Journal of Colloid and Interface Science 280 (2004) 202–211
www.elsevier.com/locate/jcis
The 3D structure of real polymer foams
Matthew D. Montminy a , Allen R. Tannenbaum b , Christopher W. Macosko a,∗
a Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455-0132, USA
b Schools of Electrical & Computer Engineering and Biomedical Engineering, Georgia Institute of Technology, 777 Atlantic Dr. NW, Atlanta,
GA 30332-0250, USA
Received 17 September 2003; accepted 26 July 2004
Available online 2 September 2004
Abstract
The intricate structure of polymeric foams may be examined using 3D imaging techniques such as MRI or X-ray tomography followed
by image processing. Using a new 3D image processing technique, six images of polyurethane foams were analyzed to create computerized
3D models of the samples. Measurements on these models yielded distributions of many microstructural features, including strut length and
window and cell shape distributions. Nearly 8000 struts, 4000 windows, and 376 cells were detected and measured in six polyurethane foam
samples. When compared against previous theories and studies, these measurements showed that the structure of real polymeric foams differs
significantly from both equilibrium models and aqueous foams. For example, previous studies of aqueous foams showed that about 70% of
foam windows were pentagons. In the polymeric sample studied here, only 55% of windows were pentagonal.
 2004 Elsevier Inc. All rights reserved.
Keywords: Foam; Microstructure; Cells; Characterization; Visualization; 3D image processing; Minimal surface
1. Introduction
The physical properties and potential applications of
foams result from the chemistry and physical properties of
the bulk material and the cell structure of the foams. The
effect of foam chemistry and processing conditions on physical properties has been researched extensively, and many
useful empirical structure–property relationships have been
developed, including many foam density/property relationships [1–5]. However, relatively little experimental work has
been done to relate foam microstructure with macroscopic
physical properties. This is in part because there are few
good structure characterization methods.
Nonetheless, many theoretical models relating the physical structure and mechanical performance of polymer foams
have been created using rules formulated by observing soap
froths. These models and discussions relied on assumptions
such as monodispersity, rhombic dodecahedral or Kelv* Corresponding author.
E-mail address: [email protected] (C.W. Macosko).
0021-9797/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcis.2004.07.032
in’s tetrakaidecahedral [6] cell shapes, tetrahedral cell intersections, and Plateau’s laws for the formation of soap
froths [2,7–10]. Other studies simplified foam shapes further, using cubic unit cells to describe foam behavior [4,11].
While these relations allowed the creation of many practically useful and fairly accurate engineering correlations,
they are abstractions which do not necessarily accurately describe real polymeric or metallic foams.
Indeed, solid foams are very different from aqueous
foams because the base materials have very different material properties. Depending on composition and processing temperature, polymeric and metallic foams are usually
formed from materials which are significantly more viscous
than water, and, during their formation, heat transfer, phase
transition, and viscoelastic effects may come into play. Because of the nature of the base materials and cooling and
phase transition effects, cell nucleation and growth in these
foams is very different than in liquid foams. Most importantly, solid foams are not equilibrium structures. Depending
on processing conditions, their high viscosity and low melting points results in quenching of the foaming process well
M.D. Montminy et al. / Journal of Colloid and Interface Science 280 (2004) 202–211
203
into the model, and can be manipulated to allow for the various styles of measurements.
Such models can be created using 3D imaging techniques
such as magnetic resonance imaging (MRI) or X-ray computerized tomography (CT) coupled with 3D image processing [19–27]. However, the difficulties involved in 3D image
analysis, including image noise and memory and processing
constraints, have prevented previous models from tackling
large datasets and achieving widespread use in industry. Pangrle et al. [25] did not model any cells, but estimated cell
density from strut and window statistics. Kose and Kashiwagi [23,24] detected and measured 8 cells in their gelatin
foam sample. Monnereau and Vignes-Adler [21] detected
and modeled 9 bubbles within wet and dry foam samples.
