible to obtain lower resolutions such as 256 256 using binning operators. Thus, the mesh index in this
0.1
0.2
0.3
0.4
0.5
0.6
Zein content (-) Fig. 4. Comparison between some popular analytical approaches giving the bounds for a two-constituent composite material and three-point bending experimental results of starch–zein composite.
case is 2 and corresponds to an element size twice larger than the original size. When analysing the results depicted in Fig. 5, it appears that the sensitivity of the composite modulus to mesh index increases when the interphase modulus is the lowest possible. For example, the scatter of the composite modulus for Ei = 2 GPa is less than 1.3% whereas for Ei = 0.1 GPa, it increases to more than 21%. When the interphase modulus is near the intrinsic moduli, the sensitivity is quite reduced because the behaviour of the composite compares to that of a two-phase model. The increase of sensitivity to mesh coarsening when the modulus is lower than the intrinsic moduli is mainly related to the loose of interface information when the resolution is changed. Analogous interpretation still holds when the interphase modulus is larger than those of intrinsic phases.
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In the following, the original resolution as obtained from the image acquisition protocol (512 512 pixels) is used for all conditions, knowing the fact that we cannot guarantee the stability of the predicted composite modulus at lower resolutions. Fig. 6a shows the evolution of the effective Young’s modulus as a function of zein content for a varied interphase modulus between 0.1 and 2.0 GPa. At any particular zein content, the increase of Ei has the consequence to increase the effective Young’s modulus, which is an obvious result. The evolution of the effective modulus seems to indicate a fairly linear trend with the zein content, irrespective of the interphase modulus. The exception for this is undoubtedly the distinguished point at a zein content of 15%. We see here that this point is totally predictable by the three-phase model, in the sense that the expected interface modulus lies within the restricted range 0.1–0.5 GPa. This point is related actually to the largest number density of zein particles in the microstructure, where the microstructure can be still described as a matrix/particle composite. At larger ratios, zein particles percolate to form a continuous phase before again the composite switches to a matrix/ particle regime, where the matrix becomes zein. There is another
2.8
Effective Modulus (GPa)
2.6 2.4 2.2 2.0 1.8 1.6
Interphase modulus (GPa) 0.1 0.5 1.0 2.0
1.4 1.2 1.0 1
2
3
4
mesh index (-) Fig. 5. Mesh sensitivity analysis for different interphase moduli. Run conditions: zein fraction = 0.39, interphase thickness 0.31 lm.
a
3.45
Ei = 0.1 GPa Ei = 0.5 GPa Ei = 1.0 GPa Ei = 2.0 GPa E experimental
3.35
Effective modulus (GPa)
3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 2.45 2.35 2.25 0
10
20
30
40
50
60
Zein content (%)
b
3.0
2.9
Ey (GPa)
2.8
2.7
2.6
2.5
Ei =0.1 GPa Ei =0.5 GPa Ei =1.0 GPa Ei =2.0 GPa Ey=1.00Ex (R²=0.90)
2.4
2.3 2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
Ex (GPa) Fig. 6. (a) Predicted composite elasticity modulus as function of zein content for different interphase moduli (interphase thickness = 0.31 lm). (b) Comparison between the modulus in X- and Y-directions for all studied conditions.
