Formulation and mechanical properties of emulsion ...

Eur. Phys. J. E (2012) 35: X

THE EUROPEAN PHYSICAL JOURNAL E

Formulation and mechanical properties of emulsion-based model polymer foams G. Ceglia1,2 , L. Mahéo3,4,5 , P. Viot3,4,5 , D. Bernard6,7 , A. Chirazi6,7 , I. Ly1,2 , O. Mondain-Monval1,2 , and V. Schmitt1,2,a 1 2 3 4 5 6 7

CNRS, CRPP, UPR 8641, F-33600 Pessac, France Université Bordeaux, CRPP, UPR 8641, F-33600 Pessac, France Université Bordeaux, I2M, UMR 5295, F-33400 Talence, France CNRS, I2M, UMR 5295, F-33400 Talence, France Arts et Metiers ParisTech, I2M, UMR 5295, F-33400 Talence, France CNRS, ICMCB, UPR 9048, F-33600 Pessac, France Université Bordeaux, ICMCB, UPR 9048, F-33600 Pessac, France Received 26 January 2012 c EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag 2012 Abstract. We produce cellular material based on the formulation of model emulsions whose drop size and composition may be continuously tuned. The obtained solid foams are characterized by narrow cell and pore size distributions in direct relation with the emulsion structure. The mechanical properties are examined, by varying independently the cell size and the foam density, and compared to theoretical predictions. Surprisingly, at constant density, Young’s modulus depends on the cell size. We believe that this observation results from the heterogeneous nature of the solid material constituting the cell walls and propose a mean-field approach that allows describing the experimental data. We discuss the possible origin of the heterogeneity and suggest that the presence of an excess of surfactant close to the interface results in a softer polymer layer near the surface and a harder layer in the bulk.

1 Introduction Cellular materials also called solid foams are used in a wide variety of applications including shock absorbers (in car bumpers, helmets, etc.), thermal or acoustic insulator devices, filters, membranes, etc. [1]. For shock absorbing devices, mechanical properties are of utmost importance and have to be controlled. Both the foam structure and the type of material, which may be of various natures as polymers, ceramics, or metals, constituting the solid phase impact the mechanical behavior. However for a given material, the relation between structure and mechanical properties is still not fully established despite the abundant literature devoted to the topic [2–9]. In particular, the evolution of the material properties as a function of the cell size has never been considered as object of systematic studies. A main reason for such a lack lies in the difficulty to synthesize sufficiently well-defined structured materials with a real and independent control over the different characterizing parameters (size, density, etc.). Such a control can be achieved using model monodisperse emulsions whose continuous phase may be polymerized. The strategy of highly concentrated emulsion polymerizaa

e-mail: [email protected]

tion has often been employed and the obtained materials are referred to as polyHIPE—for polymerized High Internal Phase Emulsions. A large amount of work focusing on polyHIPE formulation and characterization has been reported in the literature [10–16]. However, in most cases, the drop size of the initial emulsion droplets could not be carefully controlled. Herein, we propose to formulate model emulsions, with mean drop size dd and dispersed phase volume fractions φemulsion that can be tuned independently thus varying in a controlled way the foam final cell size and density. Therefore the influence of each parameter on the resulting material properties could be determined and compared to theoretical predictions. Doing so, we show that, in certain conditions, an unexpected dependence of the compression modulus on the cell size is observed. We discuss the origin of this behavior and propose a mean-field model that fits the experimental data. After a detailed description of the material preparation and structural characterization, we briefly remind the main laws governing the compression modulus as a function of foam cell size and density predicted by the Gibson-Ashby model [17]. In a second part, we present the obtained results and discuss the possible origins of the observed trends.

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Eur. Phys. J. E (2012) to be completed Table 1. Formulation parameters for mother emulsions preparation.

Average drops size (μm)