While Cenens et al. [19] modeled and examined a much
more significant number of cells, they did so using a watershed segmentation method which estimated strut locations
based on the approximated locations of cell centers. This
approach does not necessarily capture the true structure of
foams, since it implicitly assumes that all cell borders will
lie halfway between cell centers. This assumption is questionable when analyzing polydisperse foams.
In this study, computer models of real, open-celled, polydisperse polyurethane foams were created. These models are
true to the actual structure of the foam specimens modeled,
with struts, vertices, windows, and cells detected directly
from their locations in 3D X-ray tomography images of
real foams. Such computerized foam models may become
invaluable in examining the complicated relationships between the physical properties and structures of solid foams,
or in detecting subtle changes in foam structure resulting
from changes in foam formulation.
before a stable state in the liquid phase is reached. This
at least partially invalidates the use of the observations of
Plateau [7] in describing the structure of solid foams. To
further advance our knowledge of foam structure–property
relationships, better investigation of the structures of real
solid foams is needed.
Foam measurements are generally taken using visual inspection, photography, or optical microscopy. In particular, optical microscopy techniques can be used to measure
many features, including window thickness, cell diameters
and strut lengths [12,13]. Confocal microscopy can also be
used to examine foam structures [14]. However, because
of the complicated three-dimensional structures of foams,
measuring foam features manually using these techniques
is tedious, time consuming, and often requires destruction
of the sample. In addition, some distinguishing features of
foams, including cell volume, are extremely difficult to measure at all using traditional techniques. As a result, there
are few experimental studies which correlate microstructure
and macroscopic properties. Cell size and anisotropy [1] and
open-celled content [2] are known to affect some physical
properties, but because of the lack of complete structural
data, predictive modeling is not possible.
Image processing can be used in order to help ameliorate
this dilemma. Previous studies using 2D image processing
to automate foam measurement collected data in a fraction
of the time required for physical inspection. In most earlier
studies using automated image analysis to extract structural
information; however, only one or a few structural measurements were compared to physical properties [15–17].
Furthermore, the use of different standards for even simple measurements such as cell diameter [18] makes it difficult to compare the results of similar studies to develop
a body of structure–property knowledge. For these reasons,
a method for more complete characterization of foams is
needed.
Creating an accurate 3D scale model of an actual foam
sample in computer memory provides a method for more
complete characterization. Then, many spatial measurements such as strut length and cell size can easily be made
on the scale model rather than measuring the actual sample.
If sufficiently large and reliable computerized 3D models of
real foams can be created, researchers can obtain measurements from the 3D model rather than measuring features of
the original foam sample. Measurement criteria can be built
2. Materials and methods
The structures of six polyurethane foam samples were
investigated in detail over the course of this research [28].
Background information on each samples is provided in
Table 1. Samples 1a and 1b were high density foams obtained from the same foam sample provided by Air Products
and Chemicals (Allentown, PA). Samples 2a and 2b were
taken from a low-density foam sample from Air Products.
Both Samples 1 and 2 were made on a slabstock production
machine. Sample 3 is FoamEx 40, a commercial structural
Table 1
Polyurethane foam samples analyzed in this study
Sample
#
Original volume
size (voxels)
Mass density
(lb/cu ft)
Approx. cell
density (cells/cm)
Description
Source
1a
1b
2a
2b
3
4
293 × 293 × 300
226 × 226 × 300
294 × 312 × 300
275 × 284 × 300
245 × 236 × 128
213 × 223 × 100
2.1
2.1
1.1
1.1
20
20
14
14
6
8
Open-celled slabstock foam
Open-celled slabstock foam
Open-celled slabstock foam
Open-celled slabstock foam
Reticulated slabstock foam
Handmade foam bun
Air Products
Air Products
Air Products
Air Products
FoamEx 40
Created in lab
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Fig. 1. Cross-section of the 3D image of the foam sample.
foam, and Sample 4 was created in our laboratory. Samples
1, 2, and 4 were made using standard flexible polyurethane
formulations similar to those described in Refs. [1], [14],
and [25]. All images presented in this section are from Sample 1a.
