S.V. Raj

Glenn Research Center, Cleveland, Ohio

Jacob A. Kerr

Pennsylvania State University, State College, Pennsylvania 

Effect of Microstructural Parameters on the 

Relative Densities of Metal Foams

NASA/TM—2010-216797

December 2010
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S.V. Raj

Glenn Research Center, Cleveland, Ohio

Jacob A. Kerr

Pennsylvania State University, State College, Pennsylvania 

Effect of Microstructural Parameters on the 

Relative Densities of Metal Foams

NASA/TM—2010-216797

December 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This investigation, which was completed as part of Mr. Kerr’s summer internship research, was supported by NASA’s Subsonic 

Fixed Wing Program. Mr. Jacob Kerr thanks NASA’s Undergraduate Student Research Program (USRP) for funding his summer 

internship at the Glenn Research Center.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

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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. 
NASA/TM—2010-216797 1

Effect of Microstructural Parameters on the 

Relative Densities of Metal Foams

S.V. Raj

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Jacob A. Kerr

Pennsylvania State University

State College, Pennsylvania 16801

Abstract

A detailed quantitative microstructural analyses of primarily open cell FeCrAlY and 314 stainless 

steel metal foams with different relative densities and pores per inch (p.p.i.) were undertaken in the 

present investigation to determine the effect of microstructural parameters on the relative densities of 

metal foams. Several elements of the microstructure, such as major and minor cell sizes, cell areas and 

perimeters, ligament dimensions, cell shapes and area fractions of closed and open cells, were measured. 

The cross-sections of the foam ligaments showed a large number of pores, and their circularity factors and 

average sizes were determined. The area fractions of the open cells and ligaments decreased while that of 

the closed cells increased linearly with increasing relative density. The relative densities and p.p.i. were 

not significantly dependent on cell size, cell perimeter and ligament dimensions within the limits of 

experimental scatter. A phenomenological model is proposed to rationalize the present microstructural 

observations. 

1.0 Introduction

Aircraft noise is a major environmental concern especially during takeoff and landing (Ref. 1). 

Compared to the 1960’s, commercial airplanes are significantly quieter today due to several 

improvements in design and materials technology (Refs. 2 and 3). Nevertheless, further reductions in 

these noise levels are desirable. Thus, the National Aeronautics and Space Administration (NASA) has 

identified an ambitious goal of further reducing aircraft noise by -52 dB relative to the International Civil 

Aviation Organization (ICAO) 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 the development of suitable materials (Refs. 3 and 4). 

Open cell foams and other cellular materials have been successfully used for sound absorption in 

many non-aerospace applications (Refs. 5 and 6). More recently, metal foams have been proposed for use 

in jet engines as acoustic treatment over rotors (Ref. 7), fan blades (Ref. 8) and other applications (Ref. 

9). The flow resistance (Refs. 10 and 11) and sound absorption property (Refs. 12, 13, 14, and 15) 

studies on metal foams have established their effectiveness as sound absorbers especially at higher

frequencies typically above 500 Hz (Ref. 12). Unlike materials that reflect acoustic waves to isolate noise, 

foams and other cellular materials absorb sound energy. Although the physics of sound absorption in 

porous materials is fairly complex (Ref. 5), it is generally agreed that sound energy is primarily dissipated 

within the porous microstructure through viscous and thermal losses (Refs. 12 and 13).

The properties of foams, including sound absorption, 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 

(Refs. 5, 11, 13, 14, and 16). Although the wavelengths of the sound waves are much larger than the cell 

dimensions (Ref. 6), it is expected that the dimensions of the cells and their three-dimensional connectivity 

are likely to influence their sound absorption properties as energy is transferred to pumping the air columns 
NASA/TM—2010-216797 2

within the cells (Fig. 1). Simple formulae exist for correlating the foam relative densities and their 

microstructures with static flow resistance and sound attenuation (Refs. 5, 10, 11, 12, 13, and 14). Other 

factors, such as the relative area fractions of closed and open cells, are likely to influence the acoustic 

properties (Ref. 12). 

A qualitative and quantitative understanding of the role of the foam microstructure on the acoustic and 

mechanical properties of metal foams is important if they are to be used as acoustic liners in aircraft engines. 