Composition Water content

Organic phase composition

(vol%) w.r.t. total

(vol%) DVB

ST

Cs Span80

none

100

0.5

90

none

1.8

82

32.5

17.5

50

5.0

90

80.0

none

20

7.0

90

28.0

52.0

20

7.8

90

55.3

29.7

15

14.0

90

58.5

31.5

10

2 Materials and methods 2.1 Emulsion preparation Highly concentrated reverse water-in-oil emulsions are prepared in two steps. We first prepare crude polydisperse emulsions by progressively incorporating brine (aqueous phase containing 25 mM NaCl and the radical initiator potassium peroxodisulfate KPS 5 mM) into an organic phase composed of styrene (ST) as the monomer, divinylbenzene (DVB) as the crosslinker in various proportions and the surfactant namely sorbitan monooleate (SpanTM 80). The presence of salt in the dispersed phase allows eliminating Ostwald ripening and ensuring long term stability. Quasi-monodisperse emulsions are then obtained by shearing the polydisperse ones within a Couette cell made of two concentric cylinders separated by a narrow gap (200 μm). This experimental set-up allows for the preparation of large quantities of emulsions (typically on the order of one litre/sample) [18]. Previous studies performed on direct oil-in-water emulsions showed that the final droplet size is determined by the shear rate, the interfacial tension and the viscosity ratio between the dispersed and continuous phases [19–22]. For the water-inoil system under consideration, the size does not decrease continuously when increasing the shear rate so that we had to adapt the initial emulsion composition to fix the drop size dd in the range 0.5–14 μm. All details concerning the composition and average obtained diameter of the mother emulsions are provided in table 1. In all cases the applied shear was set to 4200 s−1 . Once sheared, the final composition, i.e. the dispersed phase volume fraction φemulsion and surfactant concentration Cs of the emulsion, can be fixed by controlled dilutions of the mother emulsions either with pure oil or with a mixture of surfactant and organic phase. For emulsions made initially by incorporating the aqueous phase in pure span 80 (characterized by a mean drop size equal to 0.5 μm), the final composition of the organic phase is fixed by performing several centrifugation-replacement of the continuous phase with the required one. In the following, the given compositions are those of the final emulsion, just before polymerization. Note that Cs is given with respect to the continuous phase (not to the total emulsion).

All chemicals were purchased from Sigma-Aldrich (purity > 99%) and used as received without further purification. Milli-Q water was used for all aqueous phase preparations. 2.2 Emulsion characterization The size distributions of the emulsions are deduced from static light-scattering measurements performed with a Mastersizer 2000 Hydro SM device (Malvern) using Mie theory. The organic phase used for the required dilution is hexadecane. The emulsions are characterized by their volume-averaged diameter, dd , defined as Ni Di4 dd = i (1) 3 , i Ni Di where Ni is the total number of droplets with diameter Di and the size distribution width is estimated through the polydispersity index defined as ¯ − Di 1 i N iDi3 D P = ¯ , (2) 3 D i N iDi ¯ is the median diameter, i.e. the diameter for where D which the cumulative undersized volume fraction is equal to 50%. The granulometric measurements are qualitatively checked by observing the emulsions with an optical microscope (Zeiss, Axiovert X100). Typical microscopy picture and drop size distribution are reported in figs. 1a and b. 2.3 Polymerization Polymerization of the continuous phase composed of ST/DVB/Span80 is triggered by increasing the temperature at 60 ◦ C. At this temperature, decomposition of the initiator (potassium peroxodisulfate) present in the dispersed phase occurs. Radical polymerization proceeds for a duration that depends on the surfactant concentration (5 days for 10% of surfactant to a week for 30%). During polymerization, both solidification and a slight shrinkage occur simultaneously leading to a solid monolith whose

Eur. Phys. J. E (2012) to be completed

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Fig. 2. a) Washed and dried monolith obtained after polymerization of the previous 8 μm sized emulsion φemulsion = 0.75. b) Corresponding X-ray radiography image. The uniform grey level indicates the homogeneity of the sample.

Fig. 1. a) Microscopy picture of an emulsion with average drop diameter 7 μm and polydispersity index P = 24.3% obtained following the protocol described in the text. b) Corresponding drop size distribution determined by static light scattering.

shape is given by the mould. As the film between neighboring drops breaks, connections between the aqueous drops appear, which leads to a final opened porosity (what is referred to as open-cell geometry, as opposed to the closedcell geometry).

2.4 Washing and drying The obtained materials are then washed several times in ethanol (24 hours) and acetone (8 hours) using a Soxhlet extractor in order to remove the remaining emulsifiers, monomers, crosslinkers, initiators and stabilizers. Acetone allows easy drying of the polyHIPEs monoliths at room temperature. An example of obtained material is reported in figs. 2a and b.

2.5 Solid foams structural characterization The structures of the obtained solid porous materials are characterized using scanning (SEM) and transmission (TEM) electron microscopy, mercury (Hg) porometry and X-tomography.

