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

June 2010
NASA STI Program . . . in Profi le

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientifi c and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Offi cer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major signifi cant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signifi cant 

scientifi c and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientifi c 

and technical fi ndings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientifi c and 

technical fi ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientifi c and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientifi c, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientifi c and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question via the Internet to help@

sti.nasa.gov

• Fax your question to the NASA STI Help Desk 

at 443–757–5803

• Telephone the NASA STI Help Desk at

 443–757–5802

• Write to:

 NASA Center for AeroSpace Information (CASI)

 7115 Standard Drive

 Hanover, MD 21076–1320
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

June 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The investigation was funded by NASA’s Subsonic Fixed Wing Program.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://gltrs.grc.nasa.gov

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 

Space Administration.

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.
NASA/TM—2010-216342 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 the 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% 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% of the cell faces were pentagonal while 24 to 28% were quadrilateral 

and 15 to 22% 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 3 quadrilateral, 6 pentagonal faces and 2 hexagonal faces 

consistent with the 3-6-2 cell. 

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, */s, where * and s 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 to 10). However, due to difficulties in controlling process variables, the 

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

example, some vendors specify 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. 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 6 squares and 8 hexagonal faces. In other words, 

about 43% of the faces are squares, 0% faces are pentagonal and 57% 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). In reality, cells deviate from these ideal conditions, where they 

may be distorted and their sizes and shapes non-uniform. Therefore, the objectives of this paper are to 

statistically evaluate the geometrical features of the foam cells in order to identify the ideal polyhedral 

topography to use in acoustic and structural models. 

2.0 Experimental Procedures 

Several FeCrAlY foam panels approximately 210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 by a proprietary process from 

precursor polyurethane foams dipped in metal powder slurries followed by sintering of the powder and 

burning off the polymer foams. The nominal c.p.mm. varied between 0.2 (5 p.p.i.) and 3.9 (100 p.p.i.), 

whereas */S varied between 3 and 15%. Square specimens ~25.4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. 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. A specialized digital optical microscope with a large depth of focus was 

used in making these in-situ measurements. The number of edges per face was measured 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. 
NASA/TM—2010-216342 3 

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 */S = 3.3%; Figure 1(b) shows the corresponding polygonal representations of the faces. Since portions of the inner layer were covered by ligaments of the top layer, the 

outline of these cells were demarcated as carefully as possible. The numbers identify the faces for 

tracking purposes. The complex nature of the foam microstructures is self evident in these figures. 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. 

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 */s = 3.3%; (b) polygonal 

representations of the faces forming the cells. 

(a)

(b)
NASA/TM—2010-216342 4 

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 establish that 97% 

of the faces were either four, n4, five, n5, or six, n6,-sided with over 50% of the faces being five-sided. 

Less than 1% of the faces were triangular and less than 2% were heptagonal except in the case for foams 

with 2.4 c.p.mm (60 p.p.i), which had about 4% 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 */S being 3.3% (0.2 c.p.mm.), 9.5% (2.4 c.p.mm.), 10.1% (3.1 c.p.mm.) and 9.3% 

(3.9 c.p.mm.), respectively. Significantly, these observations were not influenced by either the relative 

densities of the foams or the lineal cell densities. 

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. 

(b)

(a)
NASA/TM—2010-216342 5 

Figure 2.—Concluded. 

 

(c)

(d)
NASA/TM—2010-216342 6 

Figure 3 compares the present results with similar measurements on soap bubbles (Refs. 19 to 21) and 

polyurethane foams (Ref. 22). These literature data include measurements conducted on both surface and 

internal cells using different measurement techniques. Table 1 provides details of the percentages of four, 

five, and six-sided faces observed on the FeCrAlY foams and reported in the literature (Refs. 19 to 22), 

which were the predominant faces observed in these foams. Significantly, in all cases, more than 50% of 

the cell faces had a pentagonal geometry irrespective of the foam material and measuring technique used. 

The present results fall within the range of other observations reported in the literature. Other 

observations on plant cells (Ref. 17) and metal grains (Ref. 23),

 have also reported a predominance of 

pentagonal faces in nature. 

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

0% 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. 24). It is noted that the Kelvin model 

requires the arrangement of tetrakaidechedron 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. 24), unequal cell volumes, aperiodic spatial ordering of 

the cells (Ref. 19), and thick ligaments and triple points, can influence the cell topology. 

Figure 3.—Comparison of the frequency histograms of the distributions of the number of edges per face 

for soap bubbles (Refs. 19 and 21), polyurethane foams (Ref. 22) and a FeCrAlY foam with 0.2 c.p.mm. 