A 3D image of this foam was obtained via micro Xray computerized tomography (CT), in which a 3D image
is mathematically reconstructed from a set of 2D X-ray
shadow images [29]. CT was performed on these samples
using a SkyScan (Aartselaar, Belgium) 1072 desktop microtomograph, and a typical foam image cross-section from the
sample is presented as Fig. 1. The black voxels within this
image represent the locations of foam struts, while the white
areas represent void space. The best spot resolution currently
available on this instrument is about 5 µm [30], so this imaging method is only practical for the in-depth observation of
samples with average cell diameters of over 100 µm.
3D images obtained using X-ray CT or MRI contain a
great deal of information about foam structure, but extracting this information from these images is difficult. A 3D
image processing algorithm [31] and an image processing
software package called FoamView [28] were developed
specifically to extract structural information from 3D foam
images.1
A brief overview of the image analysis process is provided here. Once a 3D foam image has been acquired and
converted to the appropriate data format, the image analysis process can begin. The first step in the process involves
determining which voxels within the image volume represent foamed material and which voxels represent void space.
This image processing method is called segmentation. The
segmentation process converts the original 8-bit grayscale
1 FoamView, a users manual, and a 3D model of the sample described
here are available online as Supplementary Material.
Fig. 2. Surface rendering of the foam sample. Notice that this sample is very
large and contains about 100 cells. In the foam visualizations presented in
this paper, “fog” is used to convey depth. Lighter, washed-out sections of
the image are farther away from the viewer than darker sections.
image volume, which may have up to 256 different colors,
into a binary volume image. In this binary volume image,
voxels with a value of 0 (black) represent the foam structure, while those with a value of 1 (white) represent void
space. This binary volume image is the volume model. The
creation of this model is important because segmenting the
image creates a real distinction between the locations in the
image where the foam structure exists and those where void
space exists. This distinction is necessary for the detection
and analysis of shapes within the image and the creation of
the next two models. A 3D surface rendering of this volume
model is shown in Fig. 2.
From the volume model, a stick figure model of the foam
sample is created. The stick figure model is used to extract
most of the useful structural information present in the foam
image. The stick figure model contains the locations of the
struts, windows, and cells which comprise the cellular structure of the foam. The transition of the thick struts from the
volume model into a stick figure is performed using an image processing technique called volume thinning.
During volume thinning, dark voxels are successively removed from the thick foam structure until only a set of
strut centerlines remains. The volume thinning algorithm
used [32] guarantees that the resulting skeleton preserves
the connectivity of the original foam sample and is as close
to the actual centerline as possible. From this thinned volume image, a stick figure is created. Noise in the image and
thick, difficult to resolve vertices cause some artifacts to appear within the stick figure, but the stick figure can be refined
automatically or interactively using our package FoamView.
A result for this sample is presented as Fig. 3.
This resulting stick figure model consists of lists of the locations of vertices of the foam structure and the struts which
M.D. Montminy et al. / Journal of Colloid and Interface Science 280 (2004) 202–211
Fig. 3. Refined stick figure model of the sample. This model contains over
100 complete cells. In this visualization, the foam struts are shown as blue
cylinders, vertices are small red spheres, and detected cells show up as large
transparent blue spheres. Detected windows are shaded green.
connect them. From this information, foam cells and windows are detected. Since the locations of all of these features
are stored in computer memory, struts, strut intersection angles, windows, and cells are all measured directly using this
model.
This image analysis approach yields models which are
derived directly from the actual strut structure apparent in the
original 3D image, ensuring faithfulness to the actual foam
structure. These models correspond well with surface renderings showing the surface of the foam in the original 3D
image, as shown in Figs. 4 and 5 and inspection of the results
in 3D using the FoamView software.2
Thanks to the high level of automation present within the
image analysis process, this method can quickly provide data
on foam samples. Image acquisition at 1024 × 1024 × 512
voxel resolution requires about 3 h of instrument time using current X-ray CT technology, and image processing took
from 2–4 h for the above samples, depending on sample
size. Most of this time (all but about 20 min for each sample) was spent on user-assisted detection of foam features
using FoamView. The user can view and edit the 3D stick
figure structure while comparing it to the original 3D image
in FoamView, improving on the automated result and ensuring that stick figure model corresponds well to the original
3D image of the foam structure. While this prevents exact reproducibility of results, it allows models to be created from
fairly poor foam images. Noise and other artifacts can be bet2 The original foam volume and stick figure model can be compared
interactively using the FoamView software and sample datasets available
online as Supplementary Material.
