S.V. Raj

Glenn Research Center, Cleveland, Ohio

Microstructural Characterization of Metal Foams: 

An Examination of the Applicability of the

Theoretical Models for Modeling Foams

NASA/TM—2010-216342/REV1

February 2011

This publication replaces NASA/TM—2010-216342, June 2010.
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S.V. Raj

Glenn Research Center, Cleveland, Ohio

Microstructural Characterization of Metal Foams: 

An Examination of the Applicability of the

Theoretical Models for Modeling Foams

NASA/TM—2010-216342/REV1

February 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The investigation was funded by NASA’s Subsonic Fixed Wing Program. Discussions with Prof. John Russ, North Carolina State 

University, North Carolina, and Dr. Don Roth, NASA Glenn Research Center, Cleveland, Ohio, are gratefully acknowledged. 

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Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

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This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.

Document Change History

NASA/TM—2010-216342/REV1, February 2011

Microstructural Characterization of Metal Foams: An Examination of the Applicability of the

Theoretical Models for Modeling Foams

S.V. Raj

This publication replaces NASA/TM—2010-216342, June 2010.
NASA/TM—2010-216342/REV1 1 

Microstructural Characterization of Metal Foams: An Examination 

of the Applicability of the Theoretical Models for Modeling Foams

S.V. Raj 

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135 

Abstract

Establishing the geometry of foam cells is useful in developing microstructure-based acoustic and 

structural models. Since experimental data on the geometry of the foam cells are limited, most modeling 

efforts use an idealized three-dimensional, space-filling Kelvin tetrakaidecahedron. The validity of this 

assumption is investigated in the present paper. Several FeCrAlY foams with relative densities varying 

between 3 and 15 percent and cells per mm (c.p.mm.) varying between 0.2 and 3.9 c.p.mm. were 

microstructurally evaluated. The number of edges per face for each foam specimen was counted by 

approximating the cell faces by regular polygons, where the number of cell faces measured varied 

between 207 and 745. The present observations revealed that 50 to 57 percent of the cell faces were 

pentagonal while 24 to 28 percent were quadrilateral and 15 to 22 percent were hexagonal. The present 

measurements are shown to be in excellent agreement with literature data. It is demonstrated that the 

Kelvin model, as well as other proposed theoretical models, cannot accurately describe the FeCrAlY foam 

cell structure. Instead, it is suggested that the ideal foam cell geometry consists of 11 faces with three

quadrilateral, six pentagonal faces and two hexagonal faces consistent with the 3-6-2 Matzke cell. A 

compilation of 90 years of experimental data reveals that the average number of cell faces decreases 

linearly with the increasing ratio of quadrilateral to pentagonal faces. It is concluded that the Kelvin 

model is not supported by these experimental data.

1.0 Introduction

Aircraft engine noise is a major environmental concern especially in regions surrounding an airport 

during takeoff and landing (Ref. 1). Significant progress has been made since the advent of the first 

commercial jet engine-powered airplanes with current ultrahigh bypass engines being much quieter than 

the first generation engines. For example, the effective perceived noise level in decibels (EPNdB) relative 

to the International Civil Aviation Organization’s (ICAO) Chapter 3 certification standards decreased 

from about +5 EPNdB for aircraft engines developed in the 1960s to –5 EPNdB for modern engines 

(Refs. 2 and 3). Despite this large improvement in engine design, there is still a great desire among policy 

makers and designers to reduce noise much below current levels. For example, the National Aeronautics 

and Space Administration (NASA) has set ambitious goals to further reduce aircraft noise by –52 db with 

respect to the newly adapted ICAO’s Chapter 4 certification standards by the year 2020 under its 

Subsonic Fixed Wing (SFW) project (Ref. 4). It is expected that these noise reduction goals will be 

achieved through a combination of design changes and development of suitable materials (Refs. 3 and 4). 

Polymeric foams have been historically used for sound absorption in several applications (Ref. 5). 

More recently, metal foams are being investigated for their flow resistance (Refs. 6 and 7) and sound 

absorption properties (Refs. 8, 9, and 10). Metal foams have been proposed for use in jet engines as 

acoustic treatment over rotors (Ref. 11), fan blades (Ref. 12) and other applications (Ref. 13). The 

acoustic and other properties of foams are dependent on their relative density, ρ*/ρs, where ρ* and ρs are 

the densities of the foam and the solid material, respectively, and microstructure (Ref. 5). Simple 

formulae exist for correlating relative density and some elements of the microstructure, such as, ligament 

length and thickness (Refs. 5, 6, 7, 8, 9, and 10). However, due to difficulties in controlling process 

variables, the microstructures of the foams and their properties can vary by large amounts. Although 
NASA/TM—2010-216342/REV1 2 

commercially manufactured foams are specified by pores per inch (p.p.i.) and their relative densities, it is 

noted that the reported values of p.p.i. are not necessarily identical from one manufacturer to another 

(Ref. 14). For example, some vendors identify the p.p.i. of their products with that of the precursor 

polyurethane foam rather than the finished product without accounting for metal shrinkage during the 

manufacturing process. 

In the case of metal foams used as acoustic liners in aircraft engines, it is important to qualitatively 

and quantitatively understand the role their microstructures play in affecting their acoustic and 

mechanical properties. Since the complex three-dimensional microstructures of the foams help to 

dissipate the sound energy, it is evident that a quantitative analysis of the foam microstructures would 

enable important correlations to be determined between the microstructural features and the gas pressure 

flow resistance as well as the sound absorption coefficients. These correlations are essential for 

developing microstructure-based models for designing acoustic liners for aircraft engines. Particularly, 

establishing the three-dimensional topology of the cell microstructures of foams is important effectively 

to model fluid flow through them and to understand their mechanical properties. 

Modeling activities on foam cell structures fall into two broad categories: (a) idealized topological 

models based on minimizing the ratio of the surface free energy to volume free energy that can fill threedimensional (3–D) space; and (b) engineering models based on the actual reconstruction of the 3–D foam 

microstructures. Among the several possible idealized topological representations of the foam 

microstructures (Ref. 5), the three-dimensional, space-filling Kelvin tetrakaidecahedron (Refs. 5, 15, and 

16) is often favored for modeling the foam cellular network. This cell has 14 faces consisting of six 

squares and eight hexagonal faces. In other words, about 43 percent of the faces are squares, zero percent

faces are pentagonal and 57 percent of the faces are hexagonal. It is worth noting that other topological 

models have been proposed, where pentagonal faces are incorporated in the cell geometry (Refs. 17 and 

18). The Kelvin model assumes that all cells are all of the same size and volume so that the problem 

becomes one of determining the cell shape that can pack 3–D space resulting in a system with the lowest 

free energy (Ref. 15). In reality, cells deviate from these ideal conditions, where they may be distorted 

and their sizes and shapes non-uniform. Alternatively, recent computational models use actual 3–D foam 

microstructures as an input to the model. However, these models require the availability of high-powered 

computational capabilities to handle the large megabytes of input data representing the foam 

microstructures. The input data for these models are expensive to generate, and the models tend to be 

rather complex. Since foam microstructures are complex, it is necessary to develop both the relatively 

simple and elegant topological mathematical models, as well as, the complex, but realistic, computational 

engineering models in order to completely understand the microstructure-property relationships of foams. 

Several investigators have tried to evaluate the 3–D shape of fat cells (Ref. 19), soap bubbles (Refs. 

20 and 21), grains (Refs. 22, 23, 24, 25, 26, 27, 28, and 29) and foam cells (Refs. 30, 31, 32, and 33). 

The measurement techniques used in these investigations include conventional microstructural image 

analysis, serial section metallography, optical and x-ray micro-computerized tomography (μCT), 

magnetic resonance imaging (MRI), ultrasonic imaging and laser confocal microscopy (Refs. 14 and 34). 

Most of these procedures have advantages and disadvantages. The well-established quantitative 

metallography techniques (Refs. 23, 24, 25, 26, 27, 28, 29, 34, and 35) are relatively simple, inexpensive, 

and provide high resolution images which enable the acquisition of a large amount of statistically relevant 

data with relative ease. However, these procedures are destructive in nature and the 3–D information of 

the microstructure can only be inferred from the two-dimensional (2–D) sections using well-developed 

mathematical and stereological methods (Refs. 24, 25, 34, and 35). The advent of powerful computers and 

the availability of specialized software with capabilities to reconstruct 3–D images by “stitching” several 

closely-spaced 2–D images has enabled the recent development of several techniques, such as optical 

tomography (Ref. 21), MRI (Ref. 30) and μCT (Ref. 32), for accurately reproducing the complex 3–D 

foam microstructures. The primary advantages of these methods is that the resulting 3–D images along 

with quantitative information on the foam microstructures can provide a realistic image of the 3–D spatial 

distribution of the cells. In recent years, μCT is increasingly used to characterize foam microstructures

due to the advantages of using 3–D reconstructed images as input to the computational engineering 
NASA/TM—2010-216342/REV1 3 

models. Despite its relative popularity, it is time consuming, expensive and possesses a lower image 

resolution than conventional metallographic methods (Ref. 31). Typically, only a few foam cells are 

sampled so that the statistical sampling size is limited. 

The objectives of this investigation were to characterize the microstructures of PORVAIR1 metal 

foams. Quantitative information on ligament (or struts) dimensions, cell face dimensions, area fractions of 

open and closed faces, geometric shapes of the cell faces and distribution of ligament porosity were 

determined. Specifically, the present paper reports statistical data on the geometrical features of the cells 

faces to determine the validity of the Kelvin (Ref. 15) and other theoretical space-filling models (Refs. 16, 

17, and 18) in a comprehensive manner. Other quantitative details of the foam microstructure are reported 

elsewhere (Ref. 36). 

2.0 Experimental Procedures

Several FeCrAlY foam panels approximately 210 x 210 mm2 in cross-sectional area and varying in 

thicknesses between 3.2 and 25.4 mm were procured from PORVAIR Fuel Cells Technology, Inc., 

Hendersonville, North Carolina. The foam panels were manufactured from precursor polyurethane foams 

dipped in metal powder slurries followed by sintering of the powder and burning off the polymer foams. 

The c.p.mm. varied between 0.2 (5 p.p.i.) and 3.9 (100 p.p.i.), whereas ρ*/ρS varied between 3 and 

15 percent. Square specimens ~ 25.4 x 25.4 mm in cross-sectional dimensions or 50 mm in diameter 

were wire electro-discharge machined from these panels for metallographic analyses. On close 

examination, it was observed that the microstructures of these foams are extremely complicated and 

difficult to characterize. The foam microstructures consisted of interconnected cells randomly stacked in a 

three-dimensional array with the cell boundaries moving in and out of the field of view. 

Preliminary attempts to study the shapes of the foam cells using either μCT with resolutions varying 

between 20 and 100 μm or an automated serial sectioning2 of a FeCrAlY foam specimen and the 

subsequent 3–D reconstruction of the 2–D sectioned images proved to be unsatisfactory since the cell 

outline could not be clearly defined in the images. Instead, macrophotographs were obtained of the asreceived foam specimens (Fig. 1(a)). This technique allowed a 3–D visualization of the foam 

microstructure with several adjacent faces of a cell being clearly demarcated (Fig. 1(b)). It is noted that 

Figure 1(a) is similar to the 3–D reconstructed image of polymer foams (Ref. 31) except that the present 

imaging technique is faster and cheaper. Quantitative metallographic measurements were conducted on 6 

to 7 randomly selected areas for each foam specimen and a large number of faces were measured to

ensure that the measurements were representative and to minimize measurement errors. The number of 

edges per face was counted by assuming that the faces could be approximated by regular polygons with 

the number of cell faces measured varying between 207 for foams 0.2 c.p.mm. to 745 for 3.9 c.p.mm. 

This assumption was not always valid since some faces were either circular or elliptical rather than 

polygonal and the edges were often curved. In some instances, the edges of a face curved out of the plane 

of view. In addition, two adjacent edges did not meet always at a relatively sharp point but had a 

significant curvature, while adjacent faces met at triple surfaces rather than triple points in many 

instances. These issues complicated the measurements and they are likely to add to the errors in 

measurements. Nevertheless, by measuring a large number of faces, it was felt that the errors in 

measurement would be minimized. It is noted that a similar method was used by Montminy et al.

(Ref. 31) to analyze 3–D μCT images. 

 

1

PORVAIR is the trademark of PORVAIR Fuel Cells Technology, Hendersonville, North Carolina. 2

The automated sectioning of the FeCrAlY foams and the 3–D image reconstruction was conducted by UES, Inc., 

Dayton, Ohio.
NASA/TM—2010-216342/REV1 4 

Figure 1.—(a) Optical macrograph of a FeCrAlY foam with a nominal pore 

density of 0.2 c.p.mm. (5 p.p.i) and ρ*/ρs = 3.3 percent; (b) polygonal 

representations of the faces, 1, 2, 3 and 4 enclosed by the broken circle 

belong to the same cell.

3.0 Results and Discussion

Figures 1(a) shows an optical macrograph of a FeCrAlY foam specimen with a nominal cell density 

of 0.2 c.p.mm. (5 p.p.i.) and ρ*/ρS = 3.3 percent; Figure 1(b) shows the corresponding polygonal 

representations of the faces. The numbers identify the faces for tracking purposes. The complex nature of 

the foam microstructures is self evident in these figures. On close examination, it was observed that 

several neighboring faces were part of the same cell. For example, the faces numbered 1, 2, 3 and 4 

(a)

(b)
NASA/TM—2010-216342/REV1 5 

enclosed by the broken circle represent the outer faces a single cell with some of the inner faces of the cell 

visible in the background (Fig. 1(b)). The volume fractions of the open cells decreased while that of the 

closed cells increased with increasing relative density. Since it was often difficult to clearly discern the 

boundaries of closed faces, only the shapes of the open faces were demarcated in these measurements in 

order to minimize errors in measurement. The cells were generally equiaxed irrespective of c.p.mm. and 

relative density. 

Figures 2(a) to (d) show the frequency histogram and cumulative frequency plots of the number of 

edges per face for four FeCrAlY foams. An examination of Figures 2(a) to (d)clearly establishes that 

97 percent of the faces were either four, n4, five, n5, or six, n6,-sided with over 50 percent of the faces 

being five-sided. Less than 1 percent of the faces were triangular and less than 2 percent were heptagonal 

except in the case of foams with 2.4 c.p.mm (60 p.p.i), which had about 4 percent heptagonal faces. The 

average values of the number of edges per face, N , were determined to be 4.9 ± 0.7, 5.0 ± 0.8, 4.9 ± 0.8, 

and 4.9 ± 0.8 for the FeCrAlY foams with actual values of ρ*/ρS being 3.3 percent (0.2 c.p.mm.), 

9.5 percent (2.4 c.p.mm.), 10.1 percent (3.1 c.p.mm.) and 9.3 percent (3.9 c.p.mm.), respectively. The 

errors represent 95 percent confidence levels. Significantly, these observations were not influenced by 

either the relative densities of the foams or the lineal cell densities. 

 (a) (b) 

 (c) (d) 

Figure 2.—Frequency histograms and cumulative frequencies showing the distributions of the number of 

edges per face for FeCrAlY foams with different values of cells per mm and relative densities.
NASA/TM—2010-216342/REV1 6 

Figure 3.—Comparison of the frequency histograms of the distributions 

of the number of edges per face for soap bubbles (Refs. 20 and 21), 

polyurethane foams (Refs. 30 and 31), Ni foam (Refs. 32) and a 

FeCrAlY foam with 0.2 c.p.mm. (5 p.p.i.) and ρ*/ρs = 3.3 percent. 

The solid squares and associated legends represent the theoretical 

values for the Kelvin tetrakaidechedron.

Figure 3 compares the present results with similar measurements on soap bubbles (Refs. 20 and 21), 

polyurethane foams (Refs. 30 and 31) and Ni foam (Ref. 32). These literature data include measurements 

conducted on both surface and internal cells using different measurement techniques. Table 1 compares

the percentages of four, five, and six-sided faces observed on the FeCrAlY foams with those reported for 

fat cells (Ref. 19), soap bubbles (Refs. 20, 21, and 22), β-brass grains (Ref. 22), and foams (Refs. 22, 30, 

31, and 32). It is noted that the data compiled in Table 1 were obtained by several different techniques 

ranging from simple visual observations to complex NMR and μCT 3–D scans over a 90-year period. 

Significantly, in all cases, more than 50 percent of the cell faces had a pentagonal geometry irrespective 

of the material and measuring technique used (Table 1 and Fig. 3). The present results fall well within the 

range of other observations reported in the literature. 

An examination of Figure 3 shows that the Kelvin tetrakaidecahedron model (Ref. 15), which predicts 

0 percent five-sided faces, is inconsistent with the experimental observations. The fact that the Kelvin 

model fails to be consistent with the experimental results is not surprising. This model is based on a 

mathematical conjecture that soap bubbles and foam microstructures can be ideally represented by 

dividing three-dimensional space into cells of equal volume in a manner that follows Plateau’s rules for 

mechanical equilibrium and minimization of the surface area (Ref. 37). It is noted that the Kelvin model 

requires the arrangement of tetrakaidecahedron cells to be topologically ordered and spatially periodic to 

fill space. Real foams are far from this ideal configuration since factors, such as residual stresses due to 

processing methods, topological disorder (Ref. 37), unequal cell volumes, aperiodic spatial ordering of 

the cells (Ref. 20), and thick ligaments and triple points, can influence the cell topology. Matzke (Ref. 20) 

studied 400 peripheral soap bubbles and observed that the largest number of them possessed 11-hedra 

cells with three four-sided, six five-sided and two six-sided faces (3-6-2)3

 (Fig. 4). However, these soap 

bubbles only constituted 17 percent of the total number of bubble studied since 20 other shapes were 

observed. In contrast, 97 percent of the cell faces in the FeCrAlY foams were either four, five or six-

 

3

This nomenclature of identifying the cells was suggested by Kraynik et al. (Ref. 37].
NASA/TM—2010-216342/REV1 7 

sided. Therefore, it would be interesting to determine the number of faces for the ideal cell representing 

the microstructures of the FeCrAlY foams. 

TABLE 1.—COMPARISON OF THE PERCENTAGES OF FOUR, FIVE AND SIX-SIDED FACES OBSERVED IN 

FeCrAlY FOAMS WITH OBSERVATIONS ON FAT CELLS (REF. 19), SOAP BUBBLES (REFS. 20, 21, AND 22), 

β-BRASS GRAINS (REF. 22), AND FOAMS (REFS. 22, 30, 31, AND 32)

Description Measurement

technique

Percentage of polyhedral faces, n4, n5 and n6

Fat cells

(Lewis (Ref. 19)) 

Optical microscopy or visual n4 = 21%; n5 = 53%; n6 = 23% 

Soap bubbles (Matzke (Ref. 20)) Optical microscopy Peripheral: n4 = 29%; n5 = 53%; n6 = 16% ; 

Central: n4 = 11%; n5 = 67%; n6 = 22% 

Soap bubbles (Monnereau et al. (Ref. 21)) Optical tomography Upper bubbles; n4 = 29%; n5 = 52%; n6 = 18%; 

Internal bubbles: n4 = 18%; n5 = 58%; n6 = 24%

β-brass grains

Soap bubbles

Ammonium oleate foams

Gelatin foams

(Desch (Ref. 22)) 

Visual n4 = 20%; n5 = 44%; n6 = 28% 

n4 = 20%; n5 = 50%; n6 = 22% 

n4 = 21%; n5 = 50%; n6 = 25% 

n4 = 19-38%; n5 = 32-57%; n6 = 10-25% 

Polyurethane foams (Kose (Ref. 30))

(Montminy et al. (Ref. 31)) NMR

μCT

n4 = 9%; n5 = 70%; n6 = 21% 

n4 = 24%; n5 = 55%; n6 = 19%

Open cell Ni foam (Dillard et al. (Ref. 32)) μCT n4 = 18%; n5 = 57%; n6 = 22%

FeCrAlY foams 

(Present investigation)

Optical microscopy n4 = 25%; n5 = 57%; n6 = 15%

(0.2 c.p.mm.; ρ*/ρs = 3.3%)

n4 = 24%; n5 = 54%; n6 = 18%

(2.4 c.p.mm.; ρ*/ρs = 9.5%)

n4 = 28%; n5 = 52%; n6 = 18%

(3.1 c.p.mm.; ρ*/ρs = 10.1%)

n4 = 26%; n5 = 50%; n6 = 22%

(3.9 c.p.mm.; ρ*/ρs = 9.3%)

Figure 4.—A 3-6-2 11-hydra cell with three quadrilateral, 

six pentagonal and two hexagonal faces (Ref. 20). The 

numbers represent the number of edges enclosing the 

cell face. The blue solid lines representing the forward 

faces are identified by the blue lettering, while the red 

broken lines representing the back faces are identified 

by the red lettering. 
NASA/TM—2010-216342/REV1 8 

Since quantitative optical metallography gives 2–D information, the 3–D topographical characteristics

of the microstructure can be determined from well established stereology equations (Refs. 23, 24, 25, 26, 

27, 28, 29, 34, 35, 38, 39, and 40). The number of faces per cell, F, the number of edges per cell, E, and 

the number of vertices per cell, V, of the 3–D cell are related by the Euler equation (Refs. 5, 25, 34, and 

39) and they can be determined from N using the Coxeter equations (Ref. 41) 

[ ] − N = 6

12 F (1a) 

[ ] N

N

− = 6

6 E (1b) 

[ ] N

N

− = 6

4 V (1c) 

Table 2 shows the calculated values of F, E, V, and the corresponding experimental values of N4, N5

and N6 for the four FeCrAlY foams.

4 Using the measured values of N , the corresponding values of F

calculated from equation (1a) are 11.0, 11.7, 11.1 and 11.4 for foams with 0.2 (5 p.p.i.), 2.4 (60 p.p.i.), 3.1 

(80 p.p.i.) and 3.9 c.p.mm. (100 p.p.i.), respectively. Based on these results, the topological characteristics 

of the ideal PORVAIR foam cell are: F = 11, E = 27 and V = 18, which satisfy Euler’s theorem (i.e., V – 

E + F = 2) with N4 = 3, N5 = 6 and N6 = 2. These values are independent of relative density. 

TABLE 2.—CALCULATED VALUES OF F, E, V, N4, N5 AND N6 FOR FeCrAlY FOAMS

Linear cell density

(c.p.mm)

ρ*/ρs

(%)

F E V N4 N5 N6

0.2 (5 p.p.i.) 3.3 11.0 26.7 17.8 3 6 2

2.4 (60 p.p.i.) 9.5 11.7 30.0 20.0 3 6 or 7 2

3.1 (80 p.p.i.) 10.1 11.1 26.7 17.8 3 6 2

3.9 (100 p.p.i.) 9.3 11.4 26.7 17.8 3 6 2 or 3

Average 11.3 27.5 18.4 3 6 2

Table 3 compares the topological features of the FeCrAlY foams with several simple cell shapes

(Ref. 5), where C is the number of cells. The topological characteristics of the FeCrAlY foams do not 

agree with any of these simple geometries. Instead, they appear to be closer to the topological structure of 

clathrates although more detailed topological modeling needs to be conducted to establish this possibility 

(Refs. 42 and 43). As noted above, Matzke (Ref. 20) observed that most of the peripheral soap bubbles 

were eleven-hedra cells with three four-sided, six five-sided and two six-sided faces (3-6-2). Based on the 

excellent agreement between the present results and Matzke’s data on peripheral soap bubbles (Ref. 20) 

(Fig. 3) taken together with the fact that the total number of faces for the FeCrAlY foams was determined 

to be 11 (Table 2), it is reasonable to suggest that the 11-hedra 3-6-2 cell is the most representative of the 

FeCrAlY foam cellular structure. 

Table 4 shows the predicted (Refs. 15, 16, 17, and 18) and the experimental (Refs. 20, 21, and 30) 

percentage distributions of polyhedral faces and the average number of faces per cell, Faverage. As noted 

earlier, the data were obtained by different methods on several materials over a 90-year period. The 

average value of F = 11.3 determined for the FeCrAlY foam cells (Table 2) is in very good agreement 

with the experimental observations on the peripheral (Ref. 20) or upper (Ref. 21) soap bubbles and gelatin 

foams (Ref. 22) for which the average number of faces is about 11. 

 

4

In this paper, ni represents the percentage of faces with i edges, whereas Ni is number of such faces enclosing the 

cell.
NASA/TM—2010-216342/REV1 9 

TABLE 3.—COMPARISON OF THE GEOMETRIC PROPERTIES OF FeCrAlY 

FOAM CELLS WITH THOSE FOR SIMPLE POLYHEDRA (REF. 5) 

Cell shape Number of face shapes F E V C Remarks

3 4 5 6

Tetrahedron 4 -- -- -- 4 6 4 1 Regular Platonic solid

Triangular prism 2 3 -- -- 5 9 6 1 Packs to fill space

Square Prism -- 6 -- -- 6 12 8 1 Packs to fill space

Hexagonal Prism -- 6 -- 2 8 18 12 1 Packs to fill space

Octahedron 8 -- -- -- 8 12 6 1 Regular Platonic solid

Rhombic Dodecahedron -- 12 -- -- 12 24 14 1 Packs to fill space

Pentagonal Dodecahedron -- -- 12 -- 12 30 20 1 Regular Platonic solid

Tetrakaidecahedron -- 6 -- 8 14 36 24 1 Packs to fill space

Icosahedron 20 -- -- -- 20 30 12 1 Regular Platonic solid

3-6-2 cell -- 3 6 2 11 27 18 1 FeCrAlY foam

(present investigation)

TABLE 4.—COMPARISON OF THE GEOMETRIC PROPERTIES OF THE CELLS PREDICTED BY SEVERAL 

THEORETICAL MODELS (REFS. 15, 16, 17, AND 18) AND EXPERIMENTAL DATA (REFS. 20, 21, 22, 30, 31, AND 32) 

Description Percentage of polyhedral faces, n4, n5 and n6 Faverage

Kelvin cell (Ref. 15) n4 = 43%; n5 = 0%; n6 = 57% 14

Williams cell (Ref. 17) n4 = 14%; n5 = 57%; n6 = 29% 14

Weaire and Phelan model (Refs. 16 and 18) n4 = 0%; n5 = 89%; n6 = 11% 13.4

Soap bubbles (Matzke (Ref. 20)) Peripheral: n4 = 29%; n5 = 53%; n6 = 16% 

Central: n4 = 11%; n5 = 67%; n6 = 22% 

11.0 (peripheral) 

13.7 (central)

Soap bubbles (Monnereau et al. (Ref. 21)) Upper bubbles: n4 = 29%; n5 = 52%; n6 = 18% 

Internal bubbles: n4 = 18%; n5 = 58%; n6 = 24%

11.1 (upper bubbles) 

13.5 (internal bubbles)

β-brass grains

Soap bubbles

Ammonium oleate foams

Gelatin foams

(Desch (Ref. 22)) 

n4 = 20%; n5 = 44%; n6 = 28% 

n4 = 20%; n5 = 50%; n6 = 22% 

n4 = 21%; n5 = 50%; n6 = 25% 

n4 = 19-38%; n5 = 32-57%; n6 = 10-25% 

14.5

13.0

13.0

9.0-11.0

Polyurethane foam (Kose (Ref. 30))

(Montminy et al. (Ref. 31)) n4 = 9%; n5 = 70%; n6 = 21% 

n4 = 24%; n5 = 55%; n6 = 19%

13.6

13.0

Open cell Ni foam

(Dillard et al. (Ref. 32)) 

n4 = 17.6%; n5 = 56.8%; n6 = 21.8% 13.0

FeCrAlY foams (Present investigation) n4 = 24-28%; n5 = 50-57%; n6 = 15-22% 11.3

A close examination of Table 4 reveals that the present results do not agree with the predictions of the 

three topological models (Refs. 15, 16, 17, and 18). The Kelvin cell (Ref. 15) does not possess any 

pentagonal faces, whereas the Weaire-Phelan model (Refs. 16, 18, and 30) does not have any 

quadrilateral faces, with the total number of faces being either 14 or 13.4, respectively. The Williams cell

(Ref. 17) with 14 faces possesses 14 percent quadrilateral, 57 percent pentagonal and 29 percent

hexagonal faces. However, this model also does not agree with the present observations on the FeCrAlY 

foams. This difference between the experimental results and the theoretical predictions is to be expected 

since theoretical efforts mainly consider the surface and volume free energy contributions to the total free 

energy (Ref. 37). As indicated earlier, other factors can influence the final cell topology of real foams. For 

example, the effects of residual stresses developed in the foam panels during processing have not included 

in these theoretical derivations. Qualitatively, one can modify the Gibbs free energy equation as follows: 

∆G = (∆gv + ∆ge) ⋅ VC + ∆gs ⋅ SC (2) 

where, ∆G, ∆gv, ∆ge and ∆gs are the changes in the total, volume, residual strain and surface Gibbs free 

energies, respectively, VC is the cell volume and SC is the surface are of the cell. It is important to note 

that current theoretical models agree incorrectly assume that ∆ge = 0 for real foams. 
NASA/TM—2010-216342/REV1 10

Table 4 shows that Faverage varied between 9.0 and 14.5 (Refs. 20, 21, 22, and 30). On further 

examination of the data, Faverage decreases linearly with the increasing ratio, n4/n5, (Fig. 5(a)) 

Faverage = –5.1(n4/n5) + 14.3 (Rd

2

= 0.461) (3) 

where Rd

2 is the coefficient of determination. In contrast, it is independent of n6/n5 (Fig. 5(b)). The 

regression Equation (3) is represented by the solid line in Figure 5(a); the broken horizontal line in 

Figure 5(b) represents the average value of Faverage = 12.2 for all the data. Equation (3) predicts a value of 

Faverage = 14.3 for n4 = 0, Faverage = 9.2 for n4 = n5, and Faverage = ∞ for n5 = 0. Thus, the Kelvin model, for 

which n5 = 0, is not supported by the trend in the experimental data. 

(a)

(b)

Figure 5.—Variation of the average number of cell faces against 

the ratio of (a) quadrilateral to pentagonal, and (b) hexagonal to 

pentagonal faces. The regression line through the data is 

represented by the solid line in (a). The present data on 

FeCrAlY foams are compared with literature data (Refs. 20, 21, 

22, 30, 31, and 32). 
NASA/TM—2010-216342/REV1 11

4.0 Summary and Conclusions

A detailed microstructural analysis of several FeCrAlY metal foams with relative densities varying 

between 3 and 15 percent, and linear cell densities varying between 0.2 and 3.9 c.p.mm., was conducted 

to evaluate the topology of the foam cells. The shapes of cell faces were evaluated by approximating the 

faces by regular polygons. It was observed that between 24 and 28 percent of the cell faces were 

quadrilateral, 50 to 57 percent pentagonal, and 15 to 22 percent hexagonal in morphology. The present 

results are in excellent agreement with observations on soap bubbles (Refs. 20 and 21). Based on 

Matzke’s observations (Ref. 20), it is suggested that the FeCrAlY foam cells had a total of 11 faces with 

three quadrilateral, six pentagonal and two hexagonal faces. Both sets of results do not agree with the 

14-hedra Kelvin tetrakaidecahedron model (Ref. 15), which only has 43 and 57 percent quadrilateral and 

hexagonal faces, respectively. Neither do the present results agree with the Williams (Ref. 17) and 

Weaire-Phelan models (Refs. 16, 18, and 30) models. The present calculations show that the 3-6-2 cell, 

which probably best describes the FeCrAlY foam cells, has 27 edges and 18 vertices. A compilation of 

90 years of experimental data reveals that the average number of cell faces decreases linearly with the 

increasing ratio of quadrilateral to pentagonal faces. It is concluded that the Kelvin model is not supported 

by these experimental data. 

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Microstructural Characterization of Metal Foams: An Examination of the Applicability of the 

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14. ABSTRACT

Establishing the geometry of foam cells is useful in developing microstructure-based acoustic and structural models. Since experimental data 

on the geometry of the foam cells are limited, most modeling efforts use an idealized three-dimensional, space-filling Kelvin 

tetrakaidecahedron. The validity of this assumption is investigated in the present paper. Several FeCrAlY foams with relative densities 

varying between 3 and 15 percent and cells per mm (c.p.mm.) varying between 0.2 and 3.9 c.p.mm. were microstructurally evaluated. The 

number of edges per face for each foam specimen was counted by approximating the cell faces by regular polygons, where the number of 

cell faces measured varied between 207 and 745. The present observations revealed that 50 to 57 percent of the cell faces were pentagonal 

while 24 to 28 percent were quadrilateral and 15 to 22 percent were hexagonal. The present measurements are shown to be in excellent 

agreement with literature data. It is demonstrated that the Kelvin model, as well as other proposed theoretical models, cannot accurately 

describe the FeCrAlY foam cell structure. Instead, it is suggested that the ideal foam cell geometry consists of 11 faces with three 

quadrilateral, six pentagonal faces and two hexagonal faces consistent with the 3-6-2 Matzke cell. A compilation of 90 years of experimental 

data reveals that the average number of cell faces decreases linearly with the increasing ratio of quadrilateral to pentagonal faces. It is 

concluded that the Kelvin model is not supported by these experimental data.

15. SUBJECT TERMS

Metal foams; Acoustic models; Foam cell topology; Kelvin cell 

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