(5 p.p.i.) and */s. The solid squares and associated legends represent the theoretical values for the 

Kelvin tetrakaidechedron. 

 
NASA/TM—2010-216342 7 

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

FACES OBSERVED IN FECRALY FOAMS WITH OBSERVATIONS ON SOAP BUBBLES 

(REFS. 19 AND 21) AND POLYURETHANE FOAM (REF. 22). 

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

Soap bubbles (19) Peripheral: n4 = 29%; n5 = 53%; n6 = 16%; 

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

Soap bubbles (21) Upper bubbles; n4 = 29%; n5 = 52%; n6 = 18%; 

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

Polyurethane foam (22) n4 = 9%; n5 = 70%; n6 = 21% 

FeCrAlY foams 

(Present investigation) 

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

(0.2 c.p.mm.; */s = 3.3%) 

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

(2.4 c.p.mm.; */s = 9.5%) 

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

(3.1 c.p.mm.; */s = 10.1%) 

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

(3.9 c.p.mm.; */s = 9.3%) 

Although the excellent agreement between the present results and the Matzke’s data (Ref. 19) is 

encouraging, it is important to note that the present measurements were conducted on cross-sections cut 

through the three-dimensional FeCrAlY foam cells unlike Matzke’s measurements (Ref. 19), which were 

made on peripheral soap bubbles enclosed by the surface of the container. It is also noted that the present 

data are in reasonable agreement with the surface and internal cell data obtained on soap bubbles by 

Monnereau et al. (Ref. 21). Matzke (Ref. 19) studied 400 peripheral soap bubbles and observed that the 

largest number of them possessed eleven-hedra cells with 3 four-sided, 6 five-sided and 2 six-sided faces 

(3-6-2)1

 (Fig. 4). However, these soap bubbles only constituted 17% of the total number of bubbles 

studied since twenty other shapes were observed. In contrast, 97% of the cell faces in the FeCrAlY foams 

were either four, five or six-sided. 

Since quantitative optical metallography gives two-dimensional information, there is no easy and 

direct way to determine the three-dimensional topographical features of the foam cells. However, the 

number of faces per cell, F, the number of edges, E, and the number of vertices, V, of the threedimensional cell can be determined from the Coexeter equations (Refs. 5 and 25)

   n  6

12 F (1a) 

  n

n

  6

6 E (1b) 

  n

n

  6

4 V (1c)

 

1

This nomenclature of identifying the cells was suggested by Kryanik et al. (Ref. 24). 
NASA/TM—2010-216342 8 

Figure 4.—A 3-6-2 eleven-hydra cell with 3 quadrilateral, 6 pentagonal 

and 2 hexagonal faces (Ref. 19). 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. 

TABLE 2.—CALCULATED VALUES OF F, E, V, 

n4, n5, and n6 FOR FECRALY FOAMS 

Linear cell density 

(c.p.mm) 

*/s 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 

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. 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. Most of the values of F, E and V are 11, 27, and 

18, respectively, with average values of n4 = 3, n5 = 6, and n6 = 2. These values are independent of 

relative density. 

 
NASA/TM—2010-216342 9 

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. 26 and 27). As noted above, Matzke (Ref. 19) observed that most of the peripheral soap bubbles 

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

excellent agreement between the present results and Matzke’s data on peripheral soap bubbles (Ref. 19) 

(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 3-6-2 cell is the most representative of the FeCrAlY 

foam cellular structure. 

Table 4 shows the predicted (Refs. 15 to 18) and the experimental (Refs. 19, 21, and 22) percentage 

distributions of polyhedral faces and the average number of faces per cell. The percentages of n4, n5, and 

n6 for the relatively complex Weaire and Phelan model (Refs. 16 and 18) correspond to the values 

reported by Kose (Ref. 22). The present observations are in very good agreement with the experimental 

observations on the peripheral (Ref. 19) or upper (Ref. 21) soap bubbles for which the average number of 

faces is about 11. However, the average number of faces varied between 13 and 14 for the internal cells 

(Refs. 19, 21, and 22). Although it is tempting to conclude from these comparisons that the present 

observations on the FeCrAlY foam panels were influenced by surface effects much in the same way as 

the peripheral soap bubbles, it is important to note that the soap bubbles have very low stiffness compared 

to the FeCrAlY foams and they were formed dynamically. In contrast, the present observations were 

conducted on sections cut through preformed FeCrAlY panels, which are much stiffer than the soap 

bubbles. Therefore, it is doubtful that the present results can be attributed to surface effects. 

