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Elastic behavior of multi-scale, open-cell foams
Composites: Part B 44 (2013) 172–183
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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Elastic behavior of multi-scale, open-cell foams Laurent Maheo a,b,c, Philippe Viot a,b,c,⇑, Dominique Bernard d,e, Ali Chirazi d,e, Gaétane Ceglia f,g, Véronique Schmitt f,g, Olivier Mondain-Monval f,g a
Arts et Metiers ParisTech, I2M, UMR 5295, F-33400 Talence, France CNRS, I2M, UMR 5295, F-33400 Talence, France c University of Bordeaux, I2M, UMR 5295, F-33400 Talence, France d CNRS, ICMCB, UPR 9048, F-33600 Pessac, France e University of Bordeaux, ICMCB, UPR 9048, F-33600 Pessac, France f CNRS, CRPP, UPR 8641, F-33600 Pessac, France g University of Bordeaux, CRPP, UPR 8641, F-33600 Pessac, France b
a r t i c l e
i n f o
Article history: Received 5 January 2012 Received in revised form 29 May 2012 Accepted 6 June 2012 Available online 15 June 2012 Keywords: A. Foams B. Elasticity B. Microstructures C. Computational modeling Image analysis
a b s t r a c t The mechanical properties of cellular materials are still subject to numerous theoretical and experimental investigations. In particular, the impact of cell size on the foam’s elastic response has not been studied systematically mainly due to the lack of experimental techniques with which the cell size and relative density of materials can be varied independently. This paper presents the results of a study of the elastic behavior of open-cell foams as a function of relative density and the size of the interconnected, spherical pores. First, the chemical procedure allowed us to produce polystyrene open-cell foams in which the relative density and the average cell diameters were varied independently. The results of compression tests performed on these foams showed an unexpected influence of the cell diameter (at constant relative density) on the elastic response. The analysis of the microstructure of the foam revealed the presence of a complex nanostructure in the edge of the cells that appeared during the synthesis procedure. An analytical model (an extension of the Gibson–Ashby model) is presented, which takes into account the complex multi-scale structure of the foam and accurately describes the observed dependence of the measured Young’s moduli on cell size. This approach was confirmed further by a finite element numerical simulation. We concluded that the observed dependence of elastic modulus on cell size was due to the heterogeneous nature of the material that constitutes the walls of the cells. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The mechanical behavior of cellular materials is vitally important for their use in a wide variety of applications that take advantage of their ability to dissipate energy during compression. This property is affected by the structure of the foam and by the type of material that constitutes the walls of the cells. Because of the industrial importance of such materials, the relationship between their structure and mechanical properties is an ongoing topic of intense research [1,2]. The two main structural parameters of a foam are its relative density and its average cell size. The classical model developed by Gibson and Ashby [1] leads to a Young’s modulus that exhibits a square dependence with the relative density and ⇑ Corresponding author at: Arts et Metiers ParisTech, I2M-DuMAS, esplanade des Arts et Metiers, F-33405 Talence, France. E-mail addresses: [email protected] (L. Maheo), [email protected] (P. Viot), [email protected] (D. Bernard), [email protected] (A. Chirazi), [email protected] (G. Ceglia), [email protected] (V. Schmitt), [email protected] (O. Mondain-Monval). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.06.006
no dependence on cell size. The effect of foam density is well established [3], but there have been few studies related to the influence of the cell size at constant density [4–8]. This lack of data is due to the inherent difficulty associated with the synthesis of foams for which the cell size can be controlled over a sufficiently large range. From the data that have been obtained, it also is difficult to determine a general trend for the effect of cell size. Several authors [5,6] have indicated that mechanical response is independent of cell size and that density is the main parameter. Interestingly, three studies [4,7,8] referred to the existence of a dependence of Young’s modulus on the cell size, but their data were restricted to foams with millimetric cells that vary from 1 to 4 mm. Dam et al. [4] showed that the compressive properties of aluminium foam increased as a function of cell diameter. They attributed (without further insights) this dependence to a change in the microstructure of the solid struts. Gupta et al. [9] obtained similar results for syntactic foams; the cells are hollow, microspheres of a glass or epoxy balloon surrounded by a polymer matrix. Gong et al. [10] manufactured anisotropic, open-cell foams with several cell sizes and anisotropic cells. First, they studied the relationship between cell
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size and anisotropy, and they found that it was difficult to change a parameter independently of the others. Second, they showed that the cell size had no influence on the elastic modulus in the longitudinal direction; however, the elastic modulus in the transverse direction increased when the cell size decreased. Numerical simulations confirmed the results of the experiments. In their study of syntactic foams, Viot et al. [11,12] did not detect any significant effect of the diameter or the thickness of the hollow spheres, and they found that density was the most influential parameter. Thus, it appears that the influence of the cell size on the mechanical response of a cellular material is not fully understood and may depend on other variables, such as the constitutive materials and the microstructure. Very few experimental studies have linked the evolution of elastic response to the open-cell morphology of the foams. Some authors [13–15] showed the existence of nanoscale porosities in the cell walls of polymerized, high-internal-phase emulsions (polyHIPEs) [16,17], but they did not study the influence of the struts microstructure or cell size on Young’s modulus because it is very difficult to synthesize structured materials with precise control of the morphology of the foam. In this study, the elastic behavior of open-cell foams was studied as a function of cell size at constant chemical composition and relative density. The initial objective of this study is to correlate the mechanical response of open-cell foams with the calibrated diameter of the porosities. The cell size can be tuned through the polymerization of calibrated, high-internal-phase emulsions (referred to as polyHIPEs). In our materials, size change is accompanied by a change in the microstructure of the cells. The chemical procedure used to obtain calibrated cell foams generates a nanoscopic structure in the edges of the connected pores. Thus, we will show that the multiscale microstructure of the foam must be taken into account in the modeling of its mechanical properties. Finally, we will demonstrate that our materials behave mechanically as composites, with two parts that have different Young’s moduli. We will present a model that can describe the observed dependence of the size of the cells and strut morphology on the compression modulus. The paper is organized as follows: (i) first, we present a brief description of the preparation of the polymer foams that have controlled density and cell size; (ii) then, we present the results of our measurements of the elastic response of the materials as we varied those two parameters independently; (iii) next, we present our characterization of the foam microstructures using Transmission Electron Microscopy (TEM) and Scanning Electron Microscopy (SEM) images, as well as X-ray computed micro tomography (XCMT), and show that the strut structures are different at the surface and inside the strut volume; and (iv) two approaches are proposed to describe the results, i.e., (1) an analytical model (which is an extension of the Gibson–Ashby model to the case of a composite material) and (2) a finite element simulation that clarifies the reason that cells size affects the mechanical properties of the foams.
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composed of styrene (ST) monomer, and a divinylbenzene (DVB) crosslinker in various proportions in the presence of sorbitan monooleate (Span 80) surfactant. This recipe has been described extensively in the literature [17]. In the second stage (Fig. 1b), quasi-mono dispersed emulsions were obtained by shearing the poly-dispersed emulsions within a Couette device made of two concentric cylinders separated by a narrow, 200-lm gap (Ademtech, France). The size of the final droplets was imposed by the shear rate, the interfacial tension, and the ratio of the viscosity of the dispersed phase to the viscosity of the continuous phases [18]. In the present case, emulsions with an average droplet diameter ranging from 0.5 to 14 lm were obtained. The average size and poly-dispersity of the droplets were measured using static light scattering (SLS). For each droplet size, the brine volume fraction and the span 80 concentration in the organic phase can be fixed by progressive dilutions without altering the size distribution of the droplets, which allowed independent control of the sizes and densities of the foam cells to be generated later. Polymerization (stage 3) of the continuous phase composed of ST/DVB/Span80 was initiated by increasing the temperature to 60 °C (Fig. 1c). At this temperature, the initiator (potassium persulfate) in the dispersed phase decomposes. Radical polymerization depends on the surfactant concentration (5 days for 10% surfactant to 7 days for 30% surfactant). During polymerization, solidification and a slight shrinkage occurred simultaneously, leading to a solid monolith with a morphology that was determined by the mold. Even for such a low shrinkage rate, the film between adjacent drops breaks, forming connections that lead to the final, open porosity (referred to as open-cell morphology as opposed to closed-cell morphology). The resulting foams were washed several times in ethanol (24 h) and acetone (8 h) in order to remove the remaining emulsifiers, monomers, cross linkers, initiators, and stabilizers. No size variation of the monoliths could be detected using a vernier calliper, which showed that the washing steps did not significantly affect the materials at the scale of 0.1 mm. Finally, the foams were dried at room temperature (Fig. 1d). More details on the preparation and the structural characterization procedures will be published elsewhere [19]. The materials were obtained as homogeneous monoliths that showed no defect at the scale attainable by X-ray radiography with a resolution of 22 lm. Thus, the monoliths exhibited no fractures and no porosity gradients at this scale. Of course, each sample was synthesized several times to allow for the use of intrusive characterization techniques (mercury porosimetry, SEM, TEM, X-ray microtomography, and mechanical characterization) and to check the reproducibility. Mercury porosimetry was used to measure the solid fraction for each kind of samples. The values obtained were in good agreement with those deduced from the initial quantities of the organic and aqueous phases in the emulsions. 2.2. Mechanical characterization
2. Materials 2.1. Preparation of the foam The preparation of cellular material of controlled cell size is based on the formulation and polymerization of a calibrated, high-internal-phase emulsion (PolyHIPE). Calibrated, open-cell foams with controlled cell sizes and porosities were obtained in four steps, as shown in Fig. 1. The first stage of the process (Fig. 1a) was the preparation of a rough, poly-dispersed emulsion obtained by incorporating brine (an aqueous phase containing 0.5 M NaCl) and the radical initiator, 0.01 M potassium peroxodisulfate (KPS), into an organic phase
Mechanical tests were conducted on a Zwick electromechanical testing machine that had a force threshold of 250 kN and a velocity limit of 600 mm/min. A force sensor with a range of 10 kN was used for the compression tests of the cellular materials, and the displacements imposed on the specimens were determined from the punch displacement, since the rigidity of the machine was considered to be infinite in comparison to the stiffness of the porous specimen that was being tested. Since the aim of the experiment was to characterize the elastic modulus, it was sufficient to calculate the strain using the ‘‘empirical’’ expression: e ¼ hdh0 (with dh being the displacement of the punch) since the difference between the ‘‘testing-machine-based’’ strain and the true strain was
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Fig. 1. Four stages of the PolyHIPE foam preparation: (a) rough emulsion, (b) calibrated emulsion obtained after a controlled shear, (c) calibrated polymerized emulsion, and (d) final dried polymerized calibrated porous material.
negligible for low values of strain. The stress, r, was calculated from the measured force, F, per unit area, using the initial section, S0. The testing indicated that the variation of the section during compression was not significant, and the resulting error imposed on the stress calculation by using S0 was negligible. Samples with well-controlled morphologies (cylinders of initial height, H0 = 40 mm, and diameter, D = 25 mm) were compressed with an imposed displacement rate of 10 mm/min of the lower punch. Previous compression tests confirmed that the results were independent of the displacement rate in the range of 1–100 mm/ min. The initial strain rate was 4.2 103 s1. The behavior of the cellular material was studied for cell diameters of 1.8, 5, 7, 7.8, and 14 lm and for different relative densities or solid fractions, us, of 10, 15, 20, 25, and 30%. Past studies have shown the significant edge effects due to partially-cut cells at the boundaries of the samples. In our case, the volume of the sample was close to 20 103 mm3, whereas the volumes of the spherical cells were less than 1.5 106 mm3. This significant difference in the volumes allowed us to assume that the number of cut cells was not significant compared to the total number of cells in the entire sample volume. Therefore, the edge effects can be considered to be negligible. For each set of parameters, all measurements were performed in triplicate in order to check their reproducibility. The elastic modulus of the cellular material under compression was calculated using the linear dependency of the stress with respect to strain, i.e., E = re.