Scanning Electronic Microscopy (SEM) observations were performed with a SEM HITACHI TM-1000 apparatus. The monoliths were previously crushed into small pieces. All the samples were gold coated prior to observation. This characterization allows cell (cavity) size determination dc (the dimensions of about 50 cells were measured) and comparison with initial drop size. Examples of obtained materials are reported in fig. 3; the drop print is still visible. A rough estimate of the connections size dp (the holes connecting cells) is also possible. The final solid foam porosity φfoam and the pore size distribution are determined using mercury intrusion. Mercury porosimetry is performed using a Micromeritics Auto Pore IV 9500 apparatus. Samples are out-gassed under vacuum at room temperature prior to intrusion. Because mercury does not wet the surfaces, it has to be pressed in order to fill the pores, smaller pores requiring higher pressures. The intrusion mercury volume VHg is measured with an accuracy of 0.1 μL as a function of the applied pressure P in the range 2.7 · 10−3 to 207 MPa. This last value is never exceeded in order to avoid sample compression. The pore size comprised between 0.006 and 10 μm is deduced from the experimental relation VHg -P assuming a cylindrical shape and using the Laplace equation dp =

4 γHg cos(π − θHg ), P

(3)

where γHg is the surface tension of mercury equal to 485 mN m−1 , θHg is the contact angle of mercury on the surface (taken to be equal to 130◦ ). Although pores are rarely cylindrical in reality, this equation provides a value of a characteristic length in the material and its size distribution. We obtain a good agreement between the SEM determination of dp (previous paragraph on SEM) and the one deduced using the mercury intrusion data and eq. (3). An example of the pore distribution extracted from the measurement is given in fig. 4. From the total intruded mercury volume VHg and the apparent volume of the monolith

Eur. Phys. J. E (2012) to be completed Differential intrusion (ml.g-1 nm -1 )

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0.012 0.010 0.008 0.006

0.004 0.002 0.000 1

10

Pore size diameter (µm) Fig. 4. Plot of the mercury differential intrusion volume as a function of pore diameter (average drop size 14 μm and Φemulsion = 0.75).

large series of two-dimensional X-ray images taken around the vertical axis of rotation. The image resolution is determined by the source power. A home (ICMCB) set-up gives a resolution of 22 μm. A monolith radiography is reported in fig. 2b. This low-resolution picture shows that the obtained material is homogeneous at this scale with no presence of any internal fractures or cracks. Microtomography experiments using a synchrotron source (Swiss Light Source) were performed at the Paul Scherrer Institute using the TOMOCAT line. The high power source allows a resolution of 0.36 μm.

Fig. 3. Scanning Electron Micrograph of the solid dried material. The polymer appears as white or light grey: a) Average drop size 7 μm and φemulsion = 0.75. The continuous circle indicates the presence of a cell. The dashed circle indicates a connection (pore) between two cells. b) Average drop size 14 μm and φemulsion = 0.75.

(Vmonolith ), the porosity can also be deduced φfoam = VHg /Vmonolith .

(4)

Transmission Electron Microscopy (TEM, Hitachi H600) observations require the sample to be first impregnated with a liquid epoxy resin and then cut in thin slices (∼ 80 nm thick) using a diamond knife before deposition on TEM grids. This step is necessary to rigidify the thin slices and make the cutting possible. The epoxy zones appear in light grey whereas the solid part of the foam is darker. Tomography X-ray tomography is a non-destructive investigation technique based on the non-homogeneous absorption of X-rays in porous materials. Indeed the absorption depends on both the material nature and density. It is well adapted to image porous materials made of voids and bulky walls presenting a large contrast. A threedimensional image of the material is generated from a

2.6 Solid foam mechanical characterization The obtained monoliths are mechanically characterized using a ZWICK electromechanical testing machine. Samples with well-controlled morphologies (cylinders of height h0 = 40 mm and diameter d = 25 mm) are compressed with a controlled displacement rate of 10 mm/min of the upper punch (the rigidity of the machine being much larger than the sample one). We previously checked that the results were independent of the compression rate (in the range 1 to 100 mm/min). For each test, the stress σ is deduced from the measured force F (with a sensor of 10 kN range) knowing the initial section A0 and assuming that A0 does not significantly change during compression (which is reasonable in this study where we focus on the linear part of the stress-strain curve). The strain is defined as the relative height ε = h0 −h/h0 , where h0 is the initial height and h the height under compression. The compression modulus of the cellular material E ∗ is deduced from the linear part of the σ − ε curve E ∗ = σ/ε.

3 Theoretical background Gibson and Ashby [1] proposed to describe the mechanical properties of solid foams by considering the material as an assembly of open cubic cells, constituted of struts of square section t2 and length L. Adjoining cells are staggered so that their edges meet at their midpoint. Despite


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the simplicity of the representation, such an approach has shown to depict the essential features of open-cell foams in different cases [2]. In the elastic regime, when a uniaxial stress is applied so that each cell edge transmits a force F , the edges bend and the deflection δdeflection of the structure as a whole is proportional to δdeflection ∼ F L3 /(Es I),

(5)

where Es is Young’s modulus of the material constituting the solid strut and I the second moment of the area of a square strut of section t2 . For such strut geometry I ∼ t4 .

(6)

The strain ε is related to the displacement δdeflection by ε ∼ δdeflection /L

(7)

and the foam stress σ ∗ is linked to the force F through σ ∗ ∼ F/L2 .