TABLE 3.—COMPARISON OF THE GEOMETRIC PROPERTIES OF FeCrAlY 

FOAM CELLS WITH THOSE FOR SIMPLE POLYHEDRAL (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 TO 18) AND EXPERIMENTAL DATA (REFS. 19 TO 22) 

Description Percentage of polyhedral faces, n4, n5, and n6 Average number of faces 

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 22) 

n4 = 0%; n5 = 89%; n6 = 11% 13.4 

Soap bubbles (Ref. 19) Peripheral: n4 = 29%; n5 = 53%; n6 = 16% 

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

11.0 (peripheral) 

13.7 (central) 

Soap bubbles (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) 

Polyurethane foam (Ref. 22) n4 = 9%; n5 = 70%; n6 = 21% 13.6 

FeCrAlY foams 

(Present investigation) 

n4 = 24-27%; n5 = 55-60%; n6 = 14-18% 11.3 
NASA/TM—2010-216342 10 

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

three cell topological models (Refs. 15 to 18). The Kelvin cell (Ref. 15) does not possess any pentagonal 

faces, whereas the Weaire-Phelan model (Refs. 16, 18, and 22) 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 a total of 

14 faces possess 14% quadrilateral, 57% pentagonal, and 29% 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. 24). As indicated 

earlier, other factors can influence the final cell topology of real foams. For example, the effect 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: 

     ev    sSgVggG (2) 

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

energies, respectively, V is the cell volume and S 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. 

4.0 Summary and Conclusions 

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

between 3 and 15%, 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 

edges by regular polygons. It was observed that between 24 and 28% of the cell faces were quadrilateral, 

50 to 57% pentagonal, and 15 to 22% hexagonal in morphology. The present results are in excellent 

agreement with observations on soap bubbles (Refs. 19 and 21). Based on Matzke’s observations 

(Ref. 19), it is suggested that the FeCrAlY foam cells had a total of 11 faces with 3 quadrilateral, 

6 pentagonal, and 2 hexagonal faces. Both sets of results do not agree with the 14-hedra Kelvin 

tetrakaidecahedron model (Ref. 15), which only has 43 and 57% quadrilateral and hexagonal faces, 

respectively. Neither do the present results agree with the Williams (Ref. 17) and Weaire-Phelan model 

(Refs. 16, 18, and 22) 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. 

References 

1. J.L. Kerrebrock: Aircraft Engines and Gas Turbines, The MIT Press, Cambridge, MA, 1992, Ch. 9. 

2. B.L. Koff: “Gas Turbine Technology Evolutin: A Designer’s Perspective,” J. Propulsion Power, 

2004, vol. 20, pp. 577–595. 

3. D.L. Huff and E. Envia: “Jet Engine Noise Generation, Prediction, and Control,” Handbook of Noise 

and Vibration Control, edited by M.J. Crocker, John Wiley, Hoboken, NJ, 2007, pp. 1090–1102. 

4. E. Envia and R. Thomas: “Research Progress in Aircraft Noise Research,” ARMD Technical Seminar 

October 16, 2007, http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20080006600_2008004016.pdf, 

NASA Glenn Research Center, Cleveland, OH 2007. 

5. L.J. Gibson and M.F. Ashby: Cellular Solids, Cambridge University Press, Cambridge, U.K., 1997. 

6. C.Y. Zhao, T. Kim, T.J. Liu and H.P. Hodson: “Thermal Transport Phenomena in Porvair Metal 

Foams and Sintered Beds,” Final Report, August 2001, Micromechanics Centre & Whittle Lab., 

Department of Engineering, University of Cambridge, Cambridge, U.K., 2001. 

7. T.J. Lu, M. Kepets and A.P. Dowling: “Acoustic Properties of Sintered FeCrAlY Foams with Open 

Cells (I): Static Flow Resistance,” Sci. China Ser E-Tech Sci., 2008, vol. 51, 1803–1811. 
NASA/TM—2010-216342 11 

8. T.J. Lu, A. Hess and M.F. Ashby: “Sound Absorption in Metal Foams,” J. Appl. Phys., 1999, vol. 85, 

pp. 7528–7539. 

9. T.J. Lu, F. Chen and D. He: “Sound Absorption of Cellular Metals with Semiopen Cells,” J. Acoustic. 

Soc. Am., 2000, vol. 108, pp. 1697–1709. 