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Non-linear dependence of the interphase zone properties on the interphase thickness: this possibility requires a fine description of the interphase region in terms of meshing and physical properties. It should make sense to represent this effect as an elasticmodulus gradient [13,14]. However, it would probably lead to an inconclusive insertion, since we have no idea about the gradient profile that have to be implemented. In addition, Substantial improvement of the modelling technique is required which actually is beyond the scope of the present study. Microstructure anisotropy that affects interphase thickness: Because of the random distribution of the filler, this hypothesis is unlikely to provide a clear say on the large dependence of the composite effective property on zein fraction. Sensitivity of the interphase properties to filler content. At the first sight, it may appear surprising that the fraction of the filler affects the physical properties of the interphase. However, this has to deal with the nature of the biopolymer materials and mainly on different water moisture of individual species. This last hypothesis is hereafter explored. In order to put forward a mechanism explaining the approximated experimental trend, the optimal interphase modulus is plotted against zein content (Fig. 7b). We recall here that the interphase elasticity properties have to be lower than those of the intrinsic materials in order to be in line with the nanoindentation results [10]. A brief examination of Fig. 7b, some points particularly between 7% and 17.5% lead to interphase moduli larger than 3 GPa. The worst case (not shown here), is an interphase modulus beyond 100 GPa predicted for a zein content around 60%. In order to examine more deeply this result, the interphase thickness is considered as a second adjustable parameter. As a matter of fact, the results shown in Fig. 7a, through b are obtained for a constant interphase thickness of 0.31 lm. The objective in the following is
a
3.4 3.3 3.2
Effective modulus (GPa)
unpredictable point at about 45%, which exhibits the largest modulus. However, there is no particular structural feature that can be attached to this point. The orthotropy of the effective modulus is checked giving the result depicted in Fig. 6b. The scatter between moduli increases when the interphase modulus decreases. If we recall the fact that a small interface modulus highlights the interface effect, it appears now that the orthotropy of the composite elasticity is related to the quantity of interfaces parallel and perpendicular to the load direction. If these two quantities are the same, the composite will exhibit the same moduli in X- and Y-directions. In the case of Fig. 6b, the linear trend found between Ex and Ey is predicted with an acceptable correlation factor R2 = 0.90, despite the scatter in the case of Ei = 0.1 GPa. The use of interphase modulus as an adjustable variable makes it possible to derive the optimal Ei value that can explain the experimental trend. For this purpose, the experimental trend is approximated using a fitting curve (solid line in Fig. 6a). For a given zein content, the four values of interphase modulus are plotted against the corresponding composite Young’s moduli. A fitting routine is used to determine the function that best describes the correlation between the effective and interphase moduli. The optimal interphase modulus is then determined based on the knowledge of the experimental effective modulus derived from the curve in solid line. The result of the fitting procedure is shown in Fig. 7a. A fairly good agreement is found between the approximated experimental trend and that obtained using the fitting routine. We have to mention that the non-monotonic dependence of effective Young’s modulus on the zein filler is not a common result, but we do find similar behaviour, for example, in the case of POSS reinforced epoxy [11,12]. We can explore different potential explanations that can be inferred to the observed non-linear trend, among them:
3.1 3.0 2.9 2.8 2.7 2.6
Experimental Numerical
2.5 2.4 0
10
20
30
40
50
60
Zein content (%)
b
14
12
Interphase Modolus (GPa)
134
10
8
6
4
2
0 0
5
10
15
20
25
30
35
40
Zein content (%) Fig. 7. (a) Predicted effective modulus as function of zein content for an optimised interphase modulus. (b) The optimal interphase modulus as function of zein content.
to determine if the approximated experimental trend is a genuine effect or not based on the interphase thickness and modulus. Fig. 8a shows the evolution of the effective modulus for different interphase thickness giving a small interface modulus (Ei = 0.1 GPa). The effective modulus is found to decrease with the increase of zein content up to 40%. At any zein content, the effective modulus decreases with the increase of interphase thickness indicating that the interphase fraction increased in the microstructure. Because the interface modulus is lower than any of the intrinsic phase moduli, it is not surprising to observe such a trend. The same evolution is predicted when the interphase modulus is 2 GPa (Fig. 8b). In this case, the tendency of the effective modulus is, however, towards the increase because the interphase contributes better in the overall rigidity of the composite. When comparing this result with that depicted in Fig. 6a, the rigidity of the composite appears dependent on the ratio Ei/t, where t denotes the interphase thickness. The optimisation of the interphase properties similar to that shown in Fig. 7 will require the building of a complex function of the form Eeff ¼ f ðEi ; C z ; tÞ, where Cz is the zein ratio and Eeff is the effective Young’s modulus. A suitable represen-
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a
into account the increase of the ratio Ei/t and the increase of Cz, announces that the effective modulus spread increases with the increase of zein content (Fig. 7c). It seems that the lowest value of Ei/t controls this spread and thus reveals the effect of interface quantity. The spread profile shown in Fig. 7c encodes the evolution of the interface quantity as a function of zein content.
