Structure of foams modeled by Laguerre–Voronoi tessellations

Computational Materials Science 67 (2013) 216–221

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Structure of foams modeled by Laguerre–Voronoi tessellations T. Wejrzanowski a,⇑, J. Skibinski a, J. Szumbarski b, K.J. Kurzydlowski a a b

Warsaw University of Technology, Faculty of Materials Science and Engineering, Woloska 141 Str., 02-507 Warsaw, Poland Warsaw University of Technology, Faculty of Power and Aeronautical Engineering, Nowowiejska 24 Str., 00-665 Warsaw, Poland

a r t i c l e

i n f o

Article history: Received 12 April 2012 Received in revised form 9 August 2012 Accepted 31 August 2012 Available online 9 October 2012 Keywords: Foams Voronoi tessellations Porous structure Specific surface area model

a b s t r a c t Voronoi tessellations are widely used to represent various cellular structures found in the nature. In this study we propose using a model based on Laguerre–Voronoi tessellations (LVT) to simulate the geometry of engineering foams. It is demonstrated that geometrical features of the modeled foam structures, such as the number of faces per pore, are close to the ones observed experimentally. The LVT approach, used here, allows for the investigation of the influence of variations in the pore size on the specific surface area and porosity of the foams. Based on the results obtained and an analysis of the models available in literature, corrections to the volume fraction and model of specific surface area of pores is proposed, which are especially important when the foam porosity is smaller than 70%. The relationships between different structural parameters of the models of foam structures are provided. These parameters are compared with those proposed by other authors and their applicability is verified using commercially available alumina foams. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Foam structures are frequently observed in nature. Due to the high specific strength (strength related to mass) and permeability they form the most structurally demanding parts of plants and animals [1,2]. Foams are nowadays commercially used in modern biomedical and technical applications [3–6]. One of the main biomedical interests is the reconstruction of human body by the implantation of scaffolds [7,8]. In this case both mechanical and functional properties of foams (scaffolds) are of key importance. Other important fields of foam applications are filtering and catalysis. Ceramic foams are used for the filtering of liquid metals in casting processes [9,10]. They are also commonly used in diesel particulate filter or exhaust gas catalytic converters [11]. The foam structures can also be infiltrated by other materials (mainly metals) in order to produce composites [12,13]. Metallic foams, usually having a structure of enclosed porous material, are used as impact energy absorbers [14]. New manufacturing technologies are available also for open-cell metallic foams for structural applications [15,16]. All these applications require materials with specific properties. Some properties (mainly mechanical) are less sensitive to the local geometry of the foam structure [17]. In such a case porosity and

⇑ Corresponding author. Tel.: +48 22 234 8742. E-mail addresses: [email protected] (T. Wejrzanowski), j.skibinski @inmat.pw.edu.pl (J. Skibinski), [email protected] (J. Szumbarski), kjk@inmat. pw.edu.pl (K.J. Kurzydlowski). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.08.046

average pore size are the only parameters which are required to find structure–property relationships. Other properties, related to flow through the porous media, are more strongly influenced by the local environment of the particular pores and that is why they require a better understanding obtained through explicitly modeling the geometry of the foam structure [18]. There are many geometrical models proposed to capture the relevant features of real foam/grain structures. Most of these models assume equal volume of pores with their shape varying from cubic to more complex geometries (e.g. tetradecahedral) related to the so called Kelvin problem of a polyhedron with the minimum specific surface [19]. However, the experimental observations of the foams clearly reveal distribution of pore sizes [20], similar to those observed in polycrystalline metals [21]. Also, the structures obtained by replicating a single type of polyhedron exhibit anisotropy, which usually does not exist in the real foams and if does exist it has a different nature. Since growth process, associated with the foam fabrication method have a statistical nature, the Poisson–Voronoi point based tessellation (PVT) algorithm is frequently used to obtain representative cellular structures. However, it was found that geometrical parameters of real foams differ from those obtained by PVT. Specifically the average number of faces per cell for PVT (about 15.5 [22]) is higher than those of real foams (less than 14). Another drawback of PVT is that it yields structures with statistically constant variation of cell sizes. The PVT cell volume distribution, not as yet calculated analytically, is found through numerical fitting to follow a log-normal distribution with a variance of 0.424 [23]. Some modifications of PVT allow us to obtain different dispersions of cell

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217

Fig. 1. Schematic illustration of the algorithm for foam structure generation: (a) spheres of pre-determined volume distribution, (b) the same spheres after packing into lowoverall-volume aggregate, (c) cellular structure obtained via Laguerre–Voronoi tessellations, and (d) foam structure with cylindrical struts.