2.2.1. Influence of the density of the foam The influence of the density of the foam was explored while keeping the cell size constant. The mechanical response of the foam can be divided into two distinct regimes, as shown in Fig. 2. The first regime is the elastic regime, which exists until
Fig. 2. Stress–strain curves showing the effect of the solid fraction on the mechanical behavior (for a cell diameter of 7 lm).
the stress becomes equal to the yield stress. The second regime is a stress plateau that is limited by the fragile nature of the sample. The transition between the elastic behavior and the stress plateau was particularly smooth when the density of the foam was high. One can note that the failure strain decreased as the density of the foam decreased. All of the elastic moduli that were measured are given in Fig. 3. The experimental dependency of the elastic modulus on solid fraction, us, can be fitted with the power law. However, in our case, the measured exponent (2.5) was larger than the exponent that Gibson and Ashby predicted, which has been confirmed experimentally by several authors [1,3].
2.2.2. Influence of the size of the cells in the foam The influence of the diameter of the cells on the mechanical response of the foam was established for various imposed foam densities. The macroscopic behavior of the foam depends on the cell size. For the lowest cell diameter (i.e., for diameters of 1.8 lm, Fig. 4), the mechanical behavior was consistent with the classical mechanical response subdivided into the elastic domain and the stress plateau, where the stress increases slightly as a function of strain. For cells with larger diameters, the mechanical behavior of the foam was mostly of the fragile type, and the length of the stress plateau decreased drastically for cell diameters as large as 14 lm. The elastic modulus is given as a function of cell size for various values of the solid fraction us. Fig. 5 shows the typical dependency of the elastic modulus on cell size for a solid fraction of 25%. The elastic modulus increased as cell size increased, in sharp contrast with the Gibson and Ashby model, which predicts no dependency of cell size on the elastic
Fig. 3. Variation of the elastic modulus as a function of solid fraction for a material with a cell size of 7 lm.
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microtomography were in good agreement with the sizes obtained with SLS and SEM. Again, this provides evidence that the structure
Fig. 4. Stress–strain curves showing the effect of the cell diameter on the behavior of the foam (for materials with a solid fraction of 25%.
Fig. 6a. SEM picture of the structure of an open-cell foam.
Fig. 5. Variation of the elastic modulus as a function of cell diameter for materials with a solid fraction of 25%.
modulus. Thus, our results have identified an unexpected effect of cell diameter on the elastic behavior of the foams. As suggested by Dam et al. [4], such an effect might be due to a change in the strut microstructure as the cell size changes. To check this hypothesis, we performed a full micro-structural characterization of the foams using 2D and 3D imaging techniques. This work is described in the next section.
Fig. 6b. Micrograph reconstruction of a part of a foam sample (cell diameter 14 mm).
3. Microstructural characterization using imaging techniques The micro and nano structures of open-cell foams were characterized using scanning electron microscopy (SEM, Fig. 6a), transmission electron microscopy (TEM, Fig. 7), mercury porosimetry, and X-ray microtomography. The average cell size obtained via SEM was almost equal to the average cell size of the initial emulsion droplets given by Static Light Scattering (SLS) and optical microscopy observations [19], confirming that no significant changes occurred during the polymerization of the emulsion. Microtomographic measurements were conducted on the TOMCAT line of the Swiss Light Source (SLS) using a beam energy of 15.5 keV. The 1200 radiographs that were acquired for each scanned sample had 2048 2048 pixels, with a pixel size of 0.37 lm. Pre- and post-processing noise reduction algorithms applied to radiographies and reconstructed volumes, respectively, were used to enhance the quality of the images. These 3D reconstructed volumes (Fig. 6b) were used to estimate and control the diameters of the cells of the calibrated foams. The cell sizes measured using
Fig. 6c. Micrograph section of a part of a foam sample (cell diameter 14 mm).
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Fig. 6d. Zoom of the square part of the left image.
of the sample did not change during polymerization. This means that controlling the initial size of the emulsion droplets allows control of the final pore size of the solid material that is obtained. As seen in the 2D cross section of the 3D reconstructed volumes (in which the polymer zones are gray and the void cells are black (Figs. 6c and 6d), some minor gray level variations can be observed within the struts, i.e., along the white lines drawn in Figs. 6c and 6d. The corresponding gray level profiles of these lines (AB, CD, and EF) are plotted in Fig. 8a. This analysis could only be performed on the samples that had the larger cell diameter (14 lm) since the acquisition precision of 0.37 lm/pixel and the level of background noise did not allow any observable gray-level variability within the struts section of the samples that had small cell sizes. The line histograms plotted in Fig. 8a within the struts section (lines AB, CD, and EF, Fig. 6c) confirmed the existence of a gray level gradient inside the polymer struts. As seen in the line histogram shown in Fig. 8a, values close to 60 correspond to the void, while values around 100–120 relate to the struts section in which the iso-curves closer to the cell wall have slightly higher gray levels. Furthermore, for line histograms AB and EF shown in Fig. 6c, two maxima per histogram can be observed.
Fig. 7b. Zoom of the square part of the left image.
Fig. 7c. TEM Picture for a cell size of 7 mm.
These local observable gray level gradients within the struts zone can be related to a minor change in local density near the cell walls. This observation was further confirmed using TEM. To allow for TEM observations, the porous samples first had to be impregnated with a liquid epoxy resin and cut into thin slices using a diamond knife (80 nm thick) before observation using a Hitachi H600 TEM. On the 2D TEM images of Fig. 7, epoxy impregnated zones appear as smooth, light-gray areas, while the polymerized zones appear as quite darker areas. Using the enlarged pictures of Figs. 7b, 7d, and 7f, it appears that the struts indeed present a heterogeneous texture. The polymer/void interfaces appear to be rather rough, whereas the inner part of the struts exhibits a nanoporous structure. Image analysis and quantification techniques were used to quantify nanoscale heterogeneities. The process of nanoscale image quantification applied to the TEM images of various samples is as follows:
Fig. 7a. TEM Picture for a cell size of 1.8 mm.
1. In case there is a non-uniform gray level throughout the picture due to non-uniform intensity during acquisition, the first step consists of equalizing the gray level.
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2. Binarization of the image by thresholding: pixels were classified as void (respectively solid) if their gray level was lower (respectively higher) than the threshold value. This value was selected for each image from the corresponding histogram of the gray levels. 3. Segmentation of the image in two regions: the pores corresponding to the initial emulsion droplets and the polymerized zone. These pores were large (at the scale of the image) and smooth (the imprints of liquid droplets). 4. Successive dilations of the droplet pores by one pixel. At each stage, a portion of the polymerized zone located at the interface was eroded. The porosity of this eroded zone was computed, and the values were plotted as a function of the distance from the initial limit of the droplet pores. (This is the number of dilation steps multiplied by the pixel size.)
Fig. 7d. Zoom of the square part of the left image.
This process could generate variation of the typical local porosity inside the struts as a function of the distance, d, from the polymer/void interface presented in Fig. 8b. The first two points (d < 20 nm) correspond to the roughness of the interface induced by the granular texture. The following plateau (for 25 nm < d < 225 nm) provided evidence for the existence of a very dense layer, the thickness of which can be estimated providing that the 2D limiting effect of a 3D structure sectioning is taken into account
Fig. 7e. TEM Picture for a cell size of 14 mm.
Fig. 8a. Gray level histogram for lines located in strut sections.
Fig. 7f. Zoom of the square part of the left image.
Fig. 8b. Variation of the mean porosity as a function of the distance from the surface of the cell.
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Fig. 10b. Square section of the edge of the modified Gibson–Ashby cell. Fig. 9. Representative cell of Gibson–Ashby model.
for error calculation. After the plateau, the local nanoporosity slowly increases, revealing the nanoporous nature of the internal part of the polymerized zone. Such an internal structure of the cell wall is similar to the structure observed by Silverstein et al. [13] in polyHIPEs polymerized in the presence of a porogen (non-polymerizable solvent) in the continuous phase of the initial, highinternal-phase emulsions (HIPE). These observations of micro and nano structures allow us to define open-cell foams with cell walls that are composed of a bi-layered structure, as sketched in Fig. 10. A first material (material 1) located at the surface of the cell surrounds a second material (material 2), which is localized mainly in the center of the edge. Then, it can be considered that struts are constituted of an outer layer (material 1) characterized by a thickness, e, and a core (material 2). Of course, in the real material, the boundary between the two zones is not sharp, and the transition between the two zones is rather smooth. A systematic estimation of e by varying the cell size and the solid volume fraction did not provide evidence of any influence of the cell size on the thickness e, which we estimated to be 150 ± 40 nm from statistical analyses that were conducted on several samples. Furthermore, the difference of texture in the cross section of the edges seemed to indicate a density contrast, revealing a difference in the nature of the material between the surface and the inner part of the struts (Figs. 7a and 8b). The average density can be determined, but it is very difficult to estimate a real value of the density of each material. In a first approach, we considered two different densities, i.e., q1 and q2, for materials 1 and 2, respectively. In the modeling section, the influence of the local densities on the global elastic response is estimated. Obviously, it also is very difficult to directly probe the chemical characteristics and the mechanical properties (using nano indentation) of these two materials at the nano scale.
Fig. 10a. Modified Gibson–Ashby cell with two materials that constitute the cellbeams.
At this point of the manuscript, we wish to discuss possible explanations for the nanoporosity and heterogeneity of the cell wall that we observed. First, it appeared that the presence of this nanoporous structure was correlated strongly with the concentration of the surfactant in the continuous phase of the initial emulsion. The texture of the polymer part of samples that were acquired using a lower surfactant concentration was much smoother, with almost no density gradient observed on the TEM pictures (not shown here). Another possible explanation for the nanoporosity could be that the material swelled when it was washed with the ethanol and acetone solvents. Such an effect would explain the appearance of the porosity inside the cell wall, but it would not explain the behavioral differences between the surface and the inner parts of the struts. Furthermore, no macroscopic variation of the samples before and after the washing cycles could be deduced from a careful measurement of the monolith dimensions. A more likely explanation is that, because the polymerization initiator was localized initially in the water droplets of the initial emulsion, polymerization was triggered at the interface and propagated inside the sample, thus leading to a gradient in the degree of polymerization of the sample, different average chain lengths, and perhaps in the degree of crosslinking. 4. Proposed analytical and numerical models 4.1. Theoretical background Constitutive models for cellular materials already have been investigated by several authors. Gibson and Ashby [20] studied the behavior of polyurethane, polyethylene, and aluminium foams (closed or open cells) and established constitutive equations based on the analysis of the mechanical response of an ideal structure of a foam. Some more complex models using tetrakaidecahedral cells [21] or random, mono-dispersed foams [22,23] can be used to better represent the real shape of the cells and to evaluate the elastic behavior of the cellular material. However, in a first step, we wanted to check the assumption that the specific microstructure of the struts has a significant effect on the elastic behavior of this foam. For that, we decided to base the analytical modeling of the elastic behavior of the foam on the classical Gibson and Ashby’s model, which was modified to take into account the specific morphology of the foam. In a second step, these results were confirmed with complementary, finite element simulations (Section 4.3). Gibson and Ashby proposed that the mechanical properties of solid foams are described by considering the porous material as a periodic assembly of open cubic cells, constituted of edges of square section, t2, and length, l, (Fig. 9). Adjoining cells are staggered so that their edges meet at their midpoints. Despite the simplicity of the representation, such an approach has been shown to describe the essential features of open-cell foams accurately in different cases [1,3].