(8)

Since Young’s modulus E ∗ of the considered foam is defined as E ∗ = σ ∗ /ε, it comes that E ∗ /Es ∼ (t/L)4 .

(9)

Moreover, from geometrical considerations, the following relation holds: ∗

(1 − φfoam ) = ρ /ρs = (t/L) , 2

Sf 1.097 ⎢ = 2 ⎣1 − S0 3 φemulsion

(10)

where ρ∗ and ρs are the density of the foam and the solid material constituting the struts, respectively. Therefore, one can also write E ∗ /Es ∼ (t/L)4 ∼ (1 − φfoam )2 .

cells are the prints of the droplets and that emulsions did not coarsen in a detectable extent during polymerization. These results are observed in all cases. Using porosimetry and SEM imaging, the size of the connections dp between the cells may be extracted. Again the volume-averaged diameters are obtained following eq. (1). They are reported as a function of the cell size for various initial water volume fractions φemulsion (fig. 5b). The connection size is linearly connected to the cell size: for example for monoliths made from emulsions comprising initially 75 vol% of water, the pore size is about 0.30 times the initial drop size. This factor increases as the water volume fraction increases (0.58 for 90%) likely because, for more compressed drops, the contact area between drops increases. The results are independent of the organic phase composition (DVB/styrene ratio). The diameter of the contact zone dpatch between adjacent drops can be estimated and compared to dp in order to check if the connections result from the initial contact zones. For polydisperse emulsions and assuming that each drop is surrounded by 12 neighboring drops in average, Princen showed that the ratio of the deformed drops film area Sf to the non-deformed drop surface S0 only depends on the emulsion volume fraction [24–26]. Following the same argument, Arditty [27] adapted Princen’s expression for monodisperse emulsions: ⎡ ⎤

(11)

It is worth noting that, whatever the structure of the foam, the thickness of the strut t always theoretically scales linearly with the cell size L (even for a more realistic cell packing, as the tetrakaidecahedral lattice proposed by Kelvin for example [23]). As a consequence, with such a model, no dependence of Young’s modulus on the cell size is expected.

4 Results 4.1 Comparison between emulsion and solid foam structure In fig. 5a, we report, as an example, the foam cell sizes dc , deduced from SEM observations, of the monoliths made from emulsions comprising initially 75% by volume of water, as a function of the initial emulsion drop size dd . In order to allow comparison with the mean drop size, each cell size obtained from SEM or tomography is weighted by its volume so that the volume-averaged cell size is calculated using eq. (1) for both SEM and tomography techniques. These two diameters (dc and dd ) are identical showing that

=3

dpatch dd

2 .

⎥ 12 ⎦ + 1.3

1.74 φemulsion 1−φemulsion

(12)

This expression is valid for 0.635 ≤ φemulsion ≤ 0.95. Using eq. (12), for a given volume fraction, a linear relation is hence expected between dpatch and dd . The proportionality coefficient κ = dpatch /dd , deduced from eq. (12) is equal to 0.21 and 0.54 for φemulsion = 0.75 and 0.90, respectively. These values should be compared to the experimental values of κ deduced from fig. 5b: 0.3 and 0.58 (estimated from only one point), respectively, showing a good agreement. The rather good correlation, put into evidence in fig. 5c, between estimated dpatch and measured dp , confirms that it is reasonable to assume that the pores originate from the rupture of the thin films between adjacent drops inducing the open porosity. Porosimetry also allows for the determination of the real foam density. The foam density may alternatively be evaluated by weighting the monoliths once machined at a known volume. We thus obtain a good agreement between (1 − φemulsion ) and (1 − φfoam ) (fig. 5d). It is worth noting that the two terms are not perfectly equal, the foam porosity being a little larger than the emulsion dispersed phase volume fraction, thus reflecting the existence of an additional porosity inside the walls likely due to the initial presence of surfactant micelles that are extracted from the material during the washing procedure. Such an assumption is validated since the measured porosity is equal to the one estimated from adding the amount of surfactant to the initial dispersed phase volume fraction (see fig. 5d).