2.8 2.6
2.2 2.0
a
1.8
1.24
1.6 1.4 1.2 1.0 0.8 0.6
Interphase thickness (µm)
Effective Modulus (GPa)
2.4
160 x160 µm², Ei= 0.1GPa Interphase thickness (µm) 0.31 0.63 0.94 1.25 0
10
20
30
40
50
60
Zein Content (%) 2.85
2.80
Effective Modulus (GPa)
0.62
2.75
0.31
160 x160 µm², Ei = 2GPa Interphase thickness (µm) 0.31 0.63 0.94 1.25
0
2.60
20
15
20
25
30
35
40
2.6
2.65
10
10
b
2.70
0
5
Zein content (%)
Interphase modulus (GPa)
b
0.93
30
40
50
60
2.4
2.2
2.0
1.8
Zein content (%) 1.6
c
3.0 2.8
Increase of Ei/t 2.000
0
10
20
c
2.4
0.025
1.8 1.6 1.4 1.2
57 %
Increase of Zein content
0.8 0
50
50
60
Numerical Experimental
3.1
2.0
1%
40
3.2
2.2
100
150
200
250
300
350
400
FE data (-) Fig. 8. (a and b) Effect of varying the interphase thickness on effective properties of the starchy composite for two particular interphase modulus values. (c) Predicted composite Young’s modulus for all studied conditions.
Effective Modulus (GPa)
Effective modulus (GPa)
2.6
1.0
30
Zein content (%)
3.0
2.9
2.8
2.7
2.6
2.5 0
10
20
30
40
50
60
Zein content (%)
tation as that shown in Fig. 7c of all FE run results can be informative of the global tendency that can be inferred to the result of the three-phase model. Indeed, sorting out all numerical results taking
Fig. 9. (a and b) Optimised interphase parameters as function of zein content. (c) Comparison between FE and experimental Young’s modulus as function of zein content.
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The result of the optimisation process similar to that performed using one adjustable variable is shown in Fig. 9. The optimal interphase thickness exhibits a large scatter between 0.31 and 1.25 lm. Note that because of the binning operations, the interphase thickness evolves as discrete jumps. Despite there is no clear correlation between zein content and interphase thickness as depicted in Fig. 9a, the change in interphase thickness can be inferred to some physical reasons explained hereafter. Because zein and starch have different moisture contents in the composite [15,16], the water content of the composite varies as a function of zein fraction from 12% to 7%, when zein content increases from 0% to 100% [9]. Near the interface, there is undoubtedly a change in the water content that may affect the structure and consequently the mechanical properties at the interphase region. Zein fraction and phase morphology determine together the extent of the region where the water content varies from the equilibrium value in starch to the other one value in zein. The same explanation still holds in the case of the interphase modulus, where Fig. 9b shows that changing zein content in the microstructure predicts the change of Ei in the range 0.79– 2.47 GPa with no evident tendency. Note that all predicted values are lower than Young’s moduli of intrinsic phases, which means that the three-phase model is able to predict the composite effective property evolution under weak interphase conditions. The comparison between the numerical and experimental effective moduli is shown in Fig. 9c. Almost experimental tendency was reproduced using the optimal interphase parameters. The conflicting point is however the underestimation of the effective property (2.82 GPa) at the largest zein fraction (57%). All attempts to fit the experimental value, which is actually above the largest intrinsic phase modulus, were discouraging. The weak adhesion hypothesis for this particular fraction is not realistic if we consider that the experimental data are trustable. 4. Summary Analytical models giving the evolution of the effective Young’s modulus for starch–zein composite as a function of zein content fail to describe the non-linear experimental trend. A three-phase model is considered to explain such evolution taking account interphase parameters (thickness (t) and Young’s modulus (Ei)). The orthotropy of the composite is sensitive to the interface quantity in the direction of loading when the interphase modulus is quite different from that of the intrinsic phases. Combining the effect of interphase parameters shows that the effective composite property depends on the ratio Ei/t. The