Particularly, establishing the three-dimensional topology of the microstructures of open cell foams is important 

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

other microstructural features that affect sound absorption and mechanical properties is also important for 

several reasons. First, microstructural data are required to develop realistic physics-based design models to 

simulate the behavior of these foam acoustic liners under engine operating conditions. Second, the data are 

likely to be useful to foam manufacturers to enable them to optimize processing and manufacturing parameters 

in a cost effective manner. Third, foam manufacturers can use these microstructural data to ensure batch-tobatch reproducibility and maintain quality control. Fourth, the aircraft engine manufacturer can use the 

microstructural information for formulating materials design specifications. 

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. 17). For 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. It should be noted that the reticulated polymer foam precursors 

used in the manufacture of the metal foams have a large variation in p.p.i. and relative densities (Refs. 18

and 19).

Figure 1.—Schematic showing alternating layers of an open cell cellular material. The energy of sound waves of 

frequency, f, and wavelength, λ, is expended as air within the cells is pumped in and out along the y and z axes. 

The major, dL, and minor, dT, dimensions are much smaller than λ.
NASA/TM—2010-216797 3

Among the several possible idealized topological representations of the foam microstructures (Ref. 5), the 

three-dimensional, space-filling Kelvin tetrakaidecahedron (Refs. 5, 20, and 21) 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 percent of the faces are squares, 0 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. 22 and 23). In reality, foam cells deviate from these ideal conditions, 

where they may be distorted and their sizes and shapes non-uniform. Importantly, experimental observations 

do not agree with these theoretical models (Refs. 24, 25, 26, 27, and 28).

The overall objectives of this investigation were to characterize the microstructures of FeCrAlY and 

stainless steel foams with two specific goals. First, to establish the probable cell topology of the foams from 

the geometries of the cell faces in order to determine the validity of the Kelvin and other theoretical models

(Refs. 20, 22, and 23). These results are reported elsewhere, where it was shown that the cells in these foams 

possess 11 faces consisting of 3 quadrilateral, 6 pentagonal and 2 hexagonal faces (i.e., 3-6-2 cell) (Ref. 28). It 

was demonstrated that the experimental observations did not agree with the Kelvin and other theoretical 

models. Second, to determine the effects of ρ*/ρs and lineal cell density, denoted as cells per mm (c.p.mm.),

on: (a) area fractions of closed and open cells faces; (b) major, dL and minor, dT, sizes of cell faces; and (c) 

ligament lengths, l, and thicknesses, t. This paper reports the quantitative microstructural observations on 

several metal foams.

2.0 Experimental Procedures

The foam panels 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. The excess slurry was squeezed out of the slurry-filled polyurethane foams before being burnt off 

during the subsequent sintering process leaving behind the metal foam panels. Measurements were 

conducted on FeCrAlY and 314 stainless steel foam panels with the nominal c.p.mm. varying between 0.2 

(5 p.p.i.1

) and 4.7 (120 p.p.i.), and the nominal (i.e., the manufacturer supplied values) (ρ*/ρS)nominal varying 

between 3 and 15 percent. The dimensions of the panels were approximately 210 x 210 mm with the

thicknesses varying between 3.2 and 25.4 mm. 

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 (Fig. 2(a)). The magnitudes of 

(ρ*/ρS)actual were determined by measuring the geometric and immersion densities of the foam cubes 

approximately 25.4 mm long and 25.4 mm wide with thicknesses equal to that of the foam panel. The 

immersion density measurements were conducted using n-propanol. Although the relative density and other 

measurements were primarily conducted on FeCrAlY and 314 stainless steel foams, a few measurements 

were also made on Haynes 25 foams. Optical and scanning electron microscopes (SEM) were used for 

qualitative characterization of the foam microstructures. In-situ quantitative metallographic measurements 

were conducted on 6 to 7 randomly selected areas for each foam specimen at a magnification of 100x using 

a Keyence2 digital optical microscope (VHX-100K Series) equipped with a lens with a long depth of focus 

and with the ability to vary the magnification from 100x to 5000x (Fig. 2(a)). The microscope was equipped 

with a x-y stage and imaging and data acquisition software to enable in-situ quantitative measurements to be 

performed. The effective spatial resolution of the microscope was 1.9 (horizontal) x 1.9 (vertical) μm/pixel

at a magnification of 100x (Ref. 29). The microscope was calibrated using a precision scale prior to each set 

of measurements to a measurement accuracy of 0.01 mm. The complex nature of the three-dimensional

microstructures consisting of randomly stacked interconnected cells made it difficult to characterize these 

foams. As a result, a large number of faces were measured to ensure representative measurements and to 

minimize experimental errors. 