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Eur. Phys. J. E (2012) to be completed 5

pore size d p (µm)

Cell size d c (µm)

15

10

5

a)

4 3

2 1

0

5 10 Drop size d d (µm)

0

15

5

10

15

drop size d d (µm)

5

0.3

4

(1- φ emulsion )

Patch size dpatch (µm)

b)

0

0

3 2 1

c)

0.2

0.1

d) 0.0

0 0

1

2

3

4

5

0.0

Pore size dp (µm)

0.1

0.2

0.3

(1-φ foam)

Fig. 5. a) Evolution of the foam cell size dc determined from both SEM () and tomography () observations, as a function of the initial water droplets size dd determined from static light scattering. The dashed curve is a straight line with slope 1. b) Evolution of the connection size dp determined by (, •) mercury porometry and () scanning electron microscopy as a function of the cell size for φemulsion = 0.75 () and 0.90 (•). The straight dashed line is a guide to the eyes. c) Estimated patch size (see eq. (12)) as a function of the pore size. The dashed line corresponds to equality. d) Evolution of the emulsion organic phase volume fraction (1 − φemulsion ) versus foam solid fraction (1 − φfoam ) determined () by weighting the sample and () using mercury porometry. Results for the emulsion with diameter dd = 7 μm. The dashed line takes into account the surfactant volume fraction as additional porosity of the foam.

All these results show that a good control over the initial emulsions characteristics allows for a fine tuning of the monoliths structural parameters. Using this strategy, it is possible to vary the foam cell size and its density independently.

in figs. 6b and 7b. The experimental dependence on solid content clearly deviates from the expected (1 − φfoam )2 variation (see the dashed line in fig. 6b). The discrepancy becomes even larger as the solid content increases.

4.2 Influence of the cell size on the compression modulus E∗

5 Discussion

All measurements were performed in triplicate in order to check their reproducibility. Young’s modulus is reported as a function of the cell size for a fixed density (φfoam = 78%) (fig. 6a) and for various initial water volume fractions (fig. 7a). Young’s modulus is independent of the monomer/crosslinker ratio. Whatever φfoam (therefore the foam density), the modulus depends on the cell size and this dependence is stronger as the density increases. This result is in sharp contrast with the Gibson and Ashby model that predicts no influence of the cell size. 4.3 Influence of the foam density Keeping the cell size fixed, we explore the influence of the foam density. The measured Young’s moduli are reported

5.1 Heterogeneity of the foam walls The total interfacial area of emulsions is inversely proportional to the drop size. As a consequence, a dependence of E ∗ on the cell size merely reflects the influence of the amount of air/solid interface. As long as the obtained materials are homogeneous, this quantity should have no influence on Young’s modulus, only the density of the solid foam is important, in agreement with the Gibson and Ashby model. Since we experimentally observe a large dependence of E ∗ with respect to dc , we propose that the hypothesis of a homogeneous solid is not fulfilled and that the material composing the solid part of the foam (the cell wall) is heterogeneous. Such a hypothesis has already been proposed (with no further investigations) in a previous study on alumina foams [4]. It has also been reported

90 80 70 60 50 40 30 20 10 0

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a)

100

E* (MPa)

E* (MPa)


5

10

15

60 40 20

a) 0

80

0

20

0

5

cell size dc (µm)

10

15

20

Cell size dc (µm) 150

200

E* (MPa)

E* (MPa)

b) 150 100 50

100

50

b) 0

0

0.1

0.2

0.3

0.4

Solid fraction (1- φ foam)

0

0

0.1

0.2

0.3

Solid fraction (1-φ foam) ∗

Fig. 6. a) Evolution of the foam Young’s modulus E as a function of the cell diameter at a constant porosity φfoam = 75%. The solid line is the best fit of the data to eq. (17) using Ebulk /Esurf and K as free parameters. Results plotted with Ebulk /Esurf = 6 and K = 1.2. b) Evolution of the foam Young’s modulus E ∗ as a function of the foam density for a fixed emulsion drop size 7 μm, () data from Manley et al. [31] for a polydisperse emulsion with an average cell size close to 7 μm. The solid line is the fit of our data to eq. (17) with Ebulk /Esurf = 6 and K = 1.2. The dotted line is the Gibson-Ashby model obtained for homogeneous walls.

very recently that anodizing a conventional aluminium alloy [28] or oxidation of an aluminium network [29] in order to produce hybrid materials modifies significantly the mechanical properties. In our case, we hypothesize that, since the initiator is solubilized in the water phase, polymerization likely initiates at the water-oil interface and then propagates into the continuous phase, thus resulting in a heterogeneous material. Another possibility is that, due to the proximity of the water phase, polymerization does not occur in the same manner than further away from the water domains, leading to materials with locally different values of their Young’s moduli. Our hypothesis that the size effect results from a heterogeneous material implicitly assumes that at fixed volume fraction, the drop size does not modify the foam structure as can be validated from SEM images (figs. 3a and b) and Hg porometry. The open-cell structure is indeed evidenced by the Hg porometry since the measured porosity is the one that is expected from the relative amount of incorporated water in the initial emulsion, whatever the initial droplets sizes. This observation indicates that the structure is always that of an open-cell geometry since Hg would not penetrate into closed cells.