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From this description of the microstructure of open-cell foams, the solid fraction us (or the relative density) is the volume of the strut divided by the volume of a representative cell. The solid fraction is related to the two geometrical parameters l and t by:
2 q t us ¼ / l qs
ð1Þ
where q⁄ is the density of the foam, and qs is the density of the material constituting the edge. In the elastic regime, when a uniaxial stress is applied so that each cell edge transmits a force, F, the perpendicular edges (of the force axis) bend, and the deflection, d, of the representative structure is due mainly to the bending deformation of the perpendicular edges. The deflection d can be easily established as: 3
d/
Fl Es I
ð2Þ
where ES is Young’s modulus of the material that constitutes the solid edge, and I is the second moment of the area of a square strut of section t2. Considering a square section for a homogeneous edge (constituted of only one material), the second moment I is proportional to t4. From this deflection d, it is possible to calculate the elastic modulus of the foam, the Young’s modulus being the ratio of the applied stress to the strain. The stress was deduced from the force divided by the area t2 of the square section, and the macroscopic strain of the representative cell was established by dividing the deflection by the initial length of the edge. Taking all these results into account, the elastic modulus is related to the strain by:
r F=l2 F E / ¼ ¼ e d=l d l
ð3Þ
Combining Eq. (3) with Eqs. (1) and (2), Gibson and Ashby established the equation that describes the elastic modulus of an open-cell foam:
E t 4 / / Es l4
q qs
2 ð4Þ
As a consequence, with such a model, no dependence of Young’s modulus on cell size is expected, and only the influence of the relative density can be modeled with a second-order power law. It is worth noting that none of these considerations is valid in our experimental study. 4.2. Modified Gibson–Ashby model Hence, the Gibson–Ashby model must be reconsidered to take into account the microstructure of the edges of the cells, which consist of two distinct materials. From observations of the edge sections, we assumed that a first material (defined by index 1, Fig. 10) was a kind of envelope that surrounded the second material (defined by index 2). A square section of an edge is characterized by its side t and the thickness e of the skin of material 1 (Fig. 10b). The density of the cell, q⁄, and its elastic rigidity, E⁄, must be recalculated by considering the local density, qi, and the elastic modulus, Ei, of each material, i, that makes up the struts of the cellular material. 4.2.1. Constitutive equations The calculation of the actual density of the cell requires taking into account the macroscopic volume fraction at the scale of the cell (Eq. (1)) and considering the nano porosities observed in the edges of the cell. For that purpose, the density of the cell is calculated from the ratio of its mass M and its volume V = l3. It is easy to
obtain the density q⁄ of the cell as a function of the densities q1 and q2, as follows:
q /
t2 l
2
q1 1 1 2
e2 e2 þ q2 1 2 t t
ð5Þ
The relationship (Eq. (1)) of the relative density proposed by Gibson and Ashby in the case of a classical foam can be obtained from Eq. (5) when considering a strut that consists of only one material, i.e., for 2e = t, or if the densities qi of each material i are identical. Following the methodology developed by Gibson and Ashby to describe the behavior of open cells, the elastic rigidity of the foam is deduced from the deflection d of the strut bending under a force F. Then, the elastic modulus is related to the elastic rigidities Ei of constitutive materials i, the second moment I of the area, and the length l of the edge, as follows:
E /
hEIi l
4
with hEIi ¼ E1 I1 þ E2 I2
ð6Þ
Considering the characteristics of the section of the edge (Fig. 10b), the Young’s modulus of the foam reads:
E /
e4 e4 þ E E 1 1 2 1 2 1 2 4 t t l
t4
ð7Þ
Eq. (5) made it possible to determine a general formulation (Eq. (8)) of Young’s modulus that is similar to the relationship proposed by Gibson and Ashby (Eq. (4)):
4 2 1 ð1 2 et Þ4 þ EE21 1 2 et E q / E1 q1 1 1 2 e2 þ q2 1 2 e2 2 t t q
ð8Þ
1
In conclusion, we can consider that the Young’s modulus of this multi-scale, open-cell foam is given by theclassical Gibson–Ashby formula weighted by a function f et ; EE21 ; qq2 , which takes into ac1 count the specific morphology of the cell struts:
2 E q e E2 q2 e E2 q2 with f ; ; ; ; / f t E1 q1 t E1 q1 E1 q1 E2 e 4 e 4 1 12t þ E1 1 2 t ¼ 2 2 2 þ qq2 1 2 et 1 1 2 et
ð9Þ
1
4.2.2. Parameters that influence function f The function f depends on the characteristics of several parameters of the microstructure (e/t and q2/q1) of the struts and the ratio of the elastic modulus E2/E1 of the two constitutive materials. The two geometrical parameters (e and t) can be estimated approximately from the observations of the microstructure. Image analysis led to a thickness e of approximately 150 nm. For this study, the dimension t of the strut section was established from the cell size l and the solid fraction us (Eq. (1)), so that:
rffiffiffiffiffiffi t¼l
us
12
ð10Þ
The value of the dimension t calculated from this expression is close to data that have been presented in the literature [5]. It is more complex to determine the densities q2 and q1 of the two materials. The images of the microstructure (Fig. 7) of the strut section show that the density of the core (material 2) is lower than the density of the material (material 1) close to the surface of the cell. From the data obtained from image analysis, it is reasonable to consider that the ratio q2/q1 varies from 1 to 0.75, but it is impossible to determine the ratio precisely.
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The same problem is noticed for the modulus ratio E2/E1. The elastic modulus of each material cannot be identified. The elastic modulus of polystyrene is close to 1400 MPa, and it is impossible to know whether the elastic behaviors of materials 1 and 2 were modified by the preparation of the foam samples. Some attempts were performed to conduct ‘‘in situ’’ measurements using an Atomic Force Microscope (the so-called nano-indentation technique), but it appeared to be very difficult to deduce any data on the relative values of the Young’s moduli E2 and E1. Now, we evaluate the sensitivity of the function f to the two ratios, i.e., q2/q1 and E2/E1. 4.2.2.1. Influence of the ratio of Young’s moduli. The variation of f was evaluated by considering the ratio E2/E1 as variable and for different values of cell size l. In Fig. 11a, we considered different values of the ratio E2/E1, a density ratio q2/q1 of 0.75, a thickness e of 150 nm and computed f as a function of cell size. Of course, for E2/E1 = 1, we have f = 1, which can be easily established from Eq. (9). The function f increases as a function of cell diameter for (E2/ E1) > 1 (Fig. 11). This variation is particularly significant for (E2/ E1) P 5. The experimental results obtained from the compression of the multi-scale foam showed an increase in Young’s modulus as a function of cell diameter (Fig. 5). As a consequence, the case in which (E2/E1) < 1 cannot describe our results since the function f is decreasing. This means that, within the frame of our analysis, the observed mechanical behavior can be explained only by a material that is softer at the interface than it is inside the wall, which is in apparent contradiction with the TEM observation. Indeed, due to the electronic contrast, we expected a tougher material at the interface, and this point will be discussed further at the end of this section. 4.2.2.2. Influence of the density ratio. The sensitivity of f to the density ratio also was considered by keeping a fixed thickness of e = 150 nm and considering values of the density ratio to be 1 and 0.75 (Fig. 11a and b). On the whole, the variation of f with cell size was found to be much more pronounced when q2q1 was low. However, irrespective of the value of the density ratio, the function f was always an increasing function of the cell size for (E2/E1) > 1. 4.2.2.3. Discussion. Agreement between our experimental results and the model can only be achieved by considering a softer layer of material at the interface, which is in sharp contrast with what was expected from the data acquired by both TEM and X-ray microtomography, which suggested a denser surface layer.
First, it should be pointed out that, as the cell size increases, the contribution of the surface layer decreases because its area is inversely proportional to cell size. Therefore, it is not surprising to obtain E2 > E1 with our model. This observation demonstrates that the cell size parameter is coupled to another parameter, i.e., the relative amount of surface material (of Young’s modulus E1) with respect to the inner material (of Young’s modulus E2). That is why we observed such a significant dependence of the foam’s Young’s modulus on cell size. Secondly, the TEM and X-ray microtomography results suggested a denser electronic structure, which does not necessarily mean that the material is tougher. Several hypotheses can be proposed to account for this value of the E2/E1 ratio: - As mentioned at the end of Section 3, due to the initial localization of the initiator in the dispersed phase of the emulsion, polymerization starts from the surface layer and propagates inside the polymer walls. As a consequence, the concentration of radical initiator decreases as a function of the distance from the surface, which might result in the polymer chains being shorter in the surface layer. In the case of low molecular weight, Young’s modulus is expected to decrease as chain length decreases [24,25]. - In an earlier study, Williams et al. [26] showed that the Young’s modulus of polymer foams decreases as the surfactant concentration increases. Thus, one possible explanation for the lower value of Young’s modulus for the surface layer would be a surfactant concentration gradient, i.e., that an excess of surfactant is present close to the surface. Indeed, due to their amphiphilic nature, surfactants are naturally attracted by the interface, likely leading to a concentration gradient from the surface of the layer (where the local concentration is high) toward the inner part of the cell wall (where the concentration is lower). However, demonstration for the proposed hypothesis would require a deeper analysis of the materials using very precise tools and complex experiments, and, currently, we do not have any experimental proof of these two possible effects. In conclusion, this study showed that the function f is strongly dependent on the moduli ratio. The influence of the density ratio is less significant. Furthermore, it is necessary to consider (E2/ E1) > 1 to obtain agreement between our theoretical predictions and the experimental results, i.e., to obtain a Young’s modulus (and, as a consequence, the function f) that increases as the cell diameter increases. Each strut of the cell structure can be considered as a sandwich structure, but, in our case, the more rigid material constituted the core of the structure and not the external
Fig. 11. Influence of the moduli ratio and density ratio on the function f. (a) Evolution of f as a function of the cell diameter for different values of E2/E1 and for e = 0.15 and q2/ q1 = 1. (b) Evolution of f as a function of the cell diameter for different values of E2/E1 and for e = 0.15 and q2/q1 = 0.75.