 1 The nominal p.p.i. values refer to those reported by the manufacturer based on the precursor foam. 2 Keyence is the trade mark of Keyence Corporation of America, Woodcliff Lake, New Jersey.
NASA/TM—2010-216797 4

Figure 2.—(a) Schematic showing the locations of the three foam specimens on the 210 x 210 mm panels, and 

the areas of microstructural observations on each foam specimen. (b) Micrograph with a 300 x 300 μm grid 

pattern overlay used for point counting. The test lines were used for measuring c.p.mm. (c) Micrograph 

showing tracings of the perimeters, the major, dL, and minor, dT, diameters of the cell faces. The numbers 

correspond to the measured cell face. (d) Schematic showing the dimensions, l and t, of the ligaments.
NASA/TM—2010-216797 5

Figure 2.—Concluded.

The area fractions of closed and open cell faces, as well as those of the ligaments, were measured by 

the point counting method, where the point fraction, PP, is equal to the area fraction, Af, of the measured 

microstructural feature (Ref. 30). Measurements of PP were conducted by electronically overlaying a

rectangular grid with 9 rows and 11 columns on each field of view (Fig. 2(b)). The grid size was 300 x 

300 μm. The number of measurements varied between 968 and 1056. The number of counts along each 

horizontal line also gives the number of cells per unit length. The perimeters of the cell faces were 

determined by tracing the contours of each face3 (Fig. 2(c)). The imaging software calculated the 

corresponding areas of the cell faces. The equivalent diameters of the cell faces were determined from the 

perimeter data assuming that they could be represented as circles. The number of cell faces measured 

varied between 220 and 320 depending on the foam panel. The magnitudes of dL and dT were also directly 

measured on the same cell faces for which the perimeters were determined (Fig. 2(c)). The number of 

direct measurements of the cell sizes varied between 440 and 636.

Measurement of ligament lengths and thicknesses posed a special challenge because the triple points 

were not always well defined. Additionally, the ligament thickness was not always uniform. As far as 

possible, the ligament lengths were measured from the center of one triple point to the center of the other, 

and the ligament thicknesses were measured in the middle section of the ligaments (Fig. 2(d)). The total 

number of measurements varied between 315 and 790 depending on the foam panel. The perimeters of 

the porosity in the ligaments were traced as described above on different sets of specimens, and the 

corresponding pore areas were evaluated by the microscope’s imaging software. The total number of 

measurements varied between 75 and 385 counts. The error bars reported in this paper represent the 

95 percent confidence limit.

 3 Only the perimeters and diameters of the open faces were measured since it was not possible to clearly distinguish 

the boundaries between two adjacent closed faces.

l

(d)

ligament

Cell faces

t
NASA/TM—2010-216797 6

3.0 Results 

3.1 Comparison of the Nominal and Measured Parameters 

Figure 3(a) compares the measured relative densities, (ρ*/ρS)actual, of several metal foams with the 

nominal values, (ρ*/ρS)nominal, provided by the manufacturer; the broken line has a slope of unity. It is 

evident that the measured and the nominal values are scattered evenly about this line thereby indicating a 

reasonable agreement between these quantities for all p.p.i. In contrast, the measured and nominal lineal 

cell densities show no significant correlation with the data scattered about an average measured value of 

2.8 c.p.mm. (70 p.p.i.) (Fig. 3(b)).

4 This observation is consistent with other reported results in the 

literature (Ref. 17). The lack of any correlation observed in Figure 3(b) and the constant measured value 

of 2.8 c.p.mm. (70 p.p.i.) is suggestive of the fact that the metal powder slurry will shrink at about the 

same rate under similar processing conditions irrespective of p.p.i. of the polyurethane foam precursors.

Thus, (ρ*/ρS)actual is the only relevant variable to be considered for these foams; therefore, the nominal

c.p.mm. values will be only used as identification parameters in this paper. 

 

Figure 3.—Plots of measured and nominal parameters for FeCrAlY, 

Haynes 25 and 314 stainless steel foams: (a) Relative density; 

(b) lineal cell density.