205
Fig. 4. Visual comparison of the surface and stick figure models for the
sample.
Fig. 5. Close-up of the foam structure showing the correlation between the
detected strut, vertex, and cell locations and the original foam volume. The
large, blue spheres in the image indicate the centers of detected foam cells.
Some struts (top middle and bottom left) were not detected because they are
incomplete struts on the edge of the image.
ter filtered by the human eye than by computer algorithms,
especially in 3D.
3. Results and discussion
As shown in Table 2, between the six samples imaged and
analyzed over the course this research, 376 cells were detected. Nearly 8000 foam struts and 4000 windows were also
detected and measured. Note that the “cell densities” listed
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Table 2
Features measured in the samples analyzed in this study
Sample
#
Approx. cell
density (cells/cm)
Struts
Vertices
Windows
Cells
Avg. cell
volume (mm3 )
Std. dev. of cell
volume (mm3 )
1a
1b
2a
2b
3
4
20
20
14
14
6
8
2066
1249
1328
1120
1176
1024
1111
682
740
620
651
568
1063
628
631
559
578
494
106
60
54
58
54
44
0.19
0.21
0.31
0.32
4.15
3.21
0.15
0.17
0.29
0.37
1.8
0.57
7963
4372
3953
376
Total
# features detected
in the table are linear cell densities estimated by counting the
number of cells in a straight line on the surface of the sample
before processing, while the “cell volumes” listed were measured using the 3D microscopy technique described above.
Each cell volume is roughly proportional to the cube of the
reciprocal of the corresponding linear cell density, resulting
in the higher average cell volumes found in Samples 3 and 4.
Because of its complexity and polydisperse cell size distribution, only the results for Sample 1a will be discussed
in detail here. See Ref. [28] for detailed information on the
remaining samples.
3.1. Overview of sample measurements and statistics
The resulting 3D model consists of a set of vertex locations and lists of the vertices composing foam struts, windows, and cells. This information is used to calculate many
statistics and perform many measurements of foam features.
Table 3 shows selected measurements and statistics describing the structure of Sample 1a, our highlighted foam sample.
As shown in Table 3, the vertex location and connectivity
information from the stick figure model generated from this
sample image provides a wealth of information. Structural
unit measurements, including strut lengths, interior angles,
window areas, and cell volumes and surface areas are available. Shape information, including cell and window shapes,
cell isoparametric quotients, and the amount of anisotropy
present within the structure can also be calculated. The
anisotropy ratio describes the elongation of the foam in the
rise direction versus that in the other major directions. The
isoparametric quotient, 36πV 2 /A3 , is a ratio of volume to
surface area which indicates how spherical a cell is; a sphere
has an isoparametric quotient of 1. Together, all this information provides a detailed snapshot of the foam sample’s
microstructure.
Table 3
Measurements and statistics describing the cellular structure of the sample
Basic measurements
Image resolution
Voxel size
Sample volume
Solid volume
Solid fraction
Anisotropy ratio
293 × 293 × 300 voxels
12.27 × 12.27 × 12.27 µm
47.30 mm3
3.18 mm3
0.0671
1.292
Window shape distribution
# features detected
9 Triangles
251 Quadrilaterals
587 Pentagons
205 Hexagons
11 Heptagons +
2066 Struts
1111 Vertices
1063 Windows
106 Cells
Cell structure statistics
Strut Interior Window Cell
Cell
Cell
length angles area
volume surface area isoparametric
(mm) (deg)
quotient
(mm2 ) (mm3 ) (mm2 )
Mean
Std. dev.