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sheets as is usually encountered in various applications [1]. This theoretical development was completed by numerical simulations of a representative porous volume. 4.3. Numerical simulations 4.3.1. Representative volume In this paragraph, our application of numerical modeling using the finite element method is discussed. The spatial discretisation of an experimental cylindrical sample that has a volume of 10 cm3 and that contains spherical porosities with 10-lm radii is inconceivable due to the computational time and memory required. Therefore, numerical samples were established to obtain a representative volume of the foam structure with a reduced number of spherical porosities. The locations of the porosities in the sample cube were determined in order to obtain symmetrical boundary conditions. Therefore, a face-centered cubic (FCC) packing was chosen for the localization of the porosities, as shown in Fig. 12. The geometrical parameters of the finite element simulation can be defined as the length, a, of the representative volume; the size of the porosity, l = 2R; the thickness of the wall material, e; and the solid fraction (or relative density), us. The size, a, of the numerical sample was defined using geometrical considerations in order to obtain a sample with a multiple of the porosity size. The length, t, was determined using Eq. (10). The geometrical characteristics of the numerical and analytical samples are summarized in Table 1. Considering the symmetrical boundary assumptions, the faces, (x = 0, x = a) and (z = 0, z = a), are fixed. The displacement imposed to simulate the experiments occured along the y-axis with no variation in the x and z directions (Fig. 12). 4.3.2. Numerical modeling The cell struts consist of two materials (Fig. 12). Poisson’s ratios were chosen to be identical (i.e., m = 0.3) for the two materials, but two different values of elastic modulus were considered. Then, we fixed the values of Young’s moduli of the two materials with respect to the values of a classical, dense, polystyrene material weighted by two coefficients, a1 and a2, the values of which are given in Table 1 for the different cases studied. For example, we used E1 = a1Es and E2 = a2Es (with Es = 1400 MPa). Samples were discretised into 300,000 linear tetrahedron elements with six degrees of freedom per node. Calculations with
Table 1 Input data for the analytical and numerical models related to the study of the influence of the cell size. 2R (lm)
us
e (lm)
l (lm)
t (lm)
a (lm)
a1
a2
1.8 5.0 7.8 14.0
0.25 0.25 0.25 0.25
0.15 0.15 0.15 0.15
1.8 5.0 7.8 14.0
0.26 0.72 1.13 2.02
2.54 7.04 10.98 19.71
0.1 0.1 0.1 0.1
1 1 1 1
Table 2 Input data for the analytical and numerical models related to the study of the influence of the relative density. 2R (lm)
us
e (lm)
l (lm)
t (lm)
a (lm)
a1
a2
7.0 7.0 7.0 7.0 7.0
0.10 0.15 0.20 0.25 0.30
0.15 0.15 0.15 0.15 0.15
7.0 7.0 7.0 7.0 7.0
0.64 0.78 0.90 1.01 1.11
9.09 9.38 9.63 9.86 10.09
0.1 0.1 0.1 0.1 0.1
1 1 1 1 1
more elements were performed to ensure that the mesh did not influence the results. All calculations were performed using LSDyna software [27]. A negative vertical displacement of 2.0 104 mm along the y-axis was imposed on the face, y = a, of the cube, and the face, y = 0, was fixed so that no y-displacement occurred. Numerical results were compared using the compressive elastic moduli obtained by:
E ¼
aF F ¼ ; a2 Da a Da
where F and Da are the nodal force and the nodal displacement of the face, y = a, respectively. 4.4. Analytical and numerical results The analytical calculations and numerical simulations were achieved for evaluating the influences of cell size and volume fraction using the parameters defined in Tables 1 and 2, respectively. The first simulations were performed with different values of q2/ q1 and Young’s moduli E1 = a1Es and E2 = a2Es (with Es = 1400 MPa). The density ratio, q2/q1, was chosen to be equal to 0.75 (in agreement with the experimental estimation), and the moduli ratio, E2/ E1, was fixed arbitrarily to 0.1 since, as discussed earlier, the Young’s moduli cannot be identified experimentally. All the calculations were obtained with the same set of values for the thickness, e (150 nm, estimated from TEM observations), density, and moduli ratios. 4.4.1. Influence of the cell size The influence of cell size was studied for a constant solid fraction, us, defined as the ratio q⁄/q with a value of 0.25, where q⁄ is the density of the foam, and q is the average density of the solid wall. The lengths of the edge of the Gibson–Ashby cell, l = 2R, were fixed at the experimental values of 1.8, 5.0, 7.8, and 14 lm. The experimental results and the analytical and numerical predictions are compared in Fig. 13. As expected, the Gibson–Ashby model (red curve1, diamond dot, Fig. 13) cannot account for the increase in the elastic modulus as a function of cell size that was established experimentally (blue curve, square dot). By contrast, when material 1 consisted of a soft wall surrounding a more rigid core, the global elastic modulus of the foam deduced from analytical
Fig. 12. Face-centred cubic packing used geometrical modeling with two materials, material 1 (wall) and material 2 (core).
1 For interpretation of color in Figs. 1–14, the reader is referred to the web version of this article.
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law was 2.64 for the experimental results, 2.79 for the results obtained with the modified model, and 2.0 for the Gibson–Ashby model. Again, in this case, the modified Gibson–Ashby model improved the description of the real macroscopic response of the open-cell foam. The numerical simulation was less efficient at predicting the mechanical response of the foam. The power coefficient of the numerical results was 2.96. We believe that this disagreement comes from the dependence of the calculations on the arrangement of the cells in the representative volume that was modeled. For the previous simulation, an FCC packing was used. Other calculations were performed using a body-centered cubic (BCC) packing for comparison with the FCC packing. The power coefficient decreased to 2.03. This result shows that the power law is sensitive to the choice of packing. In a random stacking of spheres, the packing is known to be denser than BCC packing and less compact than FCC packing. Fig. 13. Plots of the elastic modulus versus the size of porosities for the open-cell foam under compression.
and numerical calculations did increase as cell size increased. The analytical and numerical modeling (green, down triangle dot, and brown, right triangle dot curves, respectively) describe the actual macroscopic behavior of the material.
4.4.2. Influence of relative density The influence of relative density (the solid fraction) of the foam on the global Young’s modulus of the foam was studied using l = 2R (7 lm). For this part of our study, the geometrical characteristics are summarized in Table 2. The experimental results and the analytical and numerical predictions are presented in Fig. 14. A power law of the type E ¼ A um s was applied to fit each result in the density range that was considered. The values of the m coefficient of the power law are listed in Table 3. The elastic modulus of the foam increases as the solid fraction increases. However, differences in the results can be observed in Fig. 14. The coefficient m of the regression
Fig. 14. Plots of the elastic modulus versus the solid fraction for the open-cell foam under compression.
Table 3 Power coefficient m of the function. Result
Power coefficient m
Experiments Gibson–Ashby model Modified Gibson–Ashby model Numerical FCC packing Numerical BCC packing
2.64 2.00 2.79 2.96 2.03
5. Conclusions In this paper, the elastic behavior of an open-cell foam was studied as a function of the relative density and the size of the interconnected, spherical porosities. A chemical procedure was established to produce polystyrene open-cell foam of different relative densities, in the range of 10–30%, with controlled cell diameters ranging from 1.8 lm to 14 lm. The objective was to evaluate the effect of cell diameter on the mechanical response of the open-cell foam. Compression tests were conducted on all of the foam samples, and the results highlighted unexpected influences of relative density and cell diameter on the elastic behavior of the foam. Initially, an analysis of the microstructure of the foam was conducted. The analysis of the images (obtained by SEM, TEM, and microtomography) of the microstructure confirmed that good control of the cell size during foam processing had been achieved. At the finest scale, this analysis highlighted that the edges of the cells are constituted of two materials, i.e., a material constituting the skin at the surface of the porosities and a core material located in the center of the edges. Thus, while the relative density and the cell size of the materials could be varied independently, the variation of cell size led to a change in the relative proportion of the surface/volume materials in the cell walls. Based on these observations, the Gibson–Ashby model was modified by taking into account the specific nanostructure observed in the cross section of the edge of the cells. The new formulation considered that the elastic modulus of the multi-scale, opencell foam is given by the classical Gibson–Ashby formula weighted by a function that depends on the geometrical characteristics of the nanostructure and on the elastic moduli of the two materials that comprise this nanostructure. The analytical results obtained with this new formulation were in good agreement with the experimental data and suggested a softer skin and a more rigid core. While the difference in mechanical behavior between the two zones could not be measured directly, we proposed several hypotheses as possible explanations for the observed heterogeneity. The numerical simulations that were developed confirmed the composite nature of the effective synthesized materials. A numerical, representative volume of the open-cell foam was modeled by inserting the specific nanostructure of the cell edges. The same conclusions were established, i.e., the numerical results were in good agreement with the experimental data if the elastic modulus of the core were greater than the elastic modulus of the skin. Both analytical and numerical approaches showed that the macroscopic elastic response of the foam depended essentially on the specific nanostructure of the open-cell foam. In conclusion, the developed model, which takes into account the characteristics of the nanostructure of the edge of the cells, can describe the
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elastic behavior of this multi-scale, open-cell foam as a function of cell diameter and relative density. Acknowledgments We acknowledge the Swiss Light Source (SLS) at the Paul Scherrer Institut for allowing us to use its synchrotron radiation facilities. We acknowledge the French National Research Agency (ANR) for providing financial support for this project under Grant LISTRAC No: BLAN07-3_191646 We thank the two anonymous reviewers who made significant, helpful improvements to the final version of the paper. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge University press; 2001. [2] Mills NJ, Fitzgerald C, Gilchrist A, Verdejo R. Polymer foams for personal protection: cushions, shoes and helmets. Compos Sci Technol 2003;63:2389–400. [3] Maiti SK, Gibson LJ, Ashby MF. Deformation and energy absorption diagrams for cellular solids. Acta Metal 1984;32:1963–75. [4] Dam CQ, Brezny R, Green DJ. Compressive behavior and deformation-mode map of an open cell alumina. J Mater Res 1990;5:163–71. [5] Brezni R, Green DJ. The effect of cell size on the mechanical behavior of cellular materials. Acta Metall Mater 1990;38:2517–26. [6] Doroudiani S, Kortschot MT. Polystyrene foams. III. Structure-tensile properties relationships. J Appl Polym Sci 2003;90:1427–34. [7] Yu H, Guo Z, Li B, Yao G, Luo H, Liu Y. Research into the effect of cell diameter of aluminium foam on its compressive and energy absorption properties. Mater Sci Eng A 2007;454–455:542–6. [8] Bin J, Zejun W, Naiqin Z. Effect of pore size and relative density on the mechanical properties of open-cell aluminium foams. Scripta Mater 2007;56:169–72. [9] Gupta N, Ye R, Porfiri M. Comparison of tensile and compressive characteristics of vinyl ester/glass microballoon syntactic foams. Composite Part B Eng 2010;41:236–45. [10] Gong L, Kyriakides S, Jang WY. Compressive response of open-cell foams. Part I: morphology and elastic properties. Int J Solid Struct 2005;42:1355–79. [11] Viot P, Shankar K, Bernard D. Effect of strain rate and density on dynamic behaviour of syntactic foam. Composite Struct 2008;86:314–27.
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[12] Viot P, Chirazi A, Dumon M, Bernard D, Fascio V. Ex situ study of polymeric syntactic foams’ mechanical response under compression loading: effects of foam microstructure using microtomography techniques. Adv Mater Res 2011;146–147:42–62. [13] Silverstein MS, Tai H, Sergienko A, Lumelsky Y, Pavlovsky S. PolyHIPE: IPNs, hybrids, nanoscale porosity, silica monoliths and ICP-based sensors. Polymer 2005;46:6682–94. [14] Zhang S, Chen J, Perchyonok VT. Stability of high internal phase emulsions with sole cationic surfactant and its tailoring morphology of porous polymers based on the emulsions. Polymer 2009;50:1723–31. [15] Cameron NR. High internal phase emulsion templating as a route to welldefined porous polymers. Polymer 2005;46:1439–49. [16] Cameron NR, Sherrington DC. High internal phase emulsions (HIPEs) – structure, properties and use in polymer preparation. Biopolym Liquid Cryst Polym Phase Emulsion, Book Ser: Adv Polym Sci 1996;126:163–214. [17] Williams JM, Wrobleski DA. Spatial distribution of the phases in water-in-oil emulsions. Open and closed microcellular foams from cross-linked polystyrene. Langmuir 1988;4:656–62. [18] Mabille C, Leal-Calderon F, Bibette J, Schmitt V. Monodisperse fragmentation in emulsions: mechanisms and kinetics. Europhys Lett 2003;61: 708–14. [19] Ceglia G, Maheo L, Viot P, Bernard D, Chirazi A, Ly I, et al. Formulation and mechanical properties of emulsion-based model polymer foams. Eur Phys J – E 2012;35(4):31. [20] Gibson LJ, Zhang J, Shercliff TL. Failure surface for cellular materials under multiaxial loads – I. Modeling. Int J Mech Sci 1989;31(9):635–63. [21] Zhu H, Knott J, Mills N. Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. J Mech Phys Solids 1997;45:319–43. [22] Li K, Gao XL, Subhash G. Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams. J Mech Phys Solids 2006;54(4):783–806. [23] Kraynik A, Nielsen M. Elastic behavior of cellular solids in Buschow KHJ. In: Cahn RW, Flemings MC, Merton C, Ilschner B, Kramer EJ, Mahajan S, editors. Encyclopedia of Materials-Science and Technology. Elsevier; 2001. p. 2387–9. [24] de Gennes PG. Scaling concept in polymer physics. London: Cornell University Press; 1979. [25] Doi M, Edwards SF. The theory of polymer dynamics. Oxford: Clarendon Press; 1986. [26] Williams JM, Gray AJ, Wilkerson MH. Emulsion stability and rigid foams from styrene or divinylbenzene water-in-oil emulsions. Langmuir 1990;6: 437–44. [27] Hallquist O. LS-Dyna keyword user’s manual version 971. Livermore Software Technology Corporation; May 2007.