 4

The equivalent p.p.i. are shown in Figure 3(b) in a tabular form for easy reference.
NASA/TM—2010-216797 7

3.2 Microstructures of the Foams 

Figures 4 (a) to (c) show optical macrographs of FeCrAlY foams with (ρ*/ρS)actual varying between 

3.3 and 8.6 percent and nominal lineal cell densities varying between 0.2 and 4.7 c.p.mm. Figures 4(d) to 

(e) show SEM micrographs for FeCrAlY foams with (ρ*/ρS)actual = 9.3 and 10.1 percent and nominal 

lineal cell densities of 3.9 and 3.1 c.p.mm., respectively. A close observation of these microstructures 

reveals that the number of closed faces of the cells qualitatively increases with increasing (ρ*/ρS)actual

and c.p.mm. A cross-section of the foams showed that there was significant porosity in the ligaments 

(Fig. 4(f)).

Figure 4.—(a) to (c) Optical and (d) to (e) SEM microstructures of FeCrAlY foams; (f) optical microstructure 

of a ligament cross-section showing voids. (a) 0.2 c.p.mm.; (ρ*/ρS)actual = 3.3 percent; (b) 2.4 c.p.mm.; 

(ρ*/ρS)actual = 4.1 percent; (c) 4.7 c.p.mm.; (ρ*/ρS)actual = 8.6 percent; (d) 3.1 c.p.mm.; (ρ*/ρS)actual = 10.1

percent; (e) 3.9 c.p.mm.; (ρ*/ρS)actual = 9.3 percent; (f) 1.6 c.p.mm.; (ρ*/ρS)actual = 5.3 percent.
NASA/TM—2010-216797 8

3.3 Area Fraction Measurements 

Figures 5(a) to (c)5 show the variation of (ρ*/ρS)actual with increasing values of the area fractions of 

closed, (Af)closed faces, and open, (Af)open faces, faces, and ligaments, (Af)ligaments, respectively, where the slopes 

represent linear regression values. The data are scattered about the regression lines with relatively low 

values of coefficients of determination, Rd

2

, which were 0.448, 0.512 and 0.408 for Figures 5(a), (b) and 

(c), respectively. Despite this fact, some general trends are discernable in the data. First, the relative 

density increases with increasing area fraction of closed faces (Fig. 5(a)). Second, the relative density 

decreases with increasing area fractions of open faces (Fig. 5(b)) and ligaments (Fig. 5(c)). Third, the area

fraction of closed faces decreases linearly with increasing area fraction of open cell faces (Fig. 5(d)). In 

the latter case, the regression line is represented by

(Af)closed faces = –2.0 (Af)open faces + 1.2 (Rd

2 = 0.719) (1)

Figure 5.—(a) to (c) Variations of the measured values of relative densities of FeCrAlY and 314 stainless steel foams 

with the area fractions of (a) closed faces, (b) open faces, and (c) ligaments. (d) Inverse linear relationship 

between the area fractions of closed and open faces. The broken lines represent the regressed values.

 5 All error values reported in the text and the figures in this paper represent 95 percent confidence values.
NASA/TM—2010-216797 9

3.4 Cell Face Diameters 

Figures 6(a) to (d) shows the frequency histograms of dL for several FeCrAlY foams for different values 

of (ρ*/ρS)actual varying between 5.0 and 16.5 percent. The corresponding average face dimensions varied 

between 204.5 ± 4.5 μm and 222.6 ± 5.2 μm. Clearly, the values of dL do not vary significantly for foams of 

different relative densities. Figure 7(a) confirms that (ρ*/ρS)actual is independent of dL and dT for both the 

FeCrAlY and the stainless steel foams. The vertical broken lines represent the average measured values, 

(dL)average6 = 212.4 ± 6.4 μm and (dT)average = 156.4 ± 4.5 μm, of the face major and minor dimensions, 

respectively. Similarly, the nominal lineal cell densities (Fig. 7(b)) and (Af)open faces (Fig. 7(c)) are 

independent of the cell face dimensions, where most of the data are scattered about the vertical mean lines.

Figure 6.—Frequency histograms of dL for FeCrAlY foams with nominal lineal cell density of 3.1 c.p.mm. and actual 

relative densities of (a) 5.0 percent; (b) 6.5 percent; (c) 10.1 percent; and (d) 16.5 percent. 