Min
Max
0.281 106.72 0.133
0.104 17.78 0.080
0.079 20.70 0.006
0.716 164.57 0.520
0.193
0.145
0.012
0.937
1.749
0.834
0.323
5.151
0.646
0.075
0.407
0.823
Cell shape information
Mean # of windows per cell
13.01
3.2. Strut lengths
2066 individual struts were detected and measured within
this sample. The mean length of foam struts detected within
the sample was 0.28 mm. A strut length distribution histogram for the sample is presented as Fig. 6. Notice that
the strut lengths fall in a right-skewed distribution, a com-
Fig. 6. Strut length distribution for the foam sample.
M.D. Montminy et al. / Journal of Colloid and Interface Science 280 (2004) 202–211
Fig. 7. Strut intersection angle distribution for the foam sample.
mon distribution shape for statistics describing natural systems. The shape of this strut length distribution corresponds well with those detected in previous foam structure
studies [19,25] as well as with the strut length distributions of other samples analyzed over the course of this research [28].
3.3. Strut intersection angles
When a load is placed on a foam, it places a stress on
both the struts of the foam and the vertices at which they
intersect. Therefore, analysis of the angles at which struts
intersect may provide important insight into foam behavior.
Strut intersection angles have been previously examined and
measured using image processing approaches by Cenens et
al. [19], Kose [23], and Pangrle et al. [25]. Fig. 7 shows the
distribution of the 5273 interior angles at which the foam
struts meet within the sample. The angles appear to fall in
a nearly normal distribution. The mean strut intersection angle of 106.7◦ is close to but significantly smaller than the
expected tetrahedral angle of 109.5◦ at which four edges
meeting at a corner should intersect in an equilibrium structure, as noted by Plateau [7]. All of the other samples in
Tables 1 and 2 also had average intersection angles significantly smaller than 109.5◦ [28]. There are possible reasons
for this variation from the theoretical rule. First, as noted
in the introduction, foamed polymers are not equilibrium
structures. Structural foams are frozen before being able to
reach complete equilibrium. Second, there are stresses on
the shape of real foam systems. This sample was a slabstock
polyurethane foam, and, as such, has cells that were elongated in the rise direction during the rising process. Surface
tension is not the main driving force at work in determining cell shape for polymer foams. In addition, we observed
that in real foams, more than four edges may meet at a vertex. A few occurrences of this phenomenon within a large
sample may significantly depress the mean intersection angle.
207
Fig. 8. Window area distribution for the foam sample.
3.4. Window size and shape
Many practical characteristics of foams, including their
ability to be used as filters or catalyst supports, depend upon
the sizes of a foam’s windows. The distribution of areas for
the 1063 windows detected within the sample is provided as
Fig. 8.
Window area is calculated by partitioning each window
into a set of triangles, and calculating the area of these triangles. Each of these triangles has vertices at two adjacent
window vertices plus the window’s centroid. This is necessary since the vertices of real foam windows rarely lie within
a plane.
Notice that the window area distribution is another rightskewed distribution, but that this distribution is more significantly skewed than the strut length distribution presented
above. This suggests that there is more diversity in window
sizes than in strut lengths within the sample. The difference between the strut length and window area distributions
can be explained by two effects. First, window areas are
roughly proportional with the square of strut lengths, broadening the distribution. Second, shape may play a significant
factor in window size. When struts of like size are used to
create both quadrilateral and hexagonal windows, two different windows sizes may be created, resulting in a broader
distribution. Finally, anisotropy in the rise direction will also
widen the window size distribution.
3.5. Window shape
Window shape has previously been used to describe foam
cells and how they divide space [6,23,33,34]. Kelvin [6] proposed the tetrakaidecahedron (a 14-sided shape consisting
of six quadrilaterals and eight hexagons) as a shape which
would minimize surface area within a monodisperse foam
and should therefore be favored at equilibrium. Weire and
Phelan [34] suggested that a tessellation of equally sized
cells with pentagonal and hexagonal sides could more efficiently subdivide space. However, Matzke [33] showed that
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Table 4
Comparison of window shapes for the foam Sample 1 with window shapes expected by other models and experiments
Kelvin model [6]
Weire–Phelan model [34]
Matzke experiment [33]
Kose experiment [23]
Sample 1a
# cells
analyzed
Window shape distribution (%)
Triangles
Quadrilaterals
Pentagons
Hexagons
Heptagons +
–
–
600
8
106
0
0
0
0
1
43
0
11
9
24
0
89
67
70
55
57
11
22
21
19
0
0
<1
0
1
Note that only central bubbles from Matzke’s experiments (those in the center of his jars, unaffected by edge effects) are listed here.
in reality, even monodisperse liquid foams consist of cells
and windows of many varied shapes.