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Elastic behavior of multi-scale, open-cell foams Laurent Maheo a,b,c, Philippe Viot a,b,c,⇑, Dominique Bernard d,e, Ali Chirazi d,e, Gaétane Ceglia f,g, Véronique Schmitt f,g, Olivier Mondain-Monval f,g a
Arts et Metiers ParisTech, I2M, UMR 5295, F-33400 Talence, France CNRS, I2M, UMR 5295, F-33400 Talence, France c University of Bordeaux, I2M, UMR 5295, F-33400 Talence, France d CNRS, ICMCB, UPR 9048, F-33600 Pessac, France e University of Bordeaux, ICMCB, UPR 9048, F-33600 Pessac, France f CNRS, CRPP, UPR 8641, F-33600 Pessac, France g University of Bordeaux, CRPP, UPR 8641, F-33600 Pessac, France b
a r t i c l e
i n f o
Article history: Received 5 January 2012 Received in revised form 29 May 2012 Accepted 6 June 2012 Available online 15 June 2012 Keywords: A. Foams B. Elasticity B. Microstructures C. Computational modeling Image analysis
a b s t r a c t The mechanical properties of cellular materials are still subject to numerous theoretical and experimental investigations. In particular, the impact of cell size on the foam’s elastic response has not been studied systematically mainly due to the lack of experimental techniques with which the cell size and relative density of materials can be varied independently. This paper presents the results of a study of the elastic behavior of open-cell foams as a function of relative density and the size of the interconnected, spherical pores. First, the chemical procedure allowed us to produce polystyrene open-cell foams in which the relative density and the average cell diameters were varied independently. The results of compression tests performed on these foams showed an unexpected influence of the cell diameter (at constant relative density) on the elastic response. The analysis of the microstructure of the foam revealed the presence of a complex nanostructure in the edge of the cells that appeared during the synthesis procedure. An analytical model (an extension of the Gibson–Ashby model) is presented, which takes into account the complex multi-scale structure of the foam and accurately describes the observed dependence of the measured Young’s moduli on cell size. This approach was confirmed further by a finite element numerical simulation. We concluded that the observed dependence of elastic modulus on cell size was due to the heterogeneous nature of the material that constitutes the walls of the cells. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The mechanical behavior of cellular materials is vitally important for their use in a wide variety of applications that take advantage of their ability to dissipate energy during compression. This property is affected by the structure of the foam and by the type of material that constitutes the walls of the cells. Because of the industrial importance of such materials, the relationship between their structure and mechanical properties is an ongoing topic of intense research [1,2]. The two main structural parameters of a foam are its relative density and its average cell size. The classical model developed by Gibson and Ashby [1] leads to a Young’s modulus that exhibits a square dependence with the relative density and ⇑ Corresponding author at: Arts et Metiers ParisTech, I2M-DuMAS, esplanade des Arts et Metiers, F-33405 Talence, France. E-mail addresses: [email protected] (L. Maheo), [email protected] (P. Viot), [email protected] (D. Bernard), [email protected] (A. Chirazi), [email protected] (G. Ceglia), [email protected] (V. Schmitt), [email protected] (O. Mondain-Monval). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.06.006
no dependence on cell size. The effect of foam density is well established [3], but there have been few studies related to the influence of the cell size at constant density [4–8]. This lack of data is due to the inherent difficulty associated with the synthesis of foams for which the cell size can be controlled over a sufficiently large range. From the data that have been obtained, it also is difficult to determine a general trend for the effect of cell size. Several authors [5,6] have indicated that mechanical response is independent of cell size and that density is the main parameter. Interestingly, three studies [4,7,8] referred to the existence of a dependence of Young’s modulus on the cell size, but their data were restricted to foams with millimetric cells that vary from 1 to 4 mm. Dam et al. [4] showed that the compressive properties of aluminium foam increased as a function of cell diameter. They attributed (without further insights) this dependence to a change in the microstructure of the solid struts. Gupta et al. [9] obtained similar results for syntactic foams; the cells are hollow, microspheres of a glass or epoxy balloon surrounded by a polymer matrix. Gong et al. [10] manufactured anisotropic, open-cell foams with several cell sizes and anisotropic cells. First, they studied the relationship between cell
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size and anisotropy, and they found that it was difficult to change a parameter independently of the others. Second, they showed that the cell size had no influence on the elastic modulus in the longitudinal direction; however, the elastic modulus in the transverse direction increased when the cell size decreased. Numerical simulations confirmed the results of the experiments. In their study of syntactic foams, Viot et al. [11,12] did not detect any significant effect of the diameter or the thickness of the hollow spheres, and they found that density was the most influential parameter. Thus, it appears that the influence of the cell size on the mechanical response of a cellular material is not fully understood and may depend on other variables, such as the constitutive materials and the microstructure. Very few experimental studies have linked the evolution of elastic response to the open-cell morphology of the foams. Some authors [13–15] showed the existence of nanoscale porosities in the cell walls of polymerized, high-internal-phase emulsions (polyHIPEs) [16,17], but they did not study the influence of the struts microstructure or cell size on Young’s modulus because it is very difficult to synthesize structured materials with precise control of the morphology of the foam. In this study, the elastic behavior of open-cell foams was studied as a function of cell size at constant chemical composition and relative density. The initial objective of this study is to correlate the mechanical response of open-cell foams with the calibrated diameter of the porosities. The cell size can be tuned through the polymerization of calibrated, high-internal-phase emulsions (referred to as polyHIPEs). In our materials, size change is accompanied by a change in the microstructure of the cells. The chemical procedure used to obtain calibrated cell foams generates a nanoscopic structure in the edges of the connected pores. Thus, we will show that the multiscale microstructure of the foam must be taken into account in the modeling of its mechanical properties. Finally, we will demonstrate that our materials behave mechanically as composites, with two parts that have different Young’s moduli. We will present a model that can describe the observed dependence of the size of the cells and strut morphology on the compression modulus. The paper is organized as follows: (i) first, we present a brief description of the preparation of the polymer foams that have controlled density and cell size; (ii) then, we present the results of our measurements of the elastic response of the materials as we varied those two parameters independently; (iii) next, we present our characterization of the foam microstructures using Transmission Electron Microscopy (TEM) and Scanning Electron Microscopy (SEM) images, as well as X-ray computed micro tomography (XCMT), and show that the strut structures are different at the surface and inside the strut volume; and (iv) two approaches are proposed to describe the results, i.e., (1) an analytical model (which is an extension of the Gibson–Ashby model to the case of a composite material) and (2) a finite element simulation that clarifies the reason that cells size affects the mechanical properties of the foams.
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composed of styrene (ST) monomer, and a divinylbenzene (DVB) crosslinker in various proportions in the presence of sorbitan monooleate (Span 80) surfactant. This recipe has been described extensively in the literature [17]. In the second stage (Fig. 1b), quasi-mono dispersed emulsions were obtained by shearing the poly-dispersed emulsions within a Couette device made of two concentric cylinders separated by a narrow, 200-lm gap (Ademtech, France). The size of the final droplets was imposed by the shear rate, the interfacial tension, and the ratio of the viscosity of the dispersed phase to the viscosity of the continuous phases [18]. In the present case, emulsions with an average droplet diameter ranging from 0.5 to 14 lm were obtained. The average size and poly-dispersity of the droplets were measured using static light scattering (SLS). For each droplet size, the brine volume fraction and the span 80 concentration in the organic phase can be fixed by progressive dilutions without altering the size distribution of the droplets, which allowed independent control of the sizes and densities of the foam cells to be generated later. Polymerization (stage 3) of the continuous phase composed of ST/DVB/Span80 was initiated by increasing the temperature to 60 °C (Fig. 1c). At this temperature, the initiator (potassium persulfate) in the dispersed phase decomposes. Radical polymerization depends on the surfactant concentration (5 days for 10% surfactant to 7 days for 30% surfactant). During polymerization, solidification and a slight shrinkage occurred simultaneously, leading to a solid monolith with a morphology that was determined by the mold. Even for such a low shrinkage rate, the film between adjacent drops breaks, forming connections that lead to the final, open porosity (referred to as open-cell morphology as opposed to closed-cell morphology). The resulting foams were washed several times in ethanol (24 h) and acetone (8 h) in order to remove the remaining emulsifiers, monomers, cross linkers, initiators, and stabilizers. No size variation of the monoliths could be detected using a vernier calliper, which showed that the washing steps did not significantly affect the materials at the scale of 0.1 mm. Finally, the foams were dried at room temperature (Fig. 1d). More details on the preparation and the structural characterization procedures will be published elsewhere [19]. The materials were obtained as homogeneous monoliths that showed no defect at the scale attainable by X-ray radiography with a resolution of 22 lm. Thus, the monoliths exhibited no fractures and no porosity gradients at this scale. Of course, each sample was synthesized several times to allow for the use of intrusive characterization techniques (mercury porosimetry, SEM, TEM, X-ray microtomography, and mechanical characterization) and to check the reproducibility. Mercury porosimetry was used to measure the solid fraction for each kind of samples. The values obtained were in good agreement with those deduced from the initial quantities of the organic and aqueous phases in the emulsions. 2.2. Mechanical characterization
2. Materials 2.1. Preparation of the foam The preparation of cellular material of controlled cell size is based on the formulation and polymerization of a calibrated, high-internal-phase emulsion (PolyHIPE). Calibrated, open-cell foams with controlled cell sizes and porosities were obtained in four steps, as shown in Fig. 1. The first stage of the process (Fig. 1a) was the preparation of a rough, poly-dispersed emulsion obtained by incorporating brine (an aqueous phase containing 0.5 M NaCl) and the radical initiator, 0.01 M potassium peroxodisulfate (KPS), into an organic phase
Mechanical tests were conducted on a Zwick electromechanical testing machine that had a force threshold of 250 kN and a velocity limit of 600 mm/min. A force sensor with a range of 10 kN was used for the compression tests of the cellular materials, and the displacements imposed on the specimens were determined from the punch displacement, since the rigidity of the machine was considered to be infinite in comparison to the stiffness of the porous specimen that was being tested. Since the aim of the experiment was to characterize the elastic modulus, it was sufficient to calculate the strain using the ‘‘empirical’’ expression: e ¼ hdh0 (with dh being the displacement of the punch) since the difference between the ‘‘testing-machine-based’’ strain and the true strain was
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Fig. 1. Four stages of the PolyHIPE foam preparation: (a) rough emulsion, (b) calibrated emulsion obtained after a controlled shear, (c) calibrated polymerized emulsion, and (d) final dried polymerized calibrated porous material.
negligible for low values of strain. The stress, r, was calculated from the measured force, F, per unit area, using the initial section, S0. The testing indicated that the variation of the section during compression was not significant, and the resulting error imposed on the stress calculation by using S0 was negligible. Samples with well-controlled morphologies (cylinders of initial height, H0 = 40 mm, and diameter, D = 25 mm) were compressed with an imposed displacement rate of 10 mm/min of the lower punch. Previous compression tests confirmed that the results were independent of the displacement rate in the range of 1–100 mm/ min. The initial strain rate was 4.2 103 s1. The behavior of the cellular material was studied for cell diameters of 1.8, 5, 7, 7.8, and 14 lm and for different relative densities or solid fractions, us, of 10, 15, 20, 25, and 30%. Past studies have shown the significant edge effects due to partially-cut cells at the boundaries of the samples. In our case, the volume of the sample was close to 20 103 mm3, whereas the volumes of the spherical cells were less than 1.5 106 mm3. This significant difference in the volumes allowed us to assume that the number of cut cells was not significant compared to the total number of cells in the entire sample volume. Therefore, the edge effects can be considered to be negligible. For each set of parameters, all measurements were performed in triplicate in order to check their reproducibility. The elastic modulus of the cellular material under compression was calculated using the linear dependency of the stress with respect to strain, i.e., E = re.