 6 The two outlier data for FeCrAlY were not considered in this evaluation.
NASA/TM—2010-216797 10

Figure 7.—Variations of (a) (ρ*/ρS)actual, (b) nominal lineal cell density and (c) (Af)open faces with dL and dT. The 

vertical broken lines represent the average measured values of the major and minor face dimensions.

3.5 Cell Face Perimeter and Area 

Figure 8(a) shows the variation of the relative density as a function of the measured perimeters of 

the cell faces for the FeCrAlY and stainless steel foams with different values of nominal lineal cell 

densities. The vertical broken line represents an average value of 624 ± 12.9 μm of the data with the 

exception of the two outlier data points for the FeCrAlY foam with a nominal lineal cell density of 

2.4 c.p.mm. Similar to the observations shown in Figure 7 (a), the relative densities are independent 

of the perimeters of the cell faces of the foams. Figures 8(b) and (c) show the variation of relative 

densities with the corresponding areas evaluated by the imaging software and the equivalent diameters, 

(deq)perimeter, calculated from the perimeter data, respectively. It is worth noting that the average value of 

(deq)perimeter7 = 198.5 μm is between (dL)average and (dT)average, thereby suggesting that the two independent sets 

of experimental measurements shown in Figures 7 and 8 are consistent within the errors in measurement. 

 7 As before, the two outlier data points for the FeCrAlY foams were not considered in the evaluation of the average 

value (deq)perimeter.
NASA/TM—2010-216797 11

Figure 8.—Variations of (ρ*/ρS)actual with (a) cell face perimeters, (b) cell face areas and (c) equivalent cell face 

diameters derived from the perimeter values. The vertical broken lines show the average values.

3.6 Ligament Dimensions and Microstructures

The measured relative densities of the foams are independent of the ligament length (Fig. 9(a)) and 

ligament thickness (Fig. 9(b)). The data are vertically scattered along average values of laverage =133.9 ±

2.9 μm and taverage = 41.1 ± 1.2 μm represented by the broken vertical lines in the figures. Thus, the 

relative densities of these foams are independent of the ligament dimensions. Figure 9(c) shows the 

variation of (ρ*/ρS)actual with the ratio t/l. The figure also shows the predicted variation of ρ*/ρS with t/l 

based on the Gibson-Ashby equation (Ref. 6)

ρ*/ρS = C (t/l)q (2)

where C is a constant depending on the cell shape, and q = 1 or 2 for closed and open cells, respectively.

Specifically, the predicted curve shown in Figure 9(c) is for open cells (i.e., q = 2) with C = 1 (Ref. 6). 

Although there is an apparent dependence of (ρ*/ρS)actual on t/l with C = 1 and q = 2 in Equation (1), a 

careful examination of the results reveals that most of the data are clustered in the range 0.25 ≤ t/l ≤ 0.30, 

thereby being consistent with the lack of significant correlations between ρ*/ρS and ligament dimensions 

observed in Figures 9(a) and (b). Thus, the apparent agreement of the empirical models with the 

experimental data must be considered fortuitous for the current data consistent with the observations 

shown in Figures 9(a) and (b). 
NASA/TM—2010-216797 12

Figure 9.—Variations of (ρ*/ρS)actual with (a) ligament length, (b) ligament thickness and (c) ratio t/l. The predicted 

values are based on Equation (2) given in Reference 6.

The cross-sectional microstructures of the ligaments revealed significant porosity (Figs. 4(e)). The 

ligament thicknesses were non-uniform and showed frequent necking in several regions. Figures 10(a) to 

(c) show the distributions of the areas of the ligament cavities for several FeCrAlY foams.

8 The average 

values of the areas of the pores varied between 293.6 ± 1.4 and 831.3 ± 1.6 μm2

. 

The circularity factor (C.F.) of the ligament pores was evaluated from Equation (3), where C.F. = 1 

for a circular pore (Ref. 17). 















 = 2 perimeter

area C.F. 4π (3)

Figure 10(d) shows the frequency and cumulative distributions of C.F. for a FeCrAlY foam with 

(ρ*/ρS)actual = 5.3 percent. The frequency histogram exhibits a log normal distribution. Similar log normal 

distributions were observed for the other foams. The average values of C.F. varied between 0.5 and 0.7,

which are consistent with the fact that the pores were non-circular.