Nine triangles, 251 quadrilaterals, 587 pentagons, 205
hexagons, and 11 heptagons were detected in the sample.
Most of the windows in our sample have the expected
quadrilateral, pentagonal, and hexagonal shapes, with the
pentagonal window being the most common shape.
Table 4 provides a comparison of the distribution of window shapes found in this study to those found in area minimizing models and other foam characterization experiments.
Note that the foam studied here has a much more diverse
window shape distribution than those hypothesized in areaminimizing models. This is largely due to our sample’s polydispersity. In addition, notice that the cell shape distributions
of the Kose [23] and Matzke [33] experimental results, both
performed on liquid foams, are very similar to each other,
but quite different than the window shapes within the polymer foam examined in this study. These results highlight the
facts that polymeric foams do not follow ideal theoretical
structures. The structures found in this foam are more consistent with recent studies of random soap froths found in
the important work of Kraynik [35,36]. Modeling and study
of foams with these less ordered, more realistic polydisperse
structures may allow a better understanding of the correlation between foam physical properties and microstructure.
3.6. Cell size and shape
The polydispersity of cell sizes within this foam is most
apparent when the foam cell volume distribution is examined. This distribution, presented in Fig. 9, is skewed very
heavily to the right, indicating a high degree of polydispersity of cell sizes. In fact, of the 106 cells detected, the 10
largest cells make up 25% of the sample volume. Statistics
like this one may give a good way to quantify polydispersity from an engineering perspective, since a small number
of large gaps in the foam structure (cells that are too large)
can drastically impact a foam’s performance.
Cell volumes are calculated using a method analogous to
the window area calculation method described above. Each
cell is split into a set of pyramids with the top of each pyramid at the centroid of the cell, and the base of each pyramid
being a window. The volume of all of these pyramids for
each cell are added together to calculate the cell volume.
Fig. 9. Cell volume distribution for the foam sample.
Table 5
Distribution of cell shapes for the foam sample
# sides
# occurrences
Frequency (%)
7
8
9
10
11
12
13
14
15
16
17
18
3
4
4
7
19
12
13
14
10
4
9
5
3
4
4
7
18
11
12
13
9
4
8
5
21
1
1
24
1
1
Total: 106 cells
Average # sides: 13.01
Along with polydispersity of cell sizes comes diversity
in cell shapes. When very large and fairly small cells share
borders in a sample, the large cells can end up with very large
numbers of windows, as shown in Table 5. In this sample,
most cells detected had 11–15 sides, but cells composed of
18, 21, and 24 windows were also detected.
The average cell detected in the sample was 13-sided,
which is somewhere between a rhombic dodecahedron and
M.D. Montminy et al. / Journal of Colloid and Interface Science 280 (2004) 202–211
209
Table 6
Comparison between the highlighted sample results (Sample 1a) and those of an adjacent sample of the same foam (Sample 1b)
Sample
#
Cells
detected
Mean strut
length (mm)
Mean interior
angle (deg)
1a
1b
106
60
0.281
0.281
106.7
106.1
Difference (%)
–
0.0
0.6
Mean window
area (mm2 )
0.133
0.134
Mean cell
volume (mm3 )
0.193
0.209
−0.8
−8.3
Mean cell surface
area (mm2 )
1.749
1.84
−5.2
Mean # windows
per cell
13.01
13.37
−2.8
Table 7
Comparison between two adjacent samples of another piece of polyurethane foam (Samples 2a and 2b)
Sample
#
Cells
detected
Mean strut
length (mm)
Mean interior
angle (deg)
Mean window
area (mm2 )
Mean cell
volume (mm3 )
2a
2b
54
58
0.319
0.330
107.1
107.5
0.178
0.192
0.308
0.319
Difference (%)
–
3.4
7.9
3.6
0.4
Kelvin’s tetrakaidecahedron, two theoretically proposed and
popular approximations of foam cell shapes. This suggests
that these shapes may provide reasonable approximations of
actual foam shapes, but the diversity of cell shapes within
this sample show that these truly are simplified models of the
actual structure of real polymer foams. Polydisperse foams
naturally have more diverse cell shapes.