Fig. 7. a) Evolution of the foam Young’s modulus E ∗ as a function of the cell diameter for various porosities φfoam = () 74%, () 78%, () 82%, () 87% and (•) 91%. The solid lines are the fits of our data to eq. (17) with Ebulk /Esurf = 6 and K = 1.2. b) Evolution of Young’s modulus E ∗ as a function of the foam density for different cell sizes: dc = (•) 2.8 μm, () 6 μm, () 8 μm and () 15 μm. The solid lines are the fits of our data to eq. (17) with Ebulk /Esurf = 6 and K = 1.2.

Fig. 8. TEM pictures of sliced epoxy/polyHIPEs samples for initial water volume fraction φemulsion = 78% and different cell sizes: a) dc = 8 μm, b) dc = 2.8 μm. The dashed lines are guides to the eyes and emphasize the presence of the “skin” layer close to the interface.

5.2 Observation by TEM To check our hypothesis, we performed TEM observation of foam thin slices (see fig. 8). The epoxy zones evidencing the pores appear as smooth light grey areas [30] while the polystyrene zones appear much darker and present a granular texture. This granular texture reflects the additional porosity at the nanometric scale, due to the presence of surfactant that we already described. The large smooth

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Local porosity

1.0 0.8 0.6 0.4

δ

0.2 0.0

0

200 400 600 800 Distance form the solid - air interface [nm]

Fig. 9. Evolution of the local porosity (deduced from the analysis procedure described in the text) as a function of the distance from the solid-air interface. The very high value of the local porosity very close to the interface is due to the surface roughness that can be seen on the TEM pictures. From this curve we deduce δ ≈ 180 nm. Data obtained from the analysis of the TEM picture obtained in a material with φemulsion = 75% and dc = 8 μm.

circles correspond to 2D cut through the foam cell filled with epoxy resin. Interestingly, the darker walls, which correspond to the polystyrene domains, do present a difference in texture depending on the distance from the cell wall. This is more particularly obvious from the careful observation of fig. 8a where a kind of skin of thickness δ seems to surround the circular epoxy zone. Using image analysis, this “skin” characteristic thickness may be evaluated applying the following procedure to the TEM pictures: i) In case there is a non-uniform grey level throughout the picture due to a non-uniform intensity during acquisition, the first step consists in equalizing the grey level. ii) Binarization of the image by thresholding: pixels were classified as void (respectively, solid) if their grey level was lower (respectively, higher) than the threshold value. This value was selected for each image from the corresponding gray levels histogram. iii) 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). iv) 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.) A typical curve resulting from step iv) is presented in fig. 9. The first two points correspond to the roughness of the interface induced by the granular texture. The following plateau provided evidence of the presence of a very dense zone (highlighted by a dashed line in fig. 8a, the thickness (δ) of which can be estimated roughly (the

analysis is in 2D, whereas the structure is 3D). After the plateau, the local porosity slowly increased, revealing the nano porous nature of the polymerized zone. A systematic estimation of δ varying the cell size and the solid volume fraction does not put into evidence any influence of these parameters. The analysis of all the pictures leads to an average value δ = 210 ± 39 nm.

5.3 Mean-field approach In order to simplify the foam description, we propose that the walls are composed of two materials with two distinct Young’s moduli: Esurf close to the wall surface that expands over a distance δ and Ebulk in the wall bulk. The solid modulus in the Gibson-Ashby model should hence be modified so that it takes into account the heterogeneity of the cell walls. An even more complex wall mechanical behavior could be taken into account with, for example, a Young’s modulus gradient; however the aim here is to understand the main features of the observed trends. Moreover, to be rigorous, the modulus calculated from bending struts composed of the two materials should read E ∗ /Es ∼ t4 − (t − 2δ)4 /L4 ∗ (Esurf /Es ) +[t − 2δ]4 /L4 ∗ (Ebulk /Es ).

(13)

The reader interested in such a development can refer to [31], where this microscopic analysis has been done together with finite elements numerical simulations. Herein we aim at proposing a more macroscopic mean-field approach. The foam is composed of a fraction φfoam of air, a fraction α of solid close to the wall surface and characterized by Esurf and a fraction (with respect to total volume) β of solid inside the wall and characterized by Ebulk . From geometrical consideration, one gets α = δAint /Vtotal = 6φfoam δ/dd β = 1 − φfoam (1 + 6δ/dd ),

and (14)

where Aint is the total air/solid interfacial area in the sample and Vtotal the total foam volume. In such a macroscopic mean-field approach, we first have to determine if either the stress or the strain is the same in the whole material. For a model material made of two layers perpendicular to the compression axis, the applied stress is the same for the two layers whereas the strain is different depending on each Young’s modulus (the softer material being more deformed). As a consequence, the whole material behaves with an equivalent compliance (reverse of the modulus) that is the sum of the layers compliances (fig. 10a). This model situation can experimentally be mimicked by preparing a foam composed of two layers of emulsions with two different drops sizes in various proportions: the bottom layer is an emulsion with an average drop size of 14 μm, whereas the upper layer is an emulsion with an average drop size of 1.8 μm (fig. 11a). The experimental Young’s moduli obtained for these twolayered foams are reported in fig. 11b. As expected for