2.2.1. Influence of the density of the foam The influence of the density of the foam was explored while keeping the cell size constant. The mechanical response of the foam can be divided into two distinct regimes, as shown in Fig. 2. The first regime is the elastic regime, which exists until
Fig. 2. Stress–strain curves showing the effect of the solid fraction on the mechanical behavior (for a cell diameter of 7 lm).
the stress becomes equal to the yield stress. The second regime is a stress plateau that is limited by the fragile nature of the sample. The transition between the elastic behavior and the stress plateau was particularly smooth when the density of the foam was high. One can note that the failure strain decreased as the density of the foam decreased. All of the elastic moduli that were measured are given in Fig. 3. The experimental dependency of the elastic modulus on solid fraction, us, can be fitted with the power law. However, in our case, the measured exponent (2.5) was larger than the exponent that Gibson and Ashby predicted, which has been confirmed experimentally by several authors [1,3].
2.2.2. Influence of the size of the cells in the foam The influence of the diameter of the cells on the mechanical response of the foam was established for various imposed foam densities. The macroscopic behavior of the foam depends on the cell size. For the lowest cell diameter (i.e., for diameters of 1.8 lm, Fig. 4), the mechanical behavior was consistent with the classical mechanical response subdivided into the elastic domain and the stress plateau, where the stress increases slightly as a function of strain. For cells with larger diameters, the mechanical behavior of the foam was mostly of the fragile type, and the length of the stress plateau decreased drastically for cell diameters as large as 14 lm. The elastic modulus is given as a function of cell size for various values of the solid fraction us. Fig. 5 shows the typical dependency of the elastic modulus on cell size for a solid fraction of 25%. The elastic modulus increased as cell size increased, in sharp contrast with the Gibson and Ashby model, which predicts no dependency of cell size on the elastic
Fig. 3. Variation of the elastic modulus as a function of solid fraction for a material with a cell size of 7 lm.
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microtomography were in good agreement with the sizes obtained with SLS and SEM. Again, this provides evidence that the structure
Fig. 4. Stress–strain curves showing the effect of the cell diameter on the behavior of the foam (for materials with a solid fraction of 25%.
Fig. 6a. SEM picture of the structure of an open-cell foam.
Fig. 5. Variation of the elastic modulus as a function of cell diameter for materials with a solid fraction of 25%.
modulus. Thus, our results have identified an unexpected effect of cell diameter on the elastic behavior of the foams. As suggested by Dam et al. [4], such an effect might be due to a change in the strut microstructure as the cell size changes. To check this hypothesis, we performed a full micro-structural characterization of the foams using 2D and 3D imaging techniques. This work is described in the next section.
Fig. 6b. Micrograph reconstruction of a part of a foam sample (cell diameter 14 mm).
3. Microstructural characterization using imaging techniques The micro and nano structures of open-cell foams were characterized using scanning electron microscopy (SEM, Fig. 6a), transmission electron microscopy (TEM, Fig. 7), mercury porosimetry, and X-ray microtomography. The average cell size obtained via SEM was almost equal to the average cell size of the initial emulsion droplets given by Static Light Scattering (SLS) and optical microscopy observations [19], confirming that no significant changes occurred during the polymerization of the emulsion. Microtomographic measurements were conducted on the TOMCAT line of the Swiss Light Source (SLS) using a beam energy of 15.5 keV. The 1200 radiographs that were acquired for each scanned sample had 2048 2048 pixels, with a pixel size of 0.37 lm. Pre- and post-processing noise reduction algorithms applied to radiographies and reconstructed volumes, respectively, were used to enhance the quality of the images. These 3D reconstructed volumes (Fig. 6b) were used to estimate and control the diameters of the cells of the calibrated foams. The cell sizes measured using
Fig. 6c. Micrograph section of a part of a foam sample (cell diameter 14 mm).
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Fig. 6d. Zoom of the square part of the left image.
of the sample did not change during polymerization. This means that controlling the initial size of the emulsion droplets allows control of the final pore size of the solid material that is obtained. As seen in the 2D cross section of the 3D reconstructed volumes (in which the polymer zones are gray and the void cells are black (Figs. 6c and 6d), some minor gray level variations can be observed within the struts, i.e., along the white lines drawn in Figs. 6c and 6d. The corresponding gray level profiles of these lines (AB, CD, and EF) are plotted in Fig. 8a. This analysis could only be performed on the samples that had the larger cell diameter (14 lm) since the acquisition precision of 0.37 lm/pixel and the level of background noise did not allow any observable gray-level variability within the struts section of the samples that had small cell sizes. The line histograms plotted in Fig. 8a within the struts section (lines AB, CD, and EF, Fig. 6c) confirmed the existence of a gray level gradient inside the polymer struts. As seen in the line histogram shown in Fig. 8a, values close to 60 correspond to the void, while values around 100–120 relate to the struts section in which the iso-curves closer to the cell wall have slightly higher gray levels. Furthermore, for line histograms AB and EF shown in Fig. 6c, two maxima per histogram can be observed.
Fig. 7b. Zoom of the square part of the left image.
Fig. 7c. TEM Picture for a cell size of 7 mm.
These local observable gray level gradients within the struts zone can be related to a minor change in local density near the cell walls. This observation was further confirmed using TEM. To allow for TEM observations, the porous samples first had to be impregnated with a liquid epoxy resin and cut into thin slices using a diamond knife (80 nm thick) before observation using a Hitachi H600 TEM. On the 2D TEM images of Fig. 7, epoxy impregnated zones appear as smooth, light-gray areas, while the polymerized zones appear as quite darker areas. Using the enlarged pictures of Figs. 7b, 7d, and 7f, it appears that the struts indeed present a heterogeneous texture. The polymer/void interfaces appear to be rather rough, whereas the inner part of the struts exhibits a nanoporous structure. Image analysis and quantification techniques were used to quantify nanoscale heterogeneities. The process of nanoscale image quantification applied to the TEM images of various samples is as follows:
Fig. 7a. TEM Picture for a cell size of 1.8 mm.
1. In case there is a non-uniform gray level throughout the picture due to non-uniform intensity during acquisition, the first step consists of equalizing the gray level.
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2. Binarization of the image by thresholding: pixels were classified as void (respectively solid) if their gray level was lower (respectively higher) than the threshold value. This value was selected for each image from the corresponding histogram of the gray levels. 3. Segmentation of the image in two regions: the pores corresponding to the initial emulsion droplets and the polymerized zone. These pores were large (at the scale of the image) and smooth (the imprints of liquid droplets). 4. Successive dilations of the droplet pores by one pixel. At each stage, a portion of the polymerized zone located at the interface was eroded. The porosity of this eroded zone was computed, and the values were plotted as a function of the distance from the initial limit of the droplet pores. (This is the number of dilation steps multiplied by the pixel size.)
Fig. 7d. Zoom of the square part of the left image.
This process could generate variation of the typical local porosity inside the struts as a function of the distance, d, from the polymer/void interface presented in Fig. 8b. The first two points (d < 20 nm) correspond to the roughness of the interface induced by the granular texture. The following plateau (for 25 nm < d < 225 nm) provided evidence for the existence of a very dense layer, the thickness of which can be estimated providing that the 2D limiting effect of a 3D structure sectioning is taken into account
Fig. 7e. TEM Picture for a cell size of 14 mm.
Fig. 8a. Gray level histogram for lines located in strut sections.
Fig. 7f. Zoom of the square part of the left image.
Fig. 8b. Variation of the mean porosity as a function of the distance from the surface of the cell.
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Fig. 10b. Square section of the edge of the modified Gibson–Ashby cell. Fig. 9. Representative cell of Gibson–Ashby model.
for error calculation. After the plateau, the local nanoporosity slowly increases, revealing the nanoporous nature of the internal part of the polymerized zone. Such an internal structure of the cell wall is similar to the structure observed by Silverstein et al. [13] in polyHIPEs polymerized in the presence of a porogen (non-polymerizable solvent) in the continuous phase of the initial, highinternal-phase emulsions (HIPE). These observations of micro and nano structures allow us to define open-cell foams with cell walls that are composed of a bi-layered structure, as sketched in Fig. 10. A first material (material 1) located at the surface of the cell surrounds a second material (material 2), which is localized mainly in the center of the edge. Then, it can be considered that struts are constituted of an outer layer (material 1) characterized by a thickness, e, and a core (material 2). Of course, in the real material, the boundary between the two zones is not sharp, and the transition between the two zones is rather smooth. A systematic estimation of e by varying the cell size and the solid volume fraction did not provide evidence of any influence of the cell size on the thickness e, which we estimated to be 150 ± 40 nm from statistical analyses that were conducted on several samples. Furthermore, the difference of texture in the cross section of the edges seemed to indicate a density contrast, revealing a difference in the nature of the material between the surface and the inner part of the struts (Figs. 7a and 8b). The average density can be determined, but it is very difficult to estimate a real value of the density of each material. In a first approach, we considered two different densities, i.e., q1 and q2, for materials 1 and 2, respectively. In the modeling section, the influence of the local densities on the global elastic response is estimated. Obviously, it also is very difficult to directly probe the chemical characteristics and the mechanical properties (using nano indentation) of these two materials at the nano scale.
Fig. 10a. Modified Gibson–Ashby cell with two materials that constitute the cellbeams.
At this point of the manuscript, we wish to discuss possible explanations for the nanoporosity and heterogeneity of the cell wall that we observed. First, it appeared that the presence of this nanoporous structure was correlated strongly with the concentration of the surfactant in the continuous phase of the initial emulsion. The texture of the polymer part of samples that were acquired using a lower surfactant concentration was much smoother, with almost no density gradient observed on the TEM pictures (not shown here). Another possible explanation for the nanoporosity could be that the material swelled when it was washed with the ethanol and acetone solvents. Such an effect would explain the appearance of the porosity inside the cell wall, but it would not explain the behavioral differences between the surface and the inner parts of the struts. Furthermore, no macroscopic variation of the samples before and after the washing cycles could be deduced from a careful measurement of the monolith dimensions. A more likely explanation is that, because the polymerization initiator was localized initially in the water droplets of the initial emulsion, polymerization was triggered at the interface and propagated inside the sample, thus leading to a gradient in the degree of polymerization of the sample, different average chain lengths, and perhaps in the degree of crosslinking. 4. Proposed analytical and numerical models 4.1. Theoretical background Constitutive models for cellular materials already have been investigated by several authors. Gibson and Ashby [20] studied the behavior of polyurethane, polyethylene, and aluminium foams (closed or open cells) and established constitutive equations based on the analysis of the mechanical response of an ideal structure of a foam. Some more complex models using tetrakaidecahedral cells [21] or random, mono-dispersed foams [22,23] can be used to better represent the real shape of the cells and to evaluate the elastic behavior of the cellular material. However, in a first step, we wanted to check the assumption that the specific microstructure of the struts has a significant effect on the elastic behavior of this foam. For that, we decided to base the analytical modeling of the elastic behavior of the foam on the classical Gibson and Ashby’s model, which was modified to take into account the specific morphology of the foam. In a second step, these results were confirmed with complementary, finite element simulations (Section 4.3). Gibson and Ashby proposed that the mechanical properties of solid foams are described by considering the porous material as a periodic assembly of open cubic cells, constituted of edges of square section, t2, and length, l, (Fig. 9). Adjoining cells are staggered so that their edges meet at their midpoints. Despite the simplicity of the representation, such an approach has been shown to describe the essential features of open-cell foams accurately in different cases [1,3].