sizes, e.g. by non-random placement of seeds or seed rearrangement before tessellations. Unfortunately, the ability to shape the final distribution of cell size using these methods is very limited. Other methods, such as cellular automaton [24] or 2D/3D reconstruction give better statistics in terms of faces per cell. However, the foam geometry obtained via these methods has a rasterlike form, where the structural parameters, such as specific surface area are difficult to calculate with high precision. A better approximation to the geometry of real foams offers Laguerre–Voronoi algorithm [25,26], where tessellations are performed on the set of spheres with pre-determined size distribution. Since, the spheres are densely packed, the cells obtained by LVT are very similar to the sphere size distribution. It should be noted however, that the structures obtained via randomized tessellations are statistical in their nature. It means that any two structures obtained under the same modeling conditions differ locally in their geometry. On the other hand for relatively large structures their statistical descriptors, such as an average pore size and pore size variation, are the same. As a result it is possible to establish statistical relationships between some relevant features of these structures, such as specific pore surface area and the variability of pore volume. Another important geometrical feature of the foam’s structure is that of the shape of the strut. Most of the models assume cylindrical struts with a constant diameter along the cell edges (or edges of cells). In some models overlapping of the struts at vertices is accepted without evaluation of the related error. To overcome the aforementioned issues, a new method for the generation of foam structures, based on LVT, is proposed within the present work.

2. Methods The algorithm for the modeling of foam structures employed in the present study consists of the following four steps: 1. generation of spheres with a defined size distribution (in the computations log-normal distribution of volume was adopted with the pre-selected average value, E(V) and the variation coefficient, CV(V), 2. packing of the spheres, 3. performing Laguerre–Voronoi tessellations, and 4. generation of struts. The procedure based on the above given four-steps approach is schematically presented in Fig. 1. In the process of packing, the spheres are initially placed in a relatively big box. Subsequently, the box size is reduced via random motions of spheres to increase their density. This step of the procedure is terminated when the box size is reduced to its minimum possible dimensions. The algorithm described above was implemented in the program written by the authors of the paper. After sphere packing, the Laguerre–Voronoi tessellations are applied to obtain cellular structures. In the Laguerre–Voronoi tessellation method the cell faces are created using the intersecting planes located in the proportional distance between two neighboring spheres surfaces. For details see [25]. In the final step the cylinders with constant diameter are generated along the cell edges. Additionally, spheres that have diameters equal to those of the cylinders are placed in the cell vertices to assure dimensional conformity along the edges.

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(a) 2.5

cell edge [25] current

CV (V) cell

2

1.5

1

0.5

ds 0 0

0.5

1

1.5

2

2.5

strut

3

CV (V)sphere

(b)

Fig. 4. Schematic illustration of the struts along cell edges.

15 [25]

via manipulating the initial sphere size distribution (Fig. 2a). It was also shown that the modeled structures, despite a much lower number of cells, exhibit a relationship between the average number of faces per cell, E(F), and the coefficient of cell volume, CV(V), which is similar to that reported in the literature (Fig. 2b) [25]. Since the average number of faces per cell for a specific coefficient of variation of cell volume is constant for a statistically large number of cells, the length of cell edges in a volume can be calculated by the following equation:

current

E (F)

14.5

14

13.5 0

0.5

1

1.5

2

2.5

CV (V)cell Fig. 2. Geometrical features of the cellular structures obtained via LaguereeVoronoi tessellations: (a) cell volume vs sphere volume variation coefficients and (b) number of average faces per cell as a function of cell volume variation coefficient.

LV ¼ CEðVÞ2=3

ð1Þ

where the C coefficient is a function of CV(V). For further analysis the equivalent diameter of cell (further called cell diameter), d, is applied instead of cell volume, V, with the obvious relationship between these two: 3

V ¼ pd =6 8

ð2Þ

When the average cell diameter is used instead of average cell volume, Eq. (1) can be re-written as:

exponential fit,

LV ¼ AEðdÞ2

ð3Þ

6

A ¼ 7:38e1:35CVðdÞ

ð4Þ

A

7

5 4 3 0

0.2

0.4

0.6

CV (d) Fig. 3. Size independent coefficient for calculation of the length of cell edges in the volume.

where A is a dimensionless coefficient related to the cell diameter variation coefficient. For the structures generated in the present study the relationship between A coefficient and CV(V) can be subjected to the exponential fit as shown in Fig. 3. When the strut diameter is small, overlapping of struts can be neglected and the specific surface, SV, and volume fraction of pores, VV, can be calculated using the following formula:

SV  LV pds

ð5Þ 2

Steps 3 and 4 were performed using a script for the package ANSYS written in the APDL language. The algorithm described above was used to generate 15 structures with 300 pores each of the same average volume and variation coefficient of pore volume ranging from 0.45 to 2.0.