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Fig. 10. Schematic representation of a material a) made of two distinct layers characterized by two moduli E1 and E2 and the equivalent modulus Eeq and b) made of two randomly mixed layers. What is the equivalent modulus in this case?

this packing geometry, the compliances are additive corresponding to a homogeneous stress in the material. One can now wonder how this association law would be modified when the layers disappear and are replaced by randomly distributed domains of the two materials (fig. 10b). Is the overall mechanical behavior preserved independently of the domain sizes? In order to experimentally address this question, we prepared foams from a blend of emulsions with the two different drops sizes (1.8 and 14 μm) with various ratios and rather than segregating the emulsions so that the foam is composed of two macroscopic layers, we homogenously mixed the two emulsions, mimicking the layers mixing (fig. 11a). The measured mechanical behaviors of these foams are reported in fig. 11b. The agreement between the homogeneously mixed emulsions and the segregated ones (equality of the stress throughout the sample) clearly shows that the drop packing does not modify the mechanical behavior of the emulsions. As a consequence, by extrapolating the previous experiments to any non-homogeneous emulsions, we propose that the wall heterogeneity may be very simply taken into account, by writing the modulus of the foam solid part Es in eq. (11) as 1 1 1 α β = + (15) Es α + β Esurf α + β Ebulk or identically

1 1 6δ 1 6δ φfoam φfoam = + 1− . Es (1 − φfoam ) dd Esurf (1 − φfoam ) dd Ebulk (16) Equation (16) combined with eq. (11) gives the following expression for the foam elastic modulus: (1 − φfoam )3 dd Esurf Ebulk , 6δφfoam (Ebulk − Esurf ) + (1 − φfoam )dd Esurf (17) where K is a constant. From this equation a dependence of E ∗ on the cell size is expected. Considering that far from the wall surface one expects Young’s modulus to be equal to the polystyrene modulus reported in the literature Ebulk = 3 GPa and that δ = 210 nm from TEM image analysis, expression (17) only has two independent fitting parameters: a constant K and Esurf /Ebulk . E∗ = K

Fig. 11. Mixing of emulsions with two different drop sizes dd 14 and 1.8 μm (φemulsion = 0.75); a) schematic representation of the segregated or homogenous blends and b) experimental foam Young’s moduli as a function of the 1.8 μm drop fraction for both segregated () and homogeneous (•) blends and expected moduli for the two possible association laws (continuous line: equality of the stress; dashed line: equality of the strain throughout the sample).

It is worth noting that: – if Esurf = Ebulk (homogeneous material), the unmodified Gibson and Ashby expressions are recovered; – if δ dd , the “skin effect” is negligible and E ∗ reduces to (1 − φfoam )2 Ebulk with no size dependence; – if δ ≈ dd , the foam is almost homogeneously characterized by Esurf and E ∗ reduces to (1 − φfoam )2 Esurf , with no size dependence. The influence of the drop size can therefore only be observed in a reduced size range. It is also worth noticing that, as long as the foam modulus is independent of dd , the initial emulsion polydispersity may be ignored. This is no longer the case for heterogenous foams since the size becomes an important parameter. This approach could be generalized by taking into account the drop size distribution (in eq. (16)). However in the present case, the polydispersity of the initial emulsion is reduced as can be seen in fig. 1b and it is reasonable to characterize the emulsion by its average drop size in the present mean-field approach. As mentioned earlier, the size distribution and the drop spatial arrangements may also have an influence on the surface contact area of the films between drops therefore leading to non-connected cells in the foam. In the present study, the initial emulsion dispersed phase volume fraction was always larger than 0.71 corresponding to the random close packing of polydisperse emulsions [24–26] leading always

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to open cell foams (the foam would still be open even with more polydisperse emulsions). We then expect the effective foam module to hardly depend on the connection diameter or polydispersity (as evidenced in fig. 6b) at fixed volume fraction or cell average size. In figs. 6 and 7 the best fits of eq. (17) to experimental data have been plotted. For the sake of comparison, the (1 − φfoam )2 dependence of the Gibson-Ashby model with homogeneous walls has also been plotted for the data obtained in samples with constant average cell diameter 7 μm and varying the porosity (fig. 6b).