3.7. Other quantitative measurements
Other quantitative measurements can be made using the
model information collected from the 3D foam image, although these measurements tend to be less accurate than
those listed above due to image limitations. Solid fraction
and average strut thickness can be calculated by counting
the dark pixels from the segmented 3D image. However, as
shown in Fig. 1, the original images are somewhat blurry,
and it is difficult to decide exactly where the surface of the
strut begins. This is inconsequential when determining strut
lengths because of the connectivity of the foam structure,
but causes trouble when strut thickness is examined. Unfortunately, this issue cannot be resolved by using higher
scanning resolution because bigger, more detailed pictures
would slow down the main image processing method discussed.
When a slabstock foam rises quickly, its cells are often stretched in the direction in which the foam rises. This
anisotropy or elongation is important to the final physical
properties of the foam, since the structure is strengthened
in the direction of elongation, much like the structure of an
egg. The anisotropy ratio displayed in Table 2 gives a rough
measure of elongation within the system. It is determined
by calculating the sum of absolute values of the dot products of the vectors describing each strut with the x, y, and
z unit vectors, then dividing the largest of these values by
the smallest. That is, the x, y, and z component of each strut
within the model is calculated, and the sum of each directional component for all struts is calculated. If the sample is
placed into the imaging device with the rise direction facing
Mean cell surface
area (mm2 )
2.371
2.318
−2.2
Mean # windows
per cell
12.98
12.40
−4.5
up, as this sample was, it provides a relatively good measure
of anisotropy within the system, with more elongated foam
systems exhibiting higher anisotropy ratios.
3.8. Reproducibility
Comparison of Sample 1a’s statistical results to those of
an adjacent sample from the same foam (Sample 1b) show
good agreement, as shown in Table 6. Results for Samples
2a and 2b are also provided in Table 7. Notice that in both
cases, the statistical results for the two samples agree within
less than 10%, and that the largest errors occur for cell-based
measurements, where the number of samples averaged is
small (60 and 106 cells vs thousands of struts, angles, and
windows detected).
Ensuring reproducibility of image processing results depends upon careful inspection of the model and comparison
of the model to the original 3D image by the person using the
image analysis software. Because of the time and expense
involved in obtaining and creating 3D images of foams, and
performing multiple experiments to ensure reproducibility,
this technique is more difficult and expensive than for other
experimental techniques. In this study, each sample was sent
to an external laboratory for imaging, and imaging of each
sample cost about $500.
3.9. Method and model limitations
While this method offers a new way to create models of
foams, it has some limitations. First, while the image analysis process may be faster than traditional microscopy and
examination of foams, it is still relatively time intensive. Although the process is semi-automated, analysis of the six
samples examined over the course of this work required
2–4 h each. Most of this time was spent on user-assisted
analysis of the 3D images. While this time frame is reasonable for researchers, it prevents the use of this method
for real time quality analysis and control of manufactured
foams. In addition, processing difficulty scales with the cube
210
M.D. Montminy et al. / Journal of Colloid and Interface Science 280 (2004) 202–211
of the image size, so large images can quickly become too
slow and difficult to process. At the time of this research, the
293 × 293 × 300 voxel size of Sample 1a stretched the capabilities of the image processing hardware used, 733 MHz PC
with an nVidia GeForce 2 graphics card with its own graphics processor.