optimisation of both parameters, in the case of weak interphase hypothesis, allows the prediction of almost experimental trend. The misaligned points are those where the interphase modulus has to be larger than the moduli of the intrinsic materials to fit the experimental value. It is thought that the interphase properties variation, as predicted by the three-phase model, can be related to the change in the structure of the interphase as a function of zein content. In particular, different equilibrium values of moisture content between starch and zein phases have some implication in the variation of interphase properties. References [1] Gross RA, Kalra B. Biodegradable polymers for the environment. Science 2002;298:803–7. [2] Beg MDH, Pickering KL, Weal SJ. Corn gluten meal as a biodegradable matrix material in wood fibre reinforced composites. Mater Sci Eng A 2005;412(1– 2):7–11. } Z, Dogossy G, Réczey K, Czigány T. Reducing water absorption [3] Gáspár M, Benko in compostable starch-based plastics. Polym Degrad Stabil 2005;90(3):563–9. [4] Hull D, Clyne TW. An introduction to composite materials. Cambridge: Cambridge University press; 1996. [5] Frogley MD, Ravich D, Wagner DH. Mechanical properties of carbon nanoparticle-reinforced elastomers. Compos Sci Technol 2003;63(11):1647–54. [6] Hashin Z. Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J Mech Phys Solids 2002;50(12):2509–37. [7] Sevostianov I, Kachanov M. Effect of interphase layers on the overall elastic and conductive properties of matrix composites. Applications to nanosize inclusion. Int J Solids Struct 2007;44(3–4):1304–15. [8] Esmaeili N, Tomita Y. Micro- to macroscopic responses of a glass particleblended polymer in the presence of an interphase layer. Int J Mech Sci 2006;48(10):1186–95. [9] Chanvrier H, Colonna P, Della Valle G, Lourdin D. Structure and mechanical behaviour of corn flour and starch–zein based materials in the glassy state. Carbohydr Polym 2005;59(1):109–19. [10] Guessasma S, Sehaki M, Lourdin D, Bourmaud A. Viscoelasticity properties of biopolymer composite materials determined using finite element calculation and nanoindentation. Comput Mater Sci 2008;44(2):371–7. [11] Abd Rashid ES, Ariffin K, Choong Kooi C, Md Akil H. Preparation and properties of POSS/epoxy composites for electronic packaging applications. Mater Des 2009;30:1–8. [12] Zhang Z, Gu A, Liang G, Ren P, Xie J, Wang X. Thermo-oxygen degradation mechanisms of POSS/epoxy nanocomposites. Polym Degrad Stabil 2007;92(11):1986–93. [13] Pender DC, Padture NP, Giannakopoulos AE, Suresh S. Gradients in elastic modulus for improved contact-damage resistance. Part I: The silicon nitride– oxynitride glass system. Acta Mater 2001;49(16):3255–62. [14] Askadskii AA, Goleneva LM, Bychko KA, Afonicheva OV. Synthesis and mechanical behavior of functionally gradient polyisocyanurate materials based on hydroxy-terminated butadiene rubber. Polym Sci A 2008;50(7):781–91. [15] Bizot H, Le Bail P, Leroux B, Davy J, Roger P, Buleon A. Calorimetric evaluation of the glass transition in hydrated, linear and branched polyhydro glucose compounds. Carbohydr Polym 1997;32:33–50. [16] Madeka H, Kokini JL. Effect of glass transition and crosslinking on rheological properties of zein: development of a preliminary state diagram. Cereal Chem 1996;73(4):433–8.