5.4 Results and comparison with experimental data The variations of E ∗ with the cell size and foam porosity are reasonably described by eq. (17). In order to increase the sensitivity of the fit, we also used data coming from the literature [32]; this extended domain of solid fraction confirms the correct variation with φfoam . In all cases, we chose to keep the value of K constant and equal to 1.2 in good agreement with the typical expected value given in literature (K ∼ 1) [1]. The fit to experimental data does not lead to unique values of Ebulk /Esurf and K but the sensitivity is sufficient to limit their possible values in the 6–10 and 1–1.6 ranges, respectively. For example the continuous lines in figs. 6a and b have been obtained with Ebulk /Esurf = 6 and K = 1.2. At first glance, we rather expected a larger modulus close to the surface because TEM pictures seem to evidence a denser material close to the initial water domains. However since the initiator was solubilized in water, polymerization likely begins at the interface and then propagates into the continuous phase. The numerous free radicals present in the vicinity of the interface initiate therefore polymerization of many short chains. In the bulk, despite the presence of water swollen micelles, fewer free radicals are present leading to longer polymer chains. In order to determine the role of the initiator, we chose to partially or totally replace the aqueous KPS by an initiator soluble in the oil phase, AIBN (azo-bis-isobutyronitrile). Such an attempt has already been reported in the literature, however the materials obtained for the different initiators also had different drops sizes [16]. In the present study the initiator location had no influence on the drop size making the comparison possible. The results are reported in fig. 12. No changes were detectable in the TEM pictures so that the presence of the “skin” and its thickness (245 nm) seem to be independent of the initiator location. However it is worth noticing that the foam Young modulus is lower in the presence of AIBN than with KPS, showing that polymerization is affected by the initiator; the presence of initiator in the continuous phase weakens the difference between the two moduli. Indeed, the experimental data can be correctly fitted keeping the value of δ and Esurf constant (0.21 μm and 0.5 GPa, respectively) but decreasing the ratio Ebulk /Esurf to 2.5 (i.e. Ebulk = 1.25 GPa). It seems that the presence of free radicals in the bulk promotes polymerization of shorter chains leading to a lower value of Ebulk . Another

Fig. 12. Influence of the initiator localization: AIBN is in the oil phase whereas KPS is in the aqueous phase. a) TEM picture of the foam obtained with AIBN initiator φfoam = 0.78, dc = 8 μm. b) Compared evolution of the foam Young’s modulus as a function of the foam density using different initiators () KPS and () AIBN.

hypothesis, explaining that the layer close to the interface is softer, is the influence of the local surfactant concentration on the final value of the material Young’s modulus. To test this idea, we changed the amount of surfactant keeping all the other parameters constant. The mechanical results and the corresponding TEM observations are reported in fig. 13. As more surfactant is added, the material seems to be more heterogeneous (figs. 13a, b, c) and to exhibit a lower Young’s modulus (fig. 13d), as observed in previous studies [15,16]. It is clear that the presence of surfactant alters polymerization. Such a feature has already been proposed in the literature [15] and is herein confirmed. We therefore propose that the lower value of Esurf compared to Ebulk reflects a non-homogeneous distribution of surfactant through the sample, surfactant being less concentrated at the water-oil interface than in the micellar dispersed bulk phase.


Page 11 of 11 funding the LIght STRuctures And Composites (LISTRAC) project and for financial support, respectively. The authors thank H. Deleuze, M. Silverstein and M. Birot for fruitful discussions.

References

Fig. 13. Influence of the surfactant concentration Cs in the organic phase on both the foam texture visualized by TEM: a) Cs = 10% wt. b) Cs = 20% wt. c) Cs = 30% wt. and d) the foam Young’s modulus (φemulsion = 75% and dd = 7 μm).

6 Conclusion Using a controlled emulsification procedure, we have been able to synthesize porous materials of controlled pore size and density. The materials are thoroughly characterized and the relations between the different characteristic lengths of the materials are experimentally established using a variety of experimental techniques as SEM, TEM, mercury intrusion porometry, X-ray tomography and static light scattering. The elastic mechanical properties of the materials have been probed varying independently each of these parameters. The data have evidenced a quasi-linear dependence of Young’s modulus as a function of the pore size and an unusually strong dependence as a function of the materials relative density. These two behaviors are in sharp contrast with the prediction of Ashby and Gibson. We demonstrated that these peculiar features result from an inhomogeneous repartition of matter inside the foam walls as clearly evidenced by the experimental data. We believe that the presence of surfactant micelles is likely responsible for the heterogeneous polymerization of the emulsion continuous phase. A simple mean-field approach based on the Ashby and Gibson model and considering that the walls are made of two distinct materials correctly accounts for the experimental data. The French National Agency for Research (ANR) and the Regional Council of Aquitaine are gratefully acknowledged for

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