3. Results The features of individual cells as well as global parameters were calculated for the 15 structures of different volume variation coefficient. The obtained results are shown in Fig. 2. It can be noted that these results confirm that variation of cell size can be obtained

V V  1  LV pds =4

ð6Þ

(ds denotes diameter of a strut) – see also Fig. 4. For each structure with different CV(d) the strut diameter was altered to see its impact on the properties of the modeled foam. In total 40 structures were examined with the volume fraction and specific surface of pores calculated using formula 5 and 6. The same parameters were calculated for the structures with all the cylinders merged together into one solid. The results of these calculations are presented in Fig. 5. It can be noted that CV(d) influences VV and SV more strongly when the ratio of the strut diameter to pore size, ds/E(d), is higher (see Fig. 5). The error of VV and SV estimation is also higher when CV(d) and ds/E(d) increase. For ds/E(d) = 0.36 the calculated error

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ds/E(d)=0.09

(a)

ds/E(d)=0.18

ds/E(d)=0.27

ds/E(d)=0.36

6 linear fit,

1

B = 0.8CV(d) + 5.2

5.8 5.6

B

0.9

VV

0.8

5.4 5.2

0.7

5

0.6

4.8 0

0.5 0.2

0.2

0.4

0.6

CV (d) 0.3

0.4

0.5

Fig. 6. Size independent coefficient for the cell vertices in the volume.

(b)

x 10 -3

CV (d) 2.5

(a) 16

2

no correction, CV(d)=0.45

12

1.5

VV error [%]

SV [m-2]

no correction, CV(d)=0.15

1 0.5

corrected, CV(d)=0.15 corrected, CV(d)=0.45

8

4 0 0.2

0.3

0.4

0.5

0

CV (d)

(c)

0

0.1

0.2

0.3

ds/E (d)

1

(b) 80

0.9

no correction, CV(d)=0.15 no correction, CV(d)=0.45

60

VV

SV error [%]

0.8 0.7 0.6

corrected, CV(d)=0.15 corrected, CV(d)=0.45

40

20 0.5 0.2

0.3

0.4

0.5

(d)

x 10 -3

CV (d)

0 0

0.1

0.2

0.3

ds/E (d)

2.5

Fig. 7. Error produced in the estimation of: (a) VV and (b) SV.

SV [m-2]

2 1.5 1 0.5 0 0.2

0.3

0.4

0.5

CV (d) Fig. 5. Specific surface area, SV, and porosity, VV for the simulated structures: (a) VV calculated using Eq. (6), (b) SV calculated using Eq. (5), (c) VV calculated for the structure merged to one solid, and (d) SV calculated for the structure merged into one solid.

may be 16% and even 78%, for VV (see Fig. 5a and c) and SV (see Fig. 5b and d), respectively. This error is related to the fact that

Eqs. (5) and (6) overestimate SV and underestimate VV (due to the struts overlapping at cell vertices). The estimation error produced with Eqs. (5) and (6) is thus acceptable only for relatively thin struts. Another problem may appear when ds/E(d) and CV(d) are relatively high. In this case, the struts may overlap with the walls. However, in real foams, for which the volume fraction of pores is over 60%, CV(V) does not exceed 2.0 and this effect can be neglected. Based on our results a simple correction of the overlapping at vertices can be proposed. If the number of vertices, NV, divided by E(d)-3 versus CV(d) changes as indicated in Fig. 6, the corrected SV and VV can be calculated from the following equations: 2

SV ¼ LV pds  NV pds 2

ð7Þ 3 3

V V ¼ 1  LV pds =4  NV pds ds =6

ð8Þ

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Table 1 Structural parameters of the test foams obtained by 3D X-ray tomography. Parameter

Foam

Porosity, VV (%) Specific surface, SV (1/m) Average pore diameter, E(dp) (mm) Variation coefficient of pore diameter, CV(dp) Average strut diameter, E(ds) (mm) Average window diameter, E(aw) (mm)

10 ppi

20 ppi

30 ppi

40 ppi

50 ppi

74 1053 3.53 0.28 1.35 2.82

69 1476 2.44 0.26 1.17 1.95

70 1738 2.08 0.28 0.90 1.65

70 2081 1.64 0.3 0.86 1.31

75 2449 1.17 0.33 0.58 0.93

d = dp + ds, a = aw + ds, where d is the cell diameter and a is a face diameter.

Table 2 The models available in literature, which have been verified in the present study. Reference

Pore shape

Strut shape

Lu et al. [27] and Giani et al. [28]

Cube

Cylindrical

VV ¼ 1 

Richardson et al. [29] and Buciman et al. [30]

Tetrakaidekahedron

Triangular

Grosse at al. [31]

Weaire-Phelan structure

Cylindrical

Current (no correction) Current (corrected)

Laguerre–Voronoi tessellation Laguerre–Voronoi tessellation

Cylindrical Cylindrical

V V ¼ 1  2:59 s2 pffiffiffiffiffiffiffiffiffi d 4:84 1V V 2:64ð1V V Þ SV ¼ a Eqs. (3),(4), and (6) Eqs. (8)–(10)

NV ¼ BEðdÞ3 B ¼ 5:2 þ 0:8CVðdÞ

Volume fraction, VV 2

3pds 4a2

; a  0:8d d2

Specific surface, SV SV ¼ 3pa2ds ; a  0:8d SV ¼ 7:76 d2s d

Eqs. (3)–(5) Eqs. (7),(9), and (10)

ð9Þ ð10Þ

(a)

where B is a dimensionless coefficient related to the cell diameter variation coefficient. With this simple correction, the error committed in estimation of SV and VV can be reduced from 16% to 2% and from 78% to 6%, respectively (see Fig. 7). 4. Verification The model proposed here was verified using commercial alumina foams with different pores per inch (ppi). The porosity and specific surface of foams was calculated using our model and compared with the parameters obtained from other models available in literature. In order to obtain the characteristics of the foams used for verification, 3D X-ray tomography was carried out on representative samples. Subsequently, methods of image analysis were applied to measure structural parameters. The values of these parameters are listed in Table 1. The real foams used for verification had similar porosity, which varied from 69% to 75%. The models listed in Table 2 yielded a strong scatter of the estimates with the porosity differing up to 20% (see Fig. 8a). The highest error of porosity estimation is produced with the models that use cylindrical struts with no struts’ junction correction. Since porosity is typically underestimated the specific surface area is usually overestimated. The results show that the error of estimation of specific surface area might be as high as 80%.

(b)

Fig. 8. Volume fraction of: pores (a) and specific surface area (b) calculated using the formula listed in Table 2.

5. Summary An original geometrical model of random foam structure, based on Laguerre–Voronoi tessellations of randomly spaced spheres with log-normal volume distribution, was proposed. This model permits the study of the variation coefficient effect on the pore size in foams. The results show that higher diversity in the pore size results in the reduction of the average number of faces per foam cell. It was found that this parameter varies from 13.7 to 14.5 for differ-

ent pore size variation coefficients. Compared to structures obtained via Poisson–Voronoi tessellations, where the number of faces per cell is about 15.5, the Laguerre–Voronoi approach gives results, which are closer to the real ones. Further analysis has shown that the specific surface area of foams increases linearly with pore size variation, CV(d). It can be found that the impact of CV(d) on SV is stronger when the porosity of the foam decreases.

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The second issue undertaken here was the geometrical representation of foams struts. The struts of real foams, which typically exhibit thickness distribution, were simplified in the model by using cylinders of a constant diameter. This simplification is very common in the literature. However, it is shown that the error of SV and VV estimation can be significant especially when the porosity is below 70%. Thus, a correction method is proposed which reduces this error to an acceptable level. Finally, the model was tested on a series of real foam structures. The results show that the proposed model provides more accurate results compared with other models available in literature. The proposed model gives a better inside into understanding interrelationships of isotropic foam structures. It enables for calculation structural parameters difficult to measure experimentally, which might have a significant impact on foams properties. Acknowledgement The studies were financially supported by the Polish Ministry of Science and Higher Education under Grant No. N N507 273636. References [1] Z. Jian-zhong, W. Jiu-gen, M. Jia-ju, Journal of Zhejiang University – Science A 6 (10) (2005) 1095–1099. [2] S. Dhara, M. Pradhan, D. Ghosh, P. Bhargava, Advances in Applied Ceramics 104 (1) (2005) 9–21. [3] M.J. Yaszemski, R.G. Payne, W.C. Hayes, R. Langer, A.G. Mikos, Biomaterials 17 (1996) 175–185. [4] B. Borah, G.J. Gross, T.E. Dufresne, T.S. Smith, M.D. Cockman, P.A. Chmielewski, M.W. Lundy, J.R. Hartke, E.W. Sod, Anatomical Record 265 (2001) 101–110. [5] L. Ren, K. Tsuru, S. Hayakawa, A. Osaka, Biomaterials 23 (2002) 4765–4773. [6] V.I. Sikavitsas, J.S. Temenoff, A.G. Mikos, Biomaterials 22 (2001) 2581–2593. [7] W. Swieszkowski, B.H. Tuanb, K.J. Kurzydlowski, D.W. Hutmacherc, Biomolecular Engineering 24 (5) (2007) 489–495.

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