 8 As noted earlier, these measurements were conducted on different set of specimens than those evaluated for the 

other quantitative metallographic measurements.
NASA/TM—2010-216797 13

Figure 10.—Frequency histograms and cumulative frequencies showing the distribution of cross-sectional areas 

of porosity within the ligaments for FeCrAlY foams (a) (ρ*/ρS)actual = 2.3 percent; 1.6 c.p.mm.; (b) (ρ*/ρS)actual = 

4.7 percent; 2.4 c.p.mm.; (c) (ρ*/ρS)actual = 3.9 percent; 3.1 c.p.mm. (d) Distribution of the circularity factor for a 

FeCrAlY foam with (ρ*/ρS)actual = 2.3 percent; 1.6 c.p.mm.

The area fraction of the porosity in the ligaments, Afp, was determined from the ratio of the average 

area of the ligament pores to the average cross-sectional area of the ligaments.

9 Figure 11 shows that Afp

decreases with increasing (ρ*/ρS)actual through a power-law relationship given by

(ρ*/ρS) = 2.9(Afp)

–0.2 (Rd

2 = 0.862) (4)

where (ρ*/ρS)actual is expressed in percentage. 

 9 It is noted that Afp represents the fraction of the ligament cross-section occupied by porosity and it should not be 

confused with the area fraction of ligament porosity within the foam cell volume.
NASA/TM—2010-216797 14

Figure 11.—Correlation between the measured relative density and the 

area fraction of ligament porosity. 

4.0 Discussion 

The observations reported in this paper reveal several important characteristics about the 

microstructures of these metal foams:

a) The measured and nominal relative densities of the foams are in very good agreement within the 

limits of experimental scatter, whereas there is no correlation between the measured and the nominal 

lineal cell densities (Fig. 3). All the foams had an average measure lineal cell density of 2.8 c.p.mm. 

(70 p.p.i.).

b) The (ρ*/ρS)actual increases with increasing area fraction of closed faces and decreases with increasing 

area fractions of open faces and ligaments (Fig. 5). Importantly, (Af)closed faces and (Af)open faces are 

inversely related through Equation (1).

c) The measured relative densities are independent of several microstructural parameters, such as cell 

diameters, perimeters and areas, and ligament length and thicknesses. 

d) A large amount of porosity was observed in the ligaments with the pore distribution being log normal 

and the pore geometry being predominantly non-circular. 

Empirical relationships have been proposed relating cell size, d, with ligament thickness (or 

diameter), lineal cell density and relative density for PORVAIR foams (Refs. 16 and 31):10

( )





















= 0.7 p.p.i.

0.1

s ρ* ρ t 1.4 (5)

( ) 













 = −

−

0.0004

ρ* ρs

1

100

s ρ * ρ

t d e (6)

 10 It is noted that the original reference source is an internal report (Ref. 31).
NASA/TM—2010-216797 15

Similarly, Zhao et al. (Ref. 10) have also proposed that t ∝ d using a different relationship. An 

examination of Figure 9 reveals that there is no correlation between ligament thickness and relative 

density and lineal cell density. Therefore, the present data do not support the validity of Equation (5). 

Similarly, Figures 7 and 8 reveal that the cell diameter does not change with either relative density or

nominal lineal cell density thereby revealing that Equation (6) is inconsistent with the present 

observations. Importantly, Figure 3(b) shows that the measured lineal cell density is about 2.8 c.p.mm. 

independent of the different p.p.i. for the metal foams studied in this investigation. 

Gibson and Ashby (Ref. 6) proposed a simple relationship between relative density and ligament 

dimensions given by Equation (2). Figure 12(a) schematically illustrates that Gibson-Ashby model would 

require the mass of the metal to be spread primarily in the ligaments, thereby increasing ligament 

thickness and decreasing cell size with increasing relative density for constant c.p.mm. and constant area

conditions. However, Figures 9(a) and (b) reveal that (ρ*/ρS)actual is independent of t and l so that this 

model does not explain the present observations.

Figure 12(b) proposes an alternate phenomenological model, where the relative density increases with 

increasing area fraction of closed faces as observed in Figure 5(a). There is a corresponding decrease in 

the area fractions of open cell faces and ligaments. This model does not require the relative density to 

vary with cell size, cell perimeter, ligament dimensions and other microstructural parameters. Assuming 

the mass is entirely distributed in the ligaments, triple points and the closed faces of the cell, the relative 

density can be expressed as

(ρ*/ρS) = B(1 – (Af)open faces)

p (7)

where B and p are dimensionless constants. The constant B is expected to be dependent on the geometry 

of the foam cell. Figure 13 shows the double logarithmic plot of the measured relative density against 

(1–(Af))open faces. As evident from Figure 13, there is some scatter in the data about the regression line. A 

linear regression analysis of the data revealed that B = 41.5 and p = 3.0 with Rd

2 = 0.408. 

Figure 12.—Schematic foam microstructures showing that the relative density can be 

increased by either (a) increasing ligament thickness in accordance to the Gibson-Ashby 

model (Ref. 6) or (b) increasing the area fraction of closed faces without increasing 

ligament thickness. An increase in the ligament thickness in (a) will lead to a 

corresponding decrease in cell size so that dL > dL1 > dL2. 
NASA/TM—2010-216797 16

Figure 13.—Double logarithmic plot of (ρ*/ρS)actual against 

(1 – (Af))open faces for both FeCrAlY and stainless steel foams.

It should be noted that Equation (7) implicitly assume that the ligaments do not have any porosity. In 

reality, the PORVAIR foams show extensive porosity in the ligaments (Fig. 10). Therefore, Equation (7) 

overestimates the relative density of the foams. Thus, these equations must be suitably corrected for the 

area fraction of ligament porosity, Afp. Since Afp were determined on foams different than those on which 

the other microstructural were made, this correction could not be incorporated.

5.0 Summary and Conclusions

Quantitative metallographic data were obtained on primarily open cell FeCrAlY and 314 stainless 

steel foams with nominal relative densities varying between 3 and 15 percent, and nominal lineal cell 

densities varying between 0.2 and 4.7 c.p.mm. The present observations reveal that while the measured 

and nominal values of relative densities are in good agreement, there is no correlation between the 

measured and nominal lineal cell densities. Instead, the measured lineal cell densities were constant equal 

to 2.8 c.p.mm. for all the foam studied in this investigation. An examination of the foam microstructures 

revealed that there was a qualitative increase in the number of closed faces with increasing values of 

nominal relative density and lineal cell density. Extensive porosity was observed in the cross-sectional 

microstructures of the foam ligaments. The relative density increased with increasing area fraction of 

closed faces and decreased with increasing area fractions of open faces and ligaments. There was an 

inverse linear relationship between the area fractions of closed and open faces. In contrast, the relative 

density did not vary with other microstructural features: cell size, cell perimeter, cell area, and ligament 

dimensions. A strong inverse power-law correlation was observed between the area fractions of the 

ligament porosity and the measured relative density. 

Empirical (Refs. 10, 16, and 31) and theoretical (Ref. 6) correlations between microstructural 

features, such as ligament dimensions and cell sizes, and the relative densities do not explain the present 

observations. Instead, the proposed phenomenological model rationalizes the current results by suggesting 

that the relative density is directly dependent on the relative area fractions of closed and open cell faces.
NASA/TM—2010-216797 17

References

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Effect of Microstructural Parameters on the Relative Densities of Metal Foams 

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14. ABSTRACT

A detailed quantitative microstructural analyses of primarily open cell FeCrAlY and 314 stainless steel metal foams with different relative 

densities and pores per inch (p.p.i.) were undertaken in the present investigation to determine the effect of microstructural parameters on the 

relative densities of metal foams. Several elements of the microstructure, such as longitudinal and transverse cell sizes, cell areas and 

perimeters, ligament dimensions, cell shapes and volume fractions of closed and open cells, were measured. The cross-sections of the foam 

ligaments showed a large number of shrinkage cavities, and their circularity factors and average sizes were determined. The volume 

fractions of closed cells increased linearly with increasing relative density. In contrast, the volume fractions of the open cells and ligaments 

decreased with increasing relative density. The relative densities and p.p.i. were not significantly dependent on cell size, cell perimeter and 

ligament dimensions within the limits of experimental scatter. A phenomenological model is proposed to rationalize the present 

microstructural observations. 

15. SUBJECT TERMS

Metal foams; Aviation noise; Foam cells; Foam microstructure; Kelvin model 

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