In addition, 3D imaging equipment has limited resolution. Current micro X-ray CT has a resolution of about
5 µm [30], while recent MRI experiments [26,27] obtained
resolutions of 8 µm. Due to this constraint, for reasonably accurate measurements to be obtained, struts must be at least
100 µm long and about 20 µm thick. This precludes the
analysis of microcellular polymers, which have cell diameters of 10 µm or less [37].
Because of the limited resolution of 3D imaging equipment, some foam features cannot be accurately detected or
measured. As described above, strut thickness is difficult to
judge because of the weak CT signal generated by foam
struts and the small length to thickness ratio of the struts.
Struts are often only 5 voxels thick in a 3D foam image,
making strut width measurement quite inaccurate. Similar
problems would plague attempts at quantifying strut surface
area or foam surface roughness. Relatively thick vertices
are common in polydisperse foams, and some judgment is
required in locating the center of these vertices. This introduces some imprecision into the measurement of strut
intersection angles, as small movements to a vertex can significantly impact these angles.
Thin windows are also difficult to detect and measure
at this resolution, preventing analysis of window thickness
and percentage of open-celled content. Since the thickness
or existence of windows can drastically affect the physical properties of a foam [3,5], this method would have to
be combined with window measurement techniques such as
traditional microscopy or air flow measurements to create a
complete picture of closed-celled foam structure.
els of real polymeric foams could be used to (1) model foam
performance at many levels, including modeling of foam
compression and fluid flow through foams, and (2) analyze
many elements of foam structure at once to develop robust structure–property, structure–chemistry, and structure–
processing relationships.
Reliable computerized models of real foams have many
potential uses. There is a significant industrial demand for
improved characterization techniques for quality control and
new product development in the polymeric foams industry.
This method allows for the study of foams at a microstructure level, and may perhaps allow the detection of subtle
structural changes due to changes in foam formulations, such
as the addition of catalysts or surfactants. Study of the effects
of the microstructural properties of foams on their macroscopic physical properties may assist in the development
of new foams for various applications. Academic studies
of foam compression, fluid flows through foams, and cell
nucleation would directly benefit from 3D models of real
foams. Theories developed using ideal foam models could
be tested on real foams with irregular shapes and in nonideal
conditions.
However, 3D characterization tools must be developed
further if they are to be used for many industrial purposes.
Developing robust structure–property relationships will require the analysis of hundreds of samples, which would still
be a daunting task using this method, since sample analysis requires about 2–4 h for image analysis, plus instrument
time for image acquisition. Quality control applications also
require a quick turnover time to ensure that a minimum
amount of product is wasted, making this method useful as
a quality control check for small batch processes but infeasible for large-scale continuous processes. Future research
will hopefully yield faster image acquisition and faster, more
automated image analysis algorithms, providing for quicker
analysis of samples to serve these industrial uses.
4. Conclusions
Acknowledgments
The software and foam structural data presented in this
paper show that in-depth 3D characterization and modeling
of real structural foams is possible and can yield interesting
results. For the polydisperse polyurethane foam discussed
in this paper, feature size distributions provided significant
insight into the sample’s structure.
This research shows that real polymeric foams may have
not only polydisperse cell sizes, but rich, diverse arrays of
cell and window shapes. Since polymeric foams are not equilibrium structures, they exhibit disordered structures which
are quite different from the perfectly ordered, monodisperse
tetrakaidecahedral or cubic unit cells previously used to
model the physical properties of foams.
The development of models of real polymeric foams is
an enabling technology which will facilitate future research
into the structure and physical performance of foams. Mod-
The authors thank Dr. Mark Listemann, Air Products and
Chemicals, and Dr. Xiao Dong Zhang, Dow Chemical, for
providing foam samples and input for this paper. We thank
Andy Kraynik for his very helpful comments. This paper
is based upon work supported by a National Science Foundation Graduate Research Fellowship. This work was also
supported in part by grants from the National Science Foundation, the Air Force Office of Research, the Army Research
Office, and MURI grant.
Supplementary material
The online version of this article contains additional supplementary material.
Please visit DOI: 10.1016/j.jcis.2004.07.032.
M.D. Montminy et al. / Journal of Colloid and Interface Science 280 (2004) 202–211
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