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From this description of the microstructure of open-cell foams, the solid fraction us (or the relative density) is the volume of the strut divided by the volume of a representative cell. The solid fraction is related to the two geometrical parameters l and t by:
2 q t us ¼ / l qs
ð1Þ
where q⁄ is the density of the foam, and qs is the density of the material constituting the edge. In the elastic regime, when a uniaxial stress is applied so that each cell edge transmits a force, F, the perpendicular edges (of the force axis) bend, and the deflection, d, of the representative structure is due mainly to the bending deformation of the perpendicular edges. The deflection d can be easily established as: 3
d/
Fl Es I
ð2Þ
where ES is Young’s modulus of the material that constitutes the solid edge, and I is the second moment of the area of a square strut of section t2. Considering a square section for a homogeneous edge (constituted of only one material), the second moment I is proportional to t4. From this deflection d, it is possible to calculate the elastic modulus of the foam, the Young’s modulus being the ratio of the applied stress to the strain. The stress was deduced from the force divided by the area t2 of the square section, and the macroscopic strain of the representative cell was established by dividing the deflection by the initial length of the edge. Taking all these results into account, the elastic modulus is related to the strain by:
r F=l2 F E / ¼ ¼ e d=l d l
ð3Þ
Combining Eq. (3) with Eqs. (1) and (2), Gibson and Ashby established the equation that describes the elastic modulus of an open-cell foam:
E t 4 / / Es l4
q qs
2 ð4Þ
As a consequence, with such a model, no dependence of Young’s modulus on cell size is expected, and only the influence of the relative density can be modeled with a second-order power law. It is worth noting that none of these considerations is valid in our experimental study. 4.2. Modified Gibson–Ashby model Hence, the Gibson–Ashby model must be reconsidered to take into account the microstructure of the edges of the cells, which consist of two distinct materials. From observations of the edge sections, we assumed that a first material (defined by index 1, Fig. 10) was a kind of envelope that surrounded the second material (defined by index 2). A square section of an edge is characterized by its side t and the thickness e of the skin of material 1 (Fig. 10b). The density of the cell, q⁄, and its elastic rigidity, E⁄, must be recalculated by considering the local density, qi, and the elastic modulus, Ei, of each material, i, that makes up the struts of the cellular material. 4.2.1. Constitutive equations The calculation of the actual density of the cell requires taking into account the macroscopic volume fraction at the scale of the cell (Eq. (1)) and considering the nano porosities observed in the edges of the cell. For that purpose, the density of the cell is calculated from the ratio of its mass M and its volume V = l3. It is easy to
obtain the density q⁄ of the cell as a function of the densities q1 and q2, as follows:
q /
t2 l
2
q1 1 1 2
e2 e2 þ q2 1 2 t t
ð5Þ
The relationship (Eq. (1)) of the relative density proposed by Gibson and Ashby in the case of a classical foam can be obtained from Eq. (5) when considering a strut that consists of only one material, i.e., for 2e = t, or if the densities qi of each material i are identical. Following the methodology developed by Gibson and Ashby to describe the behavior of open cells, the elastic rigidity of the foam is deduced from the deflection d of the strut bending under a force F. Then, the elastic modulus is related to the elastic rigidities Ei of constitutive materials i, the second moment I of the area, and the length l of the edge, as follows:
E /
hEIi l
4
with hEIi ¼ E1 I1 þ E2 I2
ð6Þ
Considering the characteristics of the section of the edge (Fig. 10b), the Young’s modulus of the foam reads:
E /
e4 e4 þ E E 1 1 2 1 2 1 2 4 t t l
t4
ð7Þ
Eq. (5) made it possible to determine a general formulation (Eq. (8)) of Young’s modulus that is similar to the relationship proposed by Gibson and Ashby (Eq. (4)):
4 2 1 ð1 2 et Þ4 þ EE21 1 2 et E q / E1 q1 1 1 2 e2 þ q2 1 2 e2 2 t t q
ð8Þ
1
In conclusion, we can consider that the Young’s modulus of this multi-scale, open-cell foam is given by theclassical Gibson–Ashby formula weighted by a function f et ; EE21 ; qq2 , which takes into ac1 count the specific morphology of the cell struts:
2 E q e E2 q2 e E2 q2 with f ; ; ; ; / f t E1 q1 t E1 q1 E1 q1 E2 e 4 e 4 1 12t þ E1 1 2 t ¼ 2 2 2 þ qq2 1 2 et 1 1 2 et
ð9Þ
1
4.2.2. Parameters that influence function f The function f depends on the characteristics of several parameters of the microstructure (e/t and q2/q1) of the struts and the ratio of the elastic modulus E2/E1 of the two constitutive materials. The two geometrical parameters (e and t) can be estimated approximately from the observations of the microstructure. Image analysis led to a thickness e of approximately 150 nm. For this study, the dimension t of the strut section was established from the cell size l and the solid fraction us (Eq. (1)), so that:
rffiffiffiffiffiffi t¼l
us
12
ð10Þ
The value of the dimension t calculated from this expression is close to data that have been presented in the literature [5]. It is more complex to determine the densities q2 and q1 of the two materials. The images of the microstructure (Fig. 7) of the strut section show that the density of the core (material 2) is lower than the density of the material (material 1) close to the surface of the cell. From the data obtained from image analysis, it is reasonable to consider that the ratio q2/q1 varies from 1 to 0.75, but it is impossible to determine the ratio precisely.
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The same problem is noticed for the modulus ratio E2/E1. The elastic modulus of each material cannot be identified. The elastic modulus of polystyrene is close to 1400 MPa, and it is impossible to know whether the elastic behaviors of materials 1 and 2 were modified by the preparation of the foam samples. Some attempts were performed to conduct ‘‘in situ’’ measurements using an Atomic Force Microscope (the so-called nano-indentation technique), but it appeared to be very difficult to deduce any data on the relative values of the Young’s moduli E2 and E1. Now, we evaluate the sensitivity of the function f to the two ratios, i.e., q2/q1 and E2/E1. 4.2.2.1. Influence of the ratio of Young’s moduli. The variation of f was evaluated by considering the ratio E2/E1 as variable and for different values of cell size l. In Fig. 11a, we considered different values of the ratio E2/E1, a density ratio q2/q1 of 0.75, a thickness e of 150 nm and computed f as a function of cell size. Of course, for E2/E1 = 1, we have f = 1, which can be easily established from Eq. (9). The function f increases as a function of cell diameter for (E2/ E1) > 1 (Fig. 11). This variation is particularly significant for (E2/ E1) P 5. The experimental results obtained from the compression of the multi-scale foam showed an increase in Young’s modulus as a function of cell diameter (Fig. 5). As a consequence, the case in which (E2/E1) < 1 cannot describe our results since the function f is decreasing. This means that, within the frame of our analysis, the observed mechanical behavior can be explained only by a material that is softer at the interface than it is inside the wall, which is in apparent contradiction with the TEM observation. Indeed, due to the electronic contrast, we expected a tougher material at the interface, and this point will be discussed further at the end of this section. 4.2.2.2. Influence of the density ratio. The sensitivity of f to the density ratio also was considered by keeping a fixed thickness of e = 150 nm and considering values of the density ratio to be 1 and 0.75 (Fig. 11a and b). On the whole, the variation of f with cell size was found to be much more pronounced when q2q1 was low. However, irrespective of the value of the density ratio, the function f was always an increasing function of the cell size for (E2/E1) > 1. 4.2.2.3. Discussion. Agreement between our experimental results and the model can only be achieved by considering a softer layer of material at the interface, which is in sharp contrast with what was expected from the data acquired by both TEM and X-ray microtomography, which suggested a denser surface layer.
First, it should be pointed out that, as the cell size increases, the contribution of the surface layer decreases because its area is inversely proportional to cell size. Therefore, it is not surprising to obtain E2 > E1 with our model. This observation demonstrates that the cell size parameter is coupled to another parameter, i.e., the relative amount of surface material (of Young’s modulus E1) with respect to the inner material (of Young’s modulus E2). That is why we observed such a significant dependence of the foam’s Young’s modulus on cell size. Secondly, the TEM and X-ray microtomography results suggested a denser electronic structure, which does not necessarily mean that the material is tougher. Several hypotheses can be proposed to account for this value of the E2/E1 ratio: - As mentioned at the end of Section 3, due to the initial localization of the initiator in the dispersed phase of the emulsion, polymerization starts from the surface layer and propagates inside the polymer walls. As a consequence, the concentration of radical initiator decreases as a function of the distance from the surface, which might result in the polymer chains being shorter in the surface layer. In the case of low molecular weight, Young’s modulus is expected to decrease as chain length decreases [24,25]. - In an earlier study, Williams et al. [26] showed that the Young’s modulus of polymer foams decreases as the surfactant concentration increases. Thus, one possible explanation for the lower value of Young’s modulus for the surface layer would be a surfactant concentration gradient, i.e., that an excess of surfactant is present close to the surface. Indeed, due to their amphiphilic nature, surfactants are naturally attracted by the interface, likely leading to a concentration gradient from the surface of the layer (where the local concentration is high) toward the inner part of the cell wall (where the concentration is lower). However, demonstration for the proposed hypothesis would require a deeper analysis of the materials using very precise tools and complex experiments, and, currently, we do not have any experimental proof of these two possible effects. In conclusion, this study showed that the function f is strongly dependent on the moduli ratio. The influence of the density ratio is less significant. Furthermore, it is necessary to consider (E2/ E1) > 1 to obtain agreement between our theoretical predictions and the experimental results, i.e., to obtain a Young’s modulus (and, as a consequence, the function f) that increases as the cell diameter increases. Each strut of the cell structure can be considered as a sandwich structure, but, in our case, the more rigid material constituted the core of the structure and not the external
Fig. 11. Influence of the moduli ratio and density ratio on the function f. (a) Evolution of f as a function of the cell diameter for different values of E2/E1 and for e = 0.15 and q2/ q1 = 1. (b) Evolution of f as a function of the cell diameter for different values of E2/E1 and for e = 0.15 and q2/q1 = 0.75.
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sheets as is usually encountered in various applications [1]. This theoretical development was completed by numerical simulations of a representative porous volume. 4.3. Numerical simulations 4.3.1. Representative volume In this paragraph, our application of numerical modeling using the finite element method is discussed. The spatial discretisation of an experimental cylindrical sample that has a volume of 10 cm3 and that contains spherical porosities with 10-lm radii is inconceivable due to the computational time and memory required. Therefore, numerical samples were established to obtain a representative volume of the foam structure with a reduced number of spherical porosities. The locations of the porosities in the sample cube were determined in order to obtain symmetrical boundary conditions. Therefore, a face-centered cubic (FCC) packing was chosen for the localization of the porosities, as shown in Fig. 12. The geometrical parameters of the finite element simulation can be defined as the length, a, of the representative volume; the size of the porosity, l = 2R; the thickness of the wall material, e; and the solid fraction (or relative density), us. The size, a, of the numerical sample was defined using geometrical considerations in order to obtain a sample with a multiple of the porosity size. The length, t, was determined using Eq. (10). The geometrical characteristics of the numerical and analytical samples are summarized in Table 1. Considering the symmetrical boundary assumptions, the faces, (x = 0, x = a) and (z = 0, z = a), are fixed. The displacement imposed to simulate the experiments occured along the y-axis with no variation in the x and z directions (Fig. 12). 4.3.2. Numerical modeling The cell struts consist of two materials (Fig. 12). Poisson’s ratios were chosen to be identical (i.e., m = 0.3) for the two materials, but two different values of elastic modulus were considered. Then, we fixed the values of Young’s moduli of the two materials with respect to the values of a classical, dense, polystyrene material weighted by two coefficients, a1 and a2, the values of which are given in Table 1 for the different cases studied. For example, we used E1 = a1Es and E2 = a2Es (with Es = 1400 MPa). Samples were discretised into 300,000 linear tetrahedron elements with six degrees of freedom per node. Calculations with
Table 1 Input data for the analytical and numerical models related to the study of the influence of the cell size. 2R (lm)
us
e (lm)
l (lm)
t (lm)
a (lm)
a1
a2
1.8 5.0 7.8 14.0
0.25 0.25 0.25 0.25
0.15 0.15 0.15 0.15
1.8 5.0 7.8 14.0
0.26 0.72 1.13 2.02
2.54 7.04 10.98 19.71
0.1 0.1 0.1 0.1
1 1 1 1
Table 2 Input data for the analytical and numerical models related to the study of the influence of the relative density. 2R (lm)
us
e (lm)
l (lm)
t (lm)
a (lm)
a1
a2
7.0 7.0 7.0 7.0 7.0
0.10 0.15 0.20 0.25 0.30
0.15 0.15 0.15 0.15 0.15
7.0 7.0 7.0 7.0 7.0
0.64 0.78 0.90 1.01 1.11
9.09 9.38 9.63 9.86 10.09
0.1 0.1 0.1 0.1 0.1
1 1 1 1 1
more elements were performed to ensure that the mesh did not influence the results. All calculations were performed using LSDyna software [27]. A negative vertical displacement of 2.0 104 mm along the y-axis was imposed on the face, y = a, of the cube, and the face, y = 0, was fixed so that no y-displacement occurred. Numerical results were compared using the compressive elastic moduli obtained by:
E ¼
aF F ¼ ; a2 Da a Da
where F and Da are the nodal force and the nodal displacement of the face, y = a, respectively. 4.4. Analytical and numerical results The analytical calculations and numerical simulations were achieved for evaluating the influences of cell size and volume fraction using the parameters defined in Tables 1 and 2, respectively. The first simulations were performed with different values of q2/ q1 and Young’s moduli E1 = a1Es and E2 = a2Es (with Es = 1400 MPa). The density ratio, q2/q1, was chosen to be equal to 0.75 (in agreement with the experimental estimation), and the moduli ratio, E2/ E1, was fixed arbitrarily to 0.1 since, as discussed earlier, the Young’s moduli cannot be identified experimentally. All the calculations were obtained with the same set of values for the thickness, e (150 nm, estimated from TEM observations), density, and moduli ratios. 4.4.1. Influence of the cell size The influence of cell size was studied for a constant solid fraction, us, defined as the ratio q⁄/q with a value of 0.25, where q⁄ is the density of the foam, and q is the average density of the solid wall. The lengths of the edge of the Gibson–Ashby cell, l = 2R, were fixed at the experimental values of 1.8, 5.0, 7.8, and 14 lm. The experimental results and the analytical and numerical predictions are compared in Fig. 13. As expected, the Gibson–Ashby model (red curve1, diamond dot, Fig. 13) cannot account for the increase in the elastic modulus as a function of cell size that was established experimentally (blue curve, square dot). By contrast, when material 1 consisted of a soft wall surrounding a more rigid core, the global elastic modulus of the foam deduced from analytical
Fig. 12. Face-centred cubic packing used geometrical modeling with two materials, material 1 (wall) and material 2 (core).
1 For interpretation of color in Figs. 1–14, the reader is referred to the web version of this article.
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law was 2.64 for the experimental results, 2.79 for the results obtained with the modified model, and 2.0 for the Gibson–Ashby model. Again, in this case, the modified Gibson–Ashby model improved the description of the real macroscopic response of the open-cell foam. The numerical simulation was less efficient at predicting the mechanical response of the foam. The power coefficient of the numerical results was 2.96. We believe that this disagreement comes from the dependence of the calculations on the arrangement of the cells in the representative volume that was modeled. For the previous simulation, an FCC packing was used. Other calculations were performed using a body-centered cubic (BCC) packing for comparison with the FCC packing. The power coefficient decreased to 2.03. This result shows that the power law is sensitive to the choice of packing. In a random stacking of spheres, the packing is known to be denser than BCC packing and less compact than FCC packing. Fig. 13. Plots of the elastic modulus versus the size of porosities for the open-cell foam under compression.
and numerical calculations did increase as cell size increased. The analytical and numerical modeling (green, down triangle dot, and brown, right triangle dot curves, respectively) describe the actual macroscopic behavior of the material.
4.4.2. Influence of relative density The influence of relative density (the solid fraction) of the foam on the global Young’s modulus of the foam was studied using l = 2R (7 lm). For this part of our study, the geometrical characteristics are summarized in Table 2. The experimental results and the analytical and numerical predictions are presented in Fig. 14. A power law of the type E ¼ A um s was applied to fit each result in the density range that was considered. The values of the m coefficient of the power law are listed in Table 3. The elastic modulus of the foam increases as the solid fraction increases. However, differences in the results can be observed in Fig. 14. The coefficient m of the regression
Fig. 14. Plots of the elastic modulus versus the solid fraction for the open-cell foam under compression.
Table 3 Power coefficient m of the function. Result
Power coefficient m
Experiments Gibson–Ashby model Modified Gibson–Ashby model Numerical FCC packing Numerical BCC packing
2.64 2.00 2.79 2.96 2.03
5. Conclusions In this paper, the elastic behavior of an open-cell foam was studied as a function of the relative density and the size of the interconnected, spherical porosities. A chemical procedure was established to produce polystyrene open-cell foam of different relative densities, in the range of 10–30%, with controlled cell diameters ranging from 1.8 lm to 14 lm. The objective was to evaluate the effect of cell diameter on the mechanical response of the open-cell foam. Compression tests were conducted on all of the foam samples, and the results highlighted unexpected influences of relative density and cell diameter on the elastic behavior of the foam. Initially, an analysis of the microstructure of the foam was conducted. The analysis of the images (obtained by SEM, TEM, and microtomography) of the microstructure confirmed that good control of the cell size during foam processing had been achieved. At the finest scale, this analysis highlighted that the edges of the cells are constituted of two materials, i.e., a material constituting the skin at the surface of the porosities and a core material located in the center of the edges. Thus, while the relative density and the cell size of the materials could be varied independently, the variation of cell size led to a change in the relative proportion of the surface/volume materials in the cell walls. Based on these observations, the Gibson–Ashby model was modified by taking into account the specific nanostructure observed in the cross section of the edge of the cells. The new formulation considered that the elastic modulus of the multi-scale, opencell foam is given by the classical Gibson–Ashby formula weighted by a function that depends on the geometrical characteristics of the nanostructure and on the elastic moduli of the two materials that comprise this nanostructure. The analytical results obtained with this new formulation were in good agreement with the experimental data and suggested a softer skin and a more rigid core. While the difference in mechanical behavior between the two zones could not be measured directly, we proposed several hypotheses as possible explanations for the observed heterogeneity. The numerical simulations that were developed confirmed the composite nature of the effective synthesized materials. A numerical, representative volume of the open-cell foam was modeled by inserting the specific nanostructure of the cell edges. The same conclusions were established, i.e., the numerical results were in good agreement with the experimental data if the elastic modulus of the core were greater than the elastic modulus of the skin. Both analytical and numerical approaches showed that the macroscopic elastic response of the foam depended essentially on the specific nanostructure of the open-cell foam. In conclusion, the developed model, which takes into account the characteristics of the nanostructure of the edge of the cells, can describe the
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elastic behavior of this multi-scale, open-cell foam as a function of cell diameter and relative density. Acknowledgments We acknowledge the Swiss Light Source (SLS) at the Paul Scherrer Institut for allowing us to use its synchrotron radiation facilities. We acknowledge the French National Research Agency (ANR) for providing financial support for this project under Grant LISTRAC No: BLAN07-3_191646 We thank the two anonymous reviewers who made significant, helpful improvements to the final version of the paper. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge University press; 2001. [2] Mills NJ, Fitzgerald C, Gilchrist A, Verdejo R. Polymer foams for personal protection: cushions, shoes and helmets. Compos Sci Technol 2003;63:2389–400. [3] Maiti SK, Gibson LJ, Ashby MF. Deformation and energy absorption diagrams for cellular solids. Acta Metal 1984;32:1963–75. [4] Dam CQ, Brezny R, Green DJ. Compressive behavior and deformation-mode map of an open cell alumina. J Mater Res 1990;5:163–71. [5] Brezni R, Green DJ. The effect of cell size on the mechanical behavior of cellular materials. Acta Metall Mater 1990;38:2517–26. [6] Doroudiani S, Kortschot MT. Polystyrene foams. III. Structure-tensile properties relationships. J Appl Polym Sci 2003;90:1427–34. [7] Yu H, Guo Z, Li B, Yao G, Luo H, Liu Y. Research into the effect of cell diameter of aluminium foam on its compressive and energy absorption properties. Mater Sci Eng A 2007;454–455:542–6. [8] Bin J, Zejun W, Naiqin Z. Effect of pore size and relative density on the mechanical properties of open-cell aluminium foams. Scripta Mater 2007;56:169–72. [9] Gupta N, Ye R, Porfiri M. Comparison of tensile and compressive characteristics of vinyl ester/glass microballoon syntactic foams. Composite Part B Eng 2010;41:236–45. [10] Gong L, Kyriakides S, Jang WY. Compressive response of open-cell foams. Part I: morphology and elastic properties. Int J Solid Struct 2005;42:1355–79. [11] Viot P, Shankar K, Bernard D. Effect of strain rate and density on dynamic behaviour of syntactic foam. Composite Struct 2008;86:314–27.
183
[12] Viot P, Chirazi A, Dumon M, Bernard D, Fascio V. Ex situ study of polymeric syntactic foams’ mechanical response under compression loading: effects of foam microstructure using microtomography techniques. Adv Mater Res 2011;146–147:42–62. [13] Silverstein MS, Tai H, Sergienko A, Lumelsky Y, Pavlovsky S. PolyHIPE: IPNs, hybrids, nanoscale porosity, silica monoliths and ICP-based sensors. Polymer 2005;46:6682–94. [14] Zhang S, Chen J, Perchyonok VT. Stability of high internal phase emulsions with sole cationic surfactant and its tailoring morphology of porous polymers based on the emulsions. Polymer 2009;50:1723–31. [15] Cameron NR. High internal phase emulsion templating as a route to welldefined porous polymers. Polymer 2005;46:1439–49. [16] Cameron NR, Sherrington DC. High internal phase emulsions (HIPEs) – structure, properties and use in polymer preparation. Biopolym Liquid Cryst Polym Phase Emulsion, Book Ser: Adv Polym Sci 1996;126:163–214. [17] Williams JM, Wrobleski DA. Spatial distribution of the phases in water-in-oil emulsions. Open and closed microcellular foams from cross-linked polystyrene. Langmuir 1988;4:656–62. [18] Mabille C, Leal-Calderon F, Bibette J, Schmitt V. Monodisperse fragmentation in emulsions: mechanisms and kinetics. Europhys Lett 2003;61: 708–14. [19] Ceglia G, Maheo L, Viot P, Bernard D, Chirazi A, Ly I, et al. Formulation and mechanical properties of emulsion-based model polymer foams. Eur Phys J – E 2012;35(4):31. [20] Gibson LJ, Zhang J, Shercliff TL. Failure surface for cellular materials under multiaxial loads – I. Modeling. Int J Mech Sci 1989;31(9):635–63. [21] Zhu H, Knott J, Mills N. Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. J Mech Phys Solids 1997;45:319–43. [22] Li K, Gao XL, Subhash G. Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams. J Mech Phys Solids 2006;54(4):783–806. [23] Kraynik A, Nielsen M. Elastic behavior of cellular solids in Buschow KHJ. In: Cahn RW, Flemings MC, Merton C, Ilschner B, Kramer EJ, Mahajan S, editors. Encyclopedia of Materials-Science and Technology. Elsevier; 2001. p. 2387–9. [24] de Gennes PG. Scaling concept in polymer physics. London: Cornell University Press; 1979. [25] Doi M, Edwards SF. The theory of polymer dynamics. Oxford: Clarendon Press; 1986. [26] Williams JM, Gray AJ, Wilkerson MH. Emulsion stability and rigid foams from styrene or divinylbenzene water-in-oil emulsions. Langmuir 1990;6: 437–44. [27] Hallquist O. LS-Dyna keyword user’s manual version 971. Livermore Software Technology Corporation; May 2007.