10. T.J. Lu, M. Kepets and A.P. Dowling: “Acoustic Properties of Sintered FeCrAlY Foams With Open 

Cells (II): Sound Attenuation,” Sci. China Ser E-Tech Sci., 2008, vol. 51, 1812–1837. 

11. D.L. Sutliff, D.M. Elliott, M.G. Jones, T.C. Hartley: “Attenuation of FJ44 Turbofan Engine Noise 

With a Foam-Metal Liner Installed Over-the-Rotor,” NASA/TM—2009-215666, NASA Glenn 

Research Center, Cleveland, OH, 2009. 

12. S.V. Raj, L.J. Ghosn, B.A. Lerch, M. Hebsur, L.M. Cosgirff and M. Topolski: “An Evaluation of 

Lightweight 17-4PH Stainless Steel Foam Design Concepts for Fan and Propeller Blade 

Applications,” NASA/TM—2005-213620, NASA Glenn Research Center, Cleveland, OH, 2005. 

13. W.E. Azzi, W.L. Roberts and A. Rabiei: “Developing an Application for Refractory Open Cell Metal 

Foams in Jet Engines,” Mater. Res. Soc. Symp. Proc., 2005, vol. 851, p. NN11.3. 

14. S. Mullens, J. Luyten and J. Zeschky: “Characterization of Structure and Morphology,” Cellular 

Ceramics: Structure, Manufacturing, Properties and Applications, edited by M. Scheffler and 

P. Colombo, Wiley-VCH Verlag GmbH & Co. KGaA, 2005, pp. 227–266. 

15. W. Thompson: “On the Dimension of Space with Minimum Partitional Area,” Phil. Mag. 1887, 

vol. 24, pp. 503–514. 

16. D. Weaire and S. Hutzler: The Physics of Foams, Oxford University Press, Oxford, U.K., 1999. 

17. R.E. Williams: “Space-Filling Polyhedron: Its Relation to Aggregates of Soap Bubbles, Plant Cells, 

and Metal Crystallites,” Science, 1968, vol. 161, pp. 276–277. 

18. D. Weaire and R. Phelan: “The Structure of Monodisperse Foam,” Phil. Mag. Lett., 1994, vol. 70, 

pp. 345–350. 

19. E.B. Matzke: “The Three-Dimensional Shape of Bubbles in Foam – An Analysis of the Role of 

Surface Forces in Three-Dimensional Cell Shape Determination,” Amer. J. Botany, 1946, vol. 33, 

pp. 58–80. 

20. C. Monnereau and M. Vignes-Adler: “Optical Tomographic of Real Three-Dimensional Foams,” 

J. Coll. Interface Sci., 1998, vol. 202, pp. 45–53. 

21. C. Monnereau, B. Prunet-Foch, and M. Vignes-Adler: “Topology of Slightly Polydisperse Real 

Foams,” Phys. Rev. E, 2001, vol. 63, p. 061402. 

22. K. Kose: “3D NMR Imaging of Foam Structures,” J. Magnet. Resonance, 1996, vol. A 118, 

pp. 195–201. 

23. C.S. Smith: “Structure, Substructure, Superstructure,” Rev. Modern Phys., 1964, vol. 36, 

pp. 524–532. 

24. A.M. Kraynik, D.A. Reinelt and F.V. Swol: “Structure of Random Foam,” Phys. Rev. Lett., 2004, 

vol. 93, p. 208301. 

25. H.S. M. Coxeter: “Close-Packing and Froth,” Illinois J. Math., 1958, vol. 2, pp. 746–758. 

26. V.I. Kosyakov: “Topology of Polyhedral Clathrate Frameworks. 2. Frameworks Constructed of 

Polyhedral Stories,” J. Struct. Chemistry, 1996, vol. 37, pp. 106–113. 

27. H. Tomono, H. Eguchi, and K. Tsumuraya: “Binding Between Endohedral Na atoms in Si Clathrate I; 

A First Principles Study,” Journal of Physics: Condensed Matter, 2008, vol. 20, p. 385209, 

doi:10.1088/0953-8984/20/38/385209. 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB 

control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-06-2010 

2. REPORT TYPE

Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

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

Theoretical Models for Modeling Foams 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Raj, S., V. 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 561581.02.08.03.15.12 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-17279 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2010-216342 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Categories: 05, 07, 23, 26, and 31 

Available electronically at http://gltrs.grc.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

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 the 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 3 quadrilateral, 6 pentagonal faces and 2 

hexagonal faces consistent with the 3-6-2 cell. 

15. SUBJECT TERMS

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